Tempered Erlang Queue with Multiple Arrivals

TL;DR

This paper introduces a novel tempered Erlang queue model incorporating multiple arrivals and a time change via tempered stable subordinators, deriving explicit solutions and analyzing key system distributions.

Contribution

It presents a new fractional queue model with explicit solutions and detailed distributional analysis, extending classical Erlang queues with fractional and tempered stable processes.

Findings

Explicit expressions for state probabilities derived

Distribution of inter-arrival and sojourn times obtained

Mean queue length and system performance metrics analyzed

Abstract

In this paper, we introduce and study a time-changed variant of the Erlang queue with multiple arrivals where the time-changing component used is the first hitting time of a tempered stable subordinator. The system of fractional difference-differential equations that governs its state probabilities is derived which is solved to obtain their explicit expressions. An equivalent representation in terms of phases and the mean queue length is obtained. For a particular case, the distribution of inter-arrival times, inter-phase times, sojourn times, busy period and that of conditional waiting times are derived.