Chapter 6 - Queuing Theory © 2002 South-Western/Thomson Learning™ Slides prepared by Jeff Heyl,...

Chapter 6 -Chapter 6 -Queuing TheoryQueuing Theory

© 2002 South-Western/Thomson Learning™Slides prepared by Jeff Heyl, Lincoln University

© 2002 South-Western/Thomson Slide 6-2

6.16.1 The Modeling Process The Modeling Process for Queuing Studiesfor Queuing Studies

• Step 1: Opportunity/Problem Step 1: Opportunity/Problem RecognitionRecognition

• Step 2: Model FormulationStep 2: Model Formulation• Step 3: Data CollectionStep 3: Data Collection• Step 4: Analysis of the ModelStep 4: Analysis of the Model• Step 5: Implementation Step 5: Implementation


6.26.2 The Queuing SituationThe Queuing Situation

• Characteristics of Waiting Characteristics of Waiting Line SituationsLine Situations

• The Structure of a Queuing The Structure of a Queuing SystemSystem

• The Managerial ProblemThe Managerial Problem• The Costs Involved in a The Costs Involved in a

Queuing Situation Queuing Situation


6.36.3 Modeling QueuesModeling Queues

• Queuing Model NotationQueuing Model Notation• Deterministic Queuing Deterministic Queuing

SystemsSystems• The Arrival ProcessThe Arrival Process• The Service ProcessThe Service Process• Measures for the ServiceMeasures for the Service• The Waiting LineThe Waiting Line


6.46.4 Analysis of the Basic Analysis of the Basic Queue (M/M/1 FCFS/Queue (M/M/1 FCFS///))

• Poisson-Exponential Model Poisson-Exponential Model CharacteristicsCharacteristics

• Measure of Performance Measure of Performance (Operating Characteristics)(Operating Characteristics)

• Managerial Use of the Measures of Managerial Use of the Measures of PerformancePerformance

• Using Excel’s Goal Seek FunctionUsing Excel’s Goal Seek Function


6.56.5 More Complex More Complex Queuing SituationsQueuing Situations

• Multifacility Queuing Systems (M/M/K Multifacility Queuing Systems (M/M/K FCFS/FCFS///))

• Example: Multichannel QueueExample: Multichannel Queue• Example: Multichannel Queue at Example: Multichannel Queue at

Macro-MarketMacro-Market• Serial (Multiphase) QueuesSerial (Multiphase) Queues• Example: Serial Queue—Three-Example: Serial Queue—Three-

Station ProcessStation Process


6.66.6 Detailed Modeling Detailed Modeling ExampleExample

• Step 1: Opportunity/Problem Step 1: Opportunity/Problem RecognitionRecognition

• Step 2: Model FormulationStep 2: Model Formulation• Step 3: Data CollectionStep 3: Data Collection• Step 4: Analysis of the ModelStep 4: Analysis of the Model• Step 5: ImplementationStep 5: Implementation


QuestionsQuestionsExperiential ExercisesExperiential ExercisesModeling ExercisesModeling ExercisesCase:Case: City of HelpCity of HelpCase:Case: Newtown MaintenanceNewtown Maintenance

DivisionDivision


Printer Wait ProblemPrinter Wait Problem

Printer Printer speedspeed

Arrival Arrival rate of rate of

new rintnew rint

Waiting Waiting timetime

Waiting Waiting costscosts

Consultant’s Consultant’s salariessalaries

Printer Printer costscosts

Select printer

Minimize costs

Exhibit 6.1Exhibit 6.1





new printnew print





Select printer

Minimize costs


• “Seasonality” in the data• Highest average number of jobs

submitted in a two-hour block = 162.5• Printer can process 90 jobs per hour





new printnew print





Select printer

Minimize costs


• “Seasonality” in the data• Highest average number of jobs

submitted in a two-hour block = 162.5• Printer can process 90 jobs per hour

• Average of 8.38 jobs waiting to print• Average of 9.29 jobs either waiting or

being printed• New job will wait 0.10 hours• Printer utilization is 90%


The Modeling Process The Modeling Process for Queuing Studiesfor Queuing Studies

• Descriptive toolDescriptive tool• Used for predicting operating Used for predicting operating

characteristics or performancecharacteristics or performance• There must be a waiting line, or There must be a waiting line, or

queue, which may not always be queue, which may not always be obviousobvious


The Modeling Process The Modeling Process for Queuing Studiesfor Queuing Studies

• Models assume a steady state Models assume a steady state systemsystem

• The basic type of queuing situation The basic type of queuing situation must be describedmust be described

