Heizer mod d

© 2008 Prentice Hall, Inc. D – 1

Operations ManagementOperations ManagementModule D – Module D – Waiting-Line ModelsWaiting-Line Models

PowerPoint presentation to accompany PowerPoint presentation to accompany Heizer/Render Heizer/Render Principles of Operations Management, 7ePrinciples of Operations Management, 7eOperations Management, 9e Operations Management, 9e


OutlineOutline

Characteristics of a Waiting-Line Characteristics of a Waiting-Line SystemSystem Arrival CharacteristicsArrival Characteristics

Waiting-Line CharacteristicsWaiting-Line Characteristics

Service CharacteristicsService Characteristics

Measuring a Queue’s PerformanceMeasuring a Queue’s Performance

Queuing CostsQueuing Costs


Outline – ContinuedOutline – Continued

The Variety of Queuing ModelsThe Variety of Queuing Models Model AModel A(M/M/1)(M/M/1): Single-Channel : Single-Channel

Queuing Model with Poisson Arrivals Queuing Model with Poisson Arrivals and Exponential Service Timesand Exponential Service Times

Model BModel B(M/M/S)(M/M/S): Multiple-Channel : Multiple-Channel Queuing ModelQueuing Model

Model CModel C(M/D/1)(M/D/1): Constant-Service-Time : Constant-Service-Time ModelModel

Model D: Limited-Population ModelModel D: Limited-Population Model


Outline – ContinuedOutline – Continued

Other Queuing ApproachesOther Queuing Approaches


Learning ObjectivesLearning Objectives

When you complete this module you When you complete this module you should be able to:should be able to:

1.1. Describe the characteristics of Describe the characteristics of arrivals, waiting lines, and service arrivals, waiting lines, and service systemssystems

2.2. Apply the single-channel queuing Apply the single-channel queuing model equationsmodel equations

3.3. Conduct a cost analysis for a Conduct a cost analysis for a waiting linewaiting line


Learning ObjectivesLearning Objectives

When you complete this module you When you complete this module you should be able to:should be able to:

4.4. Apply the multiple-channel Apply the multiple-channel queuing model formulasqueuing model formulas

5.5. Apply the constant-service-time Apply the constant-service-time model equationsmodel equations

6.6. Perform a limited-population Perform a limited-population model analysismodel analysis


Waiting LinesWaiting Lines

Often called queuing theoryOften called queuing theory

Waiting lines are common situationsWaiting lines are common situations

Useful in both Useful in both manufacturing manufacturing and service and service areasareas


Common Queuing Common Queuing SituationsSituations

SituationSituation Arrivals in QueueArrivals in Queue Service ProcessService Process

SupermarketSupermarket Grocery shoppersGrocery shoppers Checkout clerks at cash Checkout clerks at cash registerregister

Highway toll boothHighway toll booth AutomobilesAutomobiles Collection of tolls at boothCollection of tolls at booth

Doctor’s officeDoctor’s office PatientsPatients Treatment by doctors and Treatment by doctors and nursesnurses

Computer systemComputer system Programs to be runPrograms to be run Computer processes jobsComputer processes jobs

Telephone companyTelephone company CallersCallers Switching equipment to Switching equipment to forward callsforward calls

BankBank CustomerCustomer Transactions handled by tellerTransactions handled by teller

Machine Machine maintenancemaintenance

Broken machinesBroken machines Repair people fix machinesRepair people fix machines

HarborHarbor Ships and bargesShips and barges Dock workers load and unloadDock workers load and unload

Table D.1Table D.1


Characteristics of Waiting-Characteristics of Waiting-Line SystemsLine Systems

1.1. Arrivals or inputs to the systemArrivals or inputs to the system Population size, behavior, statistical Population size, behavior, statistical

distributiondistribution

2.2. Queue discipline, or the waiting line Queue discipline, or the waiting line itselfitself Limited or unlimited in length, discipline Limited or unlimited in length, discipline

of people or items in itof people or items in it

3.3. The service facilityThe service facility Design, statistical distribution of service Design, statistical distribution of service

timestimes


Arrival CharacteristicsArrival Characteristics

1.1. Size of the populationSize of the population Unlimited (infinite) or limited (finite)Unlimited (infinite) or limited (finite)

