Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc.Beni AsllaniBeni Asllani
University of Tennessee at ChattanoogaUniversity of Tennessee at Chattanooga
Waiting Line AnalysisWaiting Line Analysisfor Service Improvementfor Service Improvement
Operations Management - 5th EditionOperations Management - 5th Edition
Chapter 17Chapter 17
Roberta Russell & Bernard W. Taylor, IIIRoberta Russell & Bernard W. Taylor, III
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 17-17-22
Lecture OutlineLecture Outline
Elements of Waiting Line Analysis Waiting Line Analysis and Quality Single Server Models Multiple Server Model
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Waiting Line AnalysisWaiting Line Analysis
Operating characteristicsOperating characteristics average values for characteristics that describe the average values for characteristics that describe the
performance of a waiting line systemperformance of a waiting line system QueueQueue
A single waiting lineA single waiting line Waiting line system consists ofWaiting line system consists of
ArrivalsArrivals ServersServers Waiting line structuresWaiting line structures
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Elements of a Waiting LineElements of a Waiting Line
Calling populationCalling population Source of customersSource of customers Infinite - large enough that one more customer can Infinite - large enough that one more customer can
always arrive to be servedalways arrive to be served Finite - countable number of potential customersFinite - countable number of potential customers
Arrival rate (Arrival rate (λλ)) Frequency of customer arrivals at waiting line Frequency of customer arrivals at waiting line
system system Typically follows Poisson distributionTypically follows Poisson distribution
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Elements of a Waiting Line (cont.)Elements of a Waiting Line (cont.)
Service timeService time Often follows negative exponential Often follows negative exponential
distributiondistribution Average service rate = Average service rate = μμ
Arrival rate (Arrival rate (λλ) must be less than service ) must be less than service rate (rate (μμ) or system never clears out) or system never clears out
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Elements of a Waiting Line (cont.)Elements of a Waiting Line (cont.)
Queue disciplineQueue discipline Order in which customers are servedOrder in which customers are served First come, first served is most commonFirst come, first served is most common
Length can be infinite or finiteLength can be infinite or finite Infinite is most commonInfinite is most common Finite is limited by some physicalFinite is limited by some physical
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Basic Waiting Line StructuresBasic Waiting Line Structures
Channels are the number of parallel serversChannels are the number of parallel servers Single channelSingle channel Multiple channelsMultiple channels
Phases denote number of sequential servers the Phases denote number of sequential servers the customer must go throughcustomer must go through Single phaseSingle phase Multiple phasesMultiple phases
Steady stateSteady state A constant, average value for performance characteristics that A constant, average value for performance characteristics that
system will reach after a long timesystem will reach after a long time
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Operating CharacteristicsOperating Characteristics
NOTATIONNOTATION OPERATING OPERATING CHARACTERISTICCHARACTERISTIC
LL Average number of customers in the system (waiting and Average number of customers in the system (waiting and being served)being served)
LLqq Average number of customers in the waiting lineAverage number of customers in the waiting line
WW Average time a customer spends in the system Average time a customer spends in the system (waiting and being served)(waiting and being served)
WWqq Average time a customer spends waiting in lineAverage time a customer spends waiting in line
PP00 Probability of no (zero) customers in the systemProbability of no (zero) customers in the system
PPnn Probability of Probability of nn customers in the system customers in the system
ρρ Utilization rate; the proportion of time the system is Utilization rate; the proportion of time the system is in usein use Table 16.1Table 16.1
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Cost Relationship in Waiting Cost Relationship in Waiting Line AnalysisLine Analysis
Exp
ecte
d c
ost
sE
xpec
ted
co
sts
Level of serviceLevel of service
Total costTotal cost
Service costService cost
Waiting CostsWaiting Costs
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Waiting Line Costs and Quality Waiting Line Costs and Quality ServiceService
Traditional view is that the level of Traditional view is that the level of service should coincide with minimum service should coincide with minimum point on total cost curvepoint on total cost curve
TQM approach is that absolute quality TQM approach is that absolute quality service will be the most cost-effective in service will be the most cost-effective in the long runthe long run
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Single-server Models
All assume Poisson arrival rateAll assume Poisson arrival rate VariationsVariations