• Important data must be collectedImportant data must be collected• Qualitative aspects may be difficult Qualitative aspects may be difficult

to measureto measure


The Queuing SystemThe Queuing SystemC

os

t C

os

t (( $$

))

Optimal levelOptimal level Level of serviceLevel of service

| | | | | | | | | | |00 11 22 33 44 55 66 77 88 99 1010

500500 –

400400 –

300300 –

200200 –

100100 –



The Queuing SystemThe Queuing SystemC

os

t C

os

t (( $$

))

Optimal levelOptimal level Level of serviceLevel of service

| | | | | | | | | | |00 11 22 33 44 55 66 77 88 99 1010

500500 –

400400 –

300300 –

200200 –

100100 –


Total costTotal cost

MinimumMinimumCost of providing serviceCost of providing service

Cost of waitingCost of waiting


The Structure of a The Structure of a Queuing SystemQueuing System


Arrival processArrival process

PopulationPopulation

SourceSourcexx xx

xxxxxx

Waiting area Service facility

Service

SystemSystem

xxxxxxxxxxWaiting line

(queue)

x Exit


The Structure of a The Structure of a Queuing SystemQueuing System


Arrival processArrival process


SourceSourcexx xx

xxxxxx

Waiting area Service facility

Service

SystemSystem

xxxxxxxxxxWaiting line

(queue)

x Exit

Queues (waiting lines) can be:• Single• Multiple• Priority

Queue discipline can be:• Random• By appointment• FCFS

Queues can be:• Finite• Infinite


The Managerial ProblemThe Managerial Problem

• Waiting for the printer is Waiting for the printer is costing valuable consultant costing valuable consultant timetime

• Faster service will cost more Faster service will cost more moneymoney


The Managerial ProblemThe Managerial Problem

• Waiting for the printer is Waiting for the printer is costing valuable consultant costing valuable consultant timetime

• Faster service will cost more Faster service will cost more moneymoney

What is an ‘appropriate’ What is an ‘appropriate’ level of service?level of service?


The Costs Involved The Costs Involved in a Queuing Situationin a Queuing Situation

Facility Costs:Facility Costs:• Cost of constructionCost of construction• Cost of operationCost of operation• Cost of maintenance and repairCost of maintenance and repair• Other costsOther costs


The Costs Involved The Costs Involved in a Queuing Situationin a Queuing Situation


The Cost of Waiting:The Cost of Waiting:• Customer ‘ill will’Customer ‘ill will’• Loss of salesLoss of sales• Loss of customerLoss of customer


The Costs Involved in a The Costs Involved in a Queuing SituationQueuing Situation


The Cost of Waiting:The Cost of Waiting:• Customer ‘ill will’Customer ‘ill will’• Loss of salesLoss of sales• Loss of customerLoss of customer

Compare alternatives Compare alternatives based on total costbased on total cost

TC = CTC = CFF + C + CWW


Queuing Model NotationQueuing Model Notation

Six necessary items of information:Six necessary items of information:

1.1. Arrival process: Arrival process: M, Ek, D, N, U, GM, Ek, D, N, U, G

2.2. Service process: Service process: M, Ek, D, N, U, GM, Ek, D, N, U, G

3.3. Number of servers: Number of servers: KK

4.4. Queue discipline: Queue discipline: FCFS, PRIFCFS, PRI

5.5. Maximum size permitted: Maximum size permitted: ,, n n

6.6. Size of the population: Size of the population: ,, n n


Notation for Common Notation for Common Queuing SystemsQueuing Systems


Descriptive LabelDescriptive Label CommentsComments

M/M/IM/M/I FCFS/FCFS/// Standard single-server modelStandard single-server model

M/M/KM/M/K FCFS/FCFS/// Standard multiserver modelStandard multiserver model

M/Ek/IM/Ek/I FCFS/FCFS/// Single Erlang service modelSingle Erlang service model

M/G/IM/G/I FCFS/FCFS/// Service time distribution unknownService time distribution unknown

M/M/IM/M/I PRI/PRI/// Priority servicePriority service

M/M/KM/M/K PRI/PRI/// Multiserver priority serviceMultiserver priority service

M/M/IM/M/I FCFS/n/FCFS/n/ Finite queue, single serverFinite queue, single server

M/M/KM/M/K FCFS/n/FCFS/n/ Finite queue, multiserverFinite queue, multiserver

M/M/IM/M/I FCFS/FCFS//n/n Limited source, single serverLimited source, single server

