Session 10 (Waiting Line Model) FINAL

© 2008 Prentice Hall, Inc. D – 1

WaitingWaiting--Line ModelsLine Models(Session 10)(Session 10)


OutlineOutline

Characteristics of a WaitingCharacteristics of a Waiting--Line Line SystemSystem

Arrival CharacteristicsArrival CharacteristicsWaitingWaiting--Line CharacteristicsLine CharacteristicsService CharacteristicsService CharacteristicsMeasuring a QueueMeasuring a Queue’’s Performances Performance

Queuing CostsQueuing Costs


OutlineOutlineThe Variety of Queuing ModelsThe Variety of Queuing Models

Model AModel A(M/M/1)(M/M/1): Single: Single--Channel Channel Queuing Model with Poisson Arrivals Queuing Model with Poisson Arrivals and Exponential Service Timesand Exponential Service TimesModel BModel B(M/M/S)(M/M/S): Multiple: Multiple--Channel Channel Queuing ModelQueuing ModelModel CModel C(M/D/1)(M/D/1): Constant: Constant--ServiceService--Time Time ModelModelModel D: LimitedModel D: Limited--Population ModelPopulation Model

Other Queuing ApproachesOther Queuing Approaches


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 industriesindustries


Common Queuing SituationsCommon Queuing Situations

SituationSituation Arrivals in QueueArrivals in Queue Service ProcessService ProcessSupermarketSupermarket Grocery shoppersGrocery shoppers Checkout clerks at cash Checkout clerks at cash

registerregisterHighway toll boothHighway toll booth AutomobilesAutomobiles Collection of tolls at boothCollection of tolls at boothDoctorDoctor’’s offices office PatientsPatients Treatment by doctors and Treatment by doctors and

nursesnursesComputer 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 tellerMachine Machine

maintenancemaintenanceBroken machinesBroken machines Repair people fix machinesRepair people fix machines

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


Characteristics of WaitingCharacteristics of Waiting--Line Line SystemsSystems

1.1. Arrivals or inputs to the systemArrivals or inputs to the systemPopulation 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 facilityDesign, statistical distribution of service Design, statistical distribution of service timestimes


Arrival CharacteristicsArrival Characteristics

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

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

3.3. Behavior of arrivalsBehavior of arrivalsWait in the queue and do not switch Wait in the queue and do not switch lineslinesNo balking or renegingNo balking or reneging


Parts of a Waiting LineParts of a Waiting Line

DaveDave’’s 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 CharacteristicsSize of the populationSize of the populationBehavior of arrivalsBehavior of arrivalsStatistical distribution Statistical distribution of arrivalsof arrivals

Waiting Line Waiting Line CharacteristicsCharacteristicsLimited vs. Limited vs. unlimitedunlimitedQueue disciplineQueue discipline

Service CharacteristicsService CharacteristicsService designService designStatistical 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--λλλλxx

xx!!

wherewhere P(x)P(x) == probability of x arrivalsprobability of x arrivalsxx == number of arrivals per unit of timenumber of arrivals per unit of timeλλ == average arrival rateaverage arrival rateee == 2.71832.7183 ((which is the base of the which is the base of the

natural logarithmsnatural logarithms))


Poisson DistributionPoisson DistributionProbability = PProbability = P((xx)) == ee--λλλλxx

x!x!

0.25 0.25 –

0.02 0.02 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

–

Prob

abili

tyPr

obab

ility

00 11 22 33 44 55 66 77 88 99

Distribution for Distribution for λλ = 2= 2

xx

0.25 0.25 –

0.02 0.02 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

–

Prob

abili

tyPr

obab

ility

00 11 22 33 44 55 66 77 88 99Distribution for Distribution for λλ = 4= 4

xx1010 1111


WaitingWaiting--Line CharacteristicsLine Characteristics

Limited or unlimited queue lengthLimited or unlimited queue length

Queue discipline Queue discipline -- firstfirst--in, firstin, first--out 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 designsSingleSingle--channel system, multiplechannel system, multiple--channel systemchannel systemSingleSingle--phase system, multiphase phase system, multiphase systemsystem

Service time distributionService time distributionConstant service timeConstant service timeRandom service times, usually a Random service times, usually a negative exponential distributionnegative exponential distribution


