On Mixed Time-Changed Erlang Queue

TL;DR

This paper introduces a novel mixed time-changed Erlang queue model using a stable subordinator, deriving its governing fractional differential equations, explicit state probabilities, and various distributional properties.

Contribution

It presents the first analysis of a mixed time-changed Erlang queue, including explicit formulas and distributional insights, expanding queue modeling with fractional calculus.

Findings

Derived fractional differential equations for the queue

Explicit expressions for state probabilities and Laplace transforms

Distributional properties of inter-arrival, service, and busy periods

Abstract

We study a time-changed variant of the Erlang queue by taking the first hitting time of a mixed stable subordinator as the time-changing component. We call it the mixed time-changed Erlang queue. We derive the system of fractional differential equations that governs its state probabilities. The explicit expressions for the state probabilities of mixed time-changed Erlang queue and their Laplace transform are derived. Equivalently, it is represented in terms of phases and its mean queue length is obtained. Also, some distributional properties of the mixed time-changed Erlang queue such as the distribution of its inter-arrival times, inter-phase times, service times and busy period are derived. Later, its conditional waiting time is discussed and two plots of sample paths simulation are presented.