Research ArticleModeling Mixed Bicycle Traffic Flow A Comparative Study onthe Cellular Automata Approach
Dan Zhou Sheng Jin Dongfang Ma and Dianhai Wang
College of Civil Engineering and Architecture Zhejiang University Hangzhou 310058 China
Correspondence should be addressed to Sheng Jin jinshengzjueducn
Received 10 December 2014 Revised 29 April 2015 Accepted 30 April 2015
Academic Editor Tetsuji Tokihiro
Copyright copy 2015 Dan Zhou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Simulation as a powerful tool for evaluating transportation systems has beenwidely used in transportation planningmanagementand operations Most of the simulation models are focused on motorized vehicles and the modeling of nonmotorized vehicles isignoredThe cellular automata (CA)model is a very important simulation approach and is widely used formotorized vehicle trafficThe Nagel-Schreckenberg (NS) CA model and the multivalue CA (M-CA) model are two categories of CA model that have beenused in previous studies on bicycle traffic flowThis paper improves on these twoCAmodels and also compares their characteristicsIt introduces a two-lane NS CAmodel andM-CAmodel for both regular bicycles (RBs) and electric bicycles (EBs) In the researchfor this paper many cases featuring different values for the slowing down probability lane-changing probability and proportion ofEBs were simulated while the fundamental diagrams and capacities of the proposedmodels were analyzed and compared betweenthe two models Field data were collected for the evaluation of the two models The results show that the M-CA model exhibitsmore stable performance than the two-lane NS model and provides results that are closer to real bicycle traffic
1 Introduction
Traffic flow theories are generally divided into two branchesmacroscopic and microscopic theories [1] The macroscopictraffic flow models are based on fluid dynamics and aremostly used to elucidate the relationships between densityvolume and speed (also called the fundamental diagram) invarious traffic conditions The microscopic traffic models onthe other hand describe the interaction between individualvehicles The microscopic traffic models generally includecar-following models and cellular automata (CA) modelsThe car-following model is the most important modeldescribing the detailed movements of vehicles proceedingclose together in a single lane There have been many car-following models produced in the literature over the past60 years such as stimulus-response models safety distancemodels action point models fuzzy-logic-based models andoptimal velocity models [2ndash5] For a broader review referto Brackstone and McDonald [6] and Chowdhury et al[7] Recently CA models have emerged as an efficient toolfor simulating highway traffic flow because of their easy
concept simple rule and speed in conducting numericalinvestigations The rule-184 model proposed by Wolfram[8] was the first CA model to be widely used for trafficflow Nagel and Schreckenberg [9] presented the well-knownNS CA model which is an extension of the rule-184 modelallowing the maximal speed of vehicles to be more than onecells The NS model and the many improved versions ofit reproduce some basic and complicated phenomena suchas stop and go metastable states capacity drop phenomena(which means the capacity of road experiences a large dropunder critical density conditions) and synchronized flow inreal traffic conditions
Most of the aforementioned microscopic traffic modelshave been developed only for motorized vehicles Few ofthem have been used for modeling non-motorized vehiclessuch as bicycles tricycles electric bicycles and motorcyclesbecause of the complicated characteristics of such vehiclesmovements With the increasing usage of bicycles someresearchers have begun focusing on modeling the operationof bicycle facilities Jiang et al [10] introduced two differentmultivalue CA (M-CA) models in order to model bicycle
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 420581 11 pageshttpdxdoiorg1011552015420581
2 Discrete Dynamics in Nature and Society
flow Their simulation results showed that once the ran-domization effect is considered the multiple states in deter-ministic M-CA models disappear and unique flow-densityrelations exist They found the transition from free flow tocongested flow to be smooth in onemodel but of second orderin the other Lan and Chang [11] developed inhomogeneousCA models to elucidate the interacting movements of carsand motorcycles in mixed traffic contexts The car andmotorcycle were represented by nonidentical particle sizesrespectively occupying 6 times 2 and 2 times 1 cell units each of size125 times 125 meters The CA models were validated by a set offield-observed data and the relationships between flow celloccupancy (a proxy of density) and speeds under differenttraffic mixtures and road (lane) widths were elaborated AM-CA model for mixed bicycle flow was proposed by Jiaet al [12] Two types of bicycles with different maximumspeeds (1 cells and 2 cellss) were considered in the systemDifferent results were analyzed and investigated under bothdeterministic and stochastic regimes Li et al [13] presenteda multivalue cellular model for mixed nonmotorized trafficflow composed of bicycles and tricycles A bicycle wasassumed to occupy one unit of cell space and a tricycle twounits of cell spaceThe simulation results showed themultiplestate effect of mixed traffic flow Gould and Karner [14]proposed a two-lane inhomogeneous CA simulation modelan improved version of the NS model combining a lane-changing rule for bicycle traffic and collected field data fromthree UC Davis bike paths for comparison with a simulationmodel Yang et al [15] proposed an extended multivalue CAmodel that permitted the bicycles to move at faster speedsThe simulation results showed that the mixed nonmotorizedtraffic capacity increased with an increase in the electricbicycle ratio Zhang et al [16] used an improved three-laneNSmodel to analyze the speed-density characteristics of mixedbicycle flow The simulation results of the CA model wereeffectively consistent with the actual survey data when thedensity was lower than 0225 bicm2
Summarizing the above none of the aforementioned car-following models have been devoted to mixed traffic withregular bicycles (RBs) and electric bicycles (EBs) but CAmodels have been widely used for modeling nonmotorizedtrafficThemodeling ofmixed bicycle traffic usingCAmodelscan be divided into two branches NS CA model and M-CAmodel approaches The update rules of the NS CA model forbicycle flow are the same as for motorized vehicles with onlythe cell size and bicycle speeds being different As pointedout by Jiang et al [10] and Jia et al [12] the M-CA model ismore suitable than the NS model for modeling bicycle trafficflow Because the update rules of the M-CA model do notinclude direct car-following and lane-changing behavior itmay be appropriate formodeling the nonlane-based behaviorof bicycle traffic The NS CA model and M-CA model havebeen used for modeling bicycle traffic and mixed bicycletraffic with RBs and EBs However there is no evidence in theexisting literature as to which model is better for modelingmixed traffic flow nor as to the differences between these twomodels Therefore a comparison of the NS CA model andthe M-CA model in terms of their ability to model mixed
bicycle traffic is required so that CAmodels can be improvedefficiently
This paper attempts to develop twoCAmodels to describethe behaviors of mixed bicycle traffic with RBs and EBs ona separated bicycle path and to compare the characteristicsof the NS CA model and the M-CA model The remainingparts are organized as follows Section 2 introduces thedevelopment of NS and M-CA rules Section 3 presents thesimulation results of these two CA models Section 4 furtherdiscusses differences in the simulation results Finally theconclusions and ideas for future studies are addressed
2 CA Models
21 Definition of Cell Size and Bicycle Speed The maindifferences encountered when modeling bicycle traffic asopposed to motorized vehicle traffic using a CA model arethe cell size and the speed Mixed bicycle traffic with RBsand EBs on separated bicycle paths is ubiquitous in manyAsian countries such as China Vietnam Indonesia andMalaysia Because of the different operating speeds of RBsand EBs mixed traffic produces complicated behavior andcharacteristics that are likely to lead to safety and efficiencyproblems Modeling mixed bicycle traffic is very importantfor the planning operation and management of bicyclefacilities Based on the behavior of cyclists CA models arethe best option for modeling bicycle traffic The size of cellspace and the update rules are two significant aspects of CAmodels
Bicycles are shorter and narrower than motorized vehi-cles Based on field surveys the length of most RBs and EBsis 17ndash19m Meanwhile bicycle lanes are set at 1 meter widein both China and theUSA [17 18]Therefore the size of a RBor an EB is assumed rectangular with length 2m and width1m as is widely used in other CA models [14ndash16] The otherparameter for modeling bicycle traffic is speed According tothe literature the reported free flow speed of EBs is largerthan that of RBs Accordingly in this paper speeds of 2 cellss(4ms or 144 kmh) and 3 cellss (6ms or 216 kmh) werechosen for RBs and EBs respectively
