Queues with Hyper-Poisson Input and Exponential Service Time Distribution with State Dependent Arrival and Service Rates

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Abstract
The queuing system considered in this paper is characterized by (i) hyper-Poisson input with k branches with mean arrival rates depending upon the state of the system; (ii) first-come, first-served queue discipline; (iii) exponential service time distribution with mean service rate depending upon the state of the system; and (iv) finite waiting space. The functions defining the dependence of the mean arrival and service rates upon the state of the system are assumed to be arbitrary. Assuming steady-state conditions to obtain, a recurrence relation connecting the various probabilities introduced is found. By specifying a few particular forms of the state functions, graphs depicting two queue characteristics, viz., (i) probability of no delay, and (ii) mean number of units in the system are drawn. One of the particular cases has been interpreted as “a queuing problem with reneging” and another as “a multiple server queuing problem.”

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