2. 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 evaluatingIt is extremely useful in predicting and evaluating
system performance.system performance.
Queuing theory has been used for operationsQueuing theory has been used for operations
research, manufacturing and systems analysis.research, manufacturing and systems analysis.
Traditional queuing theory problems refer toTraditional queuing theory problems refer to
customers visiting a store, analogous to requestscustomers visiting a store, analogous to requests
arriving at a device.arriving at a device.
3. Applications of Queuing TheoryApplications of Queuing Theory
TelecommunicationsTelecommunications
Traffic controlTraffic control
Determining the sequence of computerDetermining the sequence of computer
operationsoperations
Predicting computer performancePredicting computer performance
Health services (e.g.. control of hospital bedHealth services (e.g.. control of hospital bed
assignments)assignments)
Airport traffic, airline ticket salesAirport traffic, airline ticket sales
Layout of manufacturing systems.Layout of manufacturing systems.
4. 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
5. Characteristics of Queuing
Systems
Key elementsKey elements of queuing systemsof queuing systems
•• Customer:--Customer:-- refers to anything that arrives at a facilityrefers to anything that arrives at a facility
and requires service, e.g., people, machines, trucks,and requires service, e.g., people, machines, trucks,
emails.emails.
•• Server:--Server:-- refers to any resource that provides therefers to any resource that provides the
requested service, eg. repairpersons, retrieval machines,requested service, eg. repairpersons, retrieval machines,
runways at airport.runways at airport.
6. 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
7. Components of a Queuing SystemComponents of a Queuing System
Arrival Process
Servers
Queue or
Waiting Line
Service Process
Exit
8. Parts of a Waiting LineParts of a Waiting Line
Dave’sDave’s
Car WashCar Wash
enterenter exitexit
Population ofPopulation of
dirty carsdirty cars
ArrivalsArrivals
from thefrom the
generalgeneral
population …population …
QueueQueue
(waiting line)(waiting line)
ServiceService
facilityfacility
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 distributionStatistical distribution
of arrivalsof arrivals
Waiting LineWaiting Line
CharacteristicsCharacteristics
•Limited vs. unlimitedLimited vs. unlimited
•Queue disciplineQueue discipline
ServiceService
CharacteristicsCharacteristics
•Service designService design
•StatisticalStatistical
distribution ofdistribution of
serviceservice
9. 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
10. 3.3. Service systemService system
1.1. A single service system.A single service system.
Queue
ArrivalsArrivals
Service
facility
DeparturesDepartures
after serviceafter service
e.g- Your family dentist’s office, Library countere.g- Your family dentist’s office, Library counter
11. 2. Multiple, parallel server, single2. Multiple, parallel server, single
queue modelqueue model
Queue
Service
facility
Channel 1
Service
facility
Channel 2
Service
facility
Channel 3
ArrivalsArrivals
DeparturesDepartures
after serviceafter service
e.g- Booking at a service statione.g- Booking at a service station
12. 3. Multiple, parallel facilities with3. Multiple, parallel facilities with
multiple queues Modelmultiple queues Model
Service station Customers
leave
Queues
Arrivals
e.g.- Different cash counters in electricity office
13. 4. Service facilities in a series4. Service facilities in a series
Arrivals
Queues
Service station 1 Service station 2
Queues
Customers
leave
Phase 1 Phase 2
e.g.- Cutting, turning, knurling, drilling, grinding,
packaging operation of steel
14. Queuing ModelsQueuing Models
1.1. Deterministic queuing modelDeterministic queuing model
2.2. Probabilistic queuing modelProbabilistic queuing model
1.1. Deterministic queuing modelDeterministic queuing model :--:--
λλ == Mean number of arrivals per timeMean number of arrivals per time
periodperiod
µµ == Mean number of units served perMean number of units served per
time periodtime period
15. AssumptionsAssumptions
1.1. IfIf λλ > µ, then waiting line shall be formed and> µ, then waiting line shall be formed and
increased indefinitely and service system would failincreased indefinitely and service system would fail
ultimatelyultimately
2. If2. If λλ µ, there shall be no waiting lineµ, there shall be no waiting line
≤
≤
16. 2.Probabilistic queuing model2.Probabilistic queuing model
Probability that n customers will arrive in theProbability that n customers will arrive in the
system in time interval T issystem in time interval T is
( ) ( )
!n
et
nP
tn
t
λ
λ −
=
17. Single Channel ModelSingle Channel Model
λλ == Mean number of arrivals per timeMean number of arrivals per time
periodperiod
µµ == Mean number of units served perMean number of units served per
time periodtime period
LLss == Average number of unitsAverage number of units
(customers) in the system (waiting and being(customers) in the system (waiting and being
served)served)
==
WWss == Average time a unit spends in theAverage time a unit spends in the
system (waiting time plus service time)system (waiting time plus service time)
==
λλ
µ –µ – λλ
11
µ –µ – λλ
18. LLqq == Average number of units waitingAverage number of units waiting
in the queuein the queue
==
WWqq == AverageAverage time a unit spendstime a unit spends
waiting in the queuewaiting in the queue
==
pp == Utilization factor for the systemUtilization factor for the system
==
λλ22
µ(µ –µ(µ – λλ))
λλ
µ(µ –µ(µ – λλ))
λλ
µµ
19. PP00 == Probability ofProbability of 00 units in theunits in the
system (that is, the service unit is idle)system (that is, the service unit is idle)
== 1 –1 –
PPn > kn > k == Probability of more than k units in theProbability of more than k units in the
system, where n is the number of units insystem, where n is the number of units in
the systemthe system
==
λλ
µµ
λλ
µµ
kk + 1+ 1
20. Single Channel Model ExampleSingle Channel Model Example
λλ == 22 cars arriving/hourcars arriving/hour
µµ = 3= 3 cars serviced/hourcars serviced/hour
LLss = = = 2= = = 2 carscars
in the system on averagein the system on average
WWss == = = 1= = 1
hour average waiting time inhour average waiting time in
the systemthe system
LLqq == = == =
1.331.33 cars waiting in linecars waiting in line
λλ22
µ(µ –µ(µ – λλ))
λλ
µ –µ – λλ
11
µ –µ – λλ
22
3 - 23 - 2
11
3 - 23 - 2
2222
3(3 - 2)3(3 - 2)
21. Cont…Cont…
λλ == 22 cars arriving/hour,cars arriving/hour, µµ = 3= 3 carscars
serviced/hourserviced/hour
WWqq = == =
= 40= 40 minuteminute
average waiting timeaverage waiting time
pp == λλ/µ = 2/3 =/µ = 2/3 =
66.6%66.6% of time mechanicof time mechanic
is busyis busy
λλ
µ(µ –µ(µ – λλ))
22
3(3 - 2)3(3 - 2)
λλ
µµ
PP00 = 1 - = .33= 1 - = .33 probabilityprobability
there arethere are 00 cars in the systemcars in the system
22. Suggestions for Managing QueuesSuggestions for Managing Queues
1.1. Determine an acceptable waiting time forDetermine an acceptable waiting time for
your customersyour customers
2.2. Try to divert your customer’s attention whenTry 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 customersKeep employees not serving the customers
out of sightout of sight
5.5. Segment customersSegment customers
23. 6.6. Train your servers to be friendlyTrain your servers to be friendly
7.7. Encourage customers to come during theEncourage customers to come during the
slack periodsslack periods
8.8. Take a long-term perspective toward gettingTake a long-term perspective toward getting
rid of the queuesrid of the queues
24. Where the Time GoesWhere the Time Goes
In a life time, the averageIn a life time, the average
person will spendperson 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
