Queuing Models Lecture Presentation.ppt

8/16/2019 Queuing Models Lecture Presentation.ppt

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© 2008 Prentice Hall, Inc. D – 1

OperationsManagement Module D –Module D –Waiting-Line ModelsWaiting-Line Models

PowerPoint presentation to accompanyPowerPoint presentation to accompany

Heizer/RenderHeizer/Render

Principles of Operations Management !ePrinciples of Operations Management !e

Operations Management "eOperations Management "e


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Outline – #ontinued Outline – #ontinued

*$e +ariety of (ueuing Models*$e +ariety of (ueuing Models

Model &Model &(M/M/1)(M/M/1), %ingle-#$annel, %ingle-#$annel

(ueuing Model wit$ Poisson &rri'als(ueuing Model wit$ Poisson &rri'alsand .ponential %er'ice *imesand .ponential %er'ice *imes

Model Model (M/M/S)(M/M/S), Multiple-#$annel, Multiple-#$annel(ueuing Model (ueuing Model

Model # Model # (M/D/1)(M/D/1), #onstant-%er'ice-*ime, #onstant-%er'ice-*imeModel Model

Model D, Limited-Population Model Model D, Limited-Population Model


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Outline – #ontinued Outline – #ontinued

Ot$er (ueuing &pproac$esOt$er (ueuing &pproac$es


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Learning O01ecti'esLearning O01ecti'es

W$en you complete t$is module youW$en you complete t$is module yous$ould 0e a0le to,s$ould 0e a0le to,

2323 Descri0e t$e c$aracteristics ofDescri0e t$e c$aracteristics ofarri'als waiting lines and ser'icearri'als waiting lines and ser'icesystemssystems

4343 &pply t$e single-c$annel 5ueuing &pply t$e single-c$annel 5ueuing

model e5uationsmodel e5uations

6363 #onduct a cost analysis for a#onduct a cost analysis for awaiting linewaiting line


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Learning O01ecti'esLearning O01ecti'es

W$en you complete t$is module youW$en you complete t$is module yous$ould 0e a0le to,s$ould 0e a0le to,

7373 &pply t$e multiple-c$annel &pply t$e multiple-c$annel5ueuing model formulas5ueuing model formulas

8383 &pply t$e constant-ser'ice-time &pply t$e constant-ser'ice-timemodel e5uationsmodel e5uations

9393 Perform a limited-populationPerform a limited-populationmodel analysismodel analysis


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9/59© 2008 Prentice Hall, Inc. D – 9

#$aracteristics of Waiting- #$aracteristics of Waiting-

Line %ystemsLine %ystems2323 &rri'als or inputs to t$e system &rri'als or inputs to t$e system

Population size 0e$a'ior statisticalPopulation size 0e$a'ior statistical

distri0utiondistri0ution

4343 (ueue discipline or t$e waiting line(ueue discipline or t$e waiting lineitself itself

Limited or unlimited in lengt$ disciplineLimited or unlimited in lengt$ disciplineof people or items in it of people or items in it

6363 *$e ser'ice facility *$e ser'ice facility

Design statistical distri0ution of ser'iceDesign statistical distri0ution of ser'ice

timestimes



&rri'al #$aracteristics &rri'al #$aracteristics

2323 %ize of t$e population%ize of t$e population

:nlimited =infinite> or limited =finite>:nlimited =infinite> or limited =finite>

4343 Pattern of arri'alsPattern of arri'als %c$eduled or random often a Poisson%c$eduled or random often a Poisson

distri0utiondistri0ution

6363 e$a'ior of arri'alse$a'ior of arri'als Wait in t$e 5ueue and do not switc$Wait in t$e 5ueue and do not switc$

lineslines

?o 0al;ing or reneging ?o 0al;ing or reneging


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Poisson Distri0utionPoisson Distri0ution

P P (( . . )) B for .B for . = 0, 1, 2, 3, 4, = 0, 1, 2, 3, 4, ee!! . .

. . ""

w$erew$ere P=.>P=.> BB pro0a0ility of . pro0a0ility of .arri'alsarri'als

. . BB num0er of arri'als pernum0er of arri'als per

unit of timeunit of time

BB a'erage arri'al ratea'erage arri'al rate

ee BB 2.71832.7183 ((w$ic$ is t$e 0asew$ic$ is t$e 0aseof t$e natural logarit$msof t$e natural logarit$ms))



Poisson Distri0utionPoisson Distri0ution

Pro0a0ility B P Pro0a0ility B P (( . . )) BBee!! . .

