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21012101--495495Advanced Topics in CE IAdvanced Topics in CE I

Intro to Queuing TheoryIntro to Queuing Theory

Manoj Lohatepanont, Sc.D.Manoj Lohatepanont, Sc.D.

Chulalongkorn UniversityChulalongkorn University

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Chulalongkorn University 2Manoj Lohatepanont 2007

OutlineOutline Introduction to QueuingIntroduction to Queuing

ApplicationsApplications Queuing SystemQueuing System

DeterministicDeterministic

StochasticStochastic Probability DistributionProbability Distribution

LittleLittles Laws Law

Queuing SystemsQueuing Systems

M/M/1M/M/1 M/G/1M/G/1

M/M/sM/M/s

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Queuing TheoryQueuing Theory What is Queuing Theory?What is Queuing Theory?

A mathematical study ofA mathematical study ofqueuesqueues QueueQueueis a waiting line especially of persons or vehiclesis a waiting line especially of persons or vehicles

awaiting processingawaiting processing

When does queue occur?When does queue occur? A queue forms when current demand for a service exceedsA queue forms when current demand for a service exceeds

the current capacity to provide that servicethe current capacity to provide that service

Why study Queuing Theory?Why study Queuing Theory? Help provide decision or guideline on system capacityHelp provide decision or guideline on system capacity

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Why Is Queuing Decision Difficult?Why Is Queuing Decision Difficult?

ProbabilisticProbabilistic nature of the problemnature of the problem

It is often impossible to predict accurately when andIt is often impossible to predict accurately when andhow many customers will show up, and how long ithow many customers will show up, and how long it

will take to service each customerwill take to service each customer

E.g., checkout queues at supermarketsE.g., checkout queues at supermarkets Providing too much capacity costs moneyProviding too much capacity costs money

Providing too little capacity leads to long wait inProviding too little capacity leads to long wait in

queues, which in turn leads to other undesirablequeues, which in turn leads to other undesirableconsequencesconsequences

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How Does Queuing Theory Help?How Does Queuing Theory Help?

Queuing theory itself does not directly provideQueuing theory itself does not directly provide

optimaloptimalsolutionsolution

Rather, it provides vital information aboutRather, it provides vital information about

various characteristics of the queue to decisionvarious characteristics of the queue to decisionmakersmakers

Average waiting timeAverage waiting time

Average queue lengthAverage queue length

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ApplicationsApplications Traffic analysisTraffic analysis

Industrial plantsIndustrial plants

Retail storesRetail stores

ServiceService--Oriented IndustriesOriented Industries Telephone switchboardTelephone switchboard

Aircraft Takeoff/Landing SequenceAircraft Takeoff/Landing Sequence Expressway Toll PlazaExpressway Toll Plaza

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QueueQueue A queue is formed whenA queue is formed when

demand exceeds service capacity for a period of timedemand exceeds service capacity for a period of time the arrival time headway is less than the service timethe arrival time headway is less than the service time

at a locationat a location

A queue needs not be physical waiting lineA queue needs not be physical waiting lineTelephone switchboardTelephone switchboard

Car repair shopCar repair shop

NonNon--Physical QueuePhysical Queue

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Queuing SystemQueuing System

QueueQueue

Server 1Server 1


Server 2Server 2

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CustomersCustomers


QueueQueue ServersServers


InputInput

SourceSource

ServedServed

CustomersCustomers

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Input SourceInput Source Size of Input Source (Calling Population)Size of Input Source (Calling Population)

Infinite (Unlimited)Infinite (Unlimited)The size of input source is relatively largeThe size of input source is relatively large

Implicit assumption of most queuing modelsImplicit assumption of most queuing models

Finite (Limited)Finite (Limited)

The size of input source is relatively smallThe size of input source is relatively small

This assumption should be used when the rate at whichThis assumption should be used when the rate at which

the input source generates new customers is significantlythe input source generates new customers is significantly

affected by the number of customers in the queuingaffected by the number of customers in the queuingsystemsystem

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Arrival ProcessArrival Process InterarrivalInterarrivalTimeTime

The time between two consecutive arrivals ofThe time between two consecutive arrivals ofcustomerscustomers

MeanMeanInterarrivalInterarrivalTimeTime

1/1/ time/customertime/customer 20 min/customer20 min/customer

Mean Arrival RateMean Arrival Rate

customers/time unitcustomers/time unit 0.05 customers/min0.05 customers/min

Time

1/ 1/ 1/

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QueueQueue Queue CapacityQueue Capacity

InfiniteInfinite Queue size can grow relatively largeQueue size can grow relatively large

Implicit assumption of most queuing modelsImplicit assumption of most queuing models

