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On Customers Acting As Servers

Author

Listed:
  • EFRAT PEREL

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

  • URI YECHIALI

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

Abstract

We consider systems comprised of two interlacing M/M/ • /• type queues, where customers of each queue are the servers of the other queue. Such systems can be found for example in file sharing programs, SETI@home project, and other applications [Arazi, A, E Ben-Jacob and U Yechiali (2005). Controlling an oscillating Jackson-type network having state-dependant service rates. Mathematical Methods of Operations Research, 62, 453–466]. Denoting by Li the number of customers in queue i (Qi), i = 1, 2, we assume that Q1 is a multi-server finite-buffer system with an overall capacity of size N, where the customers there are served by the L2 customers present in Q2. Regarding Q2, we study two different scenarios described as follows: (i) All customers present in Q1 join hands together to form a single server for the customers in Q2, with service time exponentially distributed with an overall intensity μ2L1. That is, the service rate of the customers in Q2 changes dynamically, following the state of Q1. (ii) Each of the customers present in Q1 individually acts as a server for the customers in Q2, with service time exponentially distributed with mean 1/μ2. In other words, the number of servers at Q2 changes according to the queue size fluctuations of Q1.We present a probabilistic analysis of such systems, applying both Matrix Geometric method and Probability Generating Functions (PGFs) approach, and derive the stability condition for each model, along with its two-dimensional stationary distribution function. We reveal a relationship between the roots of a given matrix, related to the PGFs, and the stability condition of the systems. In addition, we calculate the means of Li, i = 1, 2, along with their correlation coefficient, and obtain the probability of blocking at Q1. Finally, we present numerical examples and compare between the two models.

Suggested Citation

  • Efrat Perel & Uri Yechiali, 2013. "On Customers Acting As Servers," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 30(05), pages 1-23.
    • Handle: RePEc:wsi:apjorx:v:30:y:2013:i:05:n:s021759591350019x
    DOI: 10.1142/S021759591350019X
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    Citations

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    Cited by:

    1. Kostia Avrachenkov & Efrat Perel & Uri Yechiali, 2016. "Finite-buffer polling systems with threshold-based switching policy," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(3), pages 541-571, October.
    2. Mor Armony & Efrat Perel & Nir Perel & Uri Yechiali, 2019. "Exact analysis for multiserver queueing systems with cross selling," Annals of Operations Research, Springer, vol. 274(1), pages 75-100, March.
    3. Hanukov, Gabi, 2022. "A service system where junior servers approach a senior server on behalf of customers," International Journal of Production Economics, Elsevier, vol. 244(C).

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