.C .C

348348 –

3434 –

328328 –

3232 –

3838 –

–

P r o 0 a 0 i l i t y


22 4 4 66 77 8 8 9 9 ! ! E E ""

Distri0ution forDistri0ution for = 2= 2

. .

348348 –

3434 –

328328 –

3232 –

3838 –

–



22 4 4 66 77 8 8 9 9 ! ! E E ""

Distri0ution forDistri0ution for = 4= 4

. . 2 2 2222

Figure D.2Figure D.2



Waiting-Line #$aracteristicsWaiting-Line #$aracteristics

Limited or unlimited 5ueue lengt$Limited or unlimited 5ueue lengt$

(ueue discipline - first-in first-out(ueue discipline - first-in first-out=FAFO> is most common=FAFO> is most common

Ot$er priority rules may 0e used inOt$er priority rules may 0e used inspecial circumstancesspecial circumstances



%er'ice #$aracteristics%er'ice #$aracteristics

(ueuing system designs(ueuing system designs

%ingle-c$annel system multiple- %ingle-c$annel system multiple-

c$annel systemc$annel system %ingle-p$ase system multip$ase%ingle-p$ase system multip$ase

systemsystem

%er'ice time distri0ution%er'ice time distri0ution

#onstant ser'ice time#onstant ser'ice time

Random ser'ice times usually aRandom ser'ice times usually anegati'e e.ponential distri0utionnegati'e e.ponential distri0ution



(ueuing %ystem Designs(ueuing %ystem Designs


DeparturesDeparturesafter ser'iceafter ser'ice

%ingle-c$annel single-p$ase system%ingle-c$annel single-p$ase system

(ueue

&rri'als &rri'als

%ingle-c$annel multip$ase system%ingle-c$annel multip$ase system

&rri'als &rri'als DeparturesDeparturesafter ser'iceafter ser'ice

P$ase 2ser'icefacility


%er'icefacility

(ueue

& family dentist)s office & family dentist)s office

& McDonald)s dual window dri'e-t$roug$ & McDonald)s dual window dri'e-t$roug$




Figure D.3Figure D.3Multi-c$annel single-p$ase systemMulti-c$annel single-p$ase system

&rri'als &rri'als

(ueue

Most 0an; and post office ser'ice windowsMost 0an; and post office ser'ice windows


%er'icefacility

#$annel 2

%er'icefacility

#$annel 4

%er'icefacility

#$annel 6




Figure D.3Figure D.3Multi-c$annel multip$ase systemMulti-c$annel multip$ase system

&rri'als &rri'als

(ueue

%ome college registrations%ome college registrations


P$ase 4

ser'icefacility

#$annel 2


#$annel 4

P$ase 2

ser'icefacility

#$annel 2


#$annel 4


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?egati'e .ponential?egati'e .ponential

Distri0utionDistri0ution


2323 –

3"3" –

3E3E –3!3! –

3939 –

3838 –

3737 –

3636 –3434 –

3232 –

33 – P r o 0 a 0 i l i t y t $ a t s e r ' i c e t i

m e

P r o 0 a 0 i l i t y t $ a t s e r ' i c e t i m e #

1

#

1

G G G G G G G G G G G G G

3 3 348 348 38 38 3!8 3!8 23 23 2348 2348 238 238 23!8 23!8 43 43 4348 4348 438 438 43!8 43!8 63 63

*ime t =$ours>*ime t =$ours>

Pro0a0ility t$at ser'ice time is greater t$an t B ePro0a0ility t$at ser'ice time is greater t$an t B e!$!$t t for tfor t # 1# 1

$ =$ = &'erage ser'ice rate &'erage ser'ice rateee = 2.7183= 2.7183

&'erage ser'ice rate &'erage ser'ice rate ($) =($) =

2 customer per $our 2 customer per $our

&'erage ser'ice rate &'erage ser'ice rate ($) = 3($) = 3 customers per $our customers per $our ⇒

&'erage ser'ice time &'erage ser'ice time = 20= 20 minutes per customer minutes per customer


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Measuring (ueueMeasuring (ueue

