Suppose customers arrive to a queue according to a Poisson process with intensity , have independent service times with distribution , and there is an infinite number of servers available, so that service begins immediately as a customer enters the system. Assume further that the system is empty at time . Let be the number of customers in the system at time ; what is the distribution of ?
A customer who enters the system at time will still be in the system at time with probability . Let be a Poisson process with intensity and , . Then , and so for each nonnegative integer
In the case where , that is, the service times are exponentially distributed with rate , we have
The expected number of customers at time is then
Moreover, we observe that
which is the same as the stationary distribution of derived from the generalized global balance equations
(normalized by the condition ).