Post on 14-Apr-2018
transcript
7/30/2019 L15 16 Queueing
1/61
21012101--495495Advanced Topics in CE IAdvanced Topics in CE I
Intro to Queuing TheoryIntro to Queuing Theory
Manoj Lohatepanont, Sc.D.Manoj Lohatepanont, Sc.D.
Chulalongkorn UniversityChulalongkorn University
7/30/2019 L15 16 Queueing
2/61
Chulalongkorn University 2Manoj Lohatepanont 2007
OutlineOutline Introduction to QueuingIntroduction to Queuing
ApplicationsApplications Queuing SystemQueuing System
DeterministicDeterministic
StochasticStochastic Probability DistributionProbability Distribution
LittleLittles Laws Law
Queuing SystemsQueuing Systems
M/M/1M/M/1 M/G/1M/G/1
M/M/sM/M/s
7/30/2019 L15 16 Queueing
3/61
Chulalongkorn University 3Manoj Lohatepanont 2007
Queuing TheoryQueuing Theory What is Queuing Theory?What is Queuing Theory?
A mathematical study ofA mathematical study ofqueuesqueues QueueQueueis a waiting line especially of persons or vehiclesis a waiting line especially of persons or vehicles
awaiting processingawaiting processing
When does queue occur?When does queue occur? A queue forms when current demand for a service exceedsA queue forms when current demand for a service exceeds
the current capacity to provide that servicethe current capacity to provide that service
Why study Queuing Theory?Why study Queuing Theory? Help provide decision or guideline on system capacityHelp provide decision or guideline on system capacity
7/30/2019 L15 16 Queueing
4/61
Chulalongkorn University 4Manoj Lohatepanont 2007
Why Is Queuing Decision Difficult?Why Is Queuing Decision Difficult?
ProbabilisticProbabilistic nature of the problemnature of the problem
It is often impossible to predict accurately when andIt is often impossible to predict accurately when andhow many customers will show up, and how long ithow many customers will show up, and how long it
will take to service each customerwill take to service each customer
E.g., checkout queues at supermarketsE.g., checkout queues at supermarkets Providing too much capacity costs moneyProviding too much capacity costs money
Providing too little capacity leads to long wait inProviding too little capacity leads to long wait in
queues, which in turn leads to other undesirablequeues, which in turn leads to other undesirableconsequencesconsequences
7/30/2019 L15 16 Queueing
5/61
Chulalongkorn University 5Manoj Lohatepanont 2007
How Does Queuing Theory Help?How Does Queuing Theory Help?
Queuing theory itself does not directly provideQueuing theory itself does not directly provide
optimaloptimalsolutionsolution
Rather, it provides vital information aboutRather, it provides vital information about
various characteristics of the queue to decisionvarious characteristics of the queue to decisionmakersmakers
Average waiting timeAverage waiting time
Average queue lengthAverage queue length
7/30/2019 L15 16 Queueing
6/61
Chulalongkorn University 6Manoj Lohatepanont 2007
ApplicationsApplications Traffic analysisTraffic analysis
Industrial plantsIndustrial plants
Retail storesRetail stores
ServiceService--Oriented IndustriesOriented Industries Telephone switchboardTelephone switchboard
Aircraft Takeoff/Landing SequenceAircraft Takeoff/Landing Sequence Expressway Toll PlazaExpressway Toll Plaza
7/30/2019 L15 16 Queueing
7/61
Chulalongkorn University 7Manoj Lohatepanont 2007
QueueQueue A queue is formed whenA queue is formed when
demand exceeds service capacity for a period of timedemand exceeds service capacity for a period of time the arrival time headway is less than the service timethe arrival time headway is less than the service time
at a locationat a location
A queue needs not be physical waiting lineA queue needs not be physical waiting lineTelephone switchboardTelephone switchboard
Car repair shopCar repair shop
NonNon--Physical QueuePhysical Queue
7/30/2019 L15 16 Queueing
8/61
Chulalongkorn University 8Manoj Lohatepanont 2007
Queuing SystemQueuing System
QueueQueue
Server 1Server 1
Queuing SystemQueuing System
Server 2Server 2
7/30/2019 L15 16 Queueing
9/61
Chulalongkorn University 9Manoj Lohatepanont 2007
CustomersCustomers
Queuing SystemQueuing System
QueueQueue ServersServers
Queuing SystemQueuing System
InputInput
SourceSource
ServedServed
CustomersCustomers
7/30/2019 L15 16 Queueing
10/61
Chulalongkorn University 10Manoj Lohatepanont 2007
Input SourceInput Source Size of Input Source (Calling Population)Size of Input Source (Calling Population)
Infinite (Unlimited)Infinite (Unlimited)The size of input source is relatively largeThe size of input source is relatively large
Implicit assumption of most queuing modelsImplicit assumption of most queuing models
Finite (Limited)Finite (Limited)
The size of input source is relatively smallThe size of input source is relatively small
