lecture 13 (notes) manscie

7/21/2019 Lecture 13 (Notes) manscie

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Lecture Day 13

Queuing Models


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Queuing Models


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Queueis another term for a waiting line, and

a queuing system is simply a system thatinvolves a waiting line. Queuing theoryis abranch of management science that enablesthe analyst to describe the behavior ofqueuing systems.

Queuing theory does not addressoptimization problems directly. Rather, ituses elements of statistics and mathematicsfor the construction of models that describethe important descriptive statistics of aqueuing system.


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Members of a queue are known ascalling

units. The queuing system descriptive statisticsinclude such factors as the epected waitingtime of the calling units, the epected lengthof the line, and the percentage of idle time

for the service facility!the source of goodsor services for which the calling units wait".

#hen queuing theory is applied,management$s ob%ective is usually to

minimize two kinds of costs&

' Those associated with providing service

' Those associated with waiting time


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Parts of Any QueuingSystem

(alling)opulation

. . .

Queue

*ervice+acility

*erved(allingnits


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The calling population has threecharacteristics that are important toconsider when deciding on what type of

queuing model to apply&' The size of the calling population

' The pattern of arrivals at the queuingsystem

' The attitude of the calling units

(alling)opulation

. . .


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queuing model to apply&' The size of the calling population can be-nite or in-nite.

(alling)opulation

. . .

The key to determining whether anin-nite calling population can be

assumed is whether the probability ofan arrival is signi-cantly changed whena member or members of a populationare receiving service and thus cannot

arrive to the system.


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(alling)opulation

. . .

can be predetermined/scheduledor

random.0f arrivals are scheduled, analyticalqueuing models are usuallyinappropriate. 0f arrivals are random, itis necessary to determine the

probability distribution of the timebetween intervals.


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0t has been shown mathematically that if the

probability density function of the interarrival times is eponential, calling unitsarrive according to a socalled )oissonprocess.

)oisson arrivals generally eist in situationswhere the number of arrivals during a certaintime interval is independent of the number ofarrivals that have occurred in previous timearrivals.

This basic property states that the conditionalprobability of any future event depends onlyon the present state of the system and isindependent of previous states of the system.


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The )oisson )robability 1ensity +unctiongives the probability of narrivals in timeperiod t.

)n!t" 2 et!t"n n 2 3, 4, 5, . . .

n6

where&

n 2 number of arrivals

t 2 size of the time interval 2 mean arrival rate per unit of time


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' The attitude of the calling units

(alling)opulation

. . .

can be

patientor impatient.

There are two forms of impatientattitudes, namely& balkingandreneging.


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In Summary:(alling

)opulation

(haracteristics

*ize

7rrival

)attern 7ttitude

+inite 0n-nite Random)re

1etermined

)atient 0mpatien

t

)oisson 8ther 9alking Renegin

g


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The service facility has three basicproperties&

' The structure of the queuing system' The distribution of service times

' The service discipline

. . . *ervice+acility


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The service facility has three basicproperties&

'The structure of the queuing systemcan be singlephaseor multiphase.


The great ma%ority of queuing models aresinglephase models. 0t is possible,

nonetheless, to view a multiphase systemas separate, singlephase systems inwhich the output from one serverbecomes the input for another server.


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PossibleStructures:

(alling)opulation

. . .

Queue

*ervice+acility

*erved(allingnits

*ingle)hase, *ingle(hannel Queuing *ystem


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PossibleStructures:

(alling)opulation

. . .

Queue

*ervice

+acilityno. 4 *erved

(allingnits

*ingle)hase, Multi(hannel Queuing *ystem

*ervice+acility

no. 5


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PossibleStructures:

(alling)opulation

. . .

Queue

*ervice+acility

type 4

*erved(allingnits

*ervice+acility

type 5

Queue

Multi)hase, *ingle(hannel Queuing *ystem


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PossibleStructures:

(alling)opulation

. . .

Queue

*ervice

+acility

no.4type 4

*erved(allingnits

Multi)hase, Multi(hannel Queuing *ystem

*ervice

+acility

no.5type 4

*ervice

+acility

no.4type 5*ervic

e

+acility

no.5type 5


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The service facility has three basicproperties&' The structure of the queuing system

' The distribution of service times


can beconstantor random.

0f service time is a random variable, it

is necessary to determine how thatrandom variable is distributed. 0n mostcases, service times are eponentiallydistributed. 7s such, the probability ofrelatively long service times is small.


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The service facility has three basicproperties&' The structure of the queuing system

' The distribution of service times

' The service discipline


determines whichcalling unit in the queuing system

receives service.


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Classications of Service Disciplines

+irst come, -rst served

)riority

' )reemptive' :onpreemptive

Random


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In Summary:*ervice

+acility

(haracteristics

*tructure

*ervice

Times

*ervice

1iscipline

*ingle

)hase

Multi

)hase

Random(onstant

;ponenti

al

8ther*ingle

(hannel

Multi

(hannel

+(+* )riorityRandom

)reemptiv

e

:on

)reemptiv


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ElementaryQueuing Models


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Notations to be used:


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Notations to be used (cont.:

2 mean arrival rate (number of callingunits per unit of time)

2 mean service rate (number of callingunits served per unit of time)

4/2 mean service time for a calling unit

s 2 number of parallel (equivalent)service facilities in the system

)!n"2 probability of having nunits in thesystem

2 server utilization factor (that is, theproportion of timethe server

can be expected to be busy)


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The Basic Single-Server o!el

)oisson arrival process

;ponential service

times *ingle server

+(+* service discipline

0n-nite source

0n-nite queue

)atient calling units

The assumptions of this modelare&


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*tatistics to be determined& The probability of 3 calling units in the

system& )!3" 2 4 = !/"

The probability of ncalling units in thesystem& )!n" 2 )!3"!/"n

The proportion of time the server is busy&

2

/


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*tatistics to be determined!con$t."& ;pected number of calling units in the

system& B !B= 5"?2 5 / >B !5"?

2 4/B min.

2 4@ seconds


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ulti-Server o!el $ith PoissonArrivals an! "#ponential Service

Times

The assumptions of this model areidentical to those of the basic singleserver model ecept that the number ofservers is assumed to be greater than

one. 7lso, it is assumed that all servershave the same rate of service.


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*tatistics to be determined&


Times

4

The probability of 3 calling units in thesystem&

)!3" 2 4

s 4 !/"n E !/"s

n6 s6 s

n 2 3

!4 "


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*tatistics to be determined!con$t."&


Times

The probability of ncalling units in thesystem&

)!n" 2 !/"n for 3 F n F s

n6

)!3"

2 !/"n for n G s)!3"

s6sns

The proportion of time the server is busy&

2

/s

' assuming each server has the samemean service rate of units per time

period


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*tatistics to be determined!con$t."&


Times

;pected number of calling units in thequeue&

lecture 13 (notes) manscie

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