FiniteFinite

Queue size is limited by a relatively small numberQueue size is limited by a relatively small number

This assumption should be used when there is a relativelyThis assumption should be used when there is a relatively

small limit on how large the length of the queue can growsmall limit on how large the length of the queue can grow

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Queue DisciplineQueue Discipline The order in which members of the queue areThe order in which members of the queue are

selected for serviceselected for service First In, First Out (FIFO)First In, First Out (FIFO)

Most commonMost common

Last In, First Out (LIFO)Last In, First Out (LIFO)

Service in Random Order (SIRO)Service in Random Order (SIRO)

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ServersServers Number of ServersNumber of Servers

One (s = 1)One (s = 1)

Multiple (s > 1)Multiple (s > 1)

Service TimeService Time Elapsed time from the commencement of service to its completionElapsed time from the commencement of service to its completion for afor a

customercustomer

Mean Service TimeMean Service Time

1/1/ time/customertime/customer 10 min/customer10 min/customer

Mean Service RateMean Service Rate

customers/time unitcustomers/time unit 0.1 customers/min0.1 customers/min

Time

1/ 1/ 1/

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Deterministic QueueDeterministic Queue SupposeSuppose

Mean arrival rateMean arrival rate = 0.67 customer/min= 0.67 customer/min 1/1/ = 1.5 min= 1.5 min

Mean service rateMean service rate = 1 customer/min= 1 customer/min

users

1

2

3

Time(min)0 1 2 3 4 5 6 7 8

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IfIf >> What happen whenWhat happen when >> ?? = 0.67; =0.5 = 0.67; =0.5

In deterministic caseIn deterministic case

users

1

2

3

4

Time(min)0 1 2 3 4 5 6 7 8

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Probabilistic QueueProbabilistic Queue Queue can occur even whenQueue can occur even when

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Steady StateSteady State If the queuing system is allowed to operate for aIf the queuing system is allowed to operate for a

long time, it can be expected, under certainlong time, it can be expected, under certainconditions, to reachconditions, to reach an equilibriuman equilibrium oror steady statesteady state

Steady state means that the probability that youSteady state means that the probability that you

will observe a certain state of the system doeswill observe a certain state of the system does

not depend on the time at which you monitornot depend on the time at which you monitor

the systemthe system

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UtilizationUtilization We define utilization of a queuing system asWe define utilization of a queuing system as

followsfollows

wherewhere ss is the number of serversis the number of servers

ss is, therefore, TOTAL service rateis, therefore, TOTAL service rate

A steady state is achievable whenA steady state is achievable when < 1< 1

s

=

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Queuing TaxonomyQueuing Taxonomy Queuing models can be classified by threeQueuing models can be classified by three

components:components: InterarrivalInterarrivalTime Distribution (A),Time Distribution (A),

Service Time Distribution (B), andService Time Distribution (B), and

Number of Servers (m)Number of Servers (m)

using the following notation.using the following notation.

A/B/mA/B/m

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InterarrivalInterarrival/Service Time Distribution/Service Time Distribution Probability Distribution ofProbability Distribution of

InterarrivalInterarrivalTime (A)Time (A) Service Time (B)Service Time (B)

Common DistributionsCommon Distributions

D: DeterministicD: Deterministic

M:M: [[NegativeNegative]] Exponential DistributionExponential Distribution

EEkk:: kkthth--orderorder ErlangErlangDistributionDistribution

HHkk:: kkthth--orderorder HyperexponentialHyperexponential DistributionDistribution

G:G: generalgeneral DistributionDistribution

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M/M/1M/M/1 For example, M/M/1 queuing system refers toFor example, M/M/1 queuing system refers to

the queuing system that hasthe queuing system that hasExponentially distributedExponentially distributedinterarrivalinterarrivaltimetime

Times between arrivals are exponentially distributedTimes between arrivals are exponentially distributed

Exponentially distributed service timeExponentially distributed service time

Service times for customers are exponentially distributedService times for customers are exponentially distributed

One serverOne server

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0

0.5

1

1.5

2

2.5

0 1 2 3 4

t

f(t)

Exponential DistributionExponential Distribution Unsymmetrical distributionUnsymmetrical distribution

E(T) =E(T) =1/1/ ==0.50.5

= 2= 2

f(tf(t) =) =ee--tt t >= 0t >= 0

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ExampleExample If theIf the interarrivalinterarrival time is exponentiallytime is exponentially

distributed and its meandistributed and its mean interarrivalinterarrival time is 0.5time is 0.5min (arrival ratemin (arrival rate = 2 customers/min)= 2 customers/min) Customers arrive on average every 0.5 minuteCustomers arrive on average every 0.5 minute