Queuing System DesignsQueuing System Designs

DeparturesDeparturesafter serviceafter service

SingleSingle--channel, singlechannel, single--phase systemphase system

Queue

ArrivalsArrivals

SingleSingle--channel, multiphase systemchannel, multiphase system

ArrivalsArrivals DeparturesDeparturesafter serviceafter service

Phase 1 service facility


Service facility

Queue

A family dentistA family dentist’’s offices office

A McDonaldA McDonald’’s dual window drives dual window drive--throughthrough



MultiMulti--channel, singlechannel, single--phase systemphase 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



MultiMulti--channel, multiphase systemchannel, multiphase system

ArrivalsArrivals

Queue

Some college registrationsSome college registrations



Channel 1


Channel 2


Channel 1


Channel 2


Negative Exponential DistributionNegative Exponential Distribution

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 –

Prob

abili

ty th

at s

ervi

ce ti

me

Prob

abili

ty th

at s

ervi

ce ti

me ≥≥

11

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

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 3.003.00Time t (hours)Time t (hours)

Probability that service time is greater than t = eProbability that service time is greater than t = e--µµtt for t for t ≥≥ 11µµ == Average service rateAverage service ratee e = 2.7183= 2.7183

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

Average service rate Average service rate ((µµ) = 3) = 3 customers per hourcustomers per hour⇒⇒ Average service time Average service time = 20= 20 minutes per customerminutes per customer


Measuring Queue PerformanceMeasuring Queue Performance

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 length3.3. Average time each customer spends in the Average time each customer spends in the

systemsystem4.4. Average number of customers in the systemAverage number of customers in the system5.5. Probability that the service facility will be idleProbability that the service facility will be idle6.6. Utilization factor for the systemUtilization factor for the system7.7. 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 singleA single--service phaseservice phase



ModelModel NameName ExampleExample

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

NumberNumber NumberNumber ArrivalArrival ServiceServiceofof ofof RateRate TimeTime PopulationPopulation QueueQueue

ChannelsChannels PhasesPhases PatternPattern PatternPattern SizeSize DisciplineDisciplineSingleSingle SingleSingle PoissonPoisson ExponentialExponential UnlimitedUnlimited FIFOFIFO



ModelModel NameName ExampleExample

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


ChannelsChannels PhasesPhases PatternPattern PatternPattern SizeSize DisciplineDisciplineMultiMulti-- SingleSingle PoissonPoisson ExponentialExponential UnlimitedUnlimited FIFOFIFOchannelchannel



ModelModel NameName ExampleExampleCC ConstantConstant-- Automated car Automated car

service service washwash(M/D/1)(M/D/1)


ChannelsChannels PhasesPhases PatternPattern PatternPattern SizeSize DisciplineDisciplineSingleSingle SingleSingle PoissonPoisson ConstantConstant UnlimitedUnlimited FIFOFIFO



ModelModel NameName ExampleExampleDD Limited Limited Shop with only a Shop with only a

population population dozen machinesdozen machines((finite populationfinite population)) that might breakthat might break


ChannelsChannels PhasesPhases PatternPattern PatternPattern SizeSize DisciplineDisciplineSingleSingle SingleSingle PoissonPoisson ExponentialExponential LimitedLimited FIFOFIFO


Model A Model A –– SingleSingle--ChannelChannel1.1. Arrivals are served on a FIFO basis and Arrivals are served on a FIFO basis and

every arrival waits to be served regardless every arrival waits to be served regardless of the length of the queueof the length of the queue

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

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


Model A Model A –– SingleSingle--ChannelChannel

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

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 arrival The service rate is faster than the arrival raterate



λλ == Mean number of arrivals per time periodMean number of arrivals per time periodµµ == Mean number of units served per time periodMean number of units served per time period

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

==

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

==

λλµµ –– λλ

11µµ –– λλ



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 the time a unit spends waiting in the queuequeue

==

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

==

λλ22

µµ((µµ –– λλ))

λλµµ((µµ –– λλ))

λλµµ



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

==

λλµµ

λλµµ

k k + 1+ 1


SingleSingle--Channel ExampleChannel Example

λλ == 2 2 cars arriving/hourcars arriving/hour µµ = 3 = 3 cars serviced/hourcars serviced/hour

LLss = = = 2= = = 2 cars in the system on averagecars in the system on average

WWss = = = = 1= = 1 hour average waiting time in hour average waiting time in the systemthe system

