Introduction to Queueing Theory
Introduction to Queueing Theory
Motivation
Queueing Systems
Examples
Five components of a Queueing system:
ASSUME
Assumption is bad in :
Interarrival-time pdf
Service time
Number of servers
Queueing discipline
Finite Length Queues
ASSUME
A/B/m notation
A,B are chosen from the set:
Analysibility
M/M/1 system
Poisson’s Law
Poisson’s Law in Physics
Poisson’s Law in Operations Research
Poisson’s Law in Biology
Poisson’s Law in Transportation
Poisson’s Law in Optics
Poisson’s Law in Communications
- Rate parameter
Poisson’s Law and Binomial Law
Example:
We have:
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Analysis
In maple:
To obtain numbers with a Poisson pdf, you can write a program:
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Prove:
Prove:
Subst into :
Now you get:
We already knew:
So by subset:
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We have made several assumptions:
The M/M/1 queue in equilibrium
State of the system:
Memory of M/M/1:
Memoryless
Birth-death system
State-transition Diagram
Single-server queueing system
Symbles:
States
Probalility of Given State
Similar to AC
Derivation
by 3a
then:
since all prob. sum to one
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subst 7 into 6a
subst into
Mean value:
Subst (8) into (8a)
differentiate (7) wrt k
multiply both sides of (8c) by
Relationship of , N
T and
Little’s result:
For example:
Compute how long a bird waits in the Queue (on average):
Result:
Mean Queueing Time
M/G/1 Queueing System
What is
Note:
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