PerformancePerformance2323 &'erage time t$at eac$ customer or o01ect &'erage time t$at eac$ customer or o01ect

spends in t$e 5ueuespends in t$e 5ueue

4343 &'erage 5ueue lengt$ &'erage 5ueue lengt$6363 &'erage time eac$ customer spends in t$e &'erage time eac$ customer spends in t$e

systemsystem

7373 &'erage num0er of customers in t$e system &'erage num0er of customers in t$e system

8383 Pro0a0ility t$at t$e ser'ice facility will 0e idlePro0a0ility t$at t$e ser'ice facility will 0e idle

9393 :tilization factor for t$e system:tilization factor for t$e system

!3!3 Pro0a0ility of a specific num0er of customersPro0a0ility of a specific num0er of customersin t$e systemin t$e system


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(ueuing #osts(ueuing #osts


*otal e.pected cost *otal e.pected cost

#ost of pro'iding ser'ice#ost of pro'iding ser'ice

#ost #ost

Low le'el Low le'el of ser'iceof ser'ice

Hig$ le'el Hig$ le'el of ser'iceof ser'ice

#ost of waiting time#ost of waiting time

MinimumMinimum*otal *otal cost cost

Optimal Optimal ser'ice le'el ser'ice le'el


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(ueuing Models(ueuing Models

Table D.2Table D.2

Model Model ?ame?ame .ample.ample

& & %ingle-c$annel%ingle-c$annel Anformation counterAnformation counter

systemsystem at department store at department store

(M/M/1)(M/M/1)

?um0er ?um0er ?um0er ?um0er &rri'al &rri'al %er'ice%er'iceof of of of RateRate *ime*ime PopulationPopulation (ueue(ueue

#$annels#$annels P$asesP$ases PatternPattern PatternPattern %ize%ize DisciplineDiscipline

%ingle%ingle %ingle%ingle PoissonPoisson .ponential .ponential :nlimited :nlimited FAFO FAFO


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Table D.2Table D.2


Multic$annelMultic$annel &irline tic;et &irline tic;et

(M/M/S)(M/M/S) countercounter



Multi- Multi- %ingle%ingle PoissonPoisson .ponential .ponential :nlimited :nlimited FAFO FAFO c$annel c$annel


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Table D.2Table D.2


DD LimitedLimited %$op wit$ only a%$op wit$ only a population population dozen mac$ines dozen mac$ines

((finite populationfinite population

))

t$at mig$t 0rea; t$at mig$t 0rea;



%ingle%ingle %ingle%ingle PoissonPoisson .ponential .ponential Limited Limited FAFO FAFO


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Model & – %ingle-#$annel Model & – %ingle-#$annel

2323 &rri'als are ser'ed on a FAFO 0asis and &rri'als are ser'ed on a FAFO 0asis ande'ery arri'al waits to 0e ser'ede'ery arri'al waits to 0e ser'edregardless of t$e lengt$ of t$e 5ueueregardless of t$e lengt$ of t$e 5ueue

4343 &rri'als are independent of preceding &rri'als are independent of precedingarri'als 0ut t$e a'erage num0er ofarri'als 0ut t$e a'erage num0er ofarri'als does not c$ange o'er timearri'als does not c$ange o'er time

6363 &rri'als are descri0ed 0y a Poisson &rri'als are descri0ed 0y a Poisson pro0a0ility distri0ution and come from pro0a0ility distri0ution and come froman infinite populationan infinite population


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7373 %er'ice times 'ary from one customer%er'ice times 'ary from one customerto t$e ne.t and are independent of oneto t$e ne.t and are independent of oneanot$er 0ut t$eir a'erage rate isanot$er 0ut t$eir a'erage rate is

;nown;nown

8383 %er'ice times occur according to t$e%er'ice times occur according to t$enegati'e e.ponential distri0utionnegati'e e.ponential distri0ution

9393 *$e ser'ice rate is faster t$an t$e*$e ser'ice rate is faster t$an t$earri'al ratearri'al rate


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== Mean num0er of arri'als per timeMean num0er of arri'als per time period period

$$ == Mean num0er of units ser'ed perMean num0er of units ser'ed per

time period time period LLss BB &'erage num0er of units &'erage num0er of units

=customers> in t$e system =waiting and 0eing=customers> in t$e system =waiting and 0eingser'ed>ser'ed>

