On customers acting as servers

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 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 Q ₁ 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 Q ₁. (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 L_i, 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.