The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

TL;DR

This paper derives the periodic steady-state distribution for queues with Erlang arrivals and services with time-varying rates, providing explicit solutions and error bounds, advancing understanding of such non-stationary queueing systems.

Contribution

It introduces a method to compute the asymptotic periodic distribution for queues with Erlang phases and periodic rate variations, including explicit series solutions and error bounds.

Findings

Derived the asymptotic periodic distribution as a function of time within the period.

Provided explicit infinite series solutions with error bounds for waiting time and busy period distributions.

Validated the approach with examples comparing differential equation solutions to asymptotic series.

Abstract

We study a queueing system with Erlang arrivals with $k$ phases and Erlang service with $m$ phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series.