2.2. Pattern of arrivalsPattern of arrivals Scheduled or random, often a Poisson Scheduled or random, often a Poisson

distributiondistribution

3.3. Behavior of arrivalsBehavior of arrivals Wait in the queue and do not switch Wait in the queue and do not switch

lineslines

No balking or renegingNo balking or reneging


Parts of a Waiting LineParts of a Waiting Line

Figure D.1Figure D.1

Dave’s Dave’s Car WashCar Wash

EnterEnter ExitExit

Population ofPopulation ofdirty carsdirty cars

ArrivalsArrivalsfrom thefrom thegeneralgeneral

population …population …

QueueQueue(waiting line)(waiting line)

ServiceServicefacilityfacility

Exit the systemExit the system

Arrivals to the systemArrivals to the system Exit the systemExit the systemIn the systemIn the system

Arrival CharacteristicsArrival Characteristics Size of the populationSize of the population Behavior of arrivalsBehavior of arrivals Statistical distribution Statistical distribution

of arrivalsof arrivals

Waiting Line Waiting Line CharacteristicsCharacteristics

Limited vs. Limited vs. unlimitedunlimited

Queue disciplineQueue discipline

Service CharacteristicsService Characteristics Service designService design Statistical distribution Statistical distribution

of serviceof service


Poisson DistributionPoisson Distribution

PP((xx)) = for x = for x = 0, 1, 2, 3, 4, …= 0, 1, 2, 3, 4, …ee--xx

xx!!

wherewhere P(x)P(x) == probability of x arrivalsprobability of x arrivals

xx == number of arrivals per number of arrivals per unit of timeunit of time

== average arrival rateaverage arrival rate

ee == 2.71832.7183 ((which is the which is the base of the natural logarithmsbase of the natural logarithms))


Poisson DistributionPoisson DistributionProbability = PProbability = P((xx)) = =

ee--xx

x!x!

0.25 0.25 –

0.02 0.02 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

–

Pro

bab

ility

Pro

bab

ility

00 11 22 33 44 55 66 77 88 99

Distribution for Distribution for = 2 = 2

xx

0.25 0.25 –

0.02 0.02 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

–

Pro

bab

ility

Pro

bab

ility

00 11 22 33 44 55 66 77 88 99

Distribution for Distribution for = 4 = 4

xx1010 1111



Waiting-Line CharacteristicsWaiting-Line Characteristics

Limited or unlimited queue lengthLimited or unlimited queue length

Queue discipline - first-in, first-out Queue discipline - first-in, first-out (FIFO) is most common(FIFO) is most common

Other priority rules may be used in Other priority rules may be used in special circumstancesspecial circumstances


Service CharacteristicsService Characteristics

Queuing system designsQueuing system designs Single-channel system, multiple-Single-channel system, multiple-

channel systemchannel system

Single-phase system, multiphase Single-phase system, multiphase systemsystem

Service time distributionService time distribution Constant service timeConstant service time

Random service times, usually a Random service times, usually a negative exponential distributionnegative exponential distribution


Queuing System DesignsQueuing System Designs


DeparturesDeparturesafter serviceafter service

Single-channel, single-phase systemSingle-channel, single-phase system

Queue

ArrivalsArrivals

Single-channel, multiphase systemSingle-channel, multiphase system

ArrivalsArrivalsDeparturesDeparturesafter serviceafter service

Phase 1 service facility


Service facility

Queue

A family dentist’s officeA family dentist’s office

A McDonald’s dual window drive-throughA McDonald’s dual window drive-through



Figure D.3Figure D.3Multi-channel, single-phase systemMulti-channel, single-phase system

ArrivalsArrivals

Queue

Most bank and post office service windowsMost bank and post office service windows


Service facility

Channel 1

Service facility

Channel 2

Service facility

Channel 3



Figure D.3Figure D.3Multi-channel, multiphase systemMulti-channel, multiphase system

ArrivalsArrivals

Queue

Some college registrationsSome college registrations



Channel 1


Channel 2


Channel 1


Channel 2


Negative Exponential Negative Exponential DistributionDistribution


1.0 1.0 –

0.9 0.9 –

0.8 0.8 –

0.7 0.7 –

0.6 0.6 –

0.5 0.5 –

0.4 0.4 –

0.3 0.3 –

0.2 0.2 –

0.1 0.1 –

0.0 0.0 –

Pro

bab

ility

th

at s

ervi

ce t

ime

Pro

bab

ility

th

at s

ervi

ce t

ime

≥ 1

≥ 1

| | | | | | | | | | | | |

0.000.00 0.250.25 0.500.50 0.750.75 1.001.00 1.251.25 1.501.50 1.751.75 2.002.00 2.252.25 2.502.50 2.752.75 3.003.00