LLqq == = = 1.33= = 1.33 cars waiting in linecars waiting in lineλλ22

µµ((µµ –– λλ))

λλµµ –– λλ

11µµ –– λλ

223 3 -- 22

113 3 -- 22

2222

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



λλ == 2 2 cars arriving/hourcars arriving/hour µµ = 3 = 3 cars serviced/hourcars serviced/hour

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

pp = = λλ//µµ = 2/3 = 66.6% = 2/3 = 66.6% of time mechanic is busyof time mechanic is busy

λλµµ((µµ –– λλ))

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

λλµµPP00 = 1 = 1 -- = .33= .33 probability there are probability there are 00 cars in the cars in the

system system



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 = 1 -- .33.3311 .444.44422 .296.29633 .198.198 ←← Implies that there is aImplies that there is a 19.8% 19.8% chance that chance that

more thanmore than 3 3 cars are in the systemcars are in the system44 .132.13255 .088.08866 .058.05877 .039.039


SingleSingle--Channel EconomicsChannel EconomicsCustomer dissatisfactionCustomer dissatisfaction

and lost goodwilland lost goodwill = $10= $10 per hourper hourWWqq = 2/3= 2/3 hourhour

Total arrivalsTotal arrivals = 16= 16 per dayper dayMechanicMechanic’’s salarys salary = $56= $56 per dayper day

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

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

2233

Customer waitingCustomer waiting--time cost 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


Model BModel B-- MultiMulti--Channel ModelChannel Model

MM == number of channels opennumber of channels openλλ == average arrival rateaverage arrival rateµµ == average service rate at each channelaverage service rate at each channel

PP00 = f= for Mor Mµµ > > λλ11

11MM!!

11nn!!

MMµµMMµµ -- λλ

M M –– 11

n n = 0= 0

λλµµ

nnλλµµ

MM

++∑∑

LLss = P= P00 ++λλµµ((λλ//µµ))MM

((M M -- 1)!(1)!(MMµµ -- λλ)) 22λλµµ


MultiMulti--Channel ModelChannel Model

WWss = P= P00 + =+ =λλµµ((λλ//µµ))MM

((M M -- 1)!(1)!(MMµµ -- λλ)) 2211µµ

LLss

λλ

LLqq = L= Lss –– λλµµ

WWqq = W= Wss –– ==11µµ

LLqq

λλ


MultiMulti--Channel ExampleChannel Exampleλλ = 2 = 2 µµ = 3 = 3 M M = 2= 2

PP00 = = = = 11

1122!!

11nn!!

2(3)2(3)2(3) 2(3) -- 22

11

n n = 0= 0

2233

nn2233

22

++∑∑1122

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

331! 2(3) 1! 2(3) -- 22 22

1122

3344

WWqq = = .0415= = .0415.083.08322

WWss = == =3/43/422

3388

LLqq = = –– ==2233

3344

111212


MultiMulti--Channel ExampleChannel Example

Single ChannelSingle Channel Two ChannelsTwo Channels

PP00 .33.33 .5.5

LLss 22 carscars .75.75 carscars

WWss 6060 minutesminutes 22.522.5 minutesminutes

LLqq 1.331.33 carscars .083.083 carscarsWWqq 40 40 minutesminutes 2.52.5 minutesminutes


Waiting Line TablesWaiting Line Tables

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 .0147.01471.01.0 .3333.3333 .0454.0454 .0067.00671.61.6 2.84442.8444 .3128.3128 .0604.0604 .0121.01212.02.0 .8888.8888 .1739.1739 .0398.03982.62.6 4.93224.9322 .6581.6581 .1609.16093.03.0 1.52821.5282 .3541.35414.04.0 2.21642.2164


Waiting Line Table ExampleWaiting Line Table Example

Bank tellers and customersBank tellers and customersλλ = 18, = 18, µµ = 20= 20

Utilization factor Utilization factor ρρ = = λλ//µµ = .90= .90 WWqq ==LLqq

λλ

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 minutesminutes

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

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

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


ModelModel--C ConstantC Constant--Service ModelService Model

LLqq == λλ22

22µµ((µµ –– λλ))Average lengthAverage lengthof queueof queue

WWqq == λλ22µµ((µµ –– λλ))

