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A MAP/G/1 Queue with Negative Customers

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Abstract

In this paper, we consider a MAP/G/1 queue with MAP arrivals of negative customers, where there are two types of service times and two classes of removal rules: the RCA and RCH, as introduced in section 2. We provide an approach for analyzing the system. This approach is based on the classical supplementary variable method, combined with the matrix-analytic method and the censoring technique. By using this approach, we are able to relate the boundary conditions of the system of differential equations to a Markov chain of GI/G/1 type or a Markov renewal process of GI/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distributions of stationary queue length and virtual sojourn time, and the Laplace transform of the busy period. Moreover, we provide an analysis for the asymptotics of the stationary queue length of the MAP/G/1 queues with and without negative customers.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada, K1S 5B6

    Quan-Lin Li

  2. Department of Industrial Engineering, Tsinghua University, Beijing, 100084, P.R. China

    Quan-Lin Li

  3. School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada, K1S 5B6

    Yiqiang Q. Zhao

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  1. Quan-Lin Li
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Li, QL., Zhao, Y.Q. A MAP/G/1 Queue with Negative Customers. Queueing Systems 47, 5–43 (2004). https://doi.org/10.1023/B:QUES.0000032798.65858.19

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