Exponential service timesExponential service times General (or unknown) distribution of service timesGeneral (or unknown) distribution of service times Constant service timesConstant service times Exponential service times with finite queue lengthExponential service times with finite queue length Exponential service times with finite calling Exponential service times with finite calling
populationpopulation
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Basic Single-Server Basic Single-Server Model: AssumptionsModel: Assumptions
Poisson arrival ratePoisson arrival rate Exponential service timesExponential service times First-come, first-served queue disciplineFirst-come, first-served queue discipline Infinite queue lengthInfinite queue length Infinite calling populationInfinite calling population = mean arrival rate= mean arrival rate = mean service rate= mean service rate
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Formulas for Single-Formulas for Single-Server ModelServer Model
LL = =
- -
Average number of Average number of customers in the systemcustomers in the system
Probability that no customers Probability that no customers are in the system (either in the are in the system (either in the queue or being served)queue or being served)
PP00 = 1 - = 1 -
Probability of exactly Probability of exactly nn customers in the systemcustomers in the system
PPnn = • = • PP00
nn
= 1 -= 1 -
nn
Average number of Average number of customers in the waiting linecustomers in the waiting line
LLqq = =
(( - - ))
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Formulas for Single-Formulas for Single-Server Model (cont.)Server Model (cont.)
==
Probability that the server Probability that the server is busy and the customer is busy and the customer has to waithas to wait
Average time a customer Average time a customer spends in the queuing systemspends in the queuing system WW = = = =
11--
LL
Probability that the server Probability that the server is idle and a customer can is idle and a customer can be servedbe served
II = 1 - = 1 -
= 1 - == 1 - = P P00
Average time a customer Average time a customer spends waiting in line to spends waiting in line to be servedbe served
WWqq = = (( - - ))
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A Single-Server ModelA Single-Server Model
Given Given = 24 per hour, = 24 per hour, = 30 customers per hour = 30 customers per hour
Probability of no Probability of no customers in the customers in the systemsystem
PP00 = 1 - = 1 - = = 1 - = 1 - =
0.200.20
24243030
LL = = = 4 = = = 4Average number Average number of customers in of customers in the systemthe system
--
242430 - 2430 - 24
Average number Average number of customers of customers waiting in linewaiting in line
LLqq = = = 3.2 = = = 3.2(24)(24)22
30(30 - 24)30(30 - 24)22
(( - - ))
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A Single-Server ModelA Single-Server Model
Average time in the Average time in the system per customer system per customer WW = = = 0.167 hour = = = 0.167 hour
11--
1130 - 2430 - 24
Average time waiting Average time waiting in line per customer in line per customer
WWqq = = = 0.133 = = = 0.133((--))
242430(30 - 24)30(30 - 24)
Probability that the Probability that the server will be busy and server will be busy and the customer must waitthe customer must wait
= = = 0.80= = = 0.80
24243030
Probability the Probability the server will be idleserver will be idle II = 1 - = 1 - = 1 - 0.80 = 0.20 = 1 - 0.80 = 0.20
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 17-17-1717
Service Improvement AnalysisService Improvement Analysis
Possible AlternativesPossible Alternatives Another employee to pack up purchasesAnother employee to pack up purchases
service rate will increase from 30 customers to 40 service rate will increase from 30 customers to 40 customers per hourcustomers per hour
waiting time will reduce to only 2.25 minuteswaiting time will reduce to only 2.25 minutes Another checkout counterAnother checkout counter
arrival rate at each register will decrease from 24 to 12 per arrival rate at each register will decrease from 24 to 12 per hourhour
customer waiting time will be 1.33 minutescustomer waiting time will be 1.33 minutes Determining whether these improvements are worth the Determining whether these improvements are worth the
cost to achieve them is the crux of waiting line analysiscost to achieve them is the crux of waiting line analysis
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Constant Service TimesConstant Service Times
Constant service times Constant service times occur with machinery occur with machinery and automated and automated equipmentequipment
Constant service times are a special case of the single-server model with undefined service times
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Operating Characteristics for Operating Characteristics for Constant Service TimesConstant Service Times
PP00 = 1 - = 1 -Probability that no customersProbability that no customersare in systemare in system
Average number of Average number of customers in systemcustomers in system
LL = = LLqq + +
Average number of Average number of customers in queuecustomers in queue
LLqq = = 22
22(( - - ))
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Operating Characteristics for Operating Characteristics for Constant Service Times (cont.)Constant Service Times (cont.)
