““QUEUING THEORY”QUEUING THEORY”

Queuing TheoryQueuing Theory

Queuing theory is the mathematics of waiting lines.Queuing theory is the mathematics of waiting lines.

It is extremely useful in predicting and evaluating It is extremely useful in predicting and evaluating system performance.system performance.

Queuing theory has been used for operations Queuing theory has been used for operations research, manufacturing and systems analysis. research, manufacturing and systems analysis. Traditional queuing theory problems refer to Traditional queuing theory problems refer to customers visiting a store, analogous to requests customers visiting a store, analogous to requests arriving at a device.arriving at a device.

Applications of Queuing TheoryApplications of Queuing Theory

TelecommunicationsTelecommunications Traffic control Traffic control Determining the sequence of computer Determining the sequence of computer

operationsoperations Predicting computer performancePredicting computer performance Health services (e.g.. control of hospital Health services (e.g.. control of hospital

bed bed

assignments)assignments) Airport traffic, airline ticket salesAirport traffic, airline ticket sales Layout of manufacturing systems.Layout of manufacturing systems.

Queuing SystemQueuing System

Model processes in which customers arrive. Model processes in which customers arrive. Wait their turn for service.Wait their turn for service. Are serviced and then leave.Are serviced and then leave.

input

Server

Queue

output

Characteristics of Queuing Systems

Key elementsKey elements of queuing systems of queuing systems

• • Customer:--Customer:-- refers to anything that refers to anything that arrives at a facility and requires service, arrives at a facility and requires service, e.g., people, machines, trucks, emails.e.g., people, machines, trucks, emails.

•• Server:--Server:-- refers to any resource that refers to any resource that provides the requested service, eg. provides the requested service, eg. repairpersons, retrieval machines, repairpersons, retrieval machines, runways at airport.runways at airport.

System Customers Server

Reception desk People Receptionist

Hospital Patients Nurses

Airport Airplanes Runway

Road network Cars Traffic light

Grocery Shoppers Checkout station

Computer Jobs CPU, disk, CD

Queuing examplesQueuing examples

Components of a Queuing SystemComponents of a Queuing System

Arrival Process

ServersQueue or Waiting Line

Service Process

Exit

Parts of a Waiting LineParts of a Waiting Line

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

Exit the systemExit the systemArrivals to the systemArrivals to the system In 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. unlimitedLimited vs. unlimited•Queue disciplineQueue discipline

Service Service CharacteristicsCharacteristics•Service designService design•Statistical Statistical distribution of distribution of serviceservice

1. 1. Arrival ProcessArrival Process

According to sourceAccording to source According to numbersAccording to numbers According to timeAccording to time

2. Queue Structure

• First-come-first-served (FCFS) • Last-come-first-serve (LCFS)• Service-in-random-order (SIRO)• Priority service

3. 3. Service systemService system

1. 1. A single service system.A single service system.

Queue

ArrivalsArrivalsService facility

DeparturesDeparturesafter serviceafter service

e.g- Your family dentist’s office, Library countere.g- Your family dentist’s office, Library counter

2. Multiple, parallel server, single 2. Multiple, parallel server, single queue modelqueue model

Queue

Service facility

Channel 1

Service facility

Channel 2

Service facility

Channel 3

ArrivalsArrivals DeparturesDeparturesafter serviceafter service

e.g- Booking at a service statione.g- Booking at a service station

3. Multiple, parallel facilities with 3. Multiple, parallel facilities with multiple queues Modelmultiple queues Model

Service station Customers leave

QueuesArrivals

e.g.- Different cash counters in electricity office

4. Service facilities in a series4. Service facilities in a series

Arrivals

Queues

Service station 1 Service station 2

QueuesCustomers leave

Phase 1 Phase 2

e.g.- Cutting, turning, knurling, drilling, grinding, packaging operation of steel

Queuing ModelsQueuing Models1.1. Deterministic queuing modelDeterministic queuing model2.2. Probabilistic queuing modelProbabilistic queuing model

1.1. Deterministic queuing modelDeterministic queuing model :-- :--

== Mean number of arrivals per time Mean number of arrivals per time periodperiod

µµ = = Mean number of units served per Mean number of units served per time periodtime period

AssumptionsAssumptions

1.1. If If > µ, then waiting line shall be formed and > µ, then waiting line shall be formed and increased indefinitely and service system would fail increased indefinitely and service system would fail ultimatelyultimately

2. If 2. If µ, there shall be no waiting line µ, there shall be no waiting line

2.Probabilistic queuing model2.Probabilistic queuing model

Probability that n customers will arrive in the Probability that n customers will arrive in the system in time interval T is system in time interval T is

!n

etnP

tn

t

Single Channel ModelSingle Channel Model

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

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

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

==

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

==

µ – µ –

11µ – µ –

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

==

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

==

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

==

22

µ(µ – µ(µ – ))

µ(µ – µ(µ – ))

µµ

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

== 1 –1 –

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

==

µµ

µµ

k k + 1+ 1

Single Channel Model Single Channel Model ExampleExample

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

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

the system on averagethe system on average

WWss = = = = 1= = 1 hour hour

average waiting time in the average waiting time in the systemsystem

LLqq== = = 1.33 = = 1.33

cars waiting in linecars waiting in line

22

µ(µ – µ(µ – ))

µ – µ –

11µ – µ –

22

3 - 23 - 2

11

3 - 23 - 2

2222

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

Cont…Cont…

== 2 2 cars arriving/hour,cars arriving/hour, µ µ = 3 = 3 cars serviced/hourcars serviced/hour

WWqq= = = =

= 40 = 40 minute average minute average waiting timewaiting time

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

µ(µ – µ(µ – ))

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

µµ

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

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

Suggestions for Managing QueuesSuggestions for Managing Queues

1.1. Determine an acceptable waiting time for Determine an acceptable waiting time for your customersyour customers

2.2. Try to divert your customer’s attention when Try to divert your customer’s attention when waitingwaiting

3.3. Inform your customers of what to expectInform your customers of what to expect

4.4. Keep employees not serving the customers Keep employees not serving the customers out of sightout of sight

5.5. Segment customersSegment customers

6.6. Train your servers to be friendlyTrain your servers to be friendly

7.7. Encourage customers to come during the Encourage customers to come during the slack periodsslack periods

8.8. Take a long-term perspective toward getting Take a long-term perspective toward getting rid of the queuesrid of the queues

Where the Time GoesWhere the Time Goes In a life time, the average In a life time, the average person will spend person will spend ::

SIX MONTHS Waiting at stoplightsSIX MONTHS Waiting at stoplights

EIGHT MONTHS Opening junk mailEIGHT MONTHS Opening junk mail

ONE YEAR Looking for misplaced 0bjectsONE YEAR Looking for misplaced 0bjects TWO YEARS Reading E-mailTWO YEARS Reading E-mail FOUR YEARS Doing houseworkFOUR YEARS Doing housework FIVE YEARS Waiting in lineFIVE YEARS Waiting in line SIX YEARS EatingSIX YEARS Eating