This assumption should be used when the rate at whichThis assumption should be used when the rate at which
the input source generates new customers is significantlythe input source generates new customers is significantly
affected by the number of customers in the queuingaffected by the number of customers in the queuingsystemsystem
7/30/2019 L15 16 Queueing
11/61
Chulalongkorn University 11Manoj Lohatepanont 2007
Arrival ProcessArrival Process InterarrivalInterarrivalTimeTime
The time between two consecutive arrivals ofThe time between two consecutive arrivals ofcustomerscustomers
MeanMeanInterarrivalInterarrivalTimeTime
1/1/ time/customertime/customer 20 min/customer20 min/customer
Mean Arrival RateMean Arrival Rate
customers/time unitcustomers/time unit 0.05 customers/min0.05 customers/min
Time
1/ 1/ 1/
7/30/2019 L15 16 Queueing
12/61
Chulalongkorn University 12Manoj Lohatepanont 2007
QueueQueue Queue CapacityQueue Capacity
InfiniteInfinite Queue size can grow relatively largeQueue size can grow relatively large
Implicit assumption of most queuing modelsImplicit assumption of most queuing models
FiniteFinite
Queue size is limited by a relatively small numberQueue size is limited by a relatively small number
This assumption should be used when there is a relativelyThis assumption should be used when there is a relatively
small limit on how large the length of the queue can growsmall limit on how large the length of the queue can grow
7/30/2019 L15 16 Queueing
13/61
Chulalongkorn University 13Manoj Lohatepanont 2007
Queue DisciplineQueue Discipline The order in which members of the queue areThe order in which members of the queue are
selected for serviceselected for service First In, First Out (FIFO)First In, First Out (FIFO)
Most commonMost common
Last In, First Out (LIFO)Last In, First Out (LIFO)
Service in Random Order (SIRO)Service in Random Order (SIRO)
7/30/2019 L15 16 Queueing
14/61
Chulalongkorn University 14Manoj Lohatepanont 2007
ServersServers Number of ServersNumber of Servers
One (s = 1)One (s = 1)
Multiple (s > 1)Multiple (s > 1)
Service TimeService Time Elapsed time from the commencement of service to its completionElapsed time from the commencement of service to its completion for afor a
customercustomer
Mean Service TimeMean Service Time
1/1/ time/customertime/customer 10 min/customer10 min/customer
Mean Service RateMean Service Rate
customers/time unitcustomers/time unit 0.1 customers/min0.1 customers/min
Time
1/ 1/ 1/
7/30/2019 L15 16 Queueing
15/61
Chulalongkorn University 15Manoj Lohatepanont 2007
Deterministic QueueDeterministic Queue SupposeSuppose
Mean arrival rateMean arrival rate = 0.67 customer/min= 0.67 customer/min 1/1/ = 1.5 min= 1.5 min
Mean service rateMean service rate = 1 customer/min= 1 customer/min
users
1
2
3
Time(min)0 1 2 3 4 5 6 7 8
7/30/2019 L15 16 Queueing
16/61
Chulalongkorn University 16Manoj Lohatepanont 2007
IfIf >> What happen whenWhat happen when >> ?? = 0.67; =0.5 = 0.67; =0.5
In deterministic caseIn deterministic case
users
1
2
3
4
Time(min)0 1 2 3 4 5 6 7 8
7/30/2019 L15 16 Queueing
17/61
Chulalongkorn University 17Manoj Lohatepanont 2007
Probabilistic QueueProbabilistic Queue Queue can occur even whenQueue can occur even when
7/30/2019 L15 16 Queueing
18/61
Chulalongkorn University 18Manoj Lohatepanont 2007
Steady StateSteady State If the queuing system is allowed to operate for aIf the queuing system is allowed to operate for a
long time, it can be expected, under certainlong time, it can be expected, under certainconditions, to reachconditions, to reach an equilibriuman equilibrium oror steady statesteady state
Steady state means that the probability that youSteady state means that the probability that you
will observe a certain state of the system doeswill observe a certain state of the system does
not depend on the time at which you monitornot depend on the time at which you monitor
the systemthe system
7/30/2019 L15 16 Queueing
19/61
Chulalongkorn University 19Manoj Lohatepanont 2007
UtilizationUtilization We define utilization of a queuing system asWe define utilization of a queuing system as
followsfollows
wherewhere ss is the number of serversis the number of servers
ss is, therefore, TOTAL service rateis, therefore, TOTAL service rate
A steady state is achievable whenA steady state is achievable when < 1< 1
s
=
7/30/2019 L15 16 Queueing
20/61
Chulalongkorn University 20Manoj Lohatepanont 2007
Queuing TaxonomyQueuing Taxonomy Queuing models can be classified by threeQueuing models can be classified by three
components:components: InterarrivalInterarrivalTime Distribution (A),Time Distribution (A),
Service Time Distribution (B), andService Time Distribution (B), and
Number of Servers (m)Number of Servers (m)
using the following notation.using the following notation.
