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

In this paper, we consider an $M_n/PH/1$ 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 $M_n/PH/1$ 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.