Transient Behavior of Single-Server Queuing Processes with Recurrent Input and Exponentially Distributed Service Times

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Abstract
Customers arrive at a counter at the instants τ1, τ2,..., τn..., where the inter-arrival times τn-τn-1(n = 1, 2,..., τ0 = 0) are indentically distributed, independent, random variables. The customers will be served by a single server. The service times are identically distributed, independent, random variables with exponential distribution. Let ξ(t) denote the queue size at the instant t. If ξ(τn-0) = k then a transition Ek → Ek+1 is said to occur at the instant t = τn. The following probabilities are determined Gn(X) = the probability that a busy period consists of n services and its length is at most x, and further the distribution of the number of transitions Ek → Ek+1 occurring in the time interval (0, t).

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