M/M/1 model

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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 the Kendall's notation it indicates a system where:

[edit] Analysis

We can indicate the arrivals with λ and the service μ. We can then represent the system with 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 we are in state k, we go to state k + 1 with rate λ
  • A user leaves the system. So if we are in state k, we go 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 go to infinity.

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

  • The average number of users in the system
  • The average number of users in the queue
  • The throughput

[edit] Transient solution

We can define \scriptstyle \rho\,=\,\tfrac{\lambda}{\mu}.

The probability of being in state i, can be easly calculated:

\mbox{Prob}(q=i)=\pi_i=(1-\rho)\rho^i.\,

With this information, we can find the performance measures of interest, for example:

  • The average number of users in the system N is given by
N=\frac{\rho}{1-\rho}.
  • The total waiting time (queue+service) is
T=\frac{1}{\mu-\lambda}.

[edit] 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 so one queue. The customers arrive, go into the queue, they are served, and they leave the system. If the arrival process is a Poisson one, and the service time is exponential, we can use an M/M/1 model. Hence, we can easily calculate the average number of people in the queue, how long they have to wait, and so on.