Overlap Times in Tandem Queues: Identically Distributed Station Case

TL;DR

This paper analyzes the overlap times of customers in a two-station tandem queue with identical exponential service times, deriving joint and marginal distributions, moments, and tail behaviors to understand customer overlap patterns.

Contribution

It provides the first explicit derivation of joint tail distribution and moments of overlap times in a tandem queue with i.i.d. exponential service times.

Findings

Derived joint tail distribution of overlap times

Calculated moments and tail probabilities of overlap durations

Analyzed overlap behavior in serial queueing systems

Abstract

In this paper, we investigate overlap times in a two-dimensional infinite server tandem queue. Specifically, we analyze the amount of time that a pair of customers spend overlapping in any station of the two dimensional tandem network. We assume that both stations have independent and identically distributed exponential service times with the same rate parameter $\mu$. Our main contribution is the derivation of the joint tail distribution, the two marginal tail probabilities, the moments of the overlap times and the tail distribution of the sum of the overlap times in both stations. Our results shed light on how customers overlap downstream in serial queueing systems.