Time t (hours)Time t (hours)

Probability that service time is greater than t = eProbability that service time is greater than t = e-µ-µtt for t for t ≥ 1≥ 1

µ =µ = Average service rate Average service ratee e = 2.7183= 2.7183

Average service rate Average service rate (µ) = (µ) = 1 customer per hour1 customer per hour

Average service rate Average service rate (µ) = 3(µ) = 3 customers per hour customers per hour Average service time Average service time = 20= 20 minutes per customer minutes per customer


Measuring Queue Measuring Queue PerformancePerformance

1.1. Average time that each customer or object Average time that each customer or object spends in the queuespends in the queue

2.2. Average queue lengthAverage queue length

3.3. Average time each customer spends in the Average time each customer spends in the systemsystem

4.4. Average number of customers in the systemAverage number of customers in the system

5.5. Probability that the service facility will be idleProbability that the service facility will be idle

6.6. Utilization factor for the systemUtilization factor for the system

7.7. Probability of a specific number of customers Probability of a specific number of customers in the systemin the system


Queuing CostsQueuing Costs


Total expected costTotal expected cost

Cost of providing serviceCost of providing service

CostCost

Low levelLow levelof serviceof service

High levelHigh levelof serviceof service

Cost of waiting timeCost of waiting time

MinimumMinimumTotalTotalcostcost

OptimalOptimalservice levelservice level


Queuing ModelsQueuing Models

The four queuing models here all assume:The four queuing models here all assume:

Poisson distribution arrivalsPoisson distribution arrivals

FIFO disciplineFIFO discipline

A single-service phaseA single-service phase



Table D.2Table D.2

ModelModel NameName ExampleExample

AA Single-channel Single-channel Information counter Information counter system system at department store at department store(M/M/1)(M/M/1)

NumberNumber NumberNumber ArrivalArrival ServiceServiceofof ofof RateRate TimeTime PopulationPopulation QueueQueue

ChannelsChannels PhasesPhases PatternPattern PatternPattern SizeSize DisciplineDiscipline

SingleSingle SingleSingle PoissonPoisson ExponentialExponential UnlimitedUnlimited FIFOFIFO



Table D.2Table D.2


BB Multichannel Multichannel Airline ticketAirline ticket (M/M/S)(M/M/S) counter counter



Multi-Multi- SingleSingle PoissonPoisson ExponentialExponential UnlimitedUnlimited FIFOFIFO channelchannel



Table D.2Table D.2


CC Constant- Constant- Automated car Automated car service service wash wash(M/D/1)(M/D/1)



SingleSingle SingleSingle PoissonPoisson ConstantConstant UnlimitedUnlimited FIFOFIFO



Table D.2Table D.2


DD Limited Limited Shop with only a Shop with only a population population dozen machines dozen machines

((finite populationfinite population)) that might breakthat might break



SingleSingle SingleSingle PoissonPoisson ExponentialExponential LimitedLimited FIFOFIFO


Model A – Single-ChannelModel A – Single-Channel

1.1. Arrivals are served on a FIFO basis and Arrivals are served on a FIFO basis and every arrival waits to be served every arrival waits to be served regardless of the length of the queueregardless of the length of the queue

2.2. Arrivals are independent of preceding Arrivals are independent of preceding arrivals but the average number of arrivals but the average number of arrivals does not change over timearrivals does not change over time

3.3. Arrivals are described by a Poisson Arrivals are described by a Poisson probability distribution and come from probability distribution and come from an infinite populationan infinite population



4.4. Service times vary from one customer Service times vary from one customer to the next and are independent of one to the next and are independent of one another, but their average rate is another, but their average rate is knownknown

5.5. Service times occur according to the Service times occur according to the negative exponential distributionnegative exponential distribution

6.6. The service rate is faster than the The service rate is faster than the arrival ratearrival rate



== Mean number of arrivals per time periodMean number of arrivals per time period

µµ == Mean number of units served per time Mean number of units served per time periodperiod

