Queuing with Multiple Poisson Inputs and Exponential Service Times

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Abstract
This paper contains an analysis of a single-server queuing system for m different types of customers having independent Poisson arrivals with rates λı, ı = 1, …, m and exponential service times with rates μı, ı = 1, …, m. The discipline is first-come, first-served. Some derived results are (a) a recursion relation for the steady-state probability of n in queue, (b) a recursion relation for the steady-state probability that some member of a particular class is in service and that n of any class are in queue, and (c) the characteristic equation for the waiting-time distribution and its general inversion for the two-population case. The recursion relations are simple for computational purposes.

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