A/B/mA/B/m
7/30/2019 L15 16 Queueing
21/61
Chulalongkorn University 21Manoj Lohatepanont 2007
InterarrivalInterarrival/Service Time Distribution/Service Time Distribution Probability Distribution ofProbability Distribution of
InterarrivalInterarrivalTime (A)Time (A) Service Time (B)Service Time (B)
Common DistributionsCommon Distributions
D: DeterministicD: Deterministic
M:M: [[NegativeNegative]] Exponential DistributionExponential Distribution
EEkk:: kkthth--orderorder ErlangErlangDistributionDistribution
HHkk:: kkthth--orderorder HyperexponentialHyperexponential DistributionDistribution
G:G: generalgeneral DistributionDistribution
7/30/2019 L15 16 Queueing
22/61
Chulalongkorn University 22Manoj Lohatepanont 2007
M/M/1M/M/1 For example, M/M/1 queuing system refers toFor example, M/M/1 queuing system refers to
the queuing system that hasthe queuing system that hasExponentially distributedExponentially distributedinterarrivalinterarrivaltimetime
Times between arrivals are exponentially distributedTimes between arrivals are exponentially distributed
Exponentially distributed service timeExponentially distributed service time
Service times for customers are exponentially distributedService times for customers are exponentially distributed
One serverOne server
7/30/2019 L15 16 Queueing
23/61
Chulalongkorn University 23Manoj Lohatepanont 2007
0
0.5
1
1.5
2
2.5
0 1 2 3 4
t
f(t)
Exponential DistributionExponential Distribution Unsymmetrical distributionUnsymmetrical distribution
E(T) =E(T) =1/1/ ==0.50.5
= 2= 2
f(tf(t) =) =ee--tt t >= 0t >= 0
7/30/2019 L15 16 Queueing
24/61
Chulalongkorn University 24Manoj Lohatepanont 2007
ExampleExample If theIf the interarrivalinterarrival time is exponentiallytime is exponentially
distributed and its meandistributed and its mean interarrivalinterarrival time is 0.5time is 0.5min (arrival ratemin (arrival rate = 2 customers/min)= 2 customers/min) Customers arrive on average every 0.5 minuteCustomers arrive on average every 0.5 minute
~2/3 of the customers arrive less than 0.5 minute~2/3 of the customers arrive less than 0.5 minuteapartapart
~1/3 of the customers arrive more than 0.5 minute~1/3 of the customers arrive more than 0.5 minuteapartapart
7/30/2019 L15 16 Queueing
25/61
Chulalongkorn University 25Manoj Lohatepanont 2007
Memory LessMemory Less PropertyProperty The probability distribution of the remaining time untilThe probability distribution of the remaining time until
the next event (e.g., arrival) occurs is always the same,the next event (e.g., arrival) occurs is always the same,regardless of how much time has passed (i.e., it doesregardless of how much time has passed (i.e., it doesnot remember how much time has passed)not remember how much time has passed)
If theIf the interarrivalinterarrival time of a bus is exponentially distributed,time of a bus is exponentially distributed,then this property implies that the probability that a bus willthen this property implies that the probability that a bus will
arrive in the next minute is the samearrive in the next minute is the same no matter how long youno matter how long youhave waitedhave waited at the bus stopat the bus stop
7/30/2019 L15 16 Queueing
26/61
Chulalongkorn University 26Manoj Lohatepanont 2007
Relationship to Poisson DistributionRelationship to Poisson Distribution
If theIf the interarrivalinterarrival time is exponentiallytime is exponentially
distributed with parameterdistributed with parameter , then the number, then the numberof arrivals (e.g., customers) (of arrivals (e.g., customers) (XX) in a specified) in a specified
length of timelength of time TThas a Poisson distribution withhas a Poisson distribution withparameterparameter TT
Also calledAlso called Poisson Arrival ProcessPoisson Arrival Process
Prob(Prob(XX = n= n) =) =n!n!