22 NS CA Model The NS CA model used in this paperwas proposed by Nagel and Schreckenberg [9] This modelis very widely used in modeling highway traffic and bicycletraffic The NS CA model includes a car-following rule and alane-changing rule The car-following rule is based on foursteps and the lane-changing rule is based on the work ofRickert et al [19] Different vehicle behavior rules wouldlead to different simulation results With an increase in thenumber of lanes the lane-changing logic would becomemorecomplicated and make modeling more difficult Therefore inthis paper only a two-lane bicycle path is simulated and usedin the comparison In the time interval from 119905 to 119905+1 the fourbasic rules of the NS model evolve according to the followingsteps
Step 1 (longitudinal acceleration) Consider
V119894(119905 + 1) = min (V
119894(119905) + 1 V
119894max) (1)
Discrete Dynamics in Nature and Society 3
where V119894(119905) is the speed of the 119894th bicycle at updating time
119905 V119894max is the maximum speed of the 119894th bicycle This corre-
sponds to the cyclistsrsquo realistic free flow speed
Step 2 (longitudinal deceleration) Consider
V119894(119905 + 1) = min (V
119894(119905) gap
119894) (2)
where gap119894is the distance between the 119894th bicycle and the
bicycle in front of it at updating time 119905 This step ensures thatthe bicycle stays safe with no collisions
Step 3 (random slowing down) Consider
V119894(119905 + 1) = max (V
119894(119905) minus 1 0) if rand () lt 119901
119889119894 (3)
where rand() is a uniformly distributed random numberbetween 0 and 1 and 119901
119889119894is the random slowing down
probability of the 119894th bicycle The random slowing downeffect which captures one cyclistrsquos brakingmaneuver due to arandom event (eg accident road or weather related fac-tors) is one of the most significant parameters of the CAmodel This step incorporates the idea of random effects onbicycles that may cause them to slow down
Step 4 (motion) Consider
119909119894(119905 + 1) = 119909
119894(119905) + V
119894(119905 + 1) (4)
where 119909119894(119905) is the position of the 119894th bicycle at time 119905
The lane-changing logic is shown below Before the accel-eration step both lanes are examined to evaluate lane-chang-ing opportunities The following conditions are checked foreach bicycle and must be true in order for it to change lanes
(1) The speed of the bicycle currently in 119894th position islarger than or equal to the cell distance to the nextbicycle This condition ensures that this bicycle willneed to slow down at the next update
V119894(119905) ge gap
119894(119905) (5)
(2) The distance to the next bicycle in the lane adjacent tothe lane of the 119894th bicycle (gap119891
119894(119905)) is larger than the
distance to the next bicycle in its current lane(gap119894(119905)) This condition ensures that a benefit is
derived from changing lanes
gap119891119894(119905) gt gap
119894(119905) (6)
(3) The distance to backward bicycle in the lane adjacentto that of the currently 119894th bicycle (gap119887
119894(119905)) is large
enough This condition ensures that looking back-wards the closest bicycle in the adjacent lane is suf-ficiently far away
gap119887119894(119905) ge min [V119887
119894minus1 (119905) + 1 V119887
119894minus1max] (7)
(4) A uniformly distributed random number between 0and 1 is less than the probability of a lane change (119901
119905)
rand () lt 119901119905 (8)
gap119891119894(119905) and gap119887
119894(119905) can be calculated as follows
gap119891119894(119905) = 119909
119891
119894+1 (119905) minus 119909119894 (119905) minus 1
gap119887119894(119905) = 119909
119894(119905) minus 119909
119887
119894minus1 (119905) minus 1(9)
where 119909119887119894minus1(119905) V
119887
119894minus1(119905) and V119887119894minus1max(119905) are the position speed
and maximum speed of the nearest following bicycle in thelane adjacent to that of the 119894th bicycle
The new speed for the bicycle currently in the 119894th positionafter lane-changing is calculated as follows
V1015840119894(119905 + 1) = min [V
119894(119905) + 1 gap119891
119894(119905) V119894max] (10)
where V1015840119894(119905 + 1) is the speed of this bicycle after the lane-
changingThe motion of the lane-changing bicycle is
1199091015840
119894(119905 + 1) = 119909
119894(119905) + V1015840
119894(119905 + 1) (11)
where 1199091015840119894(119905 + 1) is the position of the bicycle after the lane-
changing
23 M-CA Model A family of M-CA models has recentlybeen proposed byNishinari andTakahashi [20ndash22]The basicversion of the family is obtained fromanultradiscretization ofBurgersrsquo equation Therefore it is also called the Burgers CA(BCA) Previously BCA models were proposed for highwaytraffic Recent attempts have includedBCAmodels purportedto represent bicycle flow [12 13] adapted for the unobviouscar-following and lane-changing behavior in bicycle trafficIn order to make a comparison with the NS CA model theM-CAmodel for mixed bicycle flow is improved upon in thispaper
The numbers of RBs and EBs in location 119895 at time 119905 are119880119903
119895(119905) and 119880119890
119895(119905) respectively As shown in Section 21 RBs
with a maximum speed of 2 cellss and EBs with a maximumspeed of 3 cellss are considered in the simulation systemsTherefore the updating procedures are changed as follows
(1) all bicycles in location 119895 move to their next location119895+1 if the location is not fully occupied and EBs havepriority over RBs
(2) all bicycles that moved in procedure (1) can move tolocation 119895+2 if their next location is not fully occupiedafter procedure (1) and EBs again have priority overRBs
(3) only EBsmoved in procedure (2) canmove to location119895 + 3 if their next location is not fully occupied afterprocedure (2)
The numbers of RBs and EBs that move one location onfrom location 119895 at time 119905 in procedure (1) are 119887119903
119895(119905) and 119887119890
119895(119905)
respectively The numbers of RBs and EBs that move twolocations on from location 119895 at time 119905 are 119888119903
119895(119905) and 119888119890
119895(119905) resp-
ectively 119889119890119895(119905) represents the number of EBs that move three
locations on from location 119895 at time 119905 119871 is defined as the lanenumber of the simulation bicycle path The randomization
4 Discrete Dynamics in Nature and Society
effect on the RBs is introduced as follows 119888119903119895(119905 + 1) decreases
by 1 with probability 119901119889119903
if 119888119903119895(119905 + 1) gt 0 The randomization
effect on the EBs is as follows 119889119890119895(119905 + 1) decreases by 1 with
probability 119901119889119890
if 119889119890119895(119905 + 1) gt 0 The updating rules are as
follows
Step 1 Calculation of 119887119903119895(119905 + 1) 119887119890
119895(119905 + 1) and 119887
119895(119905 + 1) (119895 =
1 2 3 119870) is as follows
119887119890
119895(119905 + 1) = min (119880119890
119895(119905) 119871 minus119880
119895+1 (119905)) (12)
119887119903
119895(119905 + 1) = min (119880119903
119895(119905) 119871 minus119880
119895+1 (119905) minus 119887119890
119895(119905 + 1)) (13)
119887119895(119905 + 1) = 119887119903
119895(119905 + 1) + 119887119890
119895(119905 + 1) (14)
Step 2 Calculation of 119888119903119895(119905 + 1) 119888119890
119895(119905 + 1) and 119888
119895(119905 + 1) is as
follows
119888119890
119895(119905 + 1) = min (119887119890
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1))
(15)
119888119903
119895(119905 + 1) = min (119887119903
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1) minus 119888
119890
119895(119905 + 1))
(16)
If rand() lt 119901119889119903 then
119888119903
119895(119905 + 1) = max (119888119903
119895(119905 + 1) minus 1 0)
119888119895(119905 + 1) = 119888119903
119895(119905 + 1) + 119888119890
119895(119905 + 1)
(17)
In (12) and (15) 119887119890119895(119905 + 1) and 119888119890
119895(119905 + 1) are calculated first
because the EBs have priority over the RBs
Step 3 Calculation of 119889119895(119905 + 1) is as follows
119889119895(119905 + 1) = min (119888119890
119895(119905 + 1) 119871 minus119880
119895+3 (119905) minus 119887119895+2 (119905 + 1)
minus 119889119895minus2 (119905 + 1) + 119889119895minus3 (119905 + 1)
119880119903
119895(119905 + 1) = 119880119903
119895(119905) minus 119887
119903
119895(119905 + 1) + 119887119903
119895minus1 (119905 + 1)
minus 119888119903
119895minus1 (119905 + 1) + 119888119903
119895minus2 (119905 + 1)
119880119895(119905 + 1) = 119880119903
119895(119905 + 1) +119880119890
119895(119905 + 1)
(20)
where rand() is a uniformly distributed random numberbetween 0 and 1
3 Simulation Results
For the comparison of the NS CA model against the M-CAmodel the simulation parameters in both models should beset to the same values In the simulations a two-lane bicyclepath (119871 = 2) was selected with length 119870 = 500 cells (equal to1000m) In the initial conditions RBs and EBs are randomlydistributed on the road using the same random number forboth modelsThe default values of the random slowing downprobability (119901
119889) the probability of a lane change (119901
119905) and the
proportion of EBs (119901119890) are 02 08 and 05 respectively for
the NS CA model (as in previous studies [14]) The defaultvalues of the random slowing down probability of RBs (119901
119889119903)