M/M/KM/M/K FCFS/FCFS//n/n Limited source, multiserverLimited source, multiserver


Deterministic Deterministic Queuing SystemsQueuing Systems

1.1. Arrival Rate Equals Service RateArrival Rate Equals Service Rate100% utilization of server and no waiting 100% utilization of server and no waiting lineslines

2.2. Arrival Rate Larger than Service RateArrival Rate Larger than Service RateWaiting line will continuously build - Waiting line will continuously build - explosive queueexplosive queue

3.3. Arrival Rate Smaller than Service RateArrival Rate Smaller than Service RateServer less than fully utilized and never a Server less than fully utilized and never a queuequeue


The Arrival ProcessThe Arrival Process

1.1. Finite Verse Infinite SourceFinite Verse Infinite SourceInfinite Infinite ((or very large) is typically assumedor very large) is typically assumed

2.2. Batch Verse Individual ArrivalsBatch Verse Individual ArrivalsIndividual arrivals is typically assumedIndividual arrivals is typically assumed

3.3. Scheduled Verses Unscheduled ArrivalsScheduled Verses Unscheduled ArrivalsUnscheduled arrivals described by average Unscheduled arrivals described by average arrival rate or average interarrival timearrival rate or average interarrival time


Measures for Unscheduled Arrivals:Measures for Unscheduled Arrivals:

Average Arrival Rate, Average Arrival Rate, Often a Poisson distributionOften a Poisson distribution

Average Interarrival Time, 1/ Average Interarrival Time, 1/ Often a negative exponential Often a negative exponential distributiondistribution



Measures for Unscheduled Arrivals:Measures for Unscheduled Arrivals:

Average Arrival Rate, Average Arrival Rate, Often a Poisson distributionOften a Poisson distribution

Average Interarrival Time, 1/ Average Interarrival Time, 1/ Often a negative exponential Often a negative exponential distributiondistribution


| | | | |7 7:11 7:33 7:51 8 A.M. Time7:03 7:13 7:53

Arrival Arrival Arrival

Arrivals

ArrivalsInterarrival time Interarrival time


The Service ProcessThe Service Process

1.1. A single facilityA single facility

2.2. Multiple, parallel, identical Multiple, parallel, identical facilities – a multifacilityfacilities – a multifacility

3.3. Multiple, parallel, non-identical Multiple, parallel, non-identical facilitiesfacilities

4.4. Service facilities arranged in a Service facilities arranged in a series – a serial arrangementseries – a serial arrangement


The Service ProcessThe Service Process((aa)) Single service facility Single service facility

((bb)) Multiple, parallel Multiple, parallel identical facilitiesidentical facilities

((cc)) Multiple facilities, Multiple facilities, multiple queuesmultiple queues

((dd)) Multiple, parallel Multiple, parallel nonidentical facilitiesnonidentical facilities

((ee)) Series of facilities Series of facilities((ff)) Combination of facilities Combination of facilities

Waiting Waiting lineline

Express lineExpress line

Regular linesRegular lines


Different Arrangements Different Arrangements of Service Facilitiesof Service Facilities


The Service ProcessThe Service Process

Measures for the Service:Measures for the Service:

Average Length of Service Average Length of Service ((Service Service TimeTime),), 1/ 1/Most commonly a negative Most commonly a negative exponential distributionexponential distribution

Average Service Rate, Average Service Rate, Often a Poisson distributionOften a Poisson distribution


The Waiting LineThe Waiting Line

1.1. Queue DisciplineQueue Discipline• Priority SystemPriority System• Emergency Emergency ((Preemptive PriorityPreemptive Priority) )

SystemSystem• Last-Come, First-Served Last-Come, First-Served ((LCFSLCFS))• First-Come, First-Served First-Come, First-Served ((FCFSFCFS))• Queue LengthQueue Length


The Waiting LineThe Waiting Line

2.2. Organization of the QueueOrganization of the Queue

3.3. Behavior in the QueueBehavior in the Queue• BalkingBalking• RenegingReneging• JockeyingJockeying• Combining or DividingCombining or Dividing• CyclingCycling


Analysis of the Basic Analysis of the Basic Queue Queue ((M/M/I FCFS/M/M/I FCFS///))

