waiting line analysis opim 310-lecture 3 instructor: jose cruz

Waiting Line AnalysisWaiting Line Analysis

OPIM 310-Lecture 3

Instructor: Jose Cruz

Elements of Waiting Elements of Waiting Line AnalysisLine Analysis

QueueQueueA single waiting lineA single waiting line

Waiting line system consists ofWaiting line system consists ofArrivalsArrivalsServersServersWaiting line structuresWaiting line structures

Common Queuing Common Queuing SituationsSituations

Dock workers load and unloadShips and bargesHarbor

Repair people fix machinesBroken machinesMachine maintenance

Transactions handled by tellerCustomerBank

Switching equipment to forward calls

CallersTelephone company

Computer processes jobsPrograms to be runComputer system

Treatment by doctors and nurses

PatientsDoctor’s office

Collection of tolls at boothAutomobilesHighway toll booth

Checkout clerks at cash register

Grocery shoppersSupermarket

Service ProcessArrivals in QueueSituation

Components of Components of Queuing SystemQueuing System

Source of customers—calling population

ServerServerArrivalsArrivals Waiting LineWaiting Lineor or

““Queue”Queue”

ServedServedcustomerscustomers

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

Elements of a Waiting LineElements of a Waiting Line Calling populationCalling population

Source of customersSource of customers Infinite - large enough that one more Infinite - large enough that one more

customer can always arrive to be servedcustomer can always arrive to be served Finite - countable number of potential Finite - countable number of potential

customerscustomers

Arrival rate (Arrival rate ()) Frequency of customer arrivals at waiting line Frequency of customer arrivals at waiting line

system system Typically follows Poisson distributionTypically follows Poisson distribution

Elements of a Waiting LineElements of a Waiting Line

Service timeService time Often follows negative exponential Often follows negative exponential

distributiondistribution Average service rate = Average service rate =

Arrival rate (Arrival rate () must be less than service ) must be less than service rate rate or system never clears outor system never clears out

Distribution Of ArrivalsDistribution Of Arrivals

• Assumption: arrivals occur randomly and Assumption: arrivals occur randomly and independently on each otherindependently on each other

• Poisson distribution provides a good description Poisson distribution provides a good description of the arrival pattern:of the arrival pattern:

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

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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

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Pro

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00 11 22 33 44 55 66 77 88 99

Distribution for Distribution for = 4 = 4

xx1010 1111

Distribution Of Service TimesDistribution Of Service Times

• In general service time can follow any arbitrary distribution

• The simplest, however, is an exponential:

Distribution Of Service TimesDistribution Of Service Times

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

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ility

th

at s

ervi

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ime

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th

at s

ervi

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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 in hoursTime t in 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

Elements of a Waiting LineElements of a Waiting Line

Queue disciplineQueue disciplineOrder in which customers are servedOrder in which customers are servedFirst come, first served is most First come, first served is most

commoncommonLength can be infinite or finiteLength can be infinite or finite

Infinite is most commonInfinite is most commonFinite is limited by some physical Finite is limited by some physical

structurestructure

Basic Waiting Line Basic Waiting Line StructuresStructures

Channels are the number of Channels are the number of parallel serversparallel servers

Phases denote number of Phases denote number of sequential servers the customer sequential servers the customer must go throughmust go through

Single-Channel StructuresSingle-Channel Structures

Single-channel, single-phaseSingle-channel, single-phase

Waiting lineWaiting line ServerServer

Single-channel, multiple phasesSingle-channel, multiple phases

ServersServersWaiting lineWaiting line

Multi-Channel StructuresMulti-Channel Structures

ServersServers

Multiple-channel, single phaseMultiple-channel, single phase

Waiting lineWaiting line

ServersServers

Waiting lineWaiting line

Multiple-channel, multiple-phaseMultiple-channel, multiple-phase

Operating Operating CharacteristicsCharacteristics

Mathematics of queuing theory does Mathematics of queuing theory does not provide optimal or best solutionsnot provide optimal or best solutions

Operating characteristics are computed Operating characteristics are computed that describe system performancethat describe system performance

Steady state is constant, average value Steady state is constant, average value for performance characteristics that the for performance characteristics that the system will reach after a long timesystem will reach after a long time


NOTATIONNOTATION OPERATING CHARACTERISTICOPERATING CHARACTERISTIC

LL Average number of customers in the Average number of customers in the system (waiting and being served)system (waiting and being served)

