Branching processes, the max-plus algebra and network calculus

Branching processes can describe the dynamics of various queueing systems, peer-to-peer systems, delay tolerant networks, etc. In this paper we study the basic stochastic recursion of multitype branching processes, but in two non-standard contexts. First, we consider this recursion in the max-plus algebra where branching corresponds to finding the maximal offspring of the current generation. Secondly, we consider network-calculus-type deterministic bounds as introduced by Cruz, which we extend to handle branching-type processes. The paper provides both qualitative and quantitative results and introduces various applications of (max-plus) branching processes in queueing theory

On Stochastic Recursive Equations and Infinite Server Queues

The purpose of this paper is to investigate some performance measures of the discrete time G/G/ queue under a general arrival process. We assume more precisely that at each time unit a batch with a random size may arrive, where the sequence of batch sizes need not be i.i.d. All we request is that it would be stationary ergodic and that the service duration has a phase type distribution. Our goal is to obtain explicit expressions for the first two moments of number of customers in steady state. We obtain this by computing the first two moments of some generic stochastic recursive equations that our system satisfies. We then show that these class of recursive equations allow to solve not only the queue but also a network of such queues. We finally investigate the process of residual activity time in a queue under general stationary ergodic assumptions, obtain the unique stationary solution and establish coupling convergence to it from any initial state. Keywords: Stochastic processes/Queueing theory. I