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Supplement DWaiting Line Models
Operations Managementby
R. Dan Reid & Nada R. Sanders2nd Edition © Wiley 2005
PowerPoint Presentation byRoger B. Grinde, University of New Hampshire
Learning Objectives
Describe the elements of a waiting line problem.
Use waiting line models to estimate system performance.
Use waiting line models to make managerial decisions.
Waiting Line System “Queue” is another name for a waiting
line. A waiting line system consists of two
components: The customer population (people or objects
to be processed) The process or service system
Whenever demand exceeds available capacity, a waiting line or queue forms
There is a tradeoff between cost and service level.
Customer Population Characteristics
Finite versus Infinite populations: Is the number of potential new customers materially
affected by the number of customers already in queue? Balking
When an arriving customer chooses not to enter a queue because it’s already too long.
Reneging When a customer already in queue gives up and exits
without being serviced. Jockeying
When a customer switches between alternate queues in an effort to reduce waiting time.
Service System
The service system is defined by: The number of waiting lines The number of servers The arrangement of servers The arrival and service patterns The service priority rules
Number of Lines Waiting lines systems can have
single or multiple queues. Single queues avoid jockeying
behavior and perceived fairness is usually high.
Multiple queues are often used when arriving customers have differing characteristics (e.g. paying with cash, less than 10 items, etc.) and can be readily segmented.
Servers Single servers or multiple, parallel
servers providing multiple channels Arrangement of servers (phases)
Multiple phase systems require customers to visit more than one server
Example of a multi-phase, multi-server system:
C C C CC Depart
Arrivals
1
2
3 6
5
4
Phase 1 Phase 2
Example Queuing Systems
Arrival & Service Patterns Arrival rate:
The average number of customers arriving per time period Modeled using the Poisson distribution Service rate usually denoted by lambda () Example: =50 customers/hour; 1/=0.02 hours between
customer arrivals (1.2 minutes between customers) Service rate:
The average number of customers that can be served during the same period of time
Service times are usually modeled using the exponential distribution
Service rate usually denoted by mu (µ) Example: µ=70 customers/hour; 1/µ=0.014 hours per customer
(0.857 minutes per customer). Even if the service rate is larger than the arrival rate, waiting
lines form! Reason is the variation in specific customer arrival and service
times.
Example Priority Rules
First come, first served Best customers first (reward loyalty) Highest profit customers first Quickest service requirements first Largest service requirements first Earliest reservation first Emergencies first Etc.
Common Performance Measures
Lq = The average number of customers waiting in queue
L = The average number of customers in the system
Wq = The average waiting time in queue W = The average time in the system p = The system utilization rate (% of
time servers are busy)
Single-Server Waiting Line Assumptions
Infinite population Customers are patient (no balking, reneging, or
jockeying) Arrivals follow a Poisson distribution with a mean
arrival rate of . This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/ .
The service rate is described by a Poisson distribution with a mean service rate of µ. This means that the service time for one customer follows an exponential distribution with an average of 1/µ.
The waiting line priority rule is first-come, first-served.
Formulas: Single-Server Case
form. eventually willline longinfinitly an
case, not the is thisIf stability. systemfor :Note
nutilizatiosystemaverage
rateservicemeanmu
ratearrivalmeanlambda
p
Formulas: Single-Server Case (continued)
in timepoint given aat
systemtheincustomersofyprobabilit1
waitingspenttimeaverage
serviceincludingsystemintimeaverage1
lineincustomersofnumberaverage
systemin customers ofnumber average
nppP
pWW
W
pLL
L
nn
q
q
Example A help desk in the computer lab serves
students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.
Based on this description, we know: µ = 20 students/hour (average service time is
3 minutes) = 15 students/hour (average time between
student arrivals is 4 minutes)
Average Utilization
%7575.020
15orp
Average Number of Studentsin the System, and in Line
studentsL 31520
15
studentspLLq 25.2375.0
Average Time in the System, and in Line
minutes12
hours2.01520
11
or
W
minutes9
hours15.02.075.0
or
pWWq
Probability of nStudents in the Line
079.075.075.011
105.075.075.011
141.075.075.011
188.075.075.011
25.0175.011
444
333
222
11
00
ppP
ppP
ppP
ppP
ppP
Single Server: Spreadsheet Approach
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10111213141516171819202122
A B C
Queuing Analysis: Single Server
InputsTime unit hourArrival Rate (lambda) 15 customers/hourService Rate (mu) 20 customers/hour
Intermediate CalculationsAverage time between arrivals 0.066667 hourAverage service time 0.05 hour
Performance MeasuresRho (average server utilization) 0.75P0 (probability the system is empty) 0.25L (average numberin the system) 3 customersLq (average number waiting in the queue) 2.25 customersW (average time in the system) 0.2 hourWq (average time in the queue) 0.15 hour
Probability of a specific number of customers in the systemNumber 2Probability 0.140625
Key Formulas
B9: =1/B5
B10: =1/B6
B13: =B5/B6
B14: =1-B13
B15: =B5/(B6-B5)
B16: =B13*B15
B17: =1/(B6-B5)
B18: =B13*B17
B22: =(1-B$13)*(B13^B21)
Use Data Table (tracking B22) to easily compute the probability of n customers in the system.