BB

W W ss BB &'erage time a unit spends in t$e &'erage time a unit spends in t$e

system =waiting time plus ser'ice time>system =waiting time plus ser'ice time>

BB

$ –$ –

11

$ –$ –

Table D.3Table D.3


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LL5 5 BB &'erage num0er of units waiting &'erage num0er of units waiting

in t$e 5ueuein t$e 5ueue

BB

W W 5 5 == %&erage%&erage time a unit spendstime a unit spends

waiting in t$e 5ueuewaiting in t$e 5ueue

BB

p p BB :tilization factor for t$e system:tilization factor for t$e system

BB

22

$($ –$($ – ))

$($ –$($ – ))

$$Table D.3Table D.3


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P P 00 BB Pro0a0ility ofPro0a0ility of 00 units in t$eunits in t$e

system =t$at is t$e ser'ice unit is idle>system =t$at is t$e ser'ice unit is idle>

BB 1 –1 –

P P n ; n ; BB Pro0a0ility of more t$an ; units in t$ePro0a0ility of more t$an ; units in t$e

system w$ere n is t$e num0er of units insystem w$ere n is t$e num0er of units int$e systemt$e system

BB

$$

$$

;; ' 1' 1

Table D.3Table D.3


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%ingle-#$annel .ample%ingle-#$annel .ample

== 22 cars arri'ing/$our cars arri'ing/$our

$$ = 3= 3 cars ser'iced/$our cars ser'iced/$our

LLss = = = 2= = = 2 carscarsin t$e system on a'eragein t$e system on a'erage

W W ss BB = = 1= = 1

$our a'erage waiting time in$our a'erage waiting time int$e systemt$e system

LL5 5 == = == =

1.331.33 cars waiting in linecars waiting in line

22

$($ –$($ – ))

$ –$ –

11

$ –$ –

22

3 ! 23 ! 2

11

3 ! 23 ! 2

2222

3(3 ! 2)3(3 ! 2)


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W W 5 5 = == =

= 2/3= 2/3 $our $our = 40= 40 minuteminute

a'erage waiting timea'erage waiting time

p p BB /$ = 2/3 = 66.6 /$ = 2/3 = 66.6

of time mec$anic is 0usy of time mec$anic is 0usy

$($ –$($ – ))

22

3(3 ! 2)3(3 ! 2)

$$P P 00 = 1 ! = .33= 1 ! = .33 pro0a0ility pro0a0ility

t$ere aret$ere are 00 cars in t$e systemcars in t$e system

== 22 cars arri'ing/$our cars arri'ing/$our

$$ = 3= 3 cars ser'iced/$our cars ser'iced/$our


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Pro0a0ility of more t$an ; #ars in t$e %ystemPro0a0ility of more t$an ; #ars in t$e %ystem

; ; P P n ; n ; = (2/3)= (2/3);; ' 1' 1

00 .667.667 ?ote t$at t$is is e5ual to?ote t$at t$is is e5ual to 1 !1 !

P P 00 = 1 ! .33= 1 ! .33

11 .444.444

22 .296.296

33

.198

.198 Amplies t$at t$ere is a

Amplies t$at t$ere is a 19.819.8

c$ance t$at more t$anc$ance t$at more t$an 33 cars are in t$ecars are in t$esystemsystem

44 .132.132

55 .088.088

66 .058.058


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%ingle-#$annel conomics%ingle-#$annel conomics

#ustomer dissatisfaction#ustomer dissatisfaction and lost goodwill and lost goodwill = 10= 10 per $our per $our

W W 5 5 = 2/3= 2/3 $our $our

*otal arri'als*otal arri'als = 16= 16 per day per day

Mec$anic)s salary Mec$anic)s salary = 56= 56 per day per day *otal $ours*otal $ourscustomers spendcustomers spendwaiting per day waiting per day

= (16) = 10= (16) = 10 $ours$ours22

3322

33

#ustomer waiting-time cost#ustomer waiting-time cost = 10 10 = 106.67= 10 10 = 106.672233

*otal e.pected costs*otal e.pected costs = 106.67 ' 56 = 162.67= 106.67 ' 56 = 162.67


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Multi-#$annel Model Multi-#$annel Model

M M == num0er of c$annelsnum0er of c$annels

openopen

== a'erage arri'al ratea'erage arri'al rate

$$ == a'erage ser'ice rate ata'erage ser'ice rate at

eac$ c$annel eac$ c$annel P P 00 B for M B for M $ *$ *11

11

M M ""

11

nn""

M M $$

M M $ !$ !

