Monotonicity and error bounds for networks of Erlang loss queues

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are (i) classical circuit switch telephone networks (loss networks) and (ii) present-day wireless mobile networks. Performance measures of interest such as loss probabilities or throughputs can be obtained from the steady state distribution. However, while this steady state distribution has a closed product form expression in the first case (loss networks), it does not have one in the second case due to blocked (and lost) handovers. Product form approximations are therefore suggested. These approximations are obtained by a combined modification of both the state space (by a hypercubic expansion) and the transition rates (by extra redial rates). It will be shown that these product form approximations lead to • upper bounds for loss probabilities and • analytic error bounds for the accuracy of the approximation for various performance measures. The proofs of these results rely upon both monotonicity results and an analytic error bound method as based on Markov reward theory. This combination and its technicalities are of interest by themselves. The technical conditions are worked out and verified for two specific applications: R.J. Boucherie ( ) Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands e-mail: [email protected] N.M. van Dijk Department of Operations Research, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands 160 Queueing Syst (2009) 62: 159–193 • pure loss networks as under (i) • GSM networks with fixed channel allocation as under (ii). The results are of practical interest for computational simplifications and, particularly, to guarantee that blocking probabilities do not exceed a given threshold such as for network dimensioning.