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:
• 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.