Modul S20 Teknik Antrian

8/16/2019 Modul S20 Teknik Antrian

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Waiting Line Models forService Improvement

Model Antrian untuk

Perbaikan Pelayanan


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Elements Of

Waiting Line Analysis Queue

a single waiting line

Waiting line system

arrivals

servers

waiting line structures


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Calling population – source of customers

– an infinite population assumes suc a largenum!er of customers tat it is always possi!lefor one more customer to arrive to !e served

–

a finite population consists of a counta!lenum!er of potential customers

Arrival rate" λ

– fre#uency of customer arrivals to system – typically follows $oisson distri!ution


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Service time – often follows negative e%ponential distri!ution

– average service rate & µ

Arrival rate must !e less tan service rate

or system never clears out

'λ < µ)


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Components Of A Queuing

System

Source of

customers Arrivals Server Served

customersWaiting Line

or #ueue


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Queue (iscipline And Lengt

Queue discipline – order in wic customers are served

– first come" first served is most common

Lengt can !e infinite or finite

– infinite is most common – finite is limited !y some pysical

structure


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)asic Waiting Line Structures

Channels are te num!er of parallel servers

Phases denote num!er of se#uentialservers te customer must go troug


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Single*Cannel Structures

ServersWaiting line

Single*cannel" multiple pases

Waiting line Server

Single*cannel" single*pase


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Multi*Cannel Structures

Servers

Servers

Waiting line

Multiple*cannel" single pase

Multiple*cannel" multiple*pase

Waiting line


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Operating Caracteristics

+e matematics of #ueuing teory does notprovide optimal or !est solutions

Instead" operating caracteristics arecomputed tat descri!e system performance

Steady state provides te average value of

performance caracteristics tat te systemwill reac after a long time


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Operating Caracteristics

Notation Description

L Average num!er of customers in te system

'waiting and !eing served,L# Average num!er of customers in te waiting line

W Average time a customer spends in te system'waiting and !eing served,

W# Average time a customer spends waiting in line


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$- $ro!a!ility of no '.ero, customers in te system

$n $ro!a!ility of n customers in te system

ρ /tili.ation rate0 te proportion of time tesystem is in use


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Cost 1elationsip In Waiting Line

Analysis

E % p e c t e d

c o s t s

Level of service

+otal cost

Servicecost

Waiting Costs


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Waiting Line Analysis

And Quality +raditional view * te level of service

sould coincide wit te minimum point on

te total cost curve

+QM view * a!solute #uality service is temost cost*effective in te long run


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Single*Cannel" Single*$ase

Models All assume $oisson arrival rate

2ariations

– e%ponential service times – general 'or un3nown, distri!ution of service times

– constant service times

– e%ponential service times wit finite #ueue lengt

– e%ponential service times wit finite callingpopulation


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)asic Single*Server Model

Assumptions4 – $oisson arrival rate

– e%ponential service times – first*come" first*served #ueue discipline

– infinite #ueue lengt

– infinite calling population

λ & mean arrival rate

µ & mean service rate


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5ormulas for Single*Server Model

$- &λ

µ'6 * ,

$n &λ

µ

n$-

&λ

µ' ,λ

µ'6 * ,

L# &

L &λ

µ − λ

λ2

µ(µ − λ)

$ro!a!ility tat no customersare in system

$ro!a!ility of e%actly n

customers in system

Average num!er ofcustomers in system

Average num!er ofcustomers in #ueue

n,'


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1

(µ − λ)

W & &L

λ

W# &λ

µ(µ − λ)

λ µ

ρ &

1 − ρΙ & & & $- λ

µ'6 * ,

Average time customer

spends in system

Average time customerspends in #ueue

$ro!a!ility tat server is !usy" utili.ation factor

$ro!a!ility tat server isidle 7 customer can !e served


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Single*Server E%ample

8iven4 λ & 9: per our" µ & ;- customers per our

$- &λ

µ

'6 * ,

L# &

L &λ

µ − λ

λ2

µ(µ − λ)

$ro!a!ility tat no customers

are in system



& 6 * '9:


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1

(µ − λ)W &

W# &λ

µ(µ − λ)

λ µ

ρ &

1 − ρI &

Average time customer spends in system



$ro!a!ility tat server isidle 7 customer can !e served

& 6';-*9:, & -=6>? r & 6- min

& 6 * -=@- & -=9-

& 9:


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Waiting Line Cost Analysis

Management wants to test two alternatives toreduce customer waiting time4

6= ire anoter employee to pac3 uppurcases

9= Open anoter cec3out counter


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Alternative 6

E%tra employee costs B6- < wee3

Eac one*minute reduction in customer waiting time avoidsB? in lost sales

E%tra employee will increase service rate to :- customers perour

1ecompute operating caracteristics

W# & -=-;@ ours & 9=9 minutes" originally was @ minutes

@=-- * 9=9 & =? minutes =? % B?


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Alternative II

Dew counter costs B>--- plus B9-- per wee3 for cec3er

Customers divide temselves !etween two cec3out lines

Arrival rate is reduced from λ & 9: to λ & 69

Service rate for eac cec3er is µ & ;- 1ecompute operating caracteristics

W# & -=-99 ours & 6=;; minutes" originally was @ minutes

@=-- * 6=;; & >=>? minutes

>=>? % B?