~2/3 of the customers arrive less than 0.5 minute~2/3 of the customers arrive less than 0.5 minuteapartapart

~1/3 of the customers arrive more than 0.5 minute~1/3 of the customers arrive more than 0.5 minuteapartapart

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Memory LessMemory Less PropertyProperty The probability distribution of the remaining time untilThe probability distribution of the remaining time until

the next event (e.g., arrival) occurs is always the same,the next event (e.g., arrival) occurs is always the same,regardless of how much time has passed (i.e., it doesregardless of how much time has passed (i.e., it doesnot remember how much time has passed)not remember how much time has passed)

If theIf the interarrivalinterarrival time of a bus is exponentially distributed,time of a bus is exponentially distributed,then this property implies that the probability that a bus willthen this property implies that the probability that a bus will

arrive in the next minute is the samearrive in the next minute is the same no matter how long youno matter how long youhave waitedhave waited at the bus stopat the bus stop

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Relationship to Poisson DistributionRelationship to Poisson Distribution

If theIf the interarrivalinterarrival time is exponentiallytime is exponentially

distributed with parameterdistributed with parameter , then the number, then the numberof arrivals (e.g., customers) (of arrivals (e.g., customers) (XX) in a specified) in a specified

length of timelength of time TThas a Poisson distribution withhas a Poisson distribution withparameterparameter TT

Also calledAlso called Poisson Arrival ProcessPoisson Arrival Process

Prob(Prob(XX = n= n) =) =n!n!

ee--TT((T)T)nn

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Poisson DistributionPoisson Distribution We can use Poisson distribution to verify ourWe can use Poisson distribution to verify our

assumption of exponentially distributedassumption of exponentially distributedinterarrivalinterarrival timetime

By comparing the number of users that arrive forBy comparing the number of users that arrive for

service during a specific period of time with theservice during a specific period of time with thenumber that the Poisson distribution suggestsnumber that the Poisson distribution suggests

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Quantities of InterestQuantities of Interest There are four major quantities of interest in aThere are four major quantities of interest in a

queuing systemqueuing system

Expected waiting time in the queueExpected waiting time in the queueWWqq ==

Expected system occupancy timeExpected system occupancy timeWW==

Expected number of users in the queueExpected number of users in the queueLLqq ==

Expected number of users in the systemExpected number of users in the systemLL==

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Basic CharacteristicsBasic Characteristics UtilizationUtilization

WhereWhere ssis the number of serversis the number of servers

LetLet PPnn be probability that there arebe probability that there are nn users in theusers in thesystem, thensystem, then

These are true for all queuing modelsThese are true for all queuing models

s

=

0

n

n

L nP

=

=

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LittleLittles Laws Law John D.C. Little at MIT is generally creditedJohn D.C. Little at MIT is generally credited

with being the first to prove thesewith being the first to prove these steadysteady--statestaterelationships formally in 1961relationships formally in 1961

L =L = WW

LLqq == WWqq

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M/M/1M/M/1

InterarrivalInterarrival time is exponentially distributedtime is exponentially distributed

Service time is exponentially distributedService time is exponentially distributed

One ServerOne Server

FCFSFCFS

Assumptions

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M/M/1M/M/1 ProbabilityProbabilitythat there arethat there are nnusers in an M/M/1users in an M/M/1

queue isqueue is

Therefore, the expected number of users in theTherefore, the expected number of users in thesystem (at steady state) issystem (at steady state) is

(1 ) nnP =

0 0

(1 )1

n

n

n n

L nP n

= =

= = = =

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M/M/1M/M/1 Once L is known all other quantities of interestOnce L is known all other quantities of interest

can be computed from Littlecan be computed from Littles Law and thes Law and thefollowing relationshipsfollowing relationships

These equations are bridges between the twoThese equations are bridges between the twoequations in Littleequations in Littles Laws Law

1q

W W

= + qL L

= +

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M/M/1M/M/1 Results can be summarized as followsResults can be summarized as follows

Note also that the probability that the system isNote also that the probability that the system isidle isidle is

L

=

2

( )qL

=

1W

=

( )qW

=

0 1P

=

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Example 1Example 1 At a small grocery store, customers arriveAt a small grocery store, customers arrive

according to Poisson process with a mean of 15according to Poisson process with a mean of 15customers per hour. The length of time it takescustomers per hour. The length of time it takes

to checkout is exponentially distributed withto checkout is exponentially distributed with

mean equals 3 minutes.mean equals 3 minutes. M/M/1M/M/1

= 15 customer/hr= 15 customer/hr = 20 customer/hr= 20 customer/hr

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Example 1Example 1 ComputeCompute

