Waiting Line Theory 2 Waiting Line Theory 2

Akhid Yulianto, SE, MSc (Log)Akhid Yulianto, SE, MSc (Log)

Poisson ProbabilityPoisson Probability

x = Tingkat kedatanganx = Tingkat kedatangan λλ = rata rata kedatangan per = rata rata kedatangan per

periodeperiode e = 2.71828e = 2.71828

!)(

x

exP

x

Eksponential ProbabilityEksponential Probability

µ =jumlah unit yang di layani per µ =jumlah unit yang di layani per periodeperiode

e = 2.71828e = 2.71828

tettimeserviceP 1)(

M/M/1M/M/1

Ls = average number of units in Ls = average number of units in the system (waiting and being the system (waiting and being served)served)

Ws = average time a unit spends Ws = average time a unit spends in the systemin the system

Lq = average number of units Lq = average number of units waiting in the queuewaiting in the queue

Wq = Average time a unit Wq = Average time a unit spends waiting in the queuespends waiting in the queue

Utilization factor for the systemUtilization factor for the system Probability of 0 units in the Probability of 0 units in the

systemsystem Probability of more than k units Probability of more than k units

in the system, where n is the in the system, where n is the number of units in the systemnumber of units in the system

1

0

2

1

1

k

kn

q

q

s

s

P

P

W

L

W

L

ExampleExample

Tom Jones, mekanik di toko Tom Jones, mekanik di toko Golden Muffler, dapat Golden Muffler, dapat memasang muffler baru dengan memasang muffler baru dengan rata rata 3/jam (mengikuti rata rata 3/jam (mengikuti eksponential distribution). eksponential distribution). Customer yang meminta service Customer yang meminta service ini dengan rata rata kedatangan ini dengan rata rata kedatangan 2/ jam (poisson distribution). 2/ jam (poisson distribution). Pelayanan FCFS dan populasi Pelayanan FCFS dan populasi yang tak terbatas.yang tak terbatas.

Analisa Waiting Line 1Analisa Waiting Line 1stst

λλ = 2 = 2 µ = 3µ = 3 Ls = rata rata 2 mobil di sistem/jamLs = rata rata 2 mobil di sistem/jam Ws = 1 jam rata rata menunggu di sistemWs = 1 jam rata rata menunggu di sistem Lq = 1.33 mobil menunggu di garis , rata Lq = 1.33 mobil menunggu di garis , rata

ratarata Wq = 40 menit waktu menunggu per Wq = 40 menit waktu menunggu per

mobil.mobil. ρρ = 66.6% mekanik sibuk = 66.6% mekanik sibuk PP0 0 = 0.33 kemungkinan tidak ada mobil di = 0.33 kemungkinan tidak ada mobil di

sistemsistem

MM//MM//kk Queuing System Queuing System

Multiple channels (with one central waiting Multiple channels (with one central waiting line)line)

Poisson arrival-rate distributionPoisson arrival-rate distribution Exponential service-time distributionExponential service-time distribution Unlimited maximum queue lengthUnlimited maximum queue length Infinite calling populationInfinite calling population Examples:Examples:

Four-teller transaction counter in bankFour-teller transaction counter in bank Two-clerk returns counter in retail storeTwo-clerk returns counter in retail store

M/M/SM/M/S

Ls = average number of Ls = average number of units in the system units in the system (waiting and being (waiting and being served)served)

Ws = average time a unit Ws = average time a unit spends in the systemspends in the system

Lq = average number of Lq = average number of units waiting in the units waiting in the queuequeue

Wq = Average time a Wq = Average time a unit spends waiting in unit spends waiting in the queuethe queue

Probability of 0 units in Probability of 0 units in the systemthe system

qsq

sq

s

M

s

mM

n

n

M

s

LWW

LL

LP

MMW

forM

MM

Mn

P

pMM

L

1

1

!1

!1

!1

1

!1

02

1

0

0

02

ExampleExample

Toko Golden Muffler Toko Golden Muffler memutuskan untuk membuka memutuskan untuk membuka garasi kedua dan menyewa garasi kedua dan menyewa mekanik kedua untuk mekanik kedua untuk menangani instalasi muffler. menangani instalasi muffler. Tingkat kedatangan dan tingkat Tingkat kedatangan dan tingkat layanan sama. Analisa? layanan sama. Analisa?

Analisa waiting line 2Analisa waiting line 2thth

Ls = 0.75 mobil di dalam sistemLs = 0.75 mobil di dalam sistem Ws = 22.5 menit sebuah mobil Ws = 22.5 menit sebuah mobil

di sistemdi sistem Lq = 0.083 mobil di antrianLq = 0.083 mobil di antrian Wq = 2.5 menit sebuah mobil di Wq = 2.5 menit sebuah mobil di

antrianantrian

M/D/1M/D/1

Constant Constant service time service time modelmodel

Contoh: Contoh: assembly assembly line/pencucian line/pencucian mobil otomatismobil otomatis

2

2

1

2

q

q

qs

qs

W

L

WW

LL

CostsCosts

Berdasar jumlah unit customerBerdasar jumlah unit customer TC = Cw L + Cs kTC = Cw L + Cs k TC = Total costTC = Total cost Cw = cost of waitingCw = cost of waiting L = jumlah rata rata units di sistemL = jumlah rata rata units di sistem Cs = Service costCs = Service cost k or s = channel numberk or s = channel number L = Lq + L = Lq +

λ

µ

Prinsip biayaPrinsip biaya

Bandingkan biaya yang Bandingkan biaya yang terendah terendah

Bisa terjadi pada perencanaan Bisa terjadi pada perencanaan untuk penambahan channeluntuk penambahan channel

Atau penambahan layananAtau penambahan layanan

TambahanTambahan

Buku lain punya rumus yang Buku lain punya rumus yang berbeda namun hasil berbeda namun hasil perhitungan perhitungan ± sama± sama

Jadi jangan bingungJadi jangan bingung

ReferenceReference

Anderson, & Sweeney, 2002, Anderson, & Sweeney, 2002, Quantitative for decision Quantitative for decision making,9making,9thth edn, Sydney edn, Sydney

Heizer, J.,& Render, B., 2006, Heizer, J.,& Render, B., 2006, Operation Management, 8Operation Management, 8thth edn, edn, Pearson Education, Singapore Pearson Education, Singapore