A numerically stable formula for the conditional distribution of the residual service time in the Mn/PH/1 queue

IF 1.4 Q2 MATHEMATICS, APPLIED

Results in Applied Mathematics Pub Date : 2025-06-27 DOI:10.1016/j.rinam.2025.100603

Yutaka Sakuma, Yan Linn Aung

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引用次数: 0

Abstract

In this paper, we consider an

M_{n} / P H / 1

queue, where arriving customers decide whether to join the queue or not join based on the queue length at arrival instants. Kerner (2008, Stochastic Models) studies the

M_{n} / G / 1

queue, and derives a recursive formula for the Laplace-Stieltjes transform (LST, for short) of the conditional distribution of the server’s residual service time, given the queue length at arrival instants. This paper aims to analyze the

M_{n} / P H / 1

queue in a much simpler way than the previous studies, and to show that our LST of the conditional distribution of the server’s residual service time is given in a more numerically stable form than that of the previous studies, specifically by avoiding the indeterminate form such as

0 / 0

. We then use the formula to compute the customers joining probabilities in Nash equilibrium.

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Mn/PH/1队列剩余服务时间条件分布的数值稳定公式

本文考虑一个Mn/PH/1队列，其中到达的顾客根据到达时刻的队列长度决定是否加入队列。Kerner （2008, Stochastic Models）研究了Mn/G/1队列，在给定到达时刻队列长度的情况下，导出了服务器剩余服务时间条件分布的Laplace-Stieltjes变换（简称LST）的递归公式。本文旨在以比以往研究简单得多的方式分析Mn/PH/1队列，并表明我们的服务器剩余服务时间条件分布的LST以比以往研究更稳定的数值形式给出，特别是避免了0/0等不确定形式。然后，我们使用该公式来计算纳什均衡中的顾客加入概率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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