LLss == Average number of units (customers) in Average number of units (customers) in the system (waiting and being served)the system (waiting and being served)

==

WWss== Average time a unit spends in the Average time a unit spends in the system (waiting time plus service time)system (waiting time plus service time)

==

µ – µ –

11µ – µ –

Table D.3Table D.3



LLqq== Average number of units waiting in the Average number of units waiting in the queuequeue

==

WWqq== Average Average time a unit spends waiting in time a unit spends waiting in the queuethe queue

==

pp == Utilization factor for the systemUtilization factor for the system

==

22

µ(µ – µ(µ – ))

µ(µ – µ(µ – ))

µµ

Table D.3Table D.3



PP00 == Probability of Probability of 00 units in the system units in the system (that is, the service unit is idle)(that is, the service unit is idle)

== 1 –1 –

PPn > kn > k ==Probability of more than k units in the Probability of more than k units in the system, where n is the number of units in system, where n is the number of units in the systemthe system

==

µµ

µµ

k k + 1+ 1

Table D.3Table D.3


Single-Channel ExampleSingle-Channel Example

== 2 2 cars arriving/hourcars arriving/hourµµ= 3 = 3 cars serviced/hourcars serviced/hour

LLss= = = 2= = = 2 cars in cars in

the system on averagethe system on average

WWss = = = = 1= = 1 hour hour

average waiting time in the average waiting time in the systemsystem

LLqq== = = 1.33 = = 1.33

cars waiting in linecars waiting in line

22

µ(µ – µ(µ – ))

µ – µ –

11µ – µ –

22

3 - 23 - 2

11

3 - 23 - 2

2222

3(3 - 2)3(3 - 2)



WWqq= = = = = =

2/3 2/3 hourhour = 40 = 40 minute minute average waiting timeaverage waiting time

pp= = /µ = 2/3 = 66.6% /µ = 2/3 = 66.6% of of time mechanic is busytime mechanic is busy

µ(µ – µ(µ – ))

223(3 - 2)3(3 - 2)

µµ

PP00= 1 - = .33= 1 - = .33 probability probability

there are there are 00 cars in the system cars in the system

== 2 2 cars arriving/hourcars arriving/hourµµ= 3 = 3 cars serviced/hourcars serviced/hour



Probability of more than k Cars in the SystemProbability of more than k Cars in the System

kk PPn > kn > k = (2/3)= (2/3)k k + 1+ 1

00 .667.667 Note that this is equal toNote that this is equal to 1 - 1 - PP00 = 1 - .33 = 1 - .33

11 .444.444

22 .296.296

33 .198.198 Implies that there is aImplies that there is a 19.8% 19.8% chance that more thanchance that more than 3 3 cars are in the cars are in the systemsystem

44 .132.132

55 .088.088

66 .058.058

77 .039.039


Single-Channel EconomicsSingle-Channel EconomicsCustomer dissatisfactionCustomer dissatisfaction

and lost goodwill and lost goodwill = $10= $10 per hour per hour

WWqq = 2/3= 2/3 hour hour

Total arrivalsTotal arrivals = 16= 16 per day per dayMechanic’s salaryMechanic’s salary = $56= $56 per day per day

Total hours Total hours customers spend customers spend waiting per daywaiting per day

= (16) = 10 = (16) = 10 hourshours2233

2233

Customer waiting-time cost Customer waiting-time cost = $10 10 = $106.67= $10 10 = $106.672233

Total expected costs Total expected costs = $106.67 + $56 = $162.67= $106.67 + $56 = $162.67


Multi-Channel ModelMulti-Channel Model

MM == number of channels opennumber of channels open == average arrival rateaverage arrival rateµµ == average service rate at each average service rate at each channelchannel

PP00 = for M = for Mµ > µ > 11

11MM!!

11nn!!

MMµµMMµ - µ -

M M – 1– 1

n n = 0= 0

µµ

nnµµ

MM

++∑∑

LLss = P = P00 + +µ(µ(/µ)/µ)MM

((M M - 1)!(- 1)!(MMµ - µ - ))22

µµ

Table D.4Table D.4


Multi-Channel ModelMulti-Channel Model

Table D.4Table D.4

WWss = P = P00 + = + =µ(µ(/µ)/µ)MM

((M M - 1)!(- 1)!(MMµ - µ - ))22

11µµ

LLss

LLqq = L = Lss – – µµ

WWqq = W = Wss – = – =11µµ

LLqq


Multi-Channel ExampleMulti-Channel Example

= 2 µ = 3 = 2 µ = 3 M M = 2= 2

PP00 = = = = 11

1122!!

