solving Queuing models

Solving Queuing models:
Queuing Systems
Queuing theory establishes a powerful tool in modeling and performance analysis of many complex systems, such as computer networks, telecommunication systems, call centers, manufacturing systems and service systems. Many real systems can be modeled as networks of queues, such as the waiting line in a bank, a bus stop, waiting line in airport, material flow in factory, and signal flow in network. So, each of these simple systems consists of three major components:
• Entities:
Something that changes the state of the system. In many cases, particularly those involving service systems, the entity may be a person. In the customer service center, the entities are the customers. Entities do not necessarily have to be people; they can also be objects. Similarly, in the factory example, the entities are components waiting to be machined; entities can be signals waiting to be received in communication networks.
• Queues:
Queues are the simulation term for lines. Entities generally wait in a queue until it is their turn to be processed.
• Resources:
Processer or server that serves the entities those are in the queue. Example factory machines, computer processer, nodes in communication networks. The relationships among these components are illustrated
1 single queue
The state diagram of birth-death process for M / M / 1

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,
For the single server
For multi-server
Then we can see that for any
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:
Mean number of entity waiting , this the number of entity in the buffer :
? Mean Response time, R,