the random slowing down probability of EBs (119901119889119890) and the
proportion of EBs are 04 04 and 05 respectively for theM-CAmodel In theM-CAmodel the slowing down probabilityis the probability that the number of bicycles (119888119903
119895(119905 + 1))
decreases which means that one bicycle decreases its speedIn this paper the simulation is based on two lanes (119871 = 2)Therefore the maximum value of 119888119903
119895(119905+1) is 2 If 119888119903
119895(119905+1) = 0
no bicycle slows down and the slowing down probability ofany bicycle is zero If 119888119903
119895(119905 + 1) = 1 only one bicycle slows
down with probability 119901 If 119888119903119895(119905+1) = 2 this means only one
bicycle may slow down with probability 119901 therefore the totalslowing down probability of bicycles is 05119901 By summing theabove three cases we assume these three cases have the samepercentage Therefore the mean of the three casesrsquo slowingdown probabilities is (0 + 119901 + 05119901)3 = 05119901 In orderto compare the two models we used a default value for therandom slowing down probability for the M-CA model ofhalf that for the NS CA model
Periodic conditions that are as close as possible to theactual conditions are used so that the bicycles ride on a circuitThe instantaneous positions and speeds for all particles areupdated in parallel per second The flow speed and densityof the mixed bicycle traffic flow can be calculated after agiven amount of time (20000 simulation steps) [15] and theaverages over the last 5000 steps are used for the calculationin order to decrease the random effect
31 Results of the NS CA Model In order to show the dif-ferent characteristics of the NS CA model under differentmodel parameters speed-density and flow-density plots (thefundamental diagram of bicycle traffic flow) were created sothat the results could be analyzed Example plots are shownin Figures 1 2 and 3 When 119901
119889= 0 it is a deterministic case
while 119901119889= 0 is a stochastic case From Figure 1 it can be seen
that with an increase in the slowing down probability 119901119889 the
fundamental diagrams drop quickly which means that thecapacity of the bicycle lane drops quickly with an increase in119901119889 When the slowing down probability 119901
119889is equal to one
the stopped RBs will lead bicycle traffic flow to jam and thespeed and flow will both be zero
Figure 2 shows the speed-density and flow-density rela-tionships under different lane-changing probabilities In thelow-density region with an increase in the lane-changingprobability the speed of bicycle flow increases In the high-density region the speeds of bicycle flow under different
Discrete Dynamics in Nature and Society 5
0
4
8
12
16
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 1 Speed-density and flow-density relationships under different slowing down probabilities when 119901119905= 08 and 119901
119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(b) Flow-density relationship
Figure 2 Speed-density and flow-density relationships under different lane-changing probabilities when 119901119889= 02 and 119901
119890= 05
6 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
flow Their simulation results showed that once the ran-domization effect is considered the multiple states in deter-ministic M-CA models disappear and unique flow-densityrelations exist They found the transition from free flow tocongested flow to be smooth in onemodel but of second orderin the other Lan and Chang [11] developed inhomogeneousCA models to elucidate the interacting movements of carsand motorcycles in mixed traffic contexts The car andmotorcycle were represented by nonidentical particle sizesrespectively occupying 6 times 2 and 2 times 1 cell units each of size125 times 125 meters The CA models were validated by a set offield-observed data and the relationships between flow celloccupancy (a proxy of density) and speeds under differenttraffic mixtures and road (lane) widths were elaborated AM-CA model for mixed bicycle flow was proposed by Jiaet al [12] Two types of bicycles with different maximumspeeds (1 cells and 2 cellss) were considered in the systemDifferent results were analyzed and investigated under bothdeterministic and stochastic regimes Li et al [13] presenteda multivalue cellular model for mixed nonmotorized trafficflow composed of bicycles and tricycles A bicycle wasassumed to occupy one unit of cell space and a tricycle twounits of cell spaceThe simulation results showed themultiplestate effect of mixed traffic flow Gould and Karner [14]proposed a two-lane inhomogeneous CA simulation modelan improved version of the NS model combining a lane-changing rule for bicycle traffic and collected field data fromthree UC Davis bike paths for comparison with a simulationmodel Yang et al [15] proposed an extended multivalue CAmodel that permitted the bicycles to move at faster speedsThe simulation results showed that the mixed nonmotorizedtraffic capacity increased with an increase in the electricbicycle ratio Zhang et al [16] used an improved three-laneNSmodel to analyze the speed-density characteristics of mixedbicycle flow The simulation results of the CA model wereeffectively consistent with the actual survey data when thedensity was lower than 0225 bicm2
Summarizing the above none of the aforementioned car-following models have been devoted to mixed traffic withregular bicycles (RBs) and electric bicycles (EBs) but CAmodels have been widely used for modeling nonmotorizedtrafficThemodeling ofmixed bicycle traffic usingCAmodelscan be divided into two branches NS CA model and M-CAmodel approaches The update rules of the NS CA model forbicycle flow are the same as for motorized vehicles with onlythe cell size and bicycle speeds being different As pointedout by Jiang et al [10] and Jia et al [12] the M-CA model ismore suitable than the NS model for modeling bicycle trafficflow Because the update rules of the M-CA model do notinclude direct car-following and lane-changing behavior itmay be appropriate formodeling the nonlane-based behaviorof bicycle traffic The NS CA model and M-CA model havebeen used for modeling bicycle traffic and mixed bicycletraffic with RBs and EBs However there is no evidence in theexisting literature as to which model is better for modelingmixed traffic flow nor as to the differences between these twomodels Therefore a comparison of the NS CA model andthe M-CA model in terms of their ability to model mixed
bicycle traffic is required so that CAmodels can be improvedefficiently
This paper attempts to develop twoCAmodels to describethe behaviors of mixed bicycle traffic with RBs and EBs ona separated bicycle path and to compare the characteristicsof the NS CA model and the M-CA model The remainingparts are organized as follows Section 2 introduces thedevelopment of NS and M-CA rules Section 3 presents thesimulation results of these two CA models Section 4 furtherdiscusses differences in the simulation results Finally theconclusions and ideas for future studies are addressed
2 CA Models
21 Definition of Cell Size and Bicycle Speed The maindifferences encountered when modeling bicycle traffic asopposed to motorized vehicle traffic using a CA model arethe cell size and the speed Mixed bicycle traffic with RBsand EBs on separated bicycle paths is ubiquitous in manyAsian countries such as China Vietnam Indonesia andMalaysia Because of the different operating speeds of RBsand EBs mixed traffic produces complicated behavior andcharacteristics that are likely to lead to safety and efficiencyproblems Modeling mixed bicycle traffic is very importantfor the planning operation and management of bicyclefacilities Based on the behavior of cyclists CA models arethe best option for modeling bicycle traffic The size of cellspace and the update rules are two significant aspects of CAmodels
Bicycles are shorter and narrower than motorized vehi-cles Based on field surveys the length of most RBs and EBsis 17ndash19m Meanwhile bicycle lanes are set at 1 meter widein both China and theUSA [17 18]Therefore the size of a RBor an EB is assumed rectangular with length 2m and width1m as is widely used in other CA models [14ndash16] The otherparameter for modeling bicycle traffic is speed According tothe literature the reported free flow speed of EBs is largerthan that of RBs Accordingly in this paper speeds of 2 cellss(4ms or 144 kmh) and 3 cellss (6ms or 216 kmh) werechosen for RBs and EBs respectively
22 NS CA Model The NS CA model used in this paperwas proposed by Nagel and Schreckenberg [9] This modelis very widely used in modeling highway traffic and bicycletraffic The NS CA model includes a car-following rule and alane-changing rule The car-following rule is based on foursteps and the lane-changing rule is based on the work ofRickert et al [19] Different vehicle behavior rules wouldlead to different simulation results With an increase in thenumber of lanes the lane-changing logic would becomemorecomplicated and make modeling more difficult Therefore inthis paper only a two-lane bicycle path is simulated and usedin the comparison In the time interval from 119905 to 119905+1 the fourbasic rules of the NS model evolve according to the followingsteps
Step 1 (longitudinal acceleration) Consider
V119894(119905 + 1) = min (V
119894(119905) + 1 V
119894max) (1)
Discrete Dynamics in Nature and Society 3
