M/M/1 model: Information from Answers.com

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

Following Kendall's notation it indicates a system where:

Arrivals are a Poisson process;
Service time is exponentially distributed;
There is one server.
The length of queue in which arriving users wait before being served is infinite
The population of users(i.e. the pool of users) available to join the system is infinite

1 Analysis
2 Stationary solution
3 Example
4 See also

Analysis

Such a system can be modelled by a birth-death process, where each state represents the number of users in the system. As the system has an infinite queue and the population is unlimited, the number of states the system can occupy is infinite: state 0 (no users in the system), state 1 (1 user), state 2 (two users), etc. As the queue will never be full and the population size being infinite, the birth rate (arrival rate), λ, is constant for every state. The death rate (service rate), μ, is also constant for all states (apart from in state 0). 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 average number of users in the queue will become infinite. I.e. the system will not have an equilibrium.

The model can reveal interesting performance measures of the system being modelled, for example:

The mean time a user spends in the system
The mean time a user spends waiting in the queue
The expected number of users in the system
The expected number of users in the queue
The throughput (Number of users served per unit time)

Stationary solution

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

The probability the system is in state i can be easily calculated:

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

With this information, the performance measures of interest can be found; for example:

The expected number of users in the system N is given by

$\overline N=\frac{\rho}{1-\rho}$ , and its variance by

$\sigma^2_N = \frac{\rho}{(1 - \rho)^2}$ .

The expected number of requests in the server

$\overline N_S = \lambda \overline x = \rho$

The expected number of requests in the queue

$\overline N_Q = \frac{\rho^2}{1 - \rho}$

The total expected waiting time (queue+service) is

$T=\frac{1}{\mu-\lambda}.$

Expected waiting time in the queue is

$W = \frac{\overline N_Q}{\lambda} = T - \overline x = T - \frac{1}{\mu} = \frac{\rho}{\mu - \lambda}$

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.

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M/M/1 model

Contents

Analysis

Stationary solution

Example

See also