ee--TT((T)T)nn
7/30/2019 L15 16 Queueing
27/61
Chulalongkorn University 27Manoj Lohatepanont 2007
Poisson DistributionPoisson Distribution We can use Poisson distribution to verify ourWe can use Poisson distribution to verify our
assumption of exponentially distributedassumption of exponentially distributedinterarrivalinterarrival timetime
By comparing the number of users that arrive forBy comparing the number of users that arrive for
service during a specific period of time with theservice during a specific period of time with thenumber that the Poisson distribution suggestsnumber that the Poisson distribution suggests
7/30/2019 L15 16 Queueing
28/61
Chulalongkorn University 28Manoj Lohatepanont 2007
Quantities of InterestQuantities of Interest There are four major quantities of interest in aThere are four major quantities of interest in a
queuing systemqueuing system
Expected waiting time in the queueExpected waiting time in the queueWWqq ==
Expected system occupancy timeExpected system occupancy timeWW==
Expected number of users in the queueExpected number of users in the queueLLqq ==
Expected number of users in the systemExpected number of users in the systemLL==
7/30/2019 L15 16 Queueing
29/61
Chulalongkorn University 29Manoj Lohatepanont 2007
Basic CharacteristicsBasic Characteristics UtilizationUtilization
WhereWhere ssis the number of serversis the number of servers
LetLet PPnn be probability that there arebe probability that there are nn users in theusers in thesystem, thensystem, then
These are true for all queuing modelsThese are true for all queuing models
s
=
0
n
n
L nP
=
=
7/30/2019 L15 16 Queueing
30/61
Chulalongkorn University 30Manoj Lohatepanont 2007
LittleLittles Laws Law John D.C. Little at MIT is generally creditedJohn D.C. Little at MIT is generally credited
with being the first to prove thesewith being the first to prove these steadysteady--statestaterelationships formally in 1961relationships formally in 1961
L =L = WW
LLqq == WWqq
7/30/2019 L15 16 Queueing
31/61
Chulalongkorn University 31Manoj Lohatepanont 2007
M/M/1M/M/1
InterarrivalInterarrival time is exponentially distributedtime is exponentially distributed
Service time is exponentially distributedService time is exponentially distributed
One ServerOne Server
FCFSFCFS
Assumptions
7/30/2019 L15 16 Queueing
32/61
Chulalongkorn University 32Manoj Lohatepanont 2007
M/M/1M/M/1 ProbabilityProbabilitythat there arethat there are nnusers in an M/M/1users in an M/M/1
queue isqueue is
Therefore, the expected number of users in theTherefore, the expected number of users in thesystem (at steady state) issystem (at steady state) is
(1 ) nnP =
0 0
(1 )1
n
n
n n
L nP n
= =
= = = =
7/30/2019 L15 16 Queueing
33/61
Chulalongkorn University 33Manoj Lohatepanont 2007
M/M/1M/M/1 Once L is known all other quantities of interestOnce L is known all other quantities of interest
can be computed from Littlecan be computed from Littles Law and thes Law and thefollowing relationshipsfollowing relationships
These equations are bridges between the twoThese equations are bridges between the twoequations in Littleequations in Littles Laws Law
1q
W W
= + qL L
= +
7/30/2019 L15 16 Queueing
34/61
Chulalongkorn University 34Manoj Lohatepanont 2007
M/M/1M/M/1 Results can be summarized as followsResults can be summarized as follows
Note also that the probability that the system isNote also that the probability that the system isidle isidle is
L
=
2
( )qL
=
1W
=
( )qW
=
0 1P
=
7/30/2019 L15 16 Queueing
35/61
Chulalongkorn University 35Manoj Lohatepanont 2007
Example 1Example 1 At a small grocery store, customers arriveAt a small grocery store, customers arrive
according to Poisson process with a mean of 15according to Poisson process with a mean of 15customers per hour. The length of time it takescustomers per hour. The length of time it takes
to checkout is exponentially distributed withto checkout is exponentially distributed with
mean equals 3 minutes.mean equals 3 minutes. M/M/1M/M/1
= 15 customer/hr= 15 customer/hr = 20 customer/hr= 20 customer/hr
7/30/2019 L15 16 Queueing
36/61
