M/M/1 model

The M/M/1 is a single-server queue model, that can be used to approximate a lot of simple systems.
M/M/1 schema

Following Kendall's notation it indicates a system where:

* Arrivals are a Poisson process;
* Service time is exponentially distributed;
* There is one server.

Analysis

If the arrival rate is λ and the service rate is μ we can represent the system by a birth-death process, where each state is the number of users. Of course we have an infinite number of states: state 0 (no users in the system), state 1 (1 user), state 2 (two users), and so on. The birth rate is every time λ and the death rate is μ. In fact, regardless of the state, we can have only two events:

* A new user arrives. So if the system is in state k, it goes to state k + 1 with rate λ
* A user leaves the system. So if the system is in state k, it goes to state k − 1 (or k if k is 0) with rate μ

It's easy now to see that the system is stable only if λ < μ. In fact if the death rate is less than the arrival rate, the number of users in the queue will become infinite.

If the system is stable, we can obtain a lot of performance measures of interest like:

* The expected number of users in the system
* The expected number of users in the queue
* The throughput

Example

There are a lot of situations where an M/M/1 model could be used. For instance, you can take a post office with only one employee, and therefore one queue. The customers arrive, go into the queue, they are served, and they leave the system. If the arrival process is Poisson, and the service time is exponential, we can use an M/M/1 model. Hence, we can easily calculate the expected number of people in the queue, the probabilities they will have to wait for a particular length of time, and so on.