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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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© 2008 Prentice Hall, Inc. D – 3
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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© 2008 Prentice Hall, Inc. D – 4
Outline – #ontinued Outline – #ontinued
Ot$er (ueuing &pproac$esOt$er (ueuing &pproac$es
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© 2008 Prentice Hall, Inc. D – 5
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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© 2008 Prentice Hall, Inc. D – 6
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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#$aracteristics of Waiting- #$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
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&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$ms))
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Poisson Distri0utionPoisson Distri0ution
Pro0a0ility B P Pro0a0ility B P (( . . )) BBee!! . .
.C .C
348348 –
3434 –
328328 –
3232 –
3838 –
–
P r o 0 a 0 i l i t y
P r o 0 a 0 i l i t y
22 4 4 66 77 8 8 9 9 ! ! E E ""
Distri0ution forDistri0ution for = 2= 2
. .
348348 –
3434 –
328328 –
3232 –
3838 –
–
P r o 0 a 0 i l i t y
P r o 0 a 0 i l i t y
22 4 4 66 77 8 8 9 9 ! ! E E ""
Distri0ution forDistri0ution for = 4= 4
. . 2 2 2222
Figure D.2Figure D.2
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Waiting-Line #$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
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%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
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(ueuing %ystem Designs(ueuing %ystem Designs
Figure D.3Figure D.3
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$ase 2ser'icefacility
P$ase 4ser'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$roug$
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(ueuing %ystem Designs(ueuing %ystem Designs
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
DeparturesDeparturesafter ser'iceafter ser'ice
%er'icefacility
#$annel 2
%er'icefacility
#$annel 4
%er'icefacility
#$annel 6
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(ueuing %ystem Designs(ueuing %ystem Designs
Figure D.3Figure D.3Multi-c$annel multip$ase systemMulti-c$annel multip$ase system
&rri'als &rri'als
(ueue
%ome college registrations%ome college registrations
DeparturesDeparturesafter ser'iceafter ser'ice
P$ase 4
ser'icefacility
#$annel 2
P$ase 4ser'icefacility
#$annel 4
P$ase 2
ser'icefacility
#$annel 2
P$ase 2ser'icefacility
#$annel 4
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?egati'e .ponential?egati'e .ponential
Distri0utionDistri0ution
Figure D.4Figure D.4
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$e system
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© 2008 Prentice Hall, Inc. D – 21
(ueuing #osts(ueuing #osts
Figure D.5Figure D.5
*otal e.pected cost *otal e.pected cost
#ost of pro'iding ser'ice#ost of pro'iding ser'ice
#ost #ost
Low le'el Low le'el of ser'iceof ser'ice
Hig$ 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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(ueuing Models(ueuing Models
Table D.2Table D.2
Model Model ?ame?ame .ample.ample
Multic$annelMultic$annel &irline tic;et &irline tic;et
(M/M/S)(M/M/S) countercounter
?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
Multi- Multi- %ingle%ingle PoissonPoisson .ponential .ponential :nlimited :nlimited FAFO FAFO c$annel c$annel
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(ueuing Models(ueuing Models
Table D.2Table D.2
Model Model ?ame?ame .ample.ample
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;
?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 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$ange o'er time
6363 &rri'als are descri0ed 0y a Poisson &rri'als are descri0ed 0y a Poisson pro0a0ility distri0ution and come from pro0a0ility distri0ution and come froman infinite populationan infinite population
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Model & – %ingle-#$annel Model & – %ingle-#$annel
7373 %er'ice times 'ary from one customer%er'ice times 'ary from one customerto t$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$earri'al ratearri'al rate
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© 2008 Prentice Hall, Inc. D – 29
Model & – %ingle-#$annel Model & – %ingle-#$annel
== Mean num0er of arri'als per timeMean num0er of arri'als per time period period
$$ == Mean num0er of units ser'ed perMean num0er of units ser'ed per
time period time period LLss BB &'erage num0er of units &'erage num0er of units
=customers> in t$e system =waiting and 0eing=customers> in t$e system =waiting and 0eingser'ed>ser'ed>
BB
W W ss BB &'erage time a unit spends in t$e &'erage time a unit spends in t$e
system =waiting time plus ser'ice time>system =waiting time plus ser'ice time>
BB
$ –$ –
11
$ –$ –
Table D.3Table D.3
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Model & – %ingle-#$annel Model & – %ingle-#$annel
LL5 5 BB &'erage num0er of units waiting &'erage num0er of units waiting
in t$e 5ueuein t$e 5ueue
BB
W W 5 5 == %&erage%&erage time a unit spendstime a unit spends
waiting in t$e 5ueuewaiting in t$e 5ueue
BB
p p BB :tilization factor for t$e system:tilization factor for t$e system
BB
22
$($ –$($ – ))
$($ –$($ – ))
$$Table D.3Table D.3
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Model & – %ingle-#$annel Model & – %ingle-#$annel
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
$$ = 3= 3 cars ser'iced/$our cars ser'iced/$our
LLss = = = 2= = = 2 carscarsin t$e system on a'eragein t$e system on a'erage
W W ss BB = = 1= = 1
$our a'erage waiting time in$our a'erage waiting time int$e systemt$e system
LL5 5 == = == =
1.331.33 cars waiting in linecars waiting in line
22
$($ –$($ – ))
$ –$ –
11
$ –$ –
22
3 ! 23 ! 2
11
3 ! 23 ! 2
2222
3(3 ! 2)3(3 ! 2)
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%ingle-#$annel .ample%ingle-#$annel .ample
W W 5 5 = == =
= 2/3= 2/3 $our $our = 40= 40 minuteminute
a'erage waiting timea'erage waiting time
p p BB /$ = 2/3 = 66.6 /$ = 2/3 = 66.6
of time mec$anic is 0usy of time mec$anic is 0usy
$($ –$($ – ))
22
3(3 ! 2)3(3 ! 2)
$$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
$$ = 3= 3 cars ser'iced/$our cars ser'iced/$our
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%ingle-#$annel .ample%ingle-#$annel .ample
Pro0a0ility of more t$an ; #ars in t$e %ystemPro0a0ility of more t$an ; #ars in t$e %ystem
; ; P P n ; n ; = (2/3)= (2/3);; ' 1' 1
00 .667.667 ?ote t$at t$is is e5ual to?ote t$at t$is is e5ual to 1 !1 !
