Queues with Poisson Input and Hyper-Exponential Output with Finite Waiting Space

The present paper deals with the steady-state solution of the queuing system in which (i) units arrive according to a Poisson distribution; (ii) the queue discipline is first-come, first-served; and (iii) the service time distribution is hyper-exponential with n branches. Assuming a finite waiting space, we derive the system-size distribution and the mean number of units in the system therefrom. Results are also deduced when an infinite waiting space is allowed. Another interesting case is discussed when only the mean service rate over all the n branches is assigned. In the end we study the case when no queue is allowed.