Probability that the checkout counter is idleProbability that the checkout counter is idle

Probability that the checkout counter is busyProbability that the checkout counter is busy

0

151 1 0.25

20P

= = =

01 1 0.25 0.75P = =

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Example 1Example 1 Probability that at least one customer is waitingProbability that at least one customer is waiting

to checkoutto checkout

(1 ) nn

P =

1

1 (1 0.75)0.75 0.1875P = =

150.75

20

= = =

0 1 0.75 0.25P = =

2 0 11 ( ) 1 (0.25 0.1875) 0.5625nP P P = + = + =

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Example 1Example 1 Average number of customers waiting toAverage number of customers waiting to

checkoutcheckout2

( )q

L

=

215 225

2.2520(20 15) 100

qL = = =

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Example 1Example 1 Suppose it costs the operator 3 Baht for eachSuppose it costs the operator 3 Baht for each

minute that a customer spends waiting in theminute that a customer spends waiting in thequeue. What is the average cost per customer?queue. What is the average cost per customer?

q qL W=

2.250.15 /

15

q

q

LW hr customer

= = =

0.15 60 3 27 /Cost baht customer = =

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Example 1Example 1 For an additional of 400 Baht per hour, the operatorFor an additional of 400 Baht per hour, the operator

can decrease the average service time to 2 minutes. Iscan decrease the average service time to 2 minutes. Isthe additional expenditure worthwhile?the additional expenditure worthwhile?

= 30 customer/hr= 30 customer/hr

15 150.033

( ) 30(30 15) 450q

W

= = = =

0.033 60 3 6 /Cost baht customer = =27 6 21 /Savings baht customer = =21 15 315 /Savings baht hr = =

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Example 1Example 1 Alternative solutionAlternative solution

Difference = 405Difference = 405 -- 90 = 315 baht/hr90 = 315 baht/hr

AFTERAFTERBEFOREBEFORE

2.25qL = 0.5qL =

2.25 60 3

405 /

Cost

baht hr

=

=

0.5 60 3

90 /

Cost

baht hr

=

=

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M/G/1M/G/1

InterarrivalInterarrival time is exponentially distributedtime is exponentially distributed

Service time is described only byService time is described only by

-- an average service time, andan average service time, and

-- a variancea varianceOne ServerOne Server

FCFSFCFS

Assumptions

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M/G/1M/G/1 In general, if the service time can be describedIn general, if the service time can be described

by two quantitiesby two quantities Mean (1/Mean (1/))

Variance (Variance (22))

the expected queue length can be estimatedthe expected queue length can be estimatedfromfrom

( )

2 2 2

2 1qL

+=

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M/G/1M/G/1

As in the previous case, all other quantities ofAs in the previous case, all other quantities of

interest can be calculated using Littleinterest can be calculated using Littles Law ands Law andthe following relationshipsthe following relationships

Note also that the probability that the system isNote also that the probability that the system isidle isidle is

1q

W W

= + qL L

= +

0 1P =

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Example 2Example 2

Aircraft arrivals and departures at BangkokAircraft arrivals and departures at BangkokInternational Airport can be approximated as a PoissonInternational Airport can be approximated as a Poissonprocess at the average rate of 40 plane/hr. It takes theprocess at the average rate of 40 plane/hr. It takes theair traffic controller 1.2 min on average to land orair traffic controller 1.2 min on average to land ordispatch an aircraft with variance equals to 1.96 mindispatch an aircraft with variance equals to 1.96 min22..

Assume FCFS queue discipline.Assume FCFS queue discipline. M/G/1M/G/1

= 40 plane/hr= 40 plane/hr

= 50 plane/hr= 50 plane/hr 22 = 0.00054444 hr= 0.00054444 hr22

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Example 2Example 2

Average number of plane waiting for landing orAverage number of plane waiting for landing or

takeofftakeoff

( )( )

( )

2 2 2

22 4050

40 50

2 1

40 0.00054

2 13.78

qL

plane

+=

+

=

=

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Example 2Example 2

Average wait timeAverage wait time

q qL W=

3.780.094 /

40q

q

LW hr plane

= = =

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Example 2Example 2

It is estimated that it costs 81,000 baht per hourIt is estimated that it costs 81,000 baht per hour

for an aircraft to wait in the queue. What is thefor an aircraft to wait in the queue. What is thetotal cost of aircraft waiting to land or takeoff attotal cost of aircraft waiting to land or takeoff atthe airport?the airport?

