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Overlap Times in Tandem Queues: Identically Distributed Station Case

Ruici Gao, Jamol Pender

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 $μ$. 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.

Overlap Times in Tandem Queues: Identically Distributed Station Case

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 . 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.
Paper Structure (48 sections, 6 theorems, 70 equations, 3 tables)

This paper contains 48 sections, 6 theorems, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Application to Quantum Computing
  3. Contributions of the Paper
  4. Organization of Paper
  5. Overlap Times in Networks of Queues
  6. Single Station Overlap Time Distribution
  7. Overlap in a Tandem Network
  8. Two Stations with the Same Customer Pair
  9. The Joint Distribution
  10. The Marginal Distribution
  11. Tail Distribution of Sum of Overlap Times
  12. Moments and Covariance
  13. Two Stations with Different Customer Pairs
  14. Joint Tail Distribution
  15. Numerical Experiments
  16. ...and 33 more sections

Key Result

Theorem 3.1

Let $O^{(1)}_{n,n+k}$ and $O^{(2)}_{n,n+k}$ be the overlap times for the $n^{th}$ and $(n+k)^{th}$ customers in the $(M/M/\infty)^2$ tandem queue with identically distributed stations, then the joint distribution of the overlap times is equal to

Theorems & Definitions (15)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 4.1
  • proof
  • ...and 5 more