On Mixed Time-Changed Erlang Queue

TL;DR

This work extends the Erlang queue by time-changing it with the inverse of a mixed stable subordinator $Y_{\alpha_{1},\alpha_{2}}(t)$, enabling long-memory dynamics through the Laplace-exponent $c_{1}z^{\alpha_{1}}+c_{2}z^{\alpha_{2}}$ (with $0<\alpha_{2}<\alpha_{1}<1$, $c_{1}+c_{2}=1$). It develops a comprehensive fractional-differential framework: state probabilities $p_{n,s}^{\alpha_{1},\alpha_{2}}(t)$ satisfy a coupled system with the operator $c_{1}\frac{d^{\alpha_{1}}}{dt^{\alpha_{1}}}+c_{2}\frac{d^{\alpha_{2}}}{dt^{\alpha_{2}}}$, while the generating function $G^{\alpha_{1},\alpha_{2}}(x,t)$ and the mean queue length $\mathcal{M}^{\alpha_{1},\alpha_{2}}(t)$ are characterized explicitly. The paper also provides the distributional properties of inter-arrival, inter-phase, and service times, the busy period, and conditional waiting time, all expressed via Mittag-Leffler functions and their convolutions, and it offers a Gillespie-type simulation scheme for sample paths. By recovering the fractional Erlang queue as a special case ($c_{1}=1$ or $c_{2}=1$) and presenting closed-form transforms and series representations, the work delivers a versatile analytical framework for queueing systems with memory and anomalous timing, with potential applications in telecommunications and finance.

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.

Figures (1)

• Figure 1: Plot (a) represents the sample path simulation of $N^{\alpha_{1},\alpha_{2}}(t)$ and Plot (b) represents the sample path simulation of $N^{\alpha_{1},\alpha_{2}}(T_{n})$ such that $T_{n}$ are the jump times of $N^{\alpha_{1},\alpha_{2}}(t)$ for parameters $c_1=0.4$, $c_2=0.6$, $\alpha_{1}=0.5$, $\alpha_{2}=0.3$, $k=4$, $\lambda=6$ and $\mu=5$.

Theorems & Definitions (16)

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