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Modul S20 Teknik Antrian

Jul 05, 2018

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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

     Average num!er ofcustomers in system

     Average num!er ofcustomers in #ueue

    & 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

     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

    & 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

     Average num!er ofcustomers in system

     Average num!er ofcustomers in #ueue

     Average time customerspends in #ueue

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

    1

    µW & W# 

     Average time customer spends in system

    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

     Average num!er ofcustomers in system

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     Average num!er of

    customers in #ueue

     Average time customer spends in system

     Average time customerspends in #ueue

     L

    W & λ '6 * $M,

    L#  &  λ '6* $M,

      µ

    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

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

     Average num!er ofcustomers in #ueue

    $-  &

    1

    (λ / µ)nDG

    'D * n,GΣD

     n & -

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    W  & W#  1

    µ

     L# L  & '6* $-,

    W# &  'D * L, λ

     L#

     Average time customer spends in system

     Average time customerspends in #ueue

     Average num!er ofcustomers in system

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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 * λ ' ,

    λ

    µ' ,λµλ µ' ,s

    's * 6) G 'sµ * λ)2 

    $- 

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

     Average num!er ofcustomers in system

    $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 $- 

     Average num!er ofcustomers in system

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

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     Average num!er ofcustomers in #ueue

     Average time customerspends in #ueue

    $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

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