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 (see e.g. Arazi, Ben-Jacob and Yechiali (2005)). 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 2-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 1 at Q1. Finally, we present numerical examples and compare between the two models.