25-09-2013, 04:29 PM
SIMULATION OF NETWORK PROTOCOLS: M/M/1 and M/M/1/N queues.
INTRODUCTION
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.
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)
Example
There are many situations in which an M/M/1 model could be applied. One example is a
post office with only one employee, and therefore one queue. The customers arrive, enter
the queue, do business with the postal worker, and leave the system. If the arrival process is
Poisson and the service time is exponential, a M/M/1 model can be used. Hence, the
expected number of people in the queue can be easily calculated, along with the
probabilities they will have to wait for a particular length of time, and so forth.