LLqq Average number of customers in the Average number of customers in the waiting linewaiting line

WW Average time a customer spends in the Average time a customer spends in the system (waiting and being served)system (waiting and being served)

WWqq Average time a customer spends Average time a customer spends waiting in linewaiting in line


NOTATIONNOTATION OPERATING CHARACTERISTICOPERATING CHARACTERISTIC

PP00 Probability of no (zero) customers in the Probability of no (zero) customers in the

systemsystem

PPnn Probability of Probability of nn customers in the system customers in the system

Utilization rate; the proportion of time Utilization rate; the proportion of time the system is in usethe system is in use

Cost Relationship in Cost Relationship in Waiting Line AnalysisWaiting Line Analysis

Exp

ecte

d c

ost

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co

sts

Level of serviceLevel of service

Total costTotal cost

Service costService cost

Waiting CostsWaiting Costs

Waiting Line Costs and Waiting Line Costs and Quality ServiceQuality Service

Traditional view is that the level of Traditional view is that the level of service should coincide with service should coincide with minimum point on total cost curveminimum point on total cost curve

TQM approach is that absolute TQM approach is that absolute quality service will be the most cost-quality service will be the most cost-effective in the long runeffective in the long run

Single-Channel, Single-Single-Channel, Single-Phase ModelsPhase Models

All assume Poisson arrival rateAll assume Poisson arrival rateVariationsVariations

Exponential service timesExponential service times General (or unknown) distribution of service General (or unknown) distribution of service

timestimes Constant service timesConstant service times Exponential service times with finite queue Exponential service times with finite queue

lengthlength Exponential service times with finite calling Exponential service times with finite calling

populationpopulation

Basic Single-Server ModelBasic Single-Server Model

Assumptions:Assumptions:Poisson arrival ratePoisson arrival rateExponential service timesExponential service timesFirst-come, first-served queue disciplineFirst-come, first-served queue disciplineInfinite queue lengthInfinite queue lengthInfinite calling populationInfinite calling population

= mean arrival rate= mean arrival rate = mean service rate= mean service rate

Formulas for Single-Formulas for Single-Server ModelServer Model

LL = =

- -

Average number of Average number of customers in the systemcustomers in the system

Probability that no customers Probability that no customers are in the system (either in the are in the system (either in the queue or being served)queue or being served)

PP00 = 1 - = 1 -

Probability of exactly Probability of exactly nn customers in the systemcustomers in the system

PPnn = • = • PP00

nn

= 1 -= 1 -

nn

Average number of Average number of customers in the waiting linecustomers in the waiting line

LLqq = =

(( - - ))

Formulas for Single-Formulas for Single-Server ModelServer Model

==

Probability that the server Probability that the server is busy and the customer is busy and the customer has to waithas to wait

Average time a customer Average time a customer spends in the queuing systemspends in the queuing system WW = = = =

11--

LL

Probability that the server Probability that the server is idle and a customer can is idle and a customer can be servedbe served

II = 1 - = 1 -

= 1 - == 1 - = P P00

Average time a customer Average time a customer spends waiting in line to spends waiting in line to be servedbe served

WWqq = =

(( - - ))

A Single-Server ModelA Single-Server ModelGiven Given = 24 per hour, = 24 per hour, = 30 customers per hour = 30 customers per hour

Probability of no Probability of no customers in the customers in the systemsystem

PP00 = 1 - = 1 - = = 1 - = 1 - =

0.200.20

24243030

LL = = = 4 = = = 4Average number Average number of customers in of customers in the systemthe system

--

242430 - 2430 - 24

Average number Average number of customers of customers waiting in linewaiting in line

LLqq = = = 3.2 = = = 3.2(24)(24)22

30(30 - 24)30(30 - 24)22

(( - - ))

A Single-Server ModelA Single-Server ModelGiven Given = 24 per hour, = 24 per hour, = 30 customers per hour = 30 customers per hour

Average time in the Average time in the system per customer system per customer WW = = = 0.167 hour = = = 0.167 hour

11--

1130 - 2430 - 24

Average time waiting Average time waiting in line per customer in line per customer

WWqq = = = 0.133 = = = 0.133((--))

242430(30 - 24)30(30 - 24)

Probability that the Probability that the server will be busy and server will be busy and the customer must waitthe customer must wait