1.1. Arrival Rate Arrival Rate (()) – random variable, – random variable, Poisson distributionPoisson distribution

2.2. Service Time Service Time (1/(1/) – negative ) – negative exponential distributionexponential distribution

3.3. Major AssumptionsMajor Assumptions• Infinite source populationInfinite source population• FCFSFCFS // < 1 = < 1 = • Steady state systemSteady state system• Unlimited queue lengthUnlimited queue length


Analysis of the Basic Analysis of the Basic Queue Queue ((M/M/I FCFS/M/M/I FCFS///))

Average Waiting Time, WAverage Waiting Time, W

Average Waiting Time in Average Waiting Time in the Queue, Wthe Queue, Wqq

Average Number of Customers Average Number of Customers in the System, Lin the System, L

Average Number of Customers Average Number of Customers in the Queue, Lin the Queue, Lqq

W =W =11

- -

WWqq = =

(( - - ))

L =L = - -

LLqq = = 22

(( - - ))

Measures of Performance:Measures of Performance:


Analysis of the Basic Analysis of the Basic Queue Queue ((M/M/I FCFS/M/M/I FCFS///))Measures of Performance:Measures of Performance:

Probability of an Empty Probability of an Empty ((IdleIdle) ) Facility, PFacility, P(0)(0)

Probability of the System Probability of the System Being Busy, PBeing Busy, Pww

Probability of Being in the Probability of Being in the System System ((Waiting and Being Waiting and Being ServedServed)) Longer than Time t Longer than Time t

PP(0)(0) = 1 -= 1 -

PPww = 1 - = 1 - PP(0) = (0) =

PP[[T T >> t t ]] = e = e -(-( - - ))tt


Analysis of the Basic Analysis of the Basic Queue Queue ((M/M/I FCFS/M/M/I FCFS///))Measures of Performance:Measures of Performance:

Probability of Waiting for Probability of Waiting for Service Longer than Time tService Longer than Time tqq

Probability of Finding Probability of Finding Exactly N Customers in Exactly N Customers in the System, Pthe System, P((NN))

Probability that the Number of Probability that the Number of Customers in the System, N, Customers in the System, N, Exceed a Given Value, nExceed a Given Value, n

PP((NN) =) = NN (1 - (1 - ))

PP[[TTqq >> t tqq]] = = ee-(-( - - )) ttqq

PP[[NN > n > n]] = = N N + 1+ 1


All-American Aviation Co.All-American Aviation Co.

Toolroom: Toolroom: = 10 = 10 per hourper hour = 12 = 12 per hourper hour

Toolroom utilizationToolroom utilization

Average waiting time at the Average waiting time at the toolroomtoolroom

Average waiting time in the lineAverage waiting time in the line

Average number of production Average number of production employees at the toolroomemployees at the toolroom

Average number of production Average number of production employees in the lineemployees in the line

= 10/12 = 0.833= 10/12 = 0.833

W W = 1/(12 - 10) = 0.5= 1/(12 - 10) = 0.5 hour hour

WWqq = 10/(12(12 - 10)) = 0.417= 10/(12(12 - 10)) = 0.417 hour hour

L L = 0.833/(1 - 0.833) = 5= 0.833/(1 - 0.833) = 5

LLqq = 0.833= 0.83322/(1 - 0.833) = 4.16/(1 - 0.833) = 4.16


All-American Aviation Co.All-American Aviation Co.

Toolroom: Toolroom: = 10 = 10 per hourper hour = 12 = 12 per hourper hour

Probability the toolroom clerk Probability the toolroom clerk will be idlewill be idle

Probability the system is busyProbability the system is busy

Probability of waiting longer Probability of waiting longer than than 0.50.5 hour in the system hour in the system

Probability of exactly four Probability of exactly four production employees in the production employees in the systemsystem

Probability of more than three Probability of more than three production employees in the production employees in the systemsystem

PP(0) = 1 - 0.833 = 0.167(0) = 1 - 0.833 = 0.167

PPww = 0.833= 0.833

PP[[T T > 0.5]> 0.5] = = ee -(12 - 10)0.5-(12 - 10)0.5 = 0.368 = 0.368

PP(4)(4) = 0.833= 0.83344(1 - 0.833) = 0.0804(1 - 0.833) = 0.0804

PP[[N N > 3]> 3] = 0.833= 0.83344 = 0.481 = 0.481


Multifacility Queuing Multifacility Queuing Systems Systems ((M/M/K FCFS/M/M/K FCFS///))


