Monotonicity and error bounds for networks of Erlang loss queues
Richard J. Boucherie
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Nico M. van Dijk
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N.M. van Dijk Department of Operations Research, Universiteit van Amsterdam
, Roetersstraat 11, 1018 WB Amsterdam,
The Netherlands
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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:
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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)