## 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 L _{i} the number of customers in queue i (Q_{i}), i = 1, 2, we assume that Q_{1} is a multi-server finite-buffer system with an overall capacity of size N, where the customers there are served by the L_{2} customers present in Q_{2}. Regarding Q_{2}, we study two different scenarios described as follows: (i) All customers present in Q _{1} join hands together to form a single server for the customers in Q_{2}, with service time exponentially distributed with an overall intensity μ_{2}L_{1}. That is, the service rate of the customers in Q_{2} changes dynamically, following the state of Q _{1}. (ii) Each of the customers present in Q_{1} individually acts as a server for the customers in Q_{2}, with service time exponentially distributed with mean 1/μ_{2}. In other words, the number of servers at Q_{2} changes according to the queue size fluctuations of Q_{1}. 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 L_{i}, i = 1, 2, along with their correlation coefficient, and obtain the probability of blocking at Q_{1}. Finally, we present numerical examples and compare between the two models.

Original language | English |
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Article number | 1350019 |

Journal | Asia-Pacific Journal of Operational Research |

Volume | 30 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2013 |

## Keywords

- Customers act as servers
- M/M/ • /N
- M/M/1/∞
- M/M/c/∞
- QBD