Approximated Transient Queue Length and Waiting Time Distributions via Steady State Analysis

We propose a method to approximate the transient performance measures of a discrete time queueing system via a steady state analysis. The main idea is to approximate the system state at time slot t or on the n-th arrival—depending on whether we are studying the transient queue length or waiting time distribution—by the system state after a negative binomially distributed number of slots or arrivals. By increasing the number of phases k of the negative binomial distribution, an accurate approximation of the transient distribution of interest can be obtained. In order to efficiently obtain the system state after a negative binomially distributed number of slots or arrivals, we introduce so-called reset Markov chains, by inserting reset events into the evolution of the queueing system under consideration. When computing the steady state vector of such a reset Markov chain, we exploit the block triangular block Toeplitz structure of the transition matrices involved and we directly obtain the approximation from its steady state vector. The concept of the reset Markov chains can be applied to a broad class of queueing systems and is demonstrated in full detail on a discrete-time queue with Markovian arrivals and phase-type services (i.e., the D-MAP/PH/1 queue). We focus on the queue length distribution at time t and the waiting time distribution of the n-th customer. Other distributions, e.g., the amount of work left behind by the n-th customer, that can be acquired in a similar way, are briefly touched upon. Using various numerical examples, it is shown that the method provides good to excellent approximations at low computational costs—as opposed to a recursive algorithm or a numerical inversion of the Laplace transform or generating function involved—offering new perspectives to the transient analysis of practical queueing systems.