where V119894(119905) is the speed of the 119894th bicycle at updating time
119905 V119894max is the maximum speed of the 119894th bicycle This corre-
sponds to the cyclistsrsquo realistic free flow speed
Step 2 (longitudinal deceleration) Consider
V119894(119905 + 1) = min (V
119894(119905) gap
119894) (2)
where gap119894is the distance between the 119894th bicycle and the
bicycle in front of it at updating time 119905 This step ensures thatthe bicycle stays safe with no collisions
Step 3 (random slowing down) Consider
V119894(119905 + 1) = max (V
119894(119905) minus 1 0) if rand () lt 119901
119889119894 (3)
where rand() is a uniformly distributed random numberbetween 0 and 1 and 119901
119889119894is the random slowing down
probability of the 119894th bicycle The random slowing downeffect which captures one cyclistrsquos brakingmaneuver due to arandom event (eg accident road or weather related fac-tors) is one of the most significant parameters of the CAmodel This step incorporates the idea of random effects onbicycles that may cause them to slow down
Step 4 (motion) Consider
119909119894(119905 + 1) = 119909
119894(119905) + V
119894(119905 + 1) (4)
where 119909119894(119905) is the position of the 119894th bicycle at time 119905
The lane-changing logic is shown below Before the accel-eration step both lanes are examined to evaluate lane-chang-ing opportunities The following conditions are checked foreach bicycle and must be true in order for it to change lanes
(1) The speed of the bicycle currently in 119894th position islarger than or equal to the cell distance to the nextbicycle This condition ensures that this bicycle willneed to slow down at the next update
V119894(119905) ge gap
119894(119905) (5)
(2) The distance to the next bicycle in the lane adjacent tothe lane of the 119894th bicycle (gap119891
119894(119905)) is larger than the
distance to the next bicycle in its current lane(gap119894(119905)) This condition ensures that a benefit is
derived from changing lanes
gap119891119894(119905) gt gap
119894(119905) (6)
(3) The distance to backward bicycle in the lane adjacentto that of the currently 119894th bicycle (gap119887
119894(119905)) is large
enough This condition ensures that looking back-wards the closest bicycle in the adjacent lane is suf-ficiently far away
gap119887119894(119905) ge min [V119887
119894minus1 (119905) + 1 V119887
119894minus1max] (7)
(4) A uniformly distributed random number between 0and 1 is less than the probability of a lane change (119901
119905)
rand () lt 119901119905 (8)
gap119891119894(119905) and gap119887
119894(119905) can be calculated as follows
gap119891119894(119905) = 119909
119891
119894+1 (119905) minus 119909119894 (119905) minus 1
gap119887119894(119905) = 119909
119894(119905) minus 119909
119887
119894minus1 (119905) minus 1(9)
where 119909119887119894minus1(119905) V
119887
119894minus1(119905) and V119887119894minus1max(119905) are the position speed
and maximum speed of the nearest following bicycle in thelane adjacent to that of the 119894th bicycle
The new speed for the bicycle currently in the 119894th positionafter lane-changing is calculated as follows
V1015840119894(119905 + 1) = min [V
119894(119905) + 1 gap119891
119894(119905) V119894max] (10)
where V1015840119894(119905 + 1) is the speed of this bicycle after the lane-
changingThe motion of the lane-changing bicycle is
1199091015840
119894(119905 + 1) = 119909
119894(119905) + V1015840
119894(119905 + 1) (11)
where 1199091015840119894(119905 + 1) is the position of the bicycle after the lane-
changing
23 M-CA Model A family of M-CA models has recentlybeen proposed byNishinari andTakahashi [20ndash22]The basicversion of the family is obtained fromanultradiscretization ofBurgersrsquo equation Therefore it is also called the Burgers CA(BCA) Previously BCA models were proposed for highwaytraffic Recent attempts have includedBCAmodels purportedto represent bicycle flow [12 13] adapted for the unobviouscar-following and lane-changing behavior in bicycle trafficIn order to make a comparison with the NS CA model theM-CAmodel for mixed bicycle flow is improved upon in thispaper
The numbers of RBs and EBs in location 119895 at time 119905 are119880119903
119895(119905) and 119880119890
119895(119905) respectively As shown in Section 21 RBs
with a maximum speed of 2 cellss and EBs with a maximumspeed of 3 cellss are considered in the simulation systemsTherefore the updating procedures are changed as follows
(1) all bicycles in location 119895 move to their next location119895+1 if the location is not fully occupied and EBs havepriority over RBs
(2) all bicycles that moved in procedure (1) can move tolocation 119895+2 if their next location is not fully occupiedafter procedure (1) and EBs again have priority overRBs
(3) only EBsmoved in procedure (2) canmove to location119895 + 3 if their next location is not fully occupied afterprocedure (2)
The numbers of RBs and EBs that move one location onfrom location 119895 at time 119905 in procedure (1) are 119887119903
119895(119905) and 119887119890
119895(119905)
respectively The numbers of RBs and EBs that move twolocations on from location 119895 at time 119905 are 119888119903
119895(119905) and 119888119890
119895(119905) resp-
ectively 119889119890119895(119905) represents the number of EBs that move three
locations on from location 119895 at time 119905 119871 is defined as the lanenumber of the simulation bicycle path The randomization
4 Discrete Dynamics in Nature and Society
effect on the RBs is introduced as follows 119888119903119895(119905 + 1) decreases
by 1 with probability 119901119889119903
if 119888119903119895(119905 + 1) gt 0 The randomization
effect on the EBs is as follows 119889119890119895(119905 + 1) decreases by 1 with
probability 119901119889119890
if 119889119890119895(119905 + 1) gt 0 The updating rules are as
follows
Step 1 Calculation of 119887119903119895(119905 + 1) 119887119890
119895(119905 + 1) and 119887
119895(119905 + 1) (119895 =
1 2 3 119870) is as follows
119887119890
119895(119905 + 1) = min (119880119890
119895(119905) 119871 minus119880
119895+1 (119905)) (12)
119887119903
119895(119905 + 1) = min (119880119903
119895(119905) 119871 minus119880
119895+1 (119905) minus 119887119890
119895(119905 + 1)) (13)
119887119895(119905 + 1) = 119887119903
119895(119905 + 1) + 119887119890
119895(119905 + 1) (14)
Step 2 Calculation of 119888119903119895(119905 + 1) 119888119890
119895(119905 + 1) and 119888
119895(119905 + 1) is as
follows
119888119890
119895(119905 + 1) = min (119887119890
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1))
(15)
119888119903
119895(119905 + 1) = min (119887119903
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1) minus 119888
119890
119895(119905 + 1))
(16)
If rand() lt 119901119889119903 then
119888119903
119895(119905 + 1) = max (119888119903
119895(119905 + 1) minus 1 0)
119888119895(119905 + 1) = 119888119903
119895(119905 + 1) + 119888119890
119895(119905 + 1)
(17)
In (12) and (15) 119887119890119895(119905 + 1) and 119888119890
119895(119905 + 1) are calculated first
because the EBs have priority over the RBs
Step 3 Calculation of 119889119895(119905 + 1) is as follows
119889119895(119905 + 1) = min (119888119890
119895(119905 + 1) 119871 minus119880
119895+3 (119905) minus 119887119895+2 (119905 + 1)
minus 119889119895minus2 (119905 + 1) + 119889119895minus3 (119905 + 1)
119880119903
119895(119905 + 1) = 119880119903
119895(119905) minus 119887
119903
119895(119905 + 1) + 119887119903
119895minus1 (119905 + 1)
minus 119888119903
119895minus1 (119905 + 1) + 119888119903
119895minus2 (119905 + 1)
119880119895(119905 + 1) = 119880119903
119895(119905 + 1) +119880119890
119895(119905 + 1)
(20)
where rand() is a uniformly distributed random numberbetween 0 and 1
3 Simulation Results
For the comparison of the NS CA model against the M-CAmodel the simulation parameters in both models should beset to the same values In the simulations a two-lane bicyclepath (119871 = 2) was selected with length 119870 = 500 cells (equal to1000m) In the initial conditions RBs and EBs are randomlydistributed on the road using the same random number forboth modelsThe default values of the random slowing downprobability (119901
119889) the probability of a lane change (119901
119905) and the
proportion of EBs (119901119890) are 02 08 and 05 respectively for
the NS CA model (as in previous studies [14]) The defaultvalues of the random slowing down probability of RBs (119901
119889119903)
the random slowing down probability of EBs (119901119889119890) and the
proportion of EBs are 04 04 and 05 respectively for theM-CAmodel In theM-CAmodel the slowing down probabilityis the probability that the number of bicycles (119888119903
119895(119905 + 1))
decreases which means that one bicycle decreases its speedIn this paper the simulation is based on two lanes (119871 = 2)Therefore the maximum value of 119888119903
119895(119905+1) is 2 If 119888119903
119895(119905+1) = 0
no bicycle slows down and the slowing down probability ofany bicycle is zero If 119888119903
119895(119905 + 1) = 1 only one bicycle slows
down with probability 119901 If 119888119903119895(119905+1) = 2 this means only one
