Abstract
In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.
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Breuer, L. Transient and Stationary Distributions for the GI/G/k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution. Queueing Systems 45, 47–57 (2003). https://doi.org/10.1023/A:1025643801208
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DOI: https://doi.org/10.1023/A:1025643801208