1. Model Antrian (Waiting Line Models ) Pertemuan 18:
Mata kuliah : K0194-Pemodelan Matematika Terapan
Tahun : 2008
Model Antrian (Waiting Line Models ) Pertemuan 18:
Bina Nusantara

2. Learning Outcomes
Mahasiswa akan dapat menjelaskan pengertian atrian, system antrian, struktur dan analisis pola kedatangan pada sistem antrian.
Bina Nusantara

3. Outline Materi: Pengertian Antrian Sistem Antrian
Struktur dasar model antrian
Bentuk-bentuk model antrian
Analisis pola kedatangan
Contoh ..
Bina Nusantara

4. 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.
Bina Nusantara

5. 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.
Bina Nusantara

6. Situation Arrivals Servers Service Process
Waiting Line Examples
Situation Arrivals Servers Service Process
Bank Customers Teller Deposit etc.
Doctor’s Patient Doctor Treatment office
Traffic Cars Traffic Controlled intersection Signal passage
Assembly line Parts Workers Assembly
Bina Nusantara

7. 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.
Bina Nusantara

8. Sistem Antrian
Bina Nusantara

9. Car Wash Example
Bina Nusantara

10. 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.
Bina Nusantara

11. 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.
Bina Nusantara

12. 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:
Time between arrivals has a negative exponential distribution.
Bina Nusantara

13. 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.
Bina Nusantara

14. 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.
Bina Nusantara

15. Single Channel, Single Phase
Arrivals
Served units
Service facility
Queue
Service system
Dock
Waiting ship line
Ships at sea
Ship unloading system
Empty ships
Bina Nusantara

16. Single Channel, Multi-Phase
Service system
Served units
Arrivals
Queue
Service facility
Service facility
McDonald’s drive-through
Cars in area
Cars & food
Waiting cars
Pay
Pick-up
Bina Nusantara

17. Multi-Channel, Single Phase
Arrivals
Served units
Service facility
Queue
Service system
Example: Bank customers wait in single line for one of several tellers.
Bina Nusantara

18. Multi-Channel, Multi-Phase
Service facility
Arrivals
Served units
Queue
Service system
Example: At a laundromat, customers use one of several washers, then one of several dryers.
Bina Nusantara

19. 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.
Bina Nusantara

20. 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.
Bina Nusantara

21. 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.
Bina Nusantara

22. Terima kasih
Semoga berhasil
Bina Nusantara