#### Abstract

The performance requirements of broadband communication networks are often expressed in terms of events with very low probability. Analytical or numerical evaluation is only possible for a very restricted class of
systems. Crude simulations require prohibitively long execution times for the accurate estimation of very low probabilities, and thus acceleration methods are necessary.
A more frequent occurrence of a formerly rare event is achieved by performing a number of simulation retrials when the process enters regions of the state space where the importance is greater, i.e., regions where the
chance of occurrence of the rare event is higher. These regions, called importance regions, are defined by comparing the value taken by a function of the system state, the importance function, with certain thresholds.
Formulas for the importance function of general Jackson networks in [1].
In [2] networks with Erlang service times with different shape parameters were studied. The rare set was defined as the number of customers in a target node exceeding a predefined threshold. Two models were studied: a network with 7 nodes all of them at \textquotedblleft distance\textquotedblright\ 1 or 2 from the target node and a 3-queue tandem network with the loads of the first and second queue much greater than the
load of the third queue. Low probabilities were accurately estimated within short computational times in both models.
In this paper we extend the simulation study made in [2] in a twofold direction. On the one hand we also simulate two additional types of networks that also could have difficulties for rare event simulation: a large network with 15 nodes, some of them at \textquotedblleft distance\textquotedblright\greater than 2, and a network with 2 nodes and very strong feedback. On the other hand we use different non-exponential distributions as hyperexponential and Erlang for modelling the interarrival and/or service times. This study will give us more insight for finding importance functions that could lead to good estimates of the probability of interest in most networks.
%\bibliography{}
[1] Vill\'{e}n-Altamirano J. 2010. Importance functions for RESTART
simulation of general Jackson networks. European Journal of Operation
Research, 203 (1): 156-165.
[2]Vill\'{e}n-Altamirano J. 2009. RESTART Simulation of Networks of Queues
with Erlang Service Times. Proc. Winter Simulation Conference, Austin (USA),
1146-1154.

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