Chulalongkorn University 36Manoj Lohatepanont 2007
Example 1Example 1 ComputeCompute
Probability that the checkout counter is idleProbability that the checkout counter is idle
Probability that the checkout counter is busyProbability that the checkout counter is busy
0
151 1 0.25
20P
= = =
01 1 0.25 0.75P = =
7/30/2019 L15 16 Queueing
37/61
Chulalongkorn University 37Manoj Lohatepanont 2007
Example 1Example 1 Probability that at least one customer is waitingProbability that at least one customer is waiting
to checkoutto checkout
(1 ) nn
P =
1
1 (1 0.75)0.75 0.1875P = =
150.75
20
= = =
0 1 0.75 0.25P = =
2 0 11 ( ) 1 (0.25 0.1875) 0.5625nP P P = + = + =
7/30/2019 L15 16 Queueing
38/61
Chulalongkorn University 38Manoj Lohatepanont 2007
Example 1Example 1 Average number of customers waiting toAverage number of customers waiting to
checkoutcheckout2
( )q
L
=
215 225
2.2520(20 15) 100
qL = = =
7/30/2019 L15 16 Queueing
39/61
Chulalongkorn University 39Manoj Lohatepanont 2007
Example 1Example 1 Suppose it costs the operator 3 Baht for eachSuppose it costs the operator 3 Baht for each
minute that a customer spends waiting in theminute that a customer spends waiting in thequeue. What is the average cost per customer?queue. What is the average cost per customer?
q qL W=
2.250.15 /
15
q
q
LW hr customer
= = =
0.15 60 3 27 /Cost baht customer = =
7/30/2019 L15 16 Queueing
40/61
Chulalongkorn University 40Manoj Lohatepanont 2007
Example 1Example 1 For an additional of 400 Baht per hour, the operatorFor an additional of 400 Baht per hour, the operator
can decrease the average service time to 2 minutes. Iscan decrease the average service time to 2 minutes. Isthe additional expenditure worthwhile?the additional expenditure worthwhile?
= 30 customer/hr= 30 customer/hr
15 150.033
( ) 30(30 15) 450q
W
= = = =
0.033 60 3 6 /Cost baht customer = =27 6 21 /Savings baht customer = =21 15 315 /Savings baht hr = =
7/30/2019 L15 16 Queueing
41/61
Chulalongkorn University 41Manoj Lohatepanont 2007
Example 1Example 1 Alternative solutionAlternative solution
Difference = 405Difference = 405 -- 90 = 315 baht/hr90 = 315 baht/hr
AFTERAFTERBEFOREBEFORE
2.25qL = 0.5qL =
2.25 60 3
405 /
Cost
baht hr
=
=
0.5 60 3
90 /
Cost
baht hr
=
=
7/30/2019 L15 16 Queueing
42/61
Chulalongkorn University 42Manoj Lohatepanont 2007
M/G/1M/G/1
InterarrivalInterarrival time is exponentially distributedtime is exponentially distributed
Service time is described only byService time is described only by
-- an average service time, andan average service time, and
-- a variancea varianceOne ServerOne Server
FCFSFCFS
Assumptions
7/30/2019 L15 16 Queueing
43/61
Chulalongkorn University 43Manoj Lohatepanont 2007
M/G/1M/G/1 In general, if the service time can be describedIn general, if the service time can be described
by two quantitiesby two quantities Mean (1/Mean (1/))
Variance (Variance (22))
the expected queue length can be estimatedthe expected queue length can be estimatedfromfrom
( )
2 2 2
2 1qL
+=
7/30/2019 L15 16 Queueing
44/61
Chulalongkorn University 44Manoj Lohatepanont 2007
M/G/1M/G/1
As in the previous case, all other quantities ofAs in the previous case, all other quantities of
interest can be calculated using Littleinterest can be calculated using Littles Law ands Law andthe following relationshipsthe following relationships
Note also that the probability that the system isNote also that the probability that the system isidle isidle is
1q
W W
= + qL L
= +
0 1P =
7/30/2019 L15 16 Queueing
45/61
Chulalongkorn University 45Manoj Lohatepanont 2007
Example 2Example 2
Aircraft arrivals and departures at BangkokAircraft arrivals and departures at BangkokInternational Airport can be approximated as a PoissonInternational Airport can be approximated as a Poissonprocess at the average rate of 40 plane/hr. It takes theprocess at the average rate of 40 plane/hr. It takes theair traffic controller 1.2 min on average to land orair traffic controller 1.2 min on average to land ordispatch an aircraft with variance equals to 1.96 mindispatch an aircraft with variance equals to 1.96 min22..
