On Erlang Queue with Multiple Arrivals and its Time-changed Variant
R. B. Pote, K. K. Kataria
TL;DR
This paper studies an Erlang queue with multiple arrivals, where arrivals follow a generalized counting process ($GCP$), and the service times are Erlang with shape parameter $k$ and rate $\mu$. It develops a comprehensive transient analysis by formulating state-phase probabilities $p_0(t)$ and $p_{n,s}(t)$ and by introducing a scalar queue-length representation $\mathcal{L}(t)$ via a bijection $g_k$, yielding a tractable linear system for moments and the busy-period distribution. The model is extended through a time-change by an independent inverse $\alpha$-stable subordinator, producing a fractional queue governed by Caputo derivatives $\frac{\mathrm{d}^{\alpha}}{dt^{\alpha}}$ with state probabilities $p_{n,s}^{\alpha}(t)$ and a mean queue length $\mathcal{M}^{\alpha}(t)$ expressed through Mittag-Leffler functions. The results unify and generalize known special cases, including the classical $M/E_k/1$ queue and fractional Erlang queues, providing analytic tools for multi-arrival dynamics under non-deterministic time scales.
Abstract
We introduce and study a queue with the Erlang service system and whose arrivals are governed by a counting process in which there is a possibility of finitely many arrivals in an infinitesimal time interval. We call it the Erlang queue with multiple arrivals. Some of its distributional properties are obtained that includes the state-phase probabilities, the mean queue length and the distribution of busy period etc. Also, we study a time-changed variant of it by subordinating it with an independent inverse stable subordinator where we obtain its state probabilities and the mean queue length.