MM – 1 – 1

nn = 0= 0

$$

nn

$$

M M

IIJJ

LLss B P B P 00 II$($(

/$) /$)M M

((MM ! 1)"(! 1)"(M M $ !$ ! ))22

$$Table D.4Table D.4


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Multi-#$annel Model Multi-#$annel Model

Table D.4Table D.4

W W ss B P B P 00 I BI B$($(

/$) /$)M M

((MM ! 1)"(! 1)"(M M $ !$ ! ))22

11

$$

LLss

LL5 5 B LB Lss – – $$

W W 5 5 B W B W ss – B – B11

$$

LL5 5


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Multi-#$annel .ampleMulti-#$annel .ample

= 2 $ = 3= 2 $ = 3 MM = 2= 2

P P 00 B BB B11

11

4 4 ""

11

nn""

2(3)2(3)

2(3) ! 22(3) ! 2

11

nn = 0= 0

2233

nn

2233

4 4

IIJJ

11

22

LLss B I B B I B(2)(3(2/3)(2)(3(2/3)22 22

331" 2(3) ! 21" 2(3) ! 2 22

11

22

33

44

W W 5 5 = = .0415= = .0415.083.083

22W W ss B B B B3/43/4

22

33

88LL5 5 B – B B – B

22

33

33

44

11

1212


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Waiting Line *a0lesWaiting Line *a0les

Table D.5Table D.5

Poisson &rri'als .ponential %er'ice *imesPoisson &rri'als .ponential %er'ice *imes?um0er of %er'ice #$annels M ?um0er of %er'ice #$annels M

K K 22 4 4 66 77 8 8

.10.10 .0111.0111

.25.25 .0833.0833 .0039.0039

.50.50 .5000.5000 .0333.0333 .0030.0030

.75.75 2.25002.2500 .1227.1227 .0147.0147

1.01.0 .3333.3333 .0454.0454 .0067.0067

1.61.6 2.84442.8444 .3128.3128 .0604.0604 .0121.0121

2.02.0 .8888.8888 .1739.1739 .0398.03982.62.6 4.93224.9322 .6581.6581 .1609.1609

3.03.0 1.52821.5282 .3541.3541

4.04.0 2.21642.2164


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Waiting Line *a0le .ampleWaiting Line *a0le .ample

an; tellers and customersan; tellers and customers

= 18,= 18, $ = 20$ = 20

From *a0le D38From *a0le D38

:tilization factor:tilization factor K B K B / /$ = .90$ = .90 W W 5 5 BBLL5 5

?um0er of?um0er ofser'ice windowsser'ice windows M M

?um0er?um0erin 5ueuein 5ueue *ime in 5ueue*ime in 5ueue

2 window 2 window 11 8.18.1 .45.45 $rs$rs 2727 minutesminutes4 windows4 windows 22 .2285.2285 .0127.0127 $rs$rs ++ minuteminute

6 windows6 windows 33 .03.03 .0017.0017 $rs$rs 66 secondsseconds

7 windows7 windows 44 .0041.0041 .0003.0003 $rs$rs 11 second second


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#onstant-%er'ice Model #onstant-%er'ice Model

Table D.6Table D.6

LL5 5 B B

22

2$($ –2$($ – )) &'erage lengt$ &'erage lengt$of 5ueueof 5ueue

W W 5 5 B B 2$($ –2$($ – )) &'erage waiting time &'erage waiting timein 5ueuein 5ueue

$$

LLss B LB L5 5 II &'erage num0er of &'erage num0er of

customers in systemcustomers in system

W W ss B W B W 5 5 II11

$$ &'erage time &'erage timein t$e systemin t$e system


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© 2008 Prentice Hall, Inc. D – 43?et sa'ings?et sa'ings = 7 /= 7 /triptrip

#onstant-%er'ice .ample#onstant-%er'ice .ample*ruc;s currently wait*ruc;s currently wait 1515 minutes on a'erageminutes on a'erage

*ruc; and dri'er cost*ruc; and dri'er cost 6060 per $our per $our

&utomated compactor ser'ice rate &utomated compactor ser'ice rate ($)($) B 24 truc;s per $our B 24 truc;s per $our