Assume FCFS queue discipline.Assume FCFS queue discipline. M/G/1M/G/1
= 40 plane/hr= 40 plane/hr
= 50 plane/hr= 50 plane/hr 22 = 0.00054444 hr= 0.00054444 hr22
7/30/2019 L15 16 Queueing
46/61
Chulalongkorn University 46Manoj Lohatepanont 2007
Example 2Example 2
Average number of plane waiting for landing orAverage number of plane waiting for landing or
takeofftakeoff
( )( )
( )
2 2 2
22 4050
40 50
2 1
40 0.00054
2 13.78
qL
plane
+=
+
=
=
7/30/2019 L15 16 Queueing
47/61
Chulalongkorn University 47Manoj Lohatepanont 2007
Example 2Example 2
Average wait timeAverage wait time
q qL W=
3.780.094 /
40q
q
LW hr plane
= = =
7/30/2019 L15 16 Queueing
48/61
Chulalongkorn University 48Manoj Lohatepanont 2007
Example 2Example 2
It is estimated that it costs 81,000 baht per hourIt is estimated that it costs 81,000 baht per hour
for an aircraft to wait in the queue. What is thefor an aircraft to wait in the queue. What is thetotal cost of aircraft waiting to land or takeoff attotal cost of aircraft waiting to land or takeoff atthe airport?the airport?
3.78 81, 000 306, 000 /Cost baht hr = =
7/30/2019 L15 16 Queueing
49/61
Chulalongkorn University 49Manoj Lohatepanont 2007
Example 2Example 2
Common mistakesCommon mistakes
0.094 81, 000 7, 650 / /Cost baht hr plane= =
3.78qL plane=
7, 650 3.78 28, 900 /Cost baht hr = =
28, 900 / 306, 000 /baht hr baht hr
7/30/2019 L15 16 Queueing
50/61
Chulalongkorn University 50Manoj Lohatepanont 2007
M/M/sM/M/sAssumptionsInterarrivalInterarrival time is exponentially distributedtime is exponentially distributed
Service time is exponentially distributedService time is exponentially distributed
There are s serversThere are s servers
FCFSFCFS
/ /
7/30/2019 L15 16 Queueing
51/61
Chulalongkorn University 51Manoj Lohatepanont 2007
M/M/sM/M/s
When there areWhen there are ssservers, all with the sameservers, all with the same
service rateservice rate
( ) ( )10 0
11/
! ! 1
n ss
n s
Pn s
=
= +
( )( ) ( )
1
0 2
1 !
s
qL Ps s
+
=
/ /
7/30/2019 L15 16 Queueing
52/61
Chulalongkorn University 52Manoj Lohatepanont 2007
M/M/sM/M/s
As in the previous cases, all other quantities ofAs in the previous cases, all other quantities of
interest can be calculated using Littleinterest can be calculated using Littles Law ands Law andthe following relationshipsthe following relationships
Note, however, thatNote, however, that
1q
W W
= +q
L L
= +
s
=
E l 3
7/30/2019 L15 16 Queueing
53/61
Chulalongkorn University 53Manoj Lohatepanont 2007
Example 3Example 3
At a Burger Mary outlet, mean arrival rate of customersAt a Burger Mary outlet, mean arrival rate of customers
is 60 customers per hour and each takes on average 3is 60 customers per hour and each takes on average 3minutes to complete order. Arriving customers formminutes to complete order. Arriving customers form
one sneak queue in front of 4 cashiers. Assumeone sneak queue in front of 4 cashiers. Assume
interarrivalinterarrival and service times are exponentiallyand service times are exponentiallydistributed.distributed.