= = Probability that the Probability that the server is busyserver is busy
Average time customer Average time customer spends in the systemspends in the system WW = = WWqq + +
11
Average time customer Average time customer spends in queuespends in queue WWqq = =
LLqq
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 17-17-2121
Constant Service Times: Constant Service Times: ExampleExample
Automated car wash with service time = 4.5 minAutomated car wash with service time = 4.5 min
Cars arrive at rate Cars arrive at rate = 10/hour (Poisson) = 10/hour (Poisson)
= 60/4.5 = 13.3/hour= 60/4.5 = 13.3/hour
WWqq = = 1.14/10 = .114 hour or 6.84 minutes = = 1.14/10 = .114 hour or 6.84 minutesLLqq
(10)(10)22
2(13.3)(13.3 - 10)2(13.3)(13.3 - 10)LLqq = = = 1.14 cars waiting = = = 1.14 cars waiting
22
22(( - - ))
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Finite Queue LengthFinite Queue Length A physical limit exists on length of queue M = maximum number in queue Service rate does not have to exceed arrival rate ()
to obtain steady-state conditions
PP00 = =Probability that no Probability that no customers are in systemcustomers are in system
1 - 1 - //
1 - (1 - (//))MM + 1 + 1
Probability of exactly Probability of exactly nn customers in systemcustomers in system
PPnn = ( = (PP00) for ) for nn ≤ ≤ MM
nn
LL = - = -Average number of Average number of customers in systemcustomers in system
//1 - 1 - //
((MM + 1)( + 1)(//))MM + 1 + 1
1 - (1 - (//))MM + 1 + 1
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Finite Queue Length (cont.)Finite Queue Length (cont.)
Let Let PPMM = probability a customer will not join system = probability a customer will not join system
Average time customerAverage time customerspends in systemspends in system WW = =
LL
(1 - (1 -
PPMM))
LLqq = = L L - - (1- (1- PPMM))
Average number of Average number of customers in queuecustomers in queue
Average time customer Average time customer spends in queuespends in queue WWqq = = W W --
11
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Finite Queue: ExampleFinite Queue: ExampleFirst National Bank has waiting space for only 3 drive in window cars. = 20, = 30, M = 4 cars (1 in service + 3 waiting)
Probability that Probability that no cars are in no cars are in the systemthe system PP00 = = = 0.38 = = = 0.38
1 - 20/301 - 20/30
1 - (20/30)1 - (20/30)55
1 - 1 - //
1 - (1 - (//))M + 1M + 1
PPnn = ( = (PP00) = (0.38) = 0.076) = (0.38) = 0.076
Probability of Probability of exactly 4 cars in exactly 4 cars in the systemthe system
2020
3030
44
nn==MM
LL = - = 1.24 = - = 1.24Average number Average number of cars in the of cars in the systemsystem
//1 - 1 - //
((MM + 1)( + 1)(//))MM + 1 + 1
1 - (1 - (//))MM + 1 + 1
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Finite Queue: Example (cont.)Finite Queue: Example (cont.)
Average time a carAverage time a carspends in the systemspends in the system W W = = 0.067 hr = = 0.067 hr
LL
(1 - (1 -
PPMM))
LLqq = = L L - = 0.62- = 0.62(1- (1- PPMM))
Average number of Average number of cars in the queuecars in the queue
Average time a carAverage time a carspends in the queuespends in the queue WWqq = = W W - = 0.033 hr- = 0.033 hr
11
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Finite Calling PopulationFinite Calling Population
Arrivals originate from a finite (countable) populationN = population size
Probability of exactly Probability of exactly n n customers in systemcustomers in system PPnn = = PP00 where where nn = 1, 2, ..., = 1, 2, ..., NN
nn
NN!!
((NN - - nn)!)!
Average number of Average number of customers in queuecustomers in queue LLqq = = NN - (1- - (1- PP00))
++
Probability that no Probability that no customers are in systemcustomers are in system
PP00 = =
nn = 0 = 0
NN!!