= = = 0.80= = = 0.80

24243030

Probability the Probability the server will be idleserver will be idle II = 1 - = 1 - = 1 - 0.80 = 0.20 = 1 - 0.80 = 0.20

1.1. Another employee to pack up Another employee to pack up purchasespurchases

2.2. Another checkout counterAnother checkout counter

Waiting Line Cost AnalysisWaiting Line Cost Analysis

To improve customer services To improve customer services management wants to test two management wants to test two alternatives to reduce customer alternatives to reduce customer waiting time:waiting time:

Waiting Line Cost AnalysisWaiting Line Cost Analysis Add extra employee to increase service rate Add extra employee to increase service rate

from 30 to 40 customers per hour from 30 to 40 customers per hour Extra employee costs $150/weekExtra employee costs $150/week Each one-minute reduction in customer Each one-minute reduction in customer

waiting time avoids $75 in lost saleswaiting time avoids $75 in lost sales Waiting time with one employee = 8 minutesWaiting time with one employee = 8 minutes

Example 2Example 2

WWqq = 0.038 hours = 2.25 minutes = 0.038 hours = 2.25 minutes

8.00 - 2.25 = 5.75 minutes reduction8.00 - 2.25 = 5.75 minutes reduction5.75 x $75/minute/week = $431.25 per week5.75 x $75/minute/week = $431.25 per week

New employee saves $431.25 - $150.00 = $281.25/wkNew employee saves $431.25 - $150.00 = $281.25/wk

Waiting Line Cost AnalysisWaiting Line Cost Analysis New counter costs $6000 plus $200 per week New counter costs $6000 plus $200 per week

for checkerfor checker Customers divide themselves between two Customers divide themselves between two

checkout linescheckout lines Arrival rate is reduced from Arrival rate is reduced from = 24 to = 24 to = 12= 12 Service rate for each checker is Service rate for each checker is = 30 = 30

Example 2Example 2

WWqq = 0.022 hours = 1.33 minutes = 0.022 hours = 1.33 minutes

8.00 - 1.33 = 6.67 minutes8.00 - 1.33 = 6.67 minutes

6.67 x $75/minute/week = $500.00/wk - $200 = $300/wk6.67 x $75/minute/week = $500.00/wk - $200 = $300/wkCounter is paid off in 6000/300 = 20 weeksCounter is paid off in 6000/300 = 20 weeks

Waiting Line Cost AnalysisWaiting Line Cost Analysis Adding an employee results in savings and Adding an employee results in savings and

improved customer serviceimproved customer service Adding a new counter results in slightly Adding a new counter results in slightly

greater savings and improved customer greater savings and improved customer service, but only after the initial investment service, but only after the initial investment has been recoveredhas been recovered

A new counter results in more idle time for A new counter results in more idle time for employeesemployees

A new counter would take up potentially A new counter would take up potentially valuable floor spacevaluable floor space

Example 2Example 2

Constant Service TimesConstant Service TimesConstant service times occur with Constant service times occur with

machinery and automated machinery and automated equipmentequipment

Constant service times Constant service times are a special case are a special case of the single-server of the single-server model with general model with general or undefined service timesor undefined service times

Finite Queue LengthFinite Queue Length A physical limit exists on length of queueA physical limit exists on length of queue MM = maximum number in queue = maximum number in queue Service rate does not have to exceed arrival Service rate does not have to exceed arrival

rate (rate () to obtain steady-state ) to obtain steady-state conditionsconditions

Single-Channel Waiting Line Model Single-Channel Waiting Line Model With Poisson Arrivals And Arbitrary With Poisson Arrivals And Arbitrary

Service Times (M/G/1)Service Times (M/G/1)

• Notation:


Service Times (M/G/1)Service Times (M/G/1)Operating Characteristics

The probability that no units are in the system

The average number of units in the waiting line

The average number of units in the system


Service Times (M/G/1)Service Times (M/G/1)Operating Characteristics

The average time a unit spends in the waiting line

The average time a unit spends in the system

The probability that an arriving unit has to wait for service

The average number of units in the waiting line

Multiple-Channel, Multiple-Channel, Single-Phase ModelsSingle-Phase Models

Two or more independent servers serve a Two or more independent servers serve a single waiting linesingle waiting line

Poisson arrivals, exponential service, Poisson arrivals, exponential service, infinite calling populationinfinite calling population

ss>>

PP00 = =11

11s!s!

ssss

ss - - nn==ss-1-1

nn=0=0

11nn!!

nn

++


Two or more independent servers serve a Two or more independent servers serve a single waiting linesingle waiting line

Poisson arrivals, exponential service, Poisson arrivals, exponential service, infinite calling populationinfinite calling population

ss>>

PP00 = =11

11s!s!

ssss

ss - - nn==ss-1-1

nn=0=0

11nn!!

nn

++

Computing P0 can be time-consuming.

