Matakuliah
Tahun
: K0414 / Riset Operasi Bisnis dan Industri
: 2008 / 2009
Model Antrian (Waiting Line Models )
Pertemuan 19
Learning Outcomes
• Mahasiswa akan dapat menjelaskan pengertian atrian,
system antrian, struktur dan analisis pola kedatangan
pada sistem antrian.
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Outline Materi:
•
•
•
•
•
•
Pengertian Antrian
Sistem Antrian
Struktur dasar model antrian
Bentuk-bentuk model antrian
Analisis pola kedatangan
Contoh.
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Pengertian
• Istilah “antrian” atau disiplin “garis tunggu” atau “waiting
lines” menunjukan pada kondisi dimana kedatangan
dipilih untuk dilayani. Prosedur yang umum diguna-kan
adalah kedatangan menempati garis tunggu atas dasar
yang : “datang pertama dilayani pertama” (FCFS),
walaupun beberapa prioritas dapat mengubah pola
pelayanan ini, namum analisis ini tidak
mempertimbangkan kemungkinan itu.
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Waiting Lines
• First studied by A. K. Erlang in 1913.
– Analyzed telephone facilities.
• Body of knowledge called queuing theory.
– Queue is another name for waiting line.
• Decision problem:
– Balance cost of providing good service with cost of customers
waiting.
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Waiting Line Examples
Situation
Arrivals
Servers
Bank
Customers
Teller
Deposit etc.
Doctor’s
office
Patient
Doctor
Treatment
Traffic
intersection
Cars
Traffic
Signal
Controlled
passage
Assembly line
Parts
Workers
Assembly
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Service Process
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Waiting Line Components
• Arrivals: Customers (people, machines, calls, etc.) that
demand service.
• Service System: Includes waiting line and servers.
• Waiting Line (Queue): Arrivals waiting for a free server.
• Servers: People or machines that provide service to the
arrivals.
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Sistem Antrian
Fasilitas pelayanan
Antrian
-----XXX
-----
Pertibaan
XXXX
langganan
O
X
X
X
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Saluran Pelayanan
P
e
Ol
a
y
a
On
XXXX ---
Keluar
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Car Wash Example
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Waiting Line Terminology
• Queue: Waiting line.
• Arrival: 1 person, machine, part, etc. that arrives and
demands service.
• Queue discipline: Rules for determining the order that
arrivals receive service.
• Channels: Parallel servers.
• Phases: Sequential stages in service.
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Input Characteristics
• Input source (population) size.
– Infinite: Number in service does not affect probability of a
new arrival.
• A very large population can be treated as infinite.
– Finite: Number in service affects probability of a new arrival.
• Example: Population = 10 aircraft that may need repair.
• Arrival pattern.
– Random: Use Poisson probability distribution.
– Non-random: Appointments.
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Poisson Distribution
• Number of events that occur in an interval of time.
– Example: Number of customers that arrive each half-hour.
• Discrete distribution with mean = 
– Example: Mean arrival rate = 5/hour .
– Probability:
e  x
P( x ) 
x
• Time between arrivals has a negative exponential distribution.
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Waiting Line Characteristics
• Line length:
– Limited: Maximum number waiting is limited.
• Example: Limited space for waiting.
– Unlimited: No limit on number waiting.
• Queue discipline:
– FIFO (FCFS): First in, First out.(First come, first served).
– Random: Select next arrival to serve at random from those waiting.
– Priority: Give some arrivals priority for service.
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Bentuk-bentuk Model Antrian
• Single channel, single phase.
– One server, one phase of service.
• Single channel, multi-phase.
– One server, multiple phases in service.
• Multi-channel, single phase.
– Multiple servers, one phase of service.
• Multi-channel, multia-phase.
– Multiple servers, multiple phases of service.
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Single Channel, Single Phase
Service system
Arrivals
Ships at
sea
Queue
Service
facility
Ship unloading system
Served
units
Empty
ships
Waiting ship line
Dock
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Single Channel, Multi-Phase
Service system
Arrivals
Cars
in area
Queue
Service
facility
McDonald’s drive-through
Cars
& food
Waiting cars
Pay
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Service
facility
Served
units
Pick-up
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Multi-Channel, Single Phase
Service system
Arrivals
Queue
Service
facility
Served
units
Service
facility
Example: Bank customers wait in single line for one
of several tellers.
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Multi-Channel, Multi-Phase
Service system
Arrivals
Queue
Service
facility
Service
facility
Service
facility
Service
facility
Served
units
Example: At a laundromat, customers use one of several
washers, then one of several dryers.
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Analisis Pola Pelayanan
• Random: Use Negative exponential probability
distribution.
– Mean service rate =
• 6 customers/hr.
– Mean service time = 1/
• 1/6 hour = 10 minutes.
• Non-random: May be constant.
– Example: Automated car wash.
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Assumptions in the Basic Model
• Customer population is homogeneous and infinite.
• Queue capacity is infinite.
• Customers are well behaved (no balking or reneging).
• Arrivals are served FCFS (FIFO).
• Poisson arrivals.
– The time between arrivals follows a negative
exponential distribution
• Service times are described by the negative exponential
distribution.
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Steady State Assumptions
• Mean arrival rate , mean service rate , and the number
of servers are constant.
• The service rate is greater than the arrival rate.
• These conditions have existed for a long time.
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