First exit time for a discrete time parallel queue

TL;DR

This paper investigates the behavior and open questions related to the steady-state of a discrete-time parallel queue with two queues, different service laws, and batch arrivals, focusing on the first exit time analysis.

Contribution

It introduces the problem of analyzing the first exit time for a discrete-time parallel queue with different service laws and discusses open questions about its steady-state behavior.

Findings

Identification of open questions related to steady-state analysis.

Framework for analyzing first exit times in discrete-time parallel queues.

Highlights differences in service laws impact on queue behavior.

Abstract

We consider a discrete time parallel queue, which is two-queue network, where at each time-slot there is a the same batch arrival to both queues and at each queue there is a random service available. The service law at each time-slot for each queue is different. Let $(Q_n^1, Q_n^2)$ be the queue length at $n$th time-slot. We present several open questions related to the steady-state of this queue.