bicycle may slow down with probability 119901 therefore the totalslowing down probability of bicycles is 05119901 By summing theabove three cases we assume these three cases have the samepercentage Therefore the mean of the three casesrsquo slowingdown probabilities is (0 + 119901 + 05119901)3 = 05119901 In orderto compare the two models we used a default value for therandom slowing down probability for the M-CA model ofhalf that for the NS CA model
Periodic conditions that are as close as possible to theactual conditions are used so that the bicycles ride on a circuitThe instantaneous positions and speeds for all particles areupdated in parallel per second The flow speed and densityof the mixed bicycle traffic flow can be calculated after agiven amount of time (20000 simulation steps) [15] and theaverages over the last 5000 steps are used for the calculationin order to decrease the random effect
31 Results of the NS CA Model In order to show the dif-ferent characteristics of the NS CA model under differentmodel parameters speed-density and flow-density plots (thefundamental diagram of bicycle traffic flow) were created sothat the results could be analyzed Example plots are shownin Figures 1 2 and 3 When 119901
119889= 0 it is a deterministic case
while 119901119889= 0 is a stochastic case From Figure 1 it can be seen
that with an increase in the slowing down probability 119901119889 the
fundamental diagrams drop quickly which means that thecapacity of the bicycle lane drops quickly with an increase in119901119889 When the slowing down probability 119901
119889is equal to one
the stopped RBs will lead bicycle traffic flow to jam and thespeed and flow will both be zero
Figure 2 shows the speed-density and flow-density rela-tionships under different lane-changing probabilities In thelow-density region with an increase in the lane-changingprobability the speed of bicycle flow increases In the high-density region the speeds of bicycle flow under different
Discrete Dynamics in Nature and Society 5
0
4
8
12
16
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 1 Speed-density and flow-density relationships under different slowing down probabilities when 119901119905= 08 and 119901
119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(b) Flow-density relationship
Figure 2 Speed-density and flow-density relationships under different lane-changing probabilities when 119901119889= 02 and 119901
119890= 05
6 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
where V119894(119905) is the speed of the 119894th bicycle at updating time
119905 V119894max is the maximum speed of the 119894th bicycle This corre-
sponds to the cyclistsrsquo realistic free flow speed
Step 2 (longitudinal deceleration) Consider
V119894(119905 + 1) = min (V
119894(119905) gap
119894) (2)
where gap119894is the distance between the 119894th bicycle and the
bicycle in front of it at updating time 119905 This step ensures thatthe bicycle stays safe with no collisions
Step 3 (random slowing down) Consider
V119894(119905 + 1) = max (V
119894(119905) minus 1 0) if rand () lt 119901
119889119894 (3)
where rand() is a uniformly distributed random numberbetween 0 and 1 and 119901
119889119894is the random slowing down
probability of the 119894th bicycle The random slowing downeffect which captures one cyclistrsquos brakingmaneuver due to arandom event (eg accident road or weather related fac-tors) is one of the most significant parameters of the CAmodel This step incorporates the idea of random effects onbicycles that may cause them to slow down
Step 4 (motion) Consider
119909119894(119905 + 1) = 119909
119894(119905) + V
119894(119905 + 1) (4)
where 119909119894(119905) is the position of the 119894th bicycle at time 119905
The lane-changing logic is shown below Before the accel-eration step both lanes are examined to evaluate lane-chang-ing opportunities The following conditions are checked foreach bicycle and must be true in order for it to change lanes
(1) The speed of the bicycle currently in 119894th position islarger than or equal to the cell distance to the nextbicycle This condition ensures that this bicycle willneed to slow down at the next update
V119894(119905) ge gap
119894(119905) (5)
(2) The distance to the next bicycle in the lane adjacent tothe lane of the 119894th bicycle (gap119891
119894(119905)) is larger than the
distance to the next bicycle in its current lane(gap119894(119905)) This condition ensures that a benefit is
derived from changing lanes
gap119891119894(119905) gt gap
119894(119905) (6)
(3) The distance to backward bicycle in the lane adjacentto that of the currently 119894th bicycle (gap119887
119894(119905)) is large
enough This condition ensures that looking back-wards the closest bicycle in the adjacent lane is suf-ficiently far away
gap119887119894(119905) ge min [V119887
119894minus1 (119905) + 1 V119887
119894minus1max] (7)
(4) A uniformly distributed random number between 0and 1 is less than the probability of a lane change (119901
119905)
rand () lt 119901119905 (8)
gap119891119894(119905) and gap119887
119894(119905) can be calculated as follows
gap119891119894(119905) = 119909
119891
119894+1 (119905) minus 119909119894 (119905) minus 1
gap119887119894(119905) = 119909
119894(119905) minus 119909
119887
119894minus1 (119905) minus 1(9)
where 119909119887119894minus1(119905) V
119887
119894minus1(119905) and V119887119894minus1max(119905) are the position speed
and maximum speed of the nearest following bicycle in thelane adjacent to that of the 119894th bicycle
The new speed for the bicycle currently in the 119894th positionafter lane-changing is calculated as follows
V1015840119894(119905 + 1) = min [V
119894(119905) + 1 gap119891
119894(119905) V119894max] (10)
where V1015840119894(119905 + 1) is the speed of this bicycle after the lane-
changingThe motion of the lane-changing bicycle is
1199091015840
119894(119905 + 1) = 119909
119894(119905) + V1015840
119894(119905 + 1) (11)
where 1199091015840119894(119905 + 1) is the position of the bicycle after the lane-
changing
23 M-CA Model A family of M-CA models has recentlybeen proposed byNishinari andTakahashi [20ndash22]The basicversion of the family is obtained fromanultradiscretization ofBurgersrsquo equation Therefore it is also called the Burgers CA(BCA) Previously BCA models were proposed for highwaytraffic Recent attempts have includedBCAmodels purportedto represent bicycle flow [12 13] adapted for the unobviouscar-following and lane-changing behavior in bicycle trafficIn order to make a comparison with the NS CA model theM-CAmodel for mixed bicycle flow is improved upon in thispaper
The numbers of RBs and EBs in location 119895 at time 119905 are119880119903
119895(119905) and 119880119890
119895(119905) respectively As shown in Section 21 RBs
with a maximum speed of 2 cellss and EBs with a maximumspeed of 3 cellss are considered in the simulation systemsTherefore the updating procedures are changed as follows
(1) all bicycles in location 119895 move to their next location119895+1 if the location is not fully occupied and EBs havepriority over RBs
(2) all bicycles that moved in procedure (1) can move tolocation 119895+2 if their next location is not fully occupiedafter procedure (1) and EBs again have priority overRBs
(3) only EBsmoved in procedure (2) canmove to location119895 + 3 if their next location is not fully occupied afterprocedure (2)
The numbers of RBs and EBs that move one location onfrom location 119895 at time 119905 in procedure (1) are 119887119903
119895(119905) and 119887119890
119895(119905)
respectively The numbers of RBs and EBs that move twolocations on from location 119895 at time 119905 are 119888119903
119895(119905) and 119888119890
119895(119905) resp-
ectively 119889119890119895(119905) represents the number of EBs that move three
locations on from location 119895 at time 119905 119871 is defined as the lanenumber of the simulation bicycle path The randomization
4 Discrete Dynamics in Nature and Society
effect on the RBs is introduced as follows 119888119903119895(119905 + 1) decreases
by 1 with probability 119901119889119903
if 119888119903119895(119905 + 1) gt 0 The randomization
effect on the EBs is as follows 119889119890119895(119905 + 1) decreases by 1 with
probability 119901119889119890
if 119889119890119895(119905 + 1) gt 0 The updating rules are as
follows
Step 1 Calculation of 119887119903119895(119905 + 1) 119887119890
119895(119905 + 1) and 119887
119895(119905 + 1) (119895 =
1 2 3 119870) is as follows
119887119890
119895(119905 + 1) = min (119880119890
119895(119905) 119871 minus119880
119895+1 (119905)) (12)
119887119903
119895(119905 + 1) = min (119880119903
119895(119905) 119871 minus119880
119895+1 (119905) minus 119887119890
119895(119905 + 1)) (13)
119887119895(119905 + 1) = 119887119903
119895(119905 + 1) + 119887119890
119895(119905 + 1) (14)
Step 2 Calculation of 119888119903119895(119905 + 1) 119888119890
119895(119905 + 1) and 119888
119895(119905 + 1) is as
follows
119888119890
119895(119905 + 1) = min (119887119890
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1))
(15)
119888119903
119895(119905 + 1) = min (119887119903
119895(119905 + 1) 119871 minus119880
119895+2 (119905) minus 119887119895+1 (119905 + 1)
+ 119887119895+2 (119905 + 1) minus 119888
119890
119895(119905 + 1))
(16)
If rand() lt 119901119889119903 then
119888119903
119895(119905 + 1) = max (119888119903
119895(119905 + 1) minus 1 0)
119888119895(119905 + 1) = 119888119903
119895(119905 + 1) + 119888119890
119895(119905 + 1)
