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Abstract <p>The <i>M</i>/<i>G</i>/1/∞ system is a non-Markov model, a single-line queueing system with waiting and a Poisson incoming flow of demands of intensity λ. However, the letter <i>G</i> in the second place of the system entry means that the service time of each demand can be distributed according to an arbitrary law <i>G</i>(<i>x</i>). If <i>G</i>(<i>x</i>) is not hyper-Erlangian, then it is impossible to construct a process η(<i>t</i>) that would describe the functioning of the system and would be a Markov process with continuous time and a discrete set of states. In particular, the number of demands in the system ν(<i>t</i>) at time <i>t</i> will not be such a process, since the distribution of the remaining service time of a demand in the system, unlike the exponential case, depends on the time that this demand has already been serviced.</p>

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The M/G/1/∞ System in the Nested Markov Chain Method in Mechanical Engineering

  • 期刊论文

  • 发表日期：

    2025年9月28日

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  • A. M. Popov

摘要

Abstract

The M/G/1/∞ system is a non-Markov model, a single-line queueing system with waiting and a Poisson incoming flow of demands of intensity λ. However, the letter G in the second place of the system entry means that the service time of each demand can be distributed according to an arbitrary law G(x). If G(x) is not hyper-Erlangian, then it is impossible to construct a process η(t) that would describe the functioning of the system and would be a Markov process with continuous time and a discrete set of states. In particular, the number of demands in the system ν(t) at time t will not be such a process, since the distribution of the remaining service time of a demand in the system, unlike the exponential case, depends on the time that this demand has already been serviced.