QUEUEING THEORY

Queueing Theory

"Queueing theory is the mathematical study of waiting lines, or queues. In queueing theory a model is constructed so that queue lengths and waiting times can be predicted."

- wikipedia

Application

telecommunications
traffic engineering
computers
shops
offices
factories
hospitals

What is a queue?

WHAT IS A QUEUE?

Why queue?

Makes economic sense
Efficient
Resources are costly
More servers -> shorter queues
Fewer servers -> full capacity
Want to maximise capacity ...
... but keep queue times manageable

WHY QUEUE?

The key:

Balancing service requests with resource utilisation.

Kendall's notation

A/S/c

A - time between arrivals to the queue.

S - size of job.

c - the number of servers

KENDALL'S NOTATION

A/S/c/K/N/D

A - time between arrivals to the queue.

S - size of job.

c - the number of servers

K - capacity of the queue

N - size of population from which the customers come

D - queueing discipline

Arrival process

How do the jobs appear?

One at time?
In batches?

How are they distributed in time?

evenly spaced?
randomly?

Service mechanism

How many servers are there?

How many queues?

SERVICE MECHANISM

How many servers are there?

How many queues?

SERVICE MECHANISM

What is the service distribution time?

Dependant on:

the amount of work required for the job
the speed of the server

Queueing discipline

Policy - how are jobs chosen?

4 most popular:

FIFO - First In First Out
LIFO - Last In First Out
Priority
SRPT - Shortest Remaining Processing Time

FiFO

First In First Out

FIFO

First In First Out

FIFO

First In First Out

LIFO

Last in first out

LIFO

Last in first out

LIFO

Last in first out

Priority

PRIORITy

PRIORITY

pre-emptive