Average waiting timeAverage waiting timein queuein queue

λλµµ

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


ConstantConstant--Service ExampleService ExampleTrucks currently wait Trucks currently wait 1515 minutes on averageminutes on averageTruck and driver cost Truck and driver cost $60$60 per hourper hourAutomated compactor service rate Automated compactor service rate ((µµ) ) = 12 trucks per hour= 12 trucks per hourArrival rate Arrival rate ((λλ)) = 8= 8 per hourper hourCompactor costs Compactor costs $3$3 per truckper truck

Current waiting cost per trip Current waiting cost per trip = (1/4= (1/4 hrhr)($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 waithr 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 /triptripNet savingsNet savings = $ 7 /= $ 7 /triptrip


ModelModel-- D LimitedD Limited--Population ModelPopulation Model

Service factor: X =Service factor: X =

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

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

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 -- FF))XFXF


LimitedLimited--Population ModelPopulation Model

Service factor: X =Service factor: X =

Average number running: J = NFAverage number running: J = NF(1 (1 -- XX))Average number waiting: L = NAverage number waiting: L = N(1 (1 -- FF))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 -- FF))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 TableXX 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 .999.99911 .237.237 .983.983

.070.070 22 .027.027 .999.99911 .275.275 .977.977

.080.080 22 .035.035 .998.99811 .313.313 .969.969

.090.090 22 .044.044 .998.99811 .350.350 .960.960

.100.100 22 .054.054 .997.99711 .386.386 .950.950


LimitedLimited--Population ExamplePopulation 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= .960For M For M = 2,= 2, D D = .044= .044 and F and F = .998= .998Average number of printers working:Average number of printers working:For M For M = 1,= 1, J J = (5)(.960)(1 = (5)(.960)(1 -- .091) = 4.36.091) = 4.36For M For M = 2,= 2, J J = (5)(.998)(1 = (5)(.998)(1 -- .091) = 4.54.091) = 4.54

222 + 202 + 20

Each of Each of 55 printers requires repair after printers requires repair after 2020 hours hours ((UU)) of useof 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


LimitedLimited--Population ExamplePopulation 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= .960For M For M = 2,= 2, D D = .044= .044 and F and F = .998= .998Average number of printers working:Average number of printers working:For M For M = 1,= 1, J J = (5)(.960)(1 = (5)(.960)(1 -- .091) = 4.36.091) = 4.36For M For M = 2,= 2, J J = (5)(.998)(1 = (5)(.998)(1 -- .091) = 4.54.091) = 4.54

222 + 202 + 20

Each of Each of 55 printers require repair after printers require repair after 2020 hours hours ((UU)) of useof 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 singleThe single--phase models cover many phase models cover many queuing situationsqueuing situations

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

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


ProblemsProblems

1) A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the probability that the waiting line is empty?

Po = 1 - λ/µ = 1 - 4/12 = 8/12 or 0.667.

2) A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a service rate of 12 per hour. What is the average time a unit spends in the system and the average time a unit spends waiting?


3) A crew of mechanics at the Highway Department garage repair vehicles that break down at an average of λ = 7.5 vehicles per day (approximately Poisson in nature). The mechanic crew can service an average of µ = 10 vehicles per day with a repair time distribution that approximates an exponential distribution.a. What is the utilization rate for this service system?b. What is the average time before the facility can return a breakdown to service?c. How much of that time is spent waiting for service?d. How many vehicles are likely to be in the system at any one time?

(a) Utilization is ρ = 7.5 / 10 = .75 or 75 percent; (b) Ws = 1 / (10 – 7.5) = 1 / 2.5 = 0.4 days; (c) Wq = 7.5 / 10*(10-7.5) = 0.3 days; (d) Ls = 7.5 / (10-7.5) = 7.5 / 2.5 = 3 units.


4) At the order fulfillment center of a major mail-order firm, customer orders, already packaged for shipment, arrive at the sorting machines to be sorted for loading onto the appropriate truck for the parcel's address. The arrival rate at the sorting machines is at the rate of 100 per hour following a Poisson distribution. The machine sorts at the constant rate of 150 per hour.a. What is the utilization rate of the system?b. What is the average number of packages waiting to be sorted?c. What is the average number of packages in the sorting system?d. How long must the average package wait until it gets sorted?e. What would Lq and Wq be if the service rate were exponential, not constant?

Session 10 (Waiting Line Model) FINAL

Documents

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average waiting

statistical

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queuing modelsmodel

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