Wiener–Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Wiener–Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

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Article ID: iaor19992894
Country: United States
Volume: 23
Issue: 1/4
Start Page Number: 301
End Page Number: 316
Publication Date: Jan 1996
Journal: Queueing Systems
Authors: ,
Keywords: M/G/1 queues
Abstract:

Two variants of an M/G/1 queue with negative customers lead to the study of a random walk Xn+1 = [Xn + ξn]+ where the integer-valued ξ;n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for 𝔼(sXn), corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrary-time queue length distribution is a mixture of two geometrical distributions.

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