11nn!!

2(3)2(3)

2(3) - 22(3) - 2

11

n n = 0= 0

2233

nn

2233

22

++∑∑

11

22

LLss = + = = + =(2)(3(2/3)(2)(3(2/3)22 22

331! 2(3) - 21! 2(3) - 2 22

11

22

33

44

WWqq = = .0415= = .0415.083.083

22WWss = = = =

3/43/4

22

33

88LLqq = – = = – =22

3333

44

11

1212


Multi-Channel ExampleMulti-Channel Example

Single ChannelSingle Channel Two ChannelsTwo Channels

PP00 .33.33 .5.5

LLss 22 cars cars .75.75 cars cars

WWss 6060 minutes minutes 22.522.5 minutes minutes

LLqq 1.331.33 cars cars .083.083 cars cars

WWqq 40 40 minutesminutes 2.52.5 minutes minutes


Waiting Line TablesWaiting Line Tables

Table D.5Table D.5

Poisson Arrivals, Exponential Service TimesPoisson Arrivals, Exponential Service TimesNumber of Service Channels, MNumber of Service Channels, M

ρρ 11 22 33 44 55

.10.10 .0111.0111

.25.25 .0833.0833 .0039.0039

.50.50 .5000.5000 .0333.0333 .0030.0030

.75.75 2.25002.2500 .1227.1227 .0147.0147

1.01.0 .3333.3333 .0454.0454 .0067.0067

1.61.6 2.84442.8444 .3128.3128 .0604.0604 .0121.0121

2.02.0 .8888.8888 .1739.1739 .0398.0398

2.62.6 4.93224.9322 .6581.6581 .1609.1609

3.03.0 1.52821.5282 .3541.3541

4.04.0 2.21642.2164


Waiting Line Table ExampleWaiting Line Table Example

Bank tellers and customersBank tellers and customers = 18, = 18, µ = 20µ = 20

From Table D.5 From Table D.5

Utilization factor Utilization factor ρρ = = //µ = .90µ = .90 WWqq = =LLqq

Number of Number of service windowsservice windows MM

Number Number in queuein queue Time in queueTime in queue

1 window1 window 11 8.18.1 .45 .45 hrs, hrs, 2727 minutes minutes

2 windows2 windows 22 .2285.2285 .0127.0127 hrs, hrs, ¾¾ minute minute

3 windows3 windows 33 .03.03 .0017.0017 hrs, hrs, 66 seconds seconds

4 windows4 windows 44 .0041.0041 .0003.0003 hrs, hrs, 11 second second


Constant-Service ModelConstant-Service Model

Table D.6Table D.6

LLqq = = 22

2µ(µ – 2µ(µ – ))Average lengthAverage lengthof queueof queue

WWqq = = 2µ(µ – 2µ(µ – ))

Average waiting timeAverage waiting timein queuein queue

µµ

LLss = L = Lqq + + Average number ofAverage number ofcustomers in systemcustomers in system

WWss = W = Wqq + + 11µµ

Average time Average time in the systemin the system


Net savingsNet savings = $ 7 /= $ 7 /triptrip

Constant-Service ExampleConstant-Service ExampleTrucks currently wait Trucks currently wait 1515 minutes on average minutes on averageTruck and driver cost Truck and driver cost $60$60 per hour per hourAutomated compactor service rate Automated compactor service rate (µ) (µ) = 12 trucks per hour= 12 trucks per hourArrival rate Arrival rate (()) = 8= 8 per hour per hourCompactor costs Compactor costs $3$3 per truck per truck

Current waiting cost per trip Current waiting cost per trip = (1/4= (1/4 hr hr)($60) = $15)($60) = $15 //triptrip

WWqq = = hour = = hour882(12)(12 2(12)(12 –– 8) 8)

111212

Waiting cost/tripWaiting cost/tripwith compactorwith compactor = (1/12= (1/12 hr wait hr wait)($60/)($60/hr costhr cost)) = $ 5 /= $ 5 /triptrip