(17)
In (12) and (15) 119887119890119895(119905 + 1) and 119888119890
119895(119905 + 1) are calculated first
because the EBs have priority over the RBs
Step 3 Calculation of 119889119895(119905 + 1) is as follows
119889119895(119905 + 1) = min (119888119890
119895(119905 + 1) 119871 minus119880
119895+3 (119905) minus 119887119895+2 (119905 + 1)
minus 119889119895minus2 (119905 + 1) + 119889119895minus3 (119905 + 1)
119880119903
119895(119905 + 1) = 119880119903
119895(119905) minus 119887
119903
119895(119905 + 1) + 119887119903
119895minus1 (119905 + 1)
minus 119888119903
119895minus1 (119905 + 1) + 119888119903
119895minus2 (119905 + 1)
119880119895(119905 + 1) = 119880119903
119895(119905 + 1) +119880119890
119895(119905 + 1)
(20)
where rand() is a uniformly distributed random numberbetween 0 and 1
3 Simulation Results
For the comparison of the NS CA model against the M-CAmodel the simulation parameters in both models should beset to the same values In the simulations a two-lane bicyclepath (119871 = 2) was selected with length 119870 = 500 cells (equal to1000m) In the initial conditions RBs and EBs are randomlydistributed on the road using the same random number forboth modelsThe default values of the random slowing downprobability (119901
119889) the probability of a lane change (119901
119905) and the
proportion of EBs (119901119890) are 02 08 and 05 respectively for
the NS CA model (as in previous studies [14]) The defaultvalues of the random slowing down probability of RBs (119901
119889119903)
the random slowing down probability of EBs (119901119889119890) and the
proportion of EBs are 04 04 and 05 respectively for theM-CAmodel In theM-CAmodel the slowing down probabilityis the probability that the number of bicycles (119888119903
119895(119905 + 1))
decreases which means that one bicycle decreases its speedIn this paper the simulation is based on two lanes (119871 = 2)Therefore the maximum value of 119888119903
119895(119905+1) is 2 If 119888119903
119895(119905+1) = 0
no bicycle slows down and the slowing down probability ofany bicycle is zero If 119888119903
119895(119905 + 1) = 1 only one bicycle slows
down with probability 119901 If 119888119903119895(119905+1) = 2 this means only one
bicycle may slow down with probability 119901 therefore the totalslowing down probability of bicycles is 05119901 By summing theabove three cases we assume these three cases have the samepercentage Therefore the mean of the three casesrsquo slowingdown probabilities is (0 + 119901 + 05119901)3 = 05119901 In orderto compare the two models we used a default value for therandom slowing down probability for the M-CA model ofhalf that for the NS CA model
Periodic conditions that are as close as possible to theactual conditions are used so that the bicycles ride on a circuitThe instantaneous positions and speeds for all particles areupdated in parallel per second The flow speed and densityof the mixed bicycle traffic flow can be calculated after agiven amount of time (20000 simulation steps) [15] and theaverages over the last 5000 steps are used for the calculationin order to decrease the random effect
31 Results of the NS CA Model In order to show the dif-ferent characteristics of the NS CA model under differentmodel parameters speed-density and flow-density plots (thefundamental diagram of bicycle traffic flow) were created sothat the results could be analyzed Example plots are shownin Figures 1 2 and 3 When 119901
119889= 0 it is a deterministic case
while 119901119889= 0 is a stochastic case From Figure 1 it can be seen
that with an increase in the slowing down probability 119901119889 the
fundamental diagrams drop quickly which means that thecapacity of the bicycle lane drops quickly with an increase in119901119889 When the slowing down probability 119901
119889is equal to one
the stopped RBs will lead bicycle traffic flow to jam and thespeed and flow will both be zero
Figure 2 shows the speed-density and flow-density rela-tionships under different lane-changing probabilities In thelow-density region with an increase in the lane-changingprobability the speed of bicycle flow increases In the high-density region the speeds of bicycle flow under different
Discrete Dynamics in Nature and Society 5
0
4
8
12
16
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 1 Speed-density and flow-density relationships under different slowing down probabilities when 119901119905= 08 and 119901
119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(b) Flow-density relationship
Figure 2 Speed-density and flow-density relationships under different lane-changing probabilities when 119901119889= 02 and 119901
119890= 05
6 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
minus 119889119895minus2 (119905 + 1) + 119889119895minus3 (119905 + 1)
119880119903
119895(119905 + 1) = 119880119903
119895(119905) minus 119887
119903
119895(119905 + 1) + 119887119903
119895minus1 (119905 + 1)
minus 119888119903
119895minus1 (119905 + 1) + 119888119903
119895minus2 (119905 + 1)
119880119895(119905 + 1) = 119880119903
119895(119905 + 1) +119880119890
119895(119905 + 1)
(20)
where rand() is a uniformly distributed random numberbetween 0 and 1
3 Simulation Results
For the comparison of the NS CA model against the M-CAmodel the simulation parameters in both models should beset to the same values In the simulations a two-lane bicyclepath (119871 = 2) was selected with length 119870 = 500 cells (equal to1000m) In the initial conditions RBs and EBs are randomlydistributed on the road using the same random number forboth modelsThe default values of the random slowing downprobability (119901
119889) the probability of a lane change (119901
119905) and the
proportion of EBs (119901119890) are 02 08 and 05 respectively for
the NS CA model (as in previous studies [14]) The defaultvalues of the random slowing down probability of RBs (119901
119889119903)
the random slowing down probability of EBs (119901119889119890) and the
proportion of EBs are 04 04 and 05 respectively for theM-CAmodel In theM-CAmodel the slowing down probabilityis the probability that the number of bicycles (119888119903
119895(119905 + 1))
decreases which means that one bicycle decreases its speedIn this paper the simulation is based on two lanes (119871 = 2)Therefore the maximum value of 119888119903
119895(119905+1) is 2 If 119888119903
119895(119905+1) = 0
no bicycle slows down and the slowing down probability ofany bicycle is zero If 119888119903
119895(119905 + 1) = 1 only one bicycle slows
down with probability 119901 If 119888119903119895(119905+1) = 2 this means only one
bicycle may slow down with probability 119901 therefore the totalslowing down probability of bicycles is 05119901 By summing theabove three cases we assume these three cases have the samepercentage Therefore the mean of the three casesrsquo slowingdown probabilities is (0 + 119901 + 05119901)3 = 05119901 In orderto compare the two models we used a default value for therandom slowing down probability for the M-CA model ofhalf that for the NS CA model
Periodic conditions that are as close as possible to theactual conditions are used so that the bicycles ride on a circuitThe instantaneous positions and speeds for all particles areupdated in parallel per second The flow speed and densityof the mixed bicycle traffic flow can be calculated after agiven amount of time (20000 simulation steps) [15] and theaverages over the last 5000 steps are used for the calculationin order to decrease the random effect
31 Results of the NS CA Model In order to show the dif-ferent characteristics of the NS CA model under differentmodel parameters speed-density and flow-density plots (thefundamental diagram of bicycle traffic flow) were created sothat the results could be analyzed Example plots are shownin Figures 1 2 and 3 When 119901
119889= 0 it is a deterministic case
while 119901119889= 0 is a stochastic case From Figure 1 it can be seen
that with an increase in the slowing down probability 119901119889 the
fundamental diagrams drop quickly which means that thecapacity of the bicycle lane drops quickly with an increase in119901119889 When the slowing down probability 119901
119889is equal to one
the stopped RBs will lead bicycle traffic flow to jam and thespeed and flow will both be zero
Figure 2 shows the speed-density and flow-density rela-tionships under different lane-changing probabilities In thelow-density region with an increase in the lane-changingprobability the speed of bicycle flow increases In the high-density region the speeds of bicycle flow under different
Discrete Dynamics in Nature and Society 5
0
4
8
12
16
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 1 Speed-density and flow-density relationships under different slowing down probabilities when 119901119905= 08 and 119901
119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(b) Flow-density relationship
Figure 2 Speed-density and flow-density relationships under different lane-changing probabilities when 119901119889= 02 and 119901
119890= 05
6 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 1 Speed-density and flow-density relationships under different slowing down probabilities when 119901119905= 08 and 119901
119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pt = 0
pt = 02
pt = 04
pt = 06
pt = 08
(b) Flow-density relationship
