Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

TL;DR

This paper analyzes the joint queue length distribution in a FIFO single-server queue with multiple batch Markovian arrival streams, providing formulas and numerical methods for computation under various service and batch size distributions.

Contribution

It introduces a new formula for the joint queue length distribution in a multi-stream Markovian queue and develops a numerical procedure for phase-type batch sizes.

Findings

Derived a formula for the vector generating function of the joint queue length.

Developed a numerically feasible procedure for phase-type batch size distributions.

Provided numerical examples illustrating the method's application.

Abstract

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.