Stochastic comparisons: A methodology for the performance evaluation of fixed and mobile networks

We propose to use a mathematical method based on stochastic comparisons of Markov chains in order to derive performance indices bounds. The main objective is to find Markovian bounding models with reduced state spaces, which are easier to solve. We apply the methodology to performance evaluation of complex telecommunication systems modelled by large size Markov chains which cannot be solved by exact methods. This methodology can be applied for continuous- or discrete-time Markov chains. In the first study, we consider an MPLS switch represented by two stages of buffers. Various kinds of traffic with different QoS levels enter the first stage, and transit in the second stage. The goal is to compute packet loss rates in the second stage. In the other study, we define a CAC scheme in a mobile network which gives the priority to the handover over the new calls. Performance evaluation of the CAC scheme consists in the computation of the dropping handover and call blocking probabilities. For the two studies, systems are represented by large state Markov chains whose resolution is difficult. We propose to define intuitively bounding systems in order to compute performance measures bounds. Using stochastic comparisons methods, we prove that the new systems represent bounds for the exact ones. Different methods can be used. For the MPLS switch, we use the coupling equivalent to the sample-path ordering, allowing the comparison of the loss rates. In the case of the CAC scheme, we apply the increasing sets formalism used to define weaker orderings, enabling the comparison of the dropping handovers and blocking probabilities. We validate stochastic comparison method by presenting some numerical results illustrating the interest of the approach.