Figure 2 Speed-density and flow-density relationships under different lane-changing probabilities when 119901119889= 02 and 119901
119890= 05
6 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 3 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 02 and 119901
119905= 08
lane-changing probabilities show smaller differences than inthe low-density region This is due to using the motorizedvehicle lane-changing rule for bicycle traffic flow The lane-changing rule proposed in this paper is very strictly based ongap acceptance theory and the looking-backward gap shouldbe large in order for lane-changing to happen However thedriving behavior of bicycles is very different from that ofmotorized vehicles The lane-changing rule for motorizedvehicles will restrict the lane-changing behavior of higher-speed bicycles (such as EBs in this case) and lead to nosignificant differences between the fundamental diagrams fordifferent lane-changing probabilities
The proportion of EBs is one of the most importantparameters for mixed bicycle traffic flow Figure 3 shows thefundamental diagrams for different proportions of EBs in themixed traffic It can be seen that with an increase in theproportion of EBs speed and capacity increase because of theEBsrsquo higher free flow speed compared to the RBs Anotherfinding observable in Figure 3 is that when 119901
119890is small the
influence on capacity is small and with the increase in 119901119890 the
influence of 119901119890on capacity becomes larger This means that
the influence of EBs on the bicycle lane capacity is not linearwhich may be due to the existence of lower-speed RBs andthe strict lane-changing rule which deter EBs from changinglanes and increasing their speed
32 Results of the M-CA Model This section presents thesimulation results of the M-CA model In order to comparethe results with those of the NS CAmodel the slowing downprobabilities of RBs and EBs are set to the same value of
04 Therefore only two parameters 119901119889and 119901
119890 are analyzed
in this simulation case Figure 4 shows the fundamentaldiagrams under different 119901
119889values Similarly to the NS CA
model the capacities drop with the increase in the slowingdown probability However the capacity drops of the M-CAmodel are smaller than those of the NS CA model as will bediscussed in detail in the next section
From Figure 5(a) in the low-density region (density lt200 bicycleskm per lane) the bicycle flow is in the free flowstate and most bicycles move independently Therefore theaverage speed of the system equals the free flow speed ofmixed bicycles which increases with the proportion of EBsAs can be seen in Figure 5(b) similar to the case of the NSCA model with an increase in the proportion of EBs thebicycle capacity also increases It is easy to see that the freeflow speed of all bicycles will increase with the proportion ofEBs The simulation results from the NS CA model and theM-CA model show the same findings
4 Discussion
Most of the previous studies on CAmodels for mixed bicycleflow discuss multiple states and the transition from free flowto congested flow However the choice of an appropriate CAmodel for simulating bicycle traffic is more important thanmodel analysis and calibrationTherefore we should comparethe simulation results of the two CAmodels presented aboveand try to draw conclusions about the selection of a CAmodel
Discrete Dynamics in Nature and Society 7
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pd = 0
pd = 02
pd = 04
pd = 06
pd = 08
pd = 10
(b) Flow-density relationship
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 4 Speed-density and flow-density relationships under different slowing down probabilities of RBs and EBs when 119901119890= 05
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
pe = 0
pe = 02
pe = 04
pe = 06
pe = 08
pe = 10
(b) Flow-density relationship
Figure 5 Speed-density and flow-density relationships under different proportions of EBs when 119901119889= 04
8 Discrete Dynamics in Nature and Society
0
4
8
12
16
20
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
TotalR-bikeE-bike
(a) NS CA model
0
4
8
12
16
20
24
0 100 200 300 400 500
Spee
d (k
mh
)Density (bicycleskm per lane)
TotalR-bikeE-bike
(b) M-CA model
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 6 Speed-density relationships for different bicycle types
Because the capacity is a significant parameter for bicyclelane planning and management capacity and fundamentaldiagrams can be used for evaluating the CAmodels [23] Thecapacity is defined as the maximum flow under particularroad conditions In the fundamental diagram the capacity isthe peak of the fundamental diagram curve For simplicitydue to the fact that the simulation densities of bicycle trafficflow cover all traffic conditions we use the maximum flow ofbicycles as the observed capacity of the bicycle path in thispaper
Figure 6 shows the comparisons of the speed-densityrelationships for the NS CA and M-CA models It can easilybe seen that for the NS CA model in the low-density region(approximately 100 bicycleskm per lane or fewer) the speedsof RBs and EBs are very different while in the high-densityregion the speeds of RBs and EBs are almost equal Similarconclusions are found for the M-CA model However thecritical density distinguishing low from high density is nearly300 bicycleskm per lane much larger than that for the NSCA model The results show that the lane-changing rule ofthe NS CAmodel enables the EBs hardly to pass the RBs andthe speeds of both bicycle types to quickly become the same
Figure 7 shows the simulated capacity values obtainedfrom the maximum values of the fundamental diagrams Itcan be seen that when the slowing down probability is zero(meaning that both models are deterministic CA models)the two CA models have the same capacity (NS CA model2387 bicyclesh per lane M-CA model 2375 bicyclesh perlane) With the increase in the slowing down probability thecapacities of both models drop linearly Linear regression
equations are also shown in Figure 7 and it can clearlybe seen that there are strongly linear relationships betweenthe capacities of the two models and the slowing downprobabilities However the difference between the regressionmodel slopes of the two models is large (minus40727 versusminus20453) Therefore when the slowing down probability ofthe M-CA model equals 1 which means that the slowingdown probability of the NS CA model is nearly 05 themaximumvolumes of the twomodels are very different (2000versus 1000 bicyclesh per lane) This means that the slowingdown probability has a greater influence on the NS CAmodelthan the M-CA model
The slowing down probability as the most significantparameter of the CA model describes the stochastic effectson bicycle traffic flow Because of the strict lane-changing rulein the NS CAmodel the EBs (fast bicycles) do not find it easyto change lanes andmust follow the RBs which leads to a verylow capacity However theM-CAmodel has an implied lane-changing rule in the update rules which has less influence oncapacity than in the NS CA model
Field bicycle data were collected at Jiaogong Road inHangzhou China The width of this bicycle path is 227mmaking it nearly a two-lane bicycle path The field data coverall traffic conditions and the EB percentages also cover awide range The average percentage of EBs is 603 andthe capacity of the bicycle path is defined as the maximumvolume with a 30-second sampling interval Figure 8 showsthe observed and simulated speed-density and flow-densityrelationships where the percentage of EBs is set to 06(equal to the field sample percentage) and other simulation
Discrete Dynamics in Nature and Society 9
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus20543x + 21656
R2 = 09779
(a) NS CA model
0
500
1000
1500
2000
2500
3000
0 02 04 06 08 1
Capa
city
(bic
ycle
sh
per l
ane)
Slowdown probability
y = minus40727x + 24001
R2 = 0982
(b) M-CA model
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 7 Simulated capacities of two CA models under different slowing down probabilities
0
5
10
15
20
25
30
35
0 100 200 300 400 500
Spee
d (k
mh
)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(a) Speed-density relationship
0
500
1000
1500
2000
2500
3000
3500
0 100 200 300 400 500
Flow
(bic
ycle
sh
per l
ane)
Density (bicycleskm per lane)
ObservationsNS modelMultivalue model
(b) Flow-density relationship
Figure 8 Observed and simulated speed-density and flow-density relationships
parameters are set to the default values presented above Theresults imply that the M-CA model performs better than theNS model in fitting the field bicycle observations
The proportion of EBs describes the proportion of fastbicycles in the mixed bicycle traffic which affects the freeflow speed and the capacity of the bicycle lane Figure 9 shows
the relationships between the proportion of EBs and the lanecapacity from observed and simulated results It can be seenthat both models produce the same nonlinear relationshipand the correlation coefficients are both very high When theproportion of EBs is low the EBs must slow down their speedand follow the RBs Therefore the capacity slowly increases
10 Discrete Dynamics in Nature and Society
1000
1500
2000
2500
3000
0 02 04 06 08 1Proportion of EBs
NS CAM-CAObservations
Capa
city
(bic
ycle
sh
per l
ane) y = 46804x2 minus 12413x + 2276
R2 = 09806
y = 40301x2 minus 13598x + 16092
R2 = 09177
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
