Modeling & Simulation- Lecture -5-
12/11/2014
Solving Queuing models:
1 single queue
The state diagram of birth-death process for M / M / 1
λ
λ
0
1
µ
λ
3
2
µ
λ
µ
µ
We have balanced equations:
the probability flow out of a state = the probability flow into that state .
We can rewrite these as shown below:
Remembering that traffic intensity,
𝜌=
1
𝜆
µ
For the single server
Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -5𝜌=
𝜆
𝑐×µ
12/11/2014
For multi-server
Then we can see that for any 𝑥𝑖
And after using normalization condition we see that:
From this the steady state of M/M/1 can be as:
Some performance measurement of M/M/1 :
 Utilisation, U the queue is being utiliesd whenever it is non-empty this mean:
 Mean number of entity in the queue, N , the entity in service facility:
2
Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -5-
12/11/2014
 Mean number of entity waiting , 𝑵𝒃 this the number of entity in the buffer :
 Mean Response time, R,
3
Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -5-
12/11/2014
Some examples for common performance measurements shown in table below:
4
Lecturer: Hawraa Sh.