Suppose that customers arrive at a single-server service station in accordance with a Poisson process having rate

λ. That is, the times between successive arrivals are independent exponential random variables having mean 1/λ. Upon arrival, each customer goes directly into service if the server is free; if not, then the customer joins the queue (that is, he waits in line). When the server finishes serving a customer, the customer leaves the system and the next customer in line, if there are any waiting, enters the service. The successive service times are assumed to be independent exponential random variables having mean 1/μ.

The preceding is known as the M/M/1 queueing system. The first M refers to the fact that the interarrival process is Markovian (since it is a Poisson process)

and the second to the fact that the service distribution is exponential (and, hence, Markovian). The 1 refers to the fact that there is a single server.

If we let X(t) denote the number in the system at time t then {X(t), t  0} is a birth and death process with μn = μ, n  1 λn = λ, n  0