Monotonicity and error bounds for networks of Erlang loss queues

Queueing Systems, Jun 2009

Networks of Erlang loss queues naturally arise when modelling finite communication systems without delays, among which, most notably are classical circuit switch telephone networks (loss networks) and 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: 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.

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Monotonicity and error bounds for networks of Erlang loss queues

Richard J. Boucherie 0 1 Nico M. van Dijk 0 1 0 N.M. van Dijk Department of Operations Research, Universiteit van Amsterdam , Roetersstraat 11, 1018 WB Amsterdam, The Netherlands 1 R.J. Boucherie ( ) Department of Applied Mathematics, University of Twente , P.O. Box 217, 7500 AE Enschede, The Netherlands 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: - 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. Mathematics Subject Classification (2000) Primary 90B22 Secondary 60K25 1 Introduction 1.1 Background The classical Erlang loss model, initially developed for a single telephone switch, is probably the most commonly known queueing model. The loss network is its generalisation to more complex circuit switched systems with multiple links, multiple switches, and multiple types of calls (see [11] for an overview). The loss network is widely used for telephone system dimensioning. The common feature of these networks is that a call arriving to the system either obtains a number of circuits from source to destination and occupies these circuits for its entire duration, or that the call is blocked and cleared because the required circuits for that call are not all available. The corresponding blocking probabilities are among the key performance measures in circuit switched telephone systems. Due to the simple structure of loss networks, their equilibrium distribution has the appealing so-called product form. This product form can be seen as a truncated multidimensional Poisson distribution, where the dimensionality is determined by the number of call types, the parameter of the Poisson distribution is determined by the load offered by all call types, and the truncation is determined by the capacity constraints of the circuits: n S, G = N nk k , k=1 nk! size of the state space considerably complicates this evaluation. To this end, various efficient numerical evaluation and approximation schemes have been developed, including Monte Carlo summation, and Erlang fixed point methods, see [11, 20]. In mobile communications networks, a call may transfer from one cell to another while in progress. As a consequence, in addition to fresh call blocking of a newly arriving call, handover blocking for a call which attempts to route to another cell, but which finds all circuits available for this cell occupied, becomes of practical interest. In that case, the blocked handover is cleared and lost. A network of Erlang loss queues with routing and common capacity restrictions is a natural representation of this network. The equilibrium distribution for a network of Erlang loss queues with handover blocking is, unfortunately, not available in closed form. Various approximations have therefore been suggested in the literature. The most appealing among these approximations is the redial rate approximation introduced in [4]. Under the redial rate approximation, an extra arrival rate of calls in cells surrounding a blocked cell is introduced. This redial rate mimics the behaviour of calls that are lost when transferring to the blocked cell. This approximation retains the call blocking structure of the original model. Under maximal redial rates, when all blocked calls attempt to redial, the equilibrium distribution is of product form, similar to that for the loss network. Moreover, the equilibrium distribution and blocking probabilities inherit the appealing insensitivity property. As the equilibrium distribution under the redial rate approximation also has a truncated multidimensional Poisson distribution, computational techniques developed for loss networks can be carried over to numerically evaluate fresh call and handover blocking probabilities. 1.2 Results The redial rate approximation of blocking probabilities introduces an (...truncated)


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Richard J. Boucherie, Nico M. van Dijk. Monotonicity and error bounds for networks of Erlang loss queues, Queueing Systems, 2009, pp. 159-193, Volume 62, Issue 1-2, DOI: 10.1007/s11134-009-9118-9