((NN - - nn)!)!NN nn
11
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Finite Calling Population (cont.)Finite Calling Population (cont.)
WWqq = = LLqq
((NN - - LL) )
Average time customer Average time customer spends in queuespends in queue
LL = = LLqq + (1 - + (1 - PP00))Average number of Average number of customers in systemcustomers in system
WW = = WWqq + +Average time customerAverage time customerspends in systemspends in system
11
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20 trucks which operate an average of 200 days before breaking down ( = 1/200 day = 0.005/day)Mean repair time = 3.6 days ( = 1/3.6 day = 0.2778/day)
Probability that no Probability that no trucks are in the systemtrucks are in the system PP00 = 0.652= 0.652
Average number of Average number of trucks in the queuetrucks in the queue LLqq = 0.169= 0.169
Average number of Average number of trucks in systemtrucks in system LL = 0.169 + (1 - 0.652) = .520 = 0.169 + (1 - 0.652) = .520
Finite Calling Population: Finite Calling Population: ExampleExample
Average time truckAverage time truckspends in queuespends in queue
WWqq = 1.74 days = 1.74 days
Average time truckAverage time truckspends in systemspends in system
WW = 5.33 days = 5.33 days
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Two or more independent servers serve a single waiting line
Poisson arrivals, exponential service, infinite calling population
s>
PP00 = =11
11s!s!
ssss
ss - - nn==ss-1-1
nn=0=0
11nn!!
nn
++
Basic Multiple-server Model
Computing P0 can be time-consuming.
Tables can used to find P0 for selected values of and s.
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Probability of exactly Probability of exactly nn customers in the customers in the systemsystem
PPnn = =
PP00, , for for n n > > ss11
ss! ! ssn-sn-s
nn
PP00, , for for n n > > ss11
nn!!
nn
Probability an arrivingProbability an arrivingcustomer must waitcustomer must wait PPww = = PP00
11
ss!!
ssss - -
ss
Average number of Average number of customers in systemcustomers in system LL = = P P00 + +
((//))ss
((ss - 1)!( - 1)!(ss - - ))22
Basic Multiple-server Model (cont.)
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WW = = LL
Average time customerAverage time customerspends in systemspends in system
= =
//ssUtilization factorUtilization factor
Average time customer Average time customer spends in queuespends in queue WWqq = = WW - = - =
11
LLqq
LLqq = = L L --
Average number of Average number of customers in queuecustomers in queue
Basic Multiple-server Model (cont.)
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Multiple-Server System: Multiple-Server System: ExampleExample
Student Health Service Waiting Room = 10 students per hour = 4 students per hour per service representatives = 3 representativess = (3)(4) = 12
PP00 = 0.045 = 0.045Probability no students Probability no students are in the systemare in the system
Number of students in Number of students in the service areathe service area LL = 6 = 6
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Multiple-Server System: Multiple-Server System: Example (cont.)Example (cont.)
LLqq = = L L - - // = 3.5 = 3.5Number of students Number of students waiting to be servedwaiting to be served
Average time students Average time students will wait in linewill wait in line WWqq = = L Lqq// = 0.35 hours = 0.35 hours
Probability that a Probability that a student must waitstudent must wait PPww = 0.703= 0.703
Waiting time in the Waiting time in the service areaservice area WW = = LL / / = 0.60 = 0.60
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Add a 4th server to improve serviceAdd a 4th server to improve service Recompute operating characteristicsRecompute operating characteristics
PP00 = 0.073 prob of no students = 0.073 prob of no students
LL = 3.0 students = 3.0 students WW = 0.30 hour, 18 min in service = 0.30 hour, 18 min in service LLqq = 0.5 students waiting = 0.5 students waiting
WWqq = 0.05 hours, 3 min waiting, versus 21 earlier = 0.05 hours, 3 min waiting, versus 21 earlier
PPww = 0.31 prob that a student must wait = 0.31 prob that a student must wait
Multiple-Server System: Multiple-Server System: Example (cont.)Example (cont.)
Copyright 2006 John Wiley & Sons, Inc.Copyright 2006 John Wiley & Sons, Inc. 17-17-3535
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