P P 00 = 1 ! .33= 1 ! .33
11 .444.444
22 .296.296
33
.198
.198 Amplies t$at t$ere is a
Amplies t$at t$ere is a 19.819.8
c$ance t$at more t$anc$ance t$at more t$an 33 cars are in t$ecars are in t$esystemsystem
44 .132.132
55 .088.088
66 .058.058
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%ingle-#$annel conomics%ingle-#$annel conomics
#ustomer dissatisfaction#ustomer dissatisfaction and lost goodwill and lost goodwill = 10= 10 per $our per $our
W W 5 5 = 2/3= 2/3 $our $our
*otal arri'als*otal arri'als = 16= 16 per day per day
Mec$anic)s salary Mec$anic)s salary = 56= 56 per day per day *otal $ours*otal $ourscustomers spendcustomers spendwaiting per day waiting per day
= (16) = 10= (16) = 10 $ours$ours22
3322
33
#ustomer waiting-time cost#ustomer waiting-time cost = 10 10 = 106.67= 10 10 = 106.672233
*otal e.pected costs*otal e.pected costs = 106.67 ' 56 = 162.67= 106.67 ' 56 = 162.67
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Multi-#$annel Model Multi-#$annel Model
M M == num0er of c$annelsnum0er of c$annels
openopen
== a'erage arri'al ratea'erage arri'al rate
$$ == 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$new e5uipment new e5uipment
= 15 (= 15 (current current )) – – 5(5(new new )) = 10= 10
/ /triptrip#ost of new e5uipment amortized #ost of new e5uipment amortized == 3 / 3 /triptri p
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Limited-Population Model Limited-Population Model
%er'ice factor, B%er'ice factor, B
&'erage num0er running, B ?F &'erage num0er running, B ?F (1 !(1 ! ))
&'erage num0er waiting, L B ? &'erage num0er waiting, L B ? (1 !(1 ! F F ))
&'erage num0er 0eing ser'iced, H B F? &'erage num0er 0eing ser'iced, H B F?
&'erage waiting time, W B &'erage waiting time, W B
?um0er of population, ? B I L I H ?um0er of population, ? B I L I H
* *
* I : * I :
* * (1 !(1 ! F F ))
F F
Table D.7Table D.7
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Limited-Population .ampleLimited-Population .ample
%er'ice factor, B%er'ice factor, B = .091 (= .091 (close toclose to .090).090)
For MFor M = 1,= 1, DD = .350= .350 and Fand F = .960= .960
For MFor M = 2,= 2, DD = .044= .044 and Fand F = .998= .998
&'erage num0er of printers wor;ing, &'erage num0er of printers wor;ing,
For MFor M = 1,= 1, = (5)(.960)(1 ! .091) = 4.36= (5)(.960)(1 ! .091) = 4.36
For MFor M = 2,= 2, = (5)(.998)(1 ! .091) = 4.54= (5)(.998)(1 ! .091) = 4.54
222 ' 202 ' 20
ac$ 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 $our
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Limited-Population .ampleLimited-Population .ample
%er'ice factor, B%er'ice factor, B = .091 (= .091 (close toclose to .090).090)
For MFor M = 1,= 1, DD = .350= .350 and Fand F = .960= .960
For MFor M = 2,= 2, DD = .044= .044 and Fand F = .998= .998
&'erage num0er of printers wor;ing, &'erage num0er of printers wor;ing,
For MFor M = 1,= 1, = (5)(.960)(1 ! .091) = 4.36= (5)(.960)(1 ! .091) = 4.36
For MFor M = 2,= 2, = (5)(.998)(1 ! .091) = 4.54= (5)(.998)(1 ! .091) = 4.54
222 ' 202 ' 20
ac$ 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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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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Demonstrati'e Pro0lemDemonstrati'e Pro0lem
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