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

Author

Listed:
  • Ruth Sagron

    (Department of Industrial Engineering and Management, Jerusalem College of Technology, Jerusalem 91160, Israel)

  • Uri Yechiali

    (Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 6997801, Israel)

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 E 2 / G /1 and C 2 / C 2 /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 E 2 / 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.

Suggested Citation

  • Ruth Sagron & Uri Yechiali, 2024. "Inter-Departure Time Correlations in PH / G /1 Queues," Mathematics, MDPI, vol. 12(9), pages 1-23, April.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1362-:d:1386446
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    References listed on IDEAS

    as
    1. Paul Beekhuizen & Dee Denteneer & Ivo Adan, 2008. "Analysis of a tandem network model of a single-router Network-on-Chip," Annals of Operations Research, Springer, vol. 162(1), pages 19-34, September.
    2. Jiashan Tang & Yiqiang Zhao, 2008. "Stationary tail asymptotics of a tandem queue with feedback," Annals of Operations Research, Springer, vol. 160(1), pages 173-189, April.
    3. Yael Perlman & Uri Yechiali, 2020. "On tandem stochastic networks with time-deteriorating product quality," International Journal of Production Research, Taylor & Francis Journals, vol. 58(13), pages 3956-3964, July.
    4. Ruth Sagron & Dean Grosbard & Gad Rabinowitz & Israel Tirkel, 2015. "Approximation of single-class queueing networks with downtime-induced traffic variability," International Journal of Production Research, Taylor & Francis Journals, vol. 53(13), pages 3871-3887, July.
    5. Gabriel R. Bitran & D. Tirupati, 1989. "Approximations for Product Departures from a Single-Server Station with Batch Processing in Multi-Product Queues," Management Science, INFORMS, vol. 35(7), pages 851-878, July.
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