3.78 81, 000 306, 000 /Cost baht hr = =

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Example 2Example 2

Common mistakesCommon mistakes

0.094 81, 000 7, 650 / /Cost baht hr plane= =

3.78qL plane=

7, 650 3.78 28, 900 /Cost baht hr = =

28, 900 / 306, 000 /baht hr baht hr

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M/M/sM/M/sAssumptionsInterarrivalInterarrival time is exponentially distributedtime is exponentially distributed

Service time is exponentially distributedService time is exponentially distributed

There are s serversThere are s servers

FCFSFCFS

/ /

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M/M/sM/M/s

When there areWhen there are ssservers, all with the sameservers, all with the same

service rateservice rate

( ) ( )10 0

11/

! ! 1

n ss

n s

Pn s

=

= +

( )( ) ( )

1

0 2

1 !

s

qL Ps s

+

=

/ /

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M/M/sM/M/s

As in the previous cases, all other quantities ofAs in the previous cases, all other quantities of

interest can be calculated using Littleinterest can be calculated using Littles Law ands Law andthe following relationshipsthe following relationships

Note, however, thatNote, however, that

1q

W W

= +q

L L

= +

s

=

E l 3

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Example 3Example 3

At a Burger Mary outlet, mean arrival rate of customersAt a Burger Mary outlet, mean arrival rate of customers

is 60 customers per hour and each takes on average 3is 60 customers per hour and each takes on average 3minutes to complete order. Arriving customers formminutes to complete order. Arriving customers form

one sneak queue in front of 4 cashiers. Assumeone sneak queue in front of 4 cashiers. Assume

interarrivalinterarrival and service times are exponentiallyand service times are exponentiallydistributed.distributed.

M/M/sM/M/s



s = 4s = 4

E l 3E l 3

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Example 3Example 3

How would the average queue length and timeHow would the average queue length and time

in the system change if a fifth cashier is opened.in the system change if a fifth cashier is opened.We first check whether steady state is achievableWe first check whether steady state is achievable

600.75

4 20s

= = =

Idl P b bili ( 4)Idl P b bili ( 4)

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Idle Probability (s=4)Idle Probability (s=4)

( ) ( )

( ) ( )

[ ]

1

0

0

43 60 60

20 20

600 4 20

11/

! ! 1

1

1/ ! 4! 1

1 / 13 3.375 4

1/26.5

0.038

n ss

n s

n

n

P

n s

n

=

=

= +

= + = +

==

E d Q L h ( 4)E d Q L h ( 4)

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Expected Queue Length (s=4)Expected Queue Length (s=4)

( )

( ) ( )( )

( )

1

0 2

56020

26020

1 !

0.038 3! 4

243

0.038 6 1

1.53

s

qL P

s s

customers

+ =

=

= =

Idl P b bili ( 5)Idl P b bilit ( 5)

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Idle Probability (s=5)Idle Probability (s=5)

( ) ( )

( ) ( )

[ ]

1

0

0

54 60 60

20 20

600 5 20

11/

! ! 1

1

1/ ! 5! 1

1/ 16.375 2.025 2.5

1/21.44

0.047

n ss

n s

n

n

P

n s

n

=

=

= +

= + = +

==

E t d Q L thE t d Q L th

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Expected Queue LengthExpected Queue Length

( )

( ) ( )( )

( )

1

0 2

66020

26020

1 !

0.047 4! 5

729

0.047 24 4

0.35

s

qL P

s s

customers

+ =

=

= =

E t d Ti i S tE p t d Tim in S t m

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Expected Time in SystemExpected Time in System

S=4S=4

S=5S=5

0.35 0.006 / 0.35 /60

qq

LW hr cus m cus

= = = =

1.530.025 / 1.53 /

60

q

q

LW hr cus m cus

= = = =

Additi n l Q in Ch t i tiAdditional Queuing Characteristics

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Additional Queuing CharacteristicsAdditional Queuing Characteristics

BalkingBalking

Occur when a customer arrives at a finite queue that is fullyOccur when a customer arrives at a finite queue that is fullyoccupiedoccupied

Or when a customer decides not to join the queue because itOr when a customer decides not to join the queue because itis too longis too long

RenegingReneging Occur when a customer leaves the system without beingOccur when a customer leaves the system without being

servedserved

JockeyingJockeying

Occur when a customer switches between queues thinkingOccur when a customer switches between queues thinkings/he will receive service faster by so doings/he will receive service faster by so doing

Final QuestionFinal Question

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Final QuestionFinal Question

Why does the use of express lanes make sense?Why does the use of express lanes make sense?

SupermarketSupermarket Expressway EntranceExpressway Entrance

Self Service CheckSelf Service Check--in at Airport (to a certain extent)in at Airport (to a certain extent)

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