Tables can used to find P0 for selected values of and s.


Probability of exactly Probability of exactly nn customers in the customers in the systemsystem

PPnn = =

PP00, , for for n n > > ss11

ss! ! ssn-sn-s

nn

PP00, , for for n n > > ss11

nn!!

nn

Probability an arrivingProbability an arrivingcustomer must waitcustomer must wait PPww = = PP00

11

ss!!

ssss - -

ss

Average number of Average number of customers in systemcustomers in system LL = = P P00 + +

((//))ss

((ss - 1)!( - 1)!(ss - - ))22


WW = = LL

Average time customerAverage time customerspends in systemspends in system

= =

//ssUtilization factorUtilization factor

Average time customer Average time customer spends in queuespends in queue WWqq = = WW - = - =

11

LLqq

LLqq = = L L --

Average number of Average number of customers in queuecustomers in queue

Multiple-Server SystemMultiple-Server SystemCustomer service areaCustomer service area = 10 customers/area= 10 customers/area = 4 customers/hour per service rep= 4 customers/hour per service repss = (3)(4) = 12 = (3)(4) = 12

PP00 = 0.045 = 0.045Probability no customers Probability no customers are in the systemare in the system

Number of customers in Number of customers in the service departmentthe service department LL = 6 = 6

Waiting time in the Waiting time in the service departmentservice department WW = = LL / / = 0.60 = 0.60

Multiple-Server SystemMultiple-Server SystemCustomer service areaCustomer service area = 10 customers/area= 10 customers/area = 4 customers/hour per service rep= 4 customers/hour per service repss = (3)(4) = 12 = (3)(4) = 12

LLqq = = L L - - // = 3.5 = 3.5Number of customers Number of customers waiting to be servedwaiting to be served

Average time customers Average time customers will wait in linewill wait in line WWqq = = L Lqq// = 0.35 hours = 0.35 hours

Probability that Probability that customers must waitcustomers must wait PPww = 0.703= 0.703

Add a 4th server to improve serviceAdd a 4th server to improve service Recompute operating characteristicsRecompute operating characteristics

PP00 = 0.073 prob of no customers = 0.073 prob of no customers

LL = 3.0 customers = 3.0 customers WW = 0.30 hour, 18 min in service = 0.30 hour, 18 min in service LLqq = 0.5 customers waiting = 0.5 customers waiting

WWqq = 0.05 hours, 3 min waiting, versus 21 earlier = 0.05 hours, 3 min waiting, versus 21 earlier

PPww = 0.31 prob that customer must wait = 0.31 prob that customer must wait

Improving ServiceImproving Service

Splitting Arrival FlowSplitting Arrival Flow• Arrival rate sometimes depends on

type of customer• Or, some customers prefer one

queue over another (when there is a choice)

• Idea: determine percentage of customers joining a queue based on type of preference

Splitting Flow ExampleSplitting Flow Example• Queue system with two lines• Line 1 is served by 2 clerks, each clerk

has an average service time of 5 minutes• Line 2 is served by a single automated

system that takes 2 minutes on average• 75% of the customers prefer the line

served by the human clerks• What is the waiting time, system size, etc?

Cost EvaluationCost Evaluation

• Service Cost = (Number of servers) x (wages per time unit) = s Cs

• Waiting Cost = (Number of customers waiting in the system) x (cost of waiting per time unit) = L Cw

• Total cost = service cost + waiting cost

Decision AreasDecision Areas Arrival RatesArrival Rates Number of Service Number of Service

FacilitiesFacilities Number of PhasesNumber of Phases Number of Servers Number of Servers

Per FacilityPer Facility Server EfficiencyServer Efficiency Priority RulePriority Rule Line ArrangementLine Arrangement

waiting line analysis opim 310-lecture 3 instructor: jose cruz

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