Inter-Departure Time Correlations in PH/G/1 Queues

Ruth Sagron*, Uri Yechiali

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In non-Markovian tandem queueing networks the output process of one site, which is the input process to the next site, is not renewal. Consequently, the correlation analysis of that output processes is essential when studying such networks. A correlation analysis in the M/G/1 queue has been studied in the literature via derivation of the joint Laplace-Stieltjes transform (LST) of the sum of two consecutive inter-departure times. That LST is obtained by considering all possible cases at departure epochs. However, those epochs are expressed via dependent variables. In this paper, we first extend the analysis to the more general PH/G/1 queue, and investigate various queues, such as E2/G/1 and C2/C2/1. Then, we consider the lag-n correlation, which requires derivation of the joint LST of sum of n + 1 consecutive inter-departure times. Yet, deriving this LST by the common approach becomes impractical for n + 1 ≥ 3, as the number of all possible cases at departure epochs increases significantly. To overcome this obstacle, we derive a corresponding single-parameter LST, which expresses the sum of n + 1 consecutive inter-departure times via the (n + 1)-st departure epoch only. Consequently, the latter LST is expressed via a much fewer number of possible cases, and not less important, as a function of independent variables only, eliminating the need to derive the corresponding joint density. Considering the M/G/1 and the E2/G/1 queues, we demonstrate that the joint LST can be reconstructed directly via the corresponding single-parameter LST when n + 1 = 2. We further conjecture that the multi-parameter joint LST can be reconstructed from the corresponding single-parameter LST in more general queues and for values of n + 1 > 2. The conjecture is validated for various PH/G/1 queues and proved for n + 1 = 3 in the M/G/1 case. The new approach facilitates the calculation of lag-n correlation of the departure process from PH/G/1 queue for n + 1 ≥ 3. Our analysis illuminates the cases when using renewal approximation of the output process provides a proper approximation when studying non-Markovian stochastic networks.

Original languageEnglish
Article number1362
JournalMathematics
Volume12
Issue number9
DOIs
StatePublished - May 2024

Keywords

  • departure process
  • lag-n correlation
  • Laplace–Stieltjes transform
  • PH/G/1 queue

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