Aggregated bounding Markov processes applied to the analysis of tandem queues

Performance evaluation of telecommunication and computer systems is essential but a complex issue in general. Quantitative analysis of systems represented by multidimensional Markov processes models is very difficult and may be intractable if there is no specific solution form. In this study, we propose an algorithm in order to derive aggregated Markov processes providing upper and lower bounds on performance measures. We prove using stochastic comparisons that these aggregated Markov processes give bounds on performance measures defined as increasing reward functions on the transient and stationary distributions. The stochastic comparison has been largely applied in performance evaluation however the state space is generally assumed to be totally ordered which induces less accurate bounds for multidimensional Markov processes. Our proposed algorithm assumes only a preorder on the state space, and is applied to the analysis of an open tandem queueing network with rejection in order to derive loss probability bounds. Numerical results are computed from two parametric aggregation schemes: a fine and a coarse in order to show the improvement of the accuracy of the bound with respect to the state space size. We propose an attractive solution to the performance study: given a performance measure threshold, we study if it is guaranteed or not by studying less complex aggregated bounding processes.