Savings withSavings withnew equipmentnew equipment

= $15 (= $15 (currentcurrent) ) –– $5( $5(newnew)) = $10 = $10 //triptrip

Cost of new equipment amortizedCost of new equipment amortized = = $ 3 /$ 3 /triptrip


Limited-Population ModelLimited-Population Model

Service factor: X =Service factor: X =

Average number running: J = NFAverage number running: J = NF(1 -(1 - X X))

Average number waiting: L = NAverage number waiting: L = N(1 -(1 - F F))

Average number being serviced: H = FNXAverage number being serviced: H = FNX

Average waiting time: W =Average waiting time: W =

Number of population: N = J + L + HNumber of population: N = J + L + H

TTT + UT + U

TT(1 -(1 - F F))XFXF

Table D.7Table D.7


Limited-Population ModelLimited-Population Model

Service factor: X =Service factor: X =

Average number running: J = NFAverage number running: J = NF(1 -(1 - X X))

Average number waiting: L = NAverage number waiting: L = N(1 -(1 - F F))

Average number being serviced: H = FNXAverage number being serviced: H = FNX

Average waiting time: W =Average waiting time: W =

Number of population: N = J + L + HNumber of population: N = J + L + H

TTT + UT + U

TT(1 -(1 - F F))XFXF

D = Probability that a unit will have to wait in queue

N = Number of potential customers

F = Efficiency factor T = Average service time

H = Average number of units being served

U = Average time between unit service requirements

J = Average number of units not in queue or in service bay

W = Average time a unit waits in line

L = Average number of units waiting for service

X = Service factor

M = Number of service channels


Finite Queuing TableFinite Queuing Table

Table D.8Table D.8

XX MM DD FF

.012.012 11 .048.048 .999.999

.025.025 11 .100.100 .997.997

.050.050 11 .198.198 .989.989

.060.060 22 .020.020 .999.999

11 .237.237 .983.983

.070.070 22 .027.027 .999.999

11 .275.275 .977.977

.080.080 22 .035.035 .998.998

11 .313.313 .969.969

.090.090 22 .044.044 .998.998

11 .350.350 .960.960

.100.100 22 .054.054 .997.997

11 .386.386 .950.950


Limited-Population ExampleLimited-Population Example

Service factor: X = Service factor: X = = .091 (= .091 (close to close to .090).090)

For M For M = 1,= 1, D D = .350= .350 and F and F = .960= .960


Average number of printers working:Average number of printers working:

For M For M = 1,= 1, J J = (5)(.960)(1 - .091) = 4.36= (5)(.960)(1 - .091) = 4.36

For M For M = 2,= 2, J J = (5)(.998)(1 - .091) = 4.54= (5)(.998)(1 - .091) = 4.54

222 + 202 + 20

Each of Each of 55 printers requires repair after printers requires repair after 2020 hours hours ((UU)) of use of useOne technician can service a printer in One technician can service a printer in 22 hours hours ((TT))Printer downtime costs Printer downtime costs $120/$120/hourhourTechnician costs Technician costs $25/$25/hourhour


Limited-Population ExampleLimited-Population Example

Service factor: X = Service factor: X = = .091 (= .091 (close to close to .090).090)



Average number of printers working:Average number of printers working:

For M For M = 1,= 1, J J = (5)(.960)(1 - .091) = 4.36= (5)(.960)(1 - .091) = 4.36

For M For M = 2,= 2, J J = (5)(.998)(1 - .091) = 4.54= (5)(.998)(1 - .091) = 4.54

222 + 202 + 20

Each of Each of 55 printers require repair after printers require repair after 2020 hours hours ((UU)) of use of useOne technician can service a printer in One technician can service a printer in 22 hours hours ((TT))Printer downtime costs Printer downtime costs $120/$120/hourhourTechnician costs Technician costs $25/$25/hourhour

Number of Technicians

Average Number Printers

Down (N - J)

Average Cost/Hr for Downtime(N - J)$120

Cost/Hr for Technicians

($25/hr)Total

Cost/Hr

1 .64 $76.80 $25.00 $101.80

2 .46 $55.20 $50.00 $105.20


Other Queuing ApproachesOther Queuing Approaches

The single-phase models cover many The single-phase models cover many queuing situationsqueuing situations

Variations of the four single-phase Variations of the four single-phase systems are possiblesystems are possible

Multiphase models Multiphase models exist for more exist for more complex situationscomplex situations

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