M/M/sM/M/s
= 60 customer/hr= 60 customer/hr
= 20 customer/hr= 20 customer/hr
s = 4s = 4
E l 3E l 3
7/30/2019 L15 16 Queueing
54/61
Chulalongkorn University 54Manoj Lohatepanont 2007
Example 3Example 3
How would the average queue length and timeHow would the average queue length and time
in the system change if a fifth cashier is opened.in the system change if a fifth cashier is opened.We first check whether steady state is achievableWe first check whether steady state is achievable
600.75
4 20s
= = =
Idl P b bili ( 4)Idl P b bili ( 4)
7/30/2019 L15 16 Queueing
55/61
Chulalongkorn University 55Manoj Lohatepanont 2007
Idle Probability (s=4)Idle Probability (s=4)
( ) ( )
( ) ( )
[ ]
1
0
0
43 60 60
20 20
600 4 20
11/
! ! 1
1
1/ ! 4! 1
1 / 13 3.375 4
1/26.5
0.038
n ss
n s
n
n
P
n s
n
=
=
= +
= + = +
==
E d Q L h ( 4)E d Q L h ( 4)
7/30/2019 L15 16 Queueing
56/61
Chulalongkorn University 56Manoj Lohatepanont 2007
Expected Queue Length (s=4)Expected Queue Length (s=4)
( )
( ) ( )( )
( )
1
0 2
56020
26020
1 !
0.038 3! 4
243
0.038 6 1
1.53
s
qL P
s s
customers
+ =
=
= =
Idl P b bili ( 5)Idl P b bilit ( 5)
7/30/2019 L15 16 Queueing
57/61
Chulalongkorn University 57Manoj Lohatepanont 2007
Idle Probability (s=5)Idle Probability (s=5)
( ) ( )
( ) ( )
[ ]
1
0
0
54 60 60
20 20
600 5 20
11/
! ! 1
1
1/ ! 5! 1
1/ 16.375 2.025 2.5
1/21.44
0.047
n ss
n s
n
n
P
n s
n
=
=
= +
= + = +
==
E t d Q L thE t d Q L th
7/30/2019 L15 16 Queueing
58/61
Chulalongkorn University 58Manoj Lohatepanont 2007
Expected Queue LengthExpected Queue Length
( )
( ) ( )( )
( )
1
0 2
66020
26020
1 !
0.047 4! 5
729
0.047 24 4
0.35
s
qL P
s s
customers
+ =
=
= =
E t d Ti i S tE p t d Tim in S t m
7/30/2019 L15 16 Queueing
59/61
Chulalongkorn University 59Manoj Lohatepanont 2007
Expected Time in SystemExpected Time in System
S=4S=4
S=5S=5
0.35 0.006 / 0.35 /60
LW hr cus m cus
= = = =
1.530.025 / 1.53 /
60
q
q
LW hr cus m cus
= = = =
Additi n l Q in Ch t i tiAdditional Queuing Characteristics
7/30/2019 L15 16 Queueing
60/61
Chulalongkorn University 60Manoj Lohatepanont 2007
Additional Queuing CharacteristicsAdditional Queuing Characteristics
BalkingBalking
Occur when a customer arrives at a finite queue that is fullyOccur when a customer arrives at a finite queue that is fullyoccupiedoccupied
Or when a customer decides not to join the queue because itOr when a customer decides not to join the queue because itis too longis too long
RenegingReneging Occur when a customer leaves the system without beingOccur when a customer leaves the system without being
servedserved
JockeyingJockeying
Occur when a customer switches between queues thinkingOccur when a customer switches between queues thinkings/he will receive service faster by so doings/he will receive service faster by so doing
Final QuestionFinal Question
7/30/2019 L15 16 Queueing
61/61
Chulalongkorn University 61Manoj Lohatepanont 2007
Final QuestionFinal Question
Why does the use of express lanes make sense?Why does the use of express lanes make sense?
SupermarketSupermarket Expressway EntranceExpressway Entrance
Self Service CheckSelf Service Check--in at Airport (to a certain extent)in at Airport (to a certain extent)