SourceSource

xxxx

xxxxxx

Waiting area

Single waiting line

xxxxxxxx

Service facilities

S1

S2

S3

S4

S5

Exit




SourceSource

xxxx

xxxxxx

Waiting area

Single waiting line

xxxxxxxx

Service facilities

S1

S2

S3

S4

S5

Exit

Assumptions:1. It is a Poisson-exponential system

2. The service facilities (channels) are identical

3. Only one waiting line exists

4. The arrival rate is smaller than the combined service rate (K) of all the service facilities



Formulas:Formulas:

= == =KK

KK

Probability of finding Probability of finding no customer in the no customer in the system:system:

PP(0) =(0) =11

KK

KK!(1 - !(1 - )) + + II

ii!!

K K - 1- 1

I I = 0= 0

Utilization factor for Utilization factor for entire systementire system::



Formulas:Formulas:

Probability of finding Probability of finding exactly N customers in exactly N customers in the system:the system:

PP((NN) =) =

PP(0)(0) when N when N K KNN

NN!!

when N when N >> K KPP(0)(0) NNKKKK

KK!!



Formulas:Formulas:

The average number of The average number of customers in the customers in the waiting line:waiting line:

Given LGiven Lqq

LLqq = =PP(0)(0) K K KK!(1 - !(1 - ))22

PP(0) =(0) =LLqqKK!(1 - !(1 - ))22

K K



Formulas:Formulas:The average number of The average number of customers in the system:customers in the system:

The average waiting The average waiting time in the queue per time in the queue per customer:customer:

The average time a The average time a customer spends in the customer spends in the system:system:

L = LL = Lqq + +

WWqq = =LLqq

W = = WW = = Wqq + +11

LL


Multichannel Queue Multichannel Queue at Macro-Marketat Macro-Market

The Multifacility Solution ProcessThe Multifacility Solution Process

KK

LLqq

LL

WWqq

WW

Exhibit 6.16Exhibit 6.16 PP(0)(0)



Multichannel Queue Multichannel Queue at Macro-Marketat Macro-Market

Service Service timetime

Rate of Rate of arrivalsarrivals

Number of stations to staff

Staffing Staffing costscosts

Total Total costscosts

Total Total revenuerevenue

Revenue/ Revenue/ customercustomer

Customer Customer wait timewait time

Customer Customer waiting costswaiting costs

Maximize profit



Multichannel Queue Multichannel Queue at Macro-Marketat Macro-MarketArrival rate: Arrival rate: = 16 = 16 per hourper hourService rate: Service rate: = 20 = 20 per hourper hour

For one station:For one station:

WWqq = 16/(20(20 - 16)) = 0.2= 16/(20(20 - 16)) = 0.2 hour per customer hour per customer

Total profits:Total profits:

Gross income: Gross income: 1616 customers customers ($15 ($15 eacheach)) == $240.00$240.00Less operating expenseLess operating expense == 15.0015.00Less waiting costsLess waiting costs == 96.0096.00

ProfitProfit == $129.00$129.00



For one station:For one station: $129.00$129.00 Profit Profit



For one station:For one station: $129.00$129.00 Profit ProfitFor two stations:For two stations:


Gross income: Gross income: 1616 customers customers ($15 ($15 eacheach)) == $240.00$240.00Less operating expenses: Less operating expenses: 2(15)2(15) == 30.0030.00Less waiting costs : Less waiting costs : 16(0.0095)(30)16(0.0095)(30) == 4.604.60




For one station:For one station: $129.00$129.00 Profit ProfitFor two stations:For two stations: $205.40$205.40 Profit Profit



For one station:For one station: $129.00$129.00 Profit ProfitFor two stations:For two stations: $205.40 $205.40 ProfitProfitFor three stations:For three stations:


Gross income: Gross income: 1616 customers customers ($15 ($15 eacheach)) == $240.00$240.00Less operating expenses: Less operating expenses: 3(15)3(15) == 45.0045.00Less waiting costs : Less waiting costs : 16(0.0011)(30)16(0.0011)(30) == .53.53




For one station:For one station: $129.00$129.00 Profit ProfitFor two stations:For two stations: $205.40 $205.40 ProfitProfitFor three stations:For three stations: $194.47 $194.47 ProfitProfit