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Constant Service +imes

Constant service times occur witmacinery and automated e#uipment

Constant service times are a specialcase of te single*server model wit

general or undefined service times


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Operating Caracteristics 5or

Constant Service +imes

W# &L#

λ

λ

µ'6 * ,$- &

L# &λ2 σ2 + (λ / µ) 2

2 ( 1 − λ / µ )

L & L# λ

µ

$ro!a!ility tat no customersare in system





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1

µW & W#


λ

µρ &

Wen service time is

constant and σ & -"formula can !e simplified

L# &λ2 σ2 + (λ / µ) 2

2 ( 1 − λ / µ )

λ2 0 + (λ / µ) 2

2 ( 1 − λ / µ )

&

& (λ / µ) 2

2 ( 1 − λ / µ )

λ 2

2 µ(µ − λ)&


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Constant Service +ime E%ample

Automated car was wit service time & := min

Cars arrive at rate λ & 6-


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5inite Queue Lengt

A pysical limit e%ists on lengt of #ueue

M & ma%imum num!er in #ueue

Service rate does not ave to e%ceed arrival rate to

o!tain steady*state conditions 'µ > λ,

$- &

L & λ / µ

1 − λ / µ

1 − λ / µ

1 − (λ / µ)M+1

$n & '$- , λµ' ,n for n ≤ M

'M 6)'λ / µ, M 6

6 * 'λ / µ ,M6

$ro!a!ility tat nocustomers are in system

$ro!a!ility of e%actly ncustomers in system



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Average num!er of

customers in #ueue



L

W & λ '6 * $M,

L# & λ '6* $M,

µ

L

1

µWW# &

Let $M & pro!a!ility a customer will not Foin te system


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5inite Queue E%ampleQuic3 Lu!e as waiting space for only ; carsλ & 9-" µ & ;-" M & : cars '6 in service ; waiting,

$ro!a!ility tat no

cars are in system

$ro!a!ility of e%actlyn cars in system

Average num!er ofcars in system

$- &1 − λ / µ

1 − (λ / µ)M+1

6 * 9-

L & λ / µ

1 − λ / µ

'M 6)'λ / µ, M 6

6 * 'λ / µ ,M6

& 9-


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Average num!er ofcars in #ueue

Average time carspends in system

Average time car

spends in #ueue

L

W & λ'6 * $M,

L# & λ '6* $M,

µ

1

µWW# &

L * 9-'6*-=-?>,

;-& 6=9: * & -=>9

6=9:

9- '6*-=-?>,& & -=>? ours

& :=-; min

1

;--=->? *& & -=-;; ours

& 9=-; min


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5inite Calling $opulation

Arrivals originate from a finite 'counta!le, population

D & population si.e

L# & λ µ

λ D *

$n & $-λ

µ' ,

nDG

'D * n,G

were n & 6" 9" ===" D

'6* $-,

$ro!a!ility tat nocustomers are in system



$- &

1

(λ / µ)nDG

'D * n,GΣD

n & -


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W & W# 1

µ

L# L & '6* $-,

W# & 'D * L, λ

L#





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5inite Calling $opHn E%ample

9- macines wic operate an average of 9-- rs!efore !rea3ing down λ & 6


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L# &

-=-- -=9??@

-=--& 9- '6* -=>9,

Average num!er ofmacines in #ueue

W & W# 6

µ

L & L# '6*$-, & -=6> '6*-=>9, & -=9-

W# & 'D * L, λ

L#

Average time macinespends in system

Average time macinespends in #ueue

Average num!er of

macines in system

& -=6>

λ µ

λD '6* $-,

'9- * -=9-, -=--

-=6>& & 6=?:

6=?: 6

-=9?@& & =;; rs


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Multiple*Cannel" Single*$ase Models

+wo or more independent servers serve a singlewaiting line

$oisson arrivals" e%ponential service" infinite callingpopulation

sµ J λ

$- &6

[ Σn & s * 6

n & -K

λ

µ' ,n6

nG

6

sG

λ

µ' ,s sµ

sµ * λ ' ,


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L &

$n & $-"λ

µ' ,n6

sG sn*sfor n J s

$n & $-"λ

µ' ,n6

nG

$w & $-λ

µ' ,s6

sG

sµ

sµ * λ ' ,

λ

µ' ,λµλ µ' ,s

's * 6) G 'sµ * λ)2

$-



$ro!a!ility an arrivingcustomer must wait

for n & s


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λ


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Multiple*Server E%ample

Customer service area

λ & 6- customers


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L &λ

µ' ,λµλ µ' ,

s

's * 6) G 'sµ * λ)2 $-


'6-,':, '6-- r & ;> min


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$w & $-λ

µ' ,s6 sµ

sµ * λ ' ,$ro!a!ility an arriving

customer must wait

W# & µ & ;= * 6-


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Improving Service

Add a :t server to improve service

1ecompute operating caracteristics

$o & -=-?; pro! of no customersL & ;=- customers

W & -=;- our" 6@ min in service

L# & -= customers waiting

W# & -=- ours" ; min waiting" versus 96 earlier

$w & -=;6 pro! tat customer must wait

Modul S20 Teknik Antrian

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