Single Server: Probability of n Students in the System
Probability of Number in System
0.0000
0.0500
0.1000
0.1500
0.2000
0.2500
0.30000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Number in System
Pro
bab
ilit
y
Changing System Performance Customer Arrival Rates
Try to smooth demand through non-peak discounts or price promotions
Number and type of service facilities Increase or decrease number of servers, or dedicate
specific servers for certain tasks (e.g., express line for under 10 items)
Change Number of Phases Can use multi-phase system instead of single phase.
This spreads the workload among more servers and may result in better flow (e.g., fast food restaurants having an order phase, pay phase, and pick-up phase during busy hours)
Changing System Performance Server efficiency
Add resources to each phase (e.g., bagger helping a checker at the grocery store)
Use technology (e.g. price scanners) to improve efficiency
Change priority rules Example: implement a reservation protocol
Change the number of lines Reduce multiple lines to single queue to
avoid jockeying Dedicate specific servers to specific
transactions
Multiple Server Case
Assumptions Same as Single-Server, except here
we have multiple, parallel servers. Single Line When server finishes with customer,
first person in line goes to the idle server.
Multiple Server Formulas
form. eventually willline longinfinitly an
case, not the is thisIf stability. systemfor :Note
nutilizatiosystemaverage
servers identical parallel, ofnumber
server for rateservicemeanmu
ratearrivalmeanlambda
s
sp
s
one
Multiple Server Formulas (continued)
in timepoint given aat system in the
customers ofy probabilit
for !
/
for !
/
in timepoint given aat system in the customers
zero ofy probabilit 1
1
!
/
!
/
0
0
11
00
n
snPss
snPnP
psnP
sn
n
n
n
s
n
sn
Multiple Server Formulas (continued)
systemin customers ofnumber average
serviceincludingsystemintimeaverage1
linein waitingspenttimeaverage
linein customers ofnumber average 1!
/2
0
WL
WW
LW
ps
pPL
q
s
q
Example: Multiple Server Computer Lab Help Desk Now 45 students/hour need help. 3 servers, each with service rate of
18 students/hour Based on this, we know:
µ = 18 students/hour s = 3 servers = 45 students/hour
Flexible Spreadsheet Approach Formulas are somewhat complex to set up initially, but
you only need to do it once!
For other multiple-server problems, can just change the input values.
This approach also makes sensitivity analysis possible.
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101112131415161718192021222324
A B C
Queuing Analysis: Multiple Servers
InputsTime unit hourArrival Rate (lambda) 45 customers/hourService Rate per Server (mu) 18 customers/hourNumber of Servers (s) 3 servers
Intermediate CalculationsAverage time between arrivals 0.022222 hourAverage service time per server 0.055556 hourCombined service rate (s*mu) 54 customers/hour
Performance MeasuresRho (average server utilization) 0.833333P0 (probability the system is empty) 0.044944L (average numberin the system) 6.011236 customersLq (average number waiting in the queue) 3.511236 customersW (average time in the system) 0.133583 hourWq (average time in the queue) 0.078027 hour
Probability of a specific number of customers in the systemNumber 5Probability 0.081279
3456789
101112131415161718192021222324252627108109
E F G HWorking Calculations, mainly for P0 Calculation
lambda/mu 2.5s! 6
n (/)^n n! Sum0 1 1 11 2.5 1 3.52 6.25 2 6.6253 15.625 6 9.2291666674 39.0625 24 10.856770835 97.65625 120 11.670572926 244.14063 720 12.009657127 610.35156 5040 12.130758628 1525.8789 40320 12.168602849 3814.6973 362880 12.17911512
10 9536.7432 3628800 12.1817431911 23841.858 39916800 12.1823404812 59604.645 479001600 12.1824649213 149011.61 6.227E+09 12.1824888514 372529.03 8.718E+10 12.1824931215 931322.57 1.308E+12 12.1824938316 2328306.4 2.092E+13 12.1824939417 5820766.1 3.557E+14 12.1824939618 14551915 6.402E+15 12.1824939699 2.489E+39 9.33E+155 12.18249396100 6.223E+39 9.33E+157 12.18249396
Key Formulas for Spreadsheet
F10: =F$5^E10 (copied down) G10: =E10*G9 (copied down) H10: =H9+(F10/G10) (copied down) F5: =B5/B6 F6: =INDEX(G9:G109,B7+1) B10: =1/B5 B11: =1/B6 B12: =B7*B6 B15: =B5/B12 B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1)
B17: =B5*B19 B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2) B19: =B20+(1/B6) B20: =B18/B5 B24: =IF(B23<=B7, ((F5^B23)*B16)/INDEX(G9:G109,B23+1),
((F5^B23)*B16)/ (INDEX(G9:G109,B7+1)*(B7^(B23-B7))))
Probability of n students in the system
Probability of Number in System
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
0.1400
0.16000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Number in System
Pro
bab
ilit
y
Supplement D Highlights
The elements of a waiting line system include the customer population source, the patience of the customer, the service system, arrival and service distributions, waiting line priority rules, and system performance measures. Understanding these elements is critical when analyzing waiting line systems.
Waiting line models allow us to estimate system performance by predicting average system utilization, average number of customers in the service system, average number of customers waiting in line, average time a customer waits in line, and the probability of n customers in the service system.
The benefit of calculating operational characteristics is to provide management with information as to whether system changes are needed. Management can change the operational performance of the waiting line system by altering any or all of the following: the customer arrival rates, the number of service facilities, the number of phases, server efficiency, the priority rule, and the number of lines in the system. Based on proposed changes, management can then evaluate the expected performance of the system.
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