Serial (Multiphase) QueuesSerial (Multiphase) Queues

Arrival rate Arrival rate = 5 = 5 per hour per hourKK11 = 1; = 1; 11 = 6 = 6 ((for station for station 1)1)

KK22 = 3; = 3; 22 = 2 = 2 ((for station for station 2)2)


Three-Station Process:Three-Station Process:


Serial (Multiphase) QueuesSerial (Multiphase) Queues

Arrival rate Arrival rate = 5 = 5 per hour per hourKK11 = 1; = 1; 11 = 6 = 6 ((for station for station 1)1)



x

x

SS33

x

x

x

SS11 SS22

xx…xx…xInfinite source

Station 1Station 1 Station 2Station 2 Station 3Station 3

xxx…xxx… xx…xx…

Queue 1Queue 1 Queue 2Queue 2 Queue 3Queue 3


Three-Station Process:Three-Station Process:


Serial (Multiphase) QueuesSerial (Multiphase) QueuesThree-Station Process:Three-Station Process:Station 1:

// = 0.833 = 0.833 LLqq11 = 4.167 = 4.167 WWqq11 = 0.833 = 0.833

Station 2:

22 = 2.5 = 2.5 22 = 0.833 = 0.833 LLqq22 = 3.333 = 3.333 WWqq22 = 0.667 = 0.667

Station 3:

33 = 1.25 = 1.25 33 = 0.625 = 0.625 LLqq33 = 0.815 = 0.815 WWqq33 = 0.163 = 0.163

Total waiting time:Total waiting time:

WWqSystemqSystem = W = Wqq11 + W + Wqq22 + W + Wqq33 = 1.663= 1.663 hours hours


Detailed Modeling ExampleDetailed Modeling Example

Truck Truck arrivalarrival

Service Service raterate

Truck Truck waiting waiting

timetime

Truck Truck operating operating

costscosts

Truck Truck unloading unloading

costscosts

Lease new equipment

Equipment Equipment costscosts

Minimize costs



Detailed Modeling ExampleDetailed Modeling Example

Truck Truck arrivalarrival


Truck Truck waiting waiting

timetime

Truck Truck operating operating

costscosts

Truck Truck unloading unloading

costscosts

Lease new equipment

Equipment Equipment costscosts

Minimize costs


Arrival rate, = 2 trucks per hourService time, 1/ = 20 minutes per truckTruck operating costs = $30 per hourLeasing additional equipment = $200 per day


Mainland Tech RegistrationMainland Tech Registration

Student Student arrivalsarrivals


Number Number of serversof servers

Minimize cost

Time in the Time in the systemsystem



• Registration is open from Registration is open from 99 A.M. to A.M. to 6:306:30 P.M. P.M. Monday thought Friday and from Monday thought Friday and from 99 A.M. to A.M. to 1212 noon on Saturday for one weeknoon on Saturday for one week

• There are There are 725725 ongoing students ongoing students• Each registration event takes Each registration event takes 15 15 minutesminutes• Time in the system should be equal to or less Time in the system should be equal to or less

than than 2525 minutes minutes



• Registration is open from Registration is open from 99 A.M. to A.M. to 6:306:30 P.M. P.M. Monday thought Friday and from Monday thought Friday and from 99 A.M. to A.M. to 1212 noon on Saturday for one weeknoon on Saturday for one week

• There are There are 725725 ongoing students ongoing students• Each registration event takes Each registration event takes 15 15 minutesminutes• Time in the system should be equal to or less Time in the system should be equal to or less

than than 2525 minutes minutes

How many staff members does Mainland How many staff members does Mainland Tech need at registration to satisfy the Tech need at registration to satisfy the student’s criteria at the lowest cost?student’s criteria at the lowest cost?



Using standard M/M/K relationships:Using standard M/M/K relationships:

Number of students Number of students = 725= 725Available registration hours Available registration hours = 50.5= 50.5Service rate per hour Service rate per hour (() = 4) = 4Arrival rate per hour Arrival rate per hour (() = 14.356) = 14.356Minimum number of servers Minimum number of servers = 4= 4

K K = 4= 4

W W = 0.713= 0.713 hrs hrsoror

42.7742.77 minutes minutes

K K = 5= 5

W W = 0.322= 0.322 hrs hrsoror

19.319.3 minutes minutes

Chapter 6 - Queuing Theory © 2002 South-Western/Thomson Learning™ Slides prepared by Jeff Heyl,...

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