&rri'al rate &rri'al rate ((

)) = 8= 8 per $our per $our

#ompactor costs#ompactor costs 33 per truc; per truc;

#urrent waiting cost per trip#urrent waiting cost per trip = (1/4= (1/4 $r $r )(60) = 15)(60) = 15 / /triptrip

W W 5 5 B B $our B B $our 88

2(12)(122(12)(12 – – 8)8)

11

1212

Waiting cost/tripWaiting cost/tripwit$ compactor wit$ compactor

= (1/12= (1/12 $r wait $r wait )(60/)(60/$r cost $r cost )) = 5 /= 5 /triptrip

%a'ings wit$%a'ings wit$new e5uipment new e5uipment

= 15 (= 15 (current current )) – – 5(5(new new )) = 10= 10

/ /triptrip#ost of new e5uipment amortized #ost of new e5uipment amortized == 3 / 3 /triptri p


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Limited-Population Model Limited-Population Model

%er'ice factor, B%er'ice factor, B

&'erage num0er running, B ?F &'erage num0er running, B ?F (1 !(1 ! ))

&'erage num0er waiting, L B ? &'erage num0er waiting, L B ? (1 !(1 ! F F ))

&'erage num0er 0eing ser'iced, H B F? &'erage num0er 0eing ser'iced, H B F?

&'erage waiting time, W B &'erage waiting time, W B

?um0er of population, ? B I L I H ?um0er of population, ? B I L I H

* *

* I : * I :

* * (1 !(1 ! F F ))

F F

Table D.7Table D.7


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Limited-Population .ampleLimited-Population .ample

%er'ice factor, B%er'ice factor, B = .091 (= .091 (close toclose to .090).090)

For MFor M = 1,= 1, DD = .350= .350 and Fand F = .960= .960


&'erage num0er of printers wor;ing, &'erage num0er of printers wor;ing,

For MFor M = 1,= 1, = (5)(.960)(1 ! .091) = 4.36= (5)(.960)(1 ! .091) = 4.36

For MFor M = 2,= 2, = (5)(.998)(1 ! .091) = 4.54= (5)(.998)(1 ! .091) = 4.54

222 ' 202 ' 20

ac$ ofac$ of 55 printers re5uires repair after printers re5uires repair after 2020 $ours$ours ((: : )) of useof use

One tec$nician can ser'ice a printer inOne tec$nician can ser'ice a printer in 22 $ours$ours ((* * ))

Printer downtime costsPrinter downtime costs 120/120/$our $our

*ec$nician costs*ec$nician costs 25/25/$our $our


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Limited-Population .ampleLimited-Population .ample

%er'ice factor, B%er'ice factor, B = .091 (= .091 (close toclose to .090).090)



&'erage num0er of printers wor;ing, &'erage num0er of printers wor;ing,

For MFor M = 1,= 1, = (5)(.960)(1 ! .091) = 4.36= (5)(.960)(1 ! .091) = 4.36

For MFor M = 2,= 2, = (5)(.998)(1 ! .091) = 4.54= (5)(.998)(1 ! .091) = 4.54

222 ' 202 ' 20

ac$ ofac$ of 55 printers re5uire repair after printers re5uire repair after 2020 $ours$ours ((: : )) of useof use

One tec$nician can ser'ice a printer inOne tec$nician can ser'ice a printer in 22 $ours$ours ((* * ))

Printer downtime costsPrinter downtime costs 120/120/$our $our

*ec$nician costs*ec$nician costs 25/25/$our $our

?um0er of*ec$nicians

&'erage?um0erPrinters

Down =? - >

&'erage#ost/Hr forDowntime=? - >N24

#ost/Hr for*ec$nicians

=N48/$r>*otal

#ost/Hr

1 .64 76.80 25.00 101.80

2 .46 55.20 50.00 105.20


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Ot$er (ueuing &pproac$esOt$er (ueuing &pproac$es

*$e single-p$ase models co'er many*$e single-p$ase models co'er many5ueuing situations5ueuing situations

+ariations of t$e four single-p$ase+ariations of t$e four single-p$asesystems are possi0lesystems are possi0le

Multip$ase modelsMultip$ase modelse.ist for moree.ist for more

comple. situationscomple. situations


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Demonstrati'e Pro0lemDemonstrati'e Pro0lem


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Queuing Models Lecture Presentation.ppt

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Transcript of Queuing Models Lecture Presentation.ppt