Figure 9 Observed and simulated capacities under different pro-portions of EBs
with the proportion of EBs When the proportion of EBs islarge lane-changing and passing occur more frequently andthe capacity increases quickly with the proportion of EBsThesimulated results of the M-CA model seem more consistentwith the observed field bicycle capacity than those of the NSCAmodelThe rootmean square error (RMSE) and themeanabsolute percentage error (MAPE) [24 25] of M-CA modelare less than those of NS CA model
Based on the above comparison and analysis some con-clusions can be drawn Firstly CA models can be used forbicycle traffic simulation because of their simple rules andquick simulation The results for the fundamental diagramsand capacities (about 2000ndash2500 bicyclesh per lane) aresimilar to those from the field data and previous studies[26] Secondly the slowing down probability has a significantinfluence on the simulation results for both the NS CAmodeland the M-CA model Meanwhile with an increase in theslowing down probability the capacity and speed drop morequickly in the NS CA model than in the M-CA model Thismay be due to the lane-changing rule of NS CA models thatrestricts EBs in changing lanes and accelerating Thirdly theproportion of EBs in the mixed traffic flow affects the criticaldensity and capacity in both the NS CAmodel and theM-CAmodel as was also reported by some researchers [27 28] Asthe proportion of EBs moves from 0 to 1 the capacities of theNSCAmodel and theM-CAmodel increase 193 and 174respectively Fourthly for the NS CA model the probabilityof lane-changing has less influence on the mixed traffic flowthan does the slowing down probability which may be due tothe strict lane-changing rule leading to less lane-changing bybicycles Lastly the NS CA model is restricted for multilanebicycle path simulation becausemore bicycle laneswill lead tomore complicated lane-changing rules that are hard to modeland calibrate In contrast with the M-CA model it is easy tosimulate a multilane bicycle path by simply setting different119871 values As can be seen from the above summary the M-CAmodel provides more effective performance for modeling
bicycle traffic and ismore consistentwith the field bicycle datathan the NS CA model
5 Conclusions
Themodeling and simulation of mixed bicycle traffic flow arebecoming increasingly significant because of the increasedpopularity of regular bicycles and electric bicycles in recentyears due to their greenness and convenienceThis paper hasproposed two improved CA models for bicycle traffic flowmodeling and simulation and has compared their character-istics The two-lane NS CA model and the multivalue CAmodel for mixed bicycle traffic flow were introduced and thesame parameters set for both models so that a comparisoncould be made under the same conditions Speed-densityand flow-density relations were obtained so as to compareand analyze the models and the capacities obtained from thesimulation results were also compared under different modelparameters Field data collected fromHangzhou China wereused for the evaluation of the proposed models The resultsshow that the M-CA model performs better than the NSCA model in simulating mixed bicycle traffic The maindifference between these twomodels is the lane-changing andslowing down probability rules making it harder or easier forEBs to change lanes and accelerate to their free flow speed
Because of the difficulty of collecting bicycle field dataespecially in congested traffic conditions the calibrationand validation of the proposed model using field data wereomitted from this paper and only simulation results wereanalyzed and compared between the two models Futurework will focus on the calibration of the slowing downprobabilities and lane-changing probabilities under differenttraffic conditions so as to further validate and evaluate theproposed models
Conflict of Interests
The authors declare that there is no conflict of commercial orassociative interests regarding the publication of this work
Acknowledgments
This work was supported by the National Natural ScienceFoundation of China (nos 51338008 51278454 51208462and 61304191) the Fundamental Research Funds for the Cen-tral Universities (2014QNA4018) the Projects in the NationalScience amp Technology Pillar Program (2014BAG03B05) andthe Key Science and Technology Innovation Team of Zhe-jiang Province (2013TD09)
References
[1] A D May Traffic Flow Fundamentals Prentice-Hall Engle-wood Cliffs NJ USA 1990
[2] D C Gazis R Herman and R W Rothery ldquoNonlinear follow-the-leadermodels of traffic flowrdquoOperations Research vol 9 no4 pp 545ndash567 1961
Discrete Dynamics in Nature and Society 11
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
[3] S Jin D-H Wang Z-Y Huang and P-F Tao ldquoVisual anglemodel for car-following theoryrdquo Physica A StatisticalMechanicsand Its Applications vol 390 no 11 pp 1931ndash1940 2011
[4] S Jin D-H Wang and X-R Yang ldquoNon-lane-based car-fol-lowing model with visual angle informationrdquo TransportationResearch Record Journal of the Transportation Research Boardno 2249 pp 7ndash14 2011
[5] S Jin D-H Wang C Xu and Z-Y Huang ldquoStaggered car-following induced by lateral separation effects in traffic flowrdquoPhysics Letters A vol 376 no 3 pp 153ndash157 2012
[6] M Brackstone and M McDonald ldquoCar-following a histori-cal reviewrdquo Transportation Research F Traffic Psychology andBehaviour vol 2 no 4 pp 181ndash196 1999
[7] D Chowdhury L Santen and A Schadschneider ldquoStatisticalphysics of vehicular traffic and some related systemsrdquo PhysicsReports vol 329 no 4ndash6 pp 199ndash329 2000
[8] S Wolfram Theory and Applications of Cellular AutomataAdvanced Series on Complex Systems World Scientific Singa-pore 1986
[9] K Nagel and M Schreckenberg ldquoA cellular automaton modelfor freeway trafficrdquo Journal de Physique I France vol 2 no 12pp 2221ndash2229 1992
[10] R Jiang B Jia and Q-S Wu ldquoStochastic multi-value cellularautomata models for bicycle flowrdquo Journal of Physics A Mathe-matical and General vol 37 no 6 pp 2063ndash2072 2004
[11] LW Lan and C-W Chang ldquoInhomogeneous cellular automatamodeling for mixed traffic with cars and motorcyclesrdquo Journalof Advanced Transportation vol 39 no 3 pp 323ndash349 2005
[12] B Jia X-G Li R Jiang and Z-Y Gao ldquoMulti-value cellularautomata model for mixed bicycle flowrdquoThe European PhysicalJournal B vol 56 no 3 pp 247ndash252 2007
[13] X-G Li Z-Y Gao X-M Zhao and B Jia ldquoMulti-value cellularautomata model for mixed non-motorized traffic flowrdquo ActaPhysica Sinica vol 57 no 8 pp 4777ndash4785 2008
[14] G Gould and A Karner ldquoModeling bicycle facility operationcellular automaton approachrdquo Transportation Research Recordno 2140 pp 157ndash164 2009
[15] X-F Yang Z-Y Niu and J-R Wang ldquoMixed non-motorizedtraffic flow capacity based on multi-value cellular automatamodelrdquo Journal of System Simulation vol 24 no 12 pp 2577ndash2581 2012
[16] S Zhang G Ren and R Yang ldquoSimulation model of speed-density characteristics for mixed bicycle flowmdashcomparisonbetween cellular automata model and gas dynamics modelrdquoPhysica A Statistical Mechanics and Its Applications vol 392no 20 pp 5110ndash5118 2013
[17] Ministry of Housing and Urban-Rural Development of China(MOHURD) ldquoCode for design of urban road engineeringrdquoTech Rep CJJ37-2012 MOHURD 2012
[18] Transportation Research Board Highway Capacity Manual(2010) National Research Council Washington DC USA2010
[19] M Rickert K Nagel M Schreckenberg and A Latour ldquoTwolane traffic simulations using cellular automatardquo Physica A vol231 no 4 pp 534ndash550 1996
[20] K Nishinari and D Takahashi ldquoAnalytical properties of ultra-discrete Burgers equation and rule-184 cellular automatonrdquoJournal of Physics A vol 31 no 24 pp 5439ndash5450 1998
[21] K Nishinari and D Takahashi ldquoA new deterministic CAmodelfor traffic flowwith multiple statesrdquo Journal of Physics A vol 32no 1 pp 93ndash104 1999
[22] K Nishinari and D Takahashi ldquoMulti-value cellular automatonmodels and metastable states in a congested phaserdquo Journal ofPhysics A Mathematical and General vol 33 no 43 pp 7709ndash7720 2000
[23] X Qu SWang and J Zhang ldquoOn the fundamental diagram forfreeway traffic a novel calibration approach for single-regimemodelsrdquo Transportation Research Part B Methodological vol73 pp 91ndash102 2015
[24] S Jin D-H Wang C Xu and D-F Ma ldquoShort-term trafficsafety forecasting using Gaussian mixture model and Kalmanfilterrdquo Journal of Zhejiang University SCIENCE A vol 14 no 4pp 231ndash243 2013
[25] Y Kuang X Qu and S Wang ldquoA tree-structured crash surro-gate measure for freewaysrdquo Accident Analysis amp Prevention vol77 pp 137ndash148 2015
[26] S Jin X Qu D Zhou C Xu D Ma and D Wang ldquoEstimatingcycleway capacity and bicycle equivalent unit for electric bicy-clesrdquo Transportation Research Part A vol 77 pp 225ndash248 2015
[27] D Wang D Zhou S Jin and D Ma ldquoCharacteristics of mixedbicycle traffic flow on conventional bicycle pathrdquo in Proceedingsof the 94thAnnualMeeting of the TransportationResearch BoardWashington DC USA 2015
[28] D Zhou C Xu D Wang and S Jin ldquoEstimating capacityof bicycle path on urban roads in Hangzhou Chinardquo inProceedings of the 94th Annual Meeting of the TransportationResearch Board Washington DC USA 2015
