Example 6.6 (A Multiserver Exponential Queueing System) Consider an exponential queueing system in which there are s servers available, each serving at rate

μ. An entering customer first waits in line and then goes to the first free server.

This is a birth and death process with parameters

μn =



nμ, 1 n  s sμ, n > s

λn = λ, n  0 To see why this is true, reason as follows: If there are n customers in the system, where n  s, then n servers will be busy. Since each of these servers works at rate μ, the total departure rate will be nμ. On the other hand, if there are n customers in the system, where n > s, then all s of the servers will be busy, and thus the total departure rate will be sμ. This is known as an M/M/s queueing model.