## Subsolution approach for the simulation of a rare event in the GI|GI|1 tandem queue

Anne Buijsrogge, UT-EWI-SOR

#### Abstract:

In this work we study the event that the total number of customers in a GI|GI|1 tandem queue reaches some high level in a busy cycle of the system. Using simulation, we want to estimate the probability of this event. As this event is rare when the level is high, we choose to use importance sampling to speed up the simulation. Therefore we need to find a change of measure that is asymptotically efficient.

For a Markovian tandem queue a state-dependent change of measure has previously been proven to be asymptotically efficient by using the so-called subsolution approach. Our goal is to generalize this to a GI|GI|1 tandem queue in order to build more realistic models.

The approach we use is similar to the existing subsolution approach. To obtain a Markov process, we add the residual inter-arrival time and the residual service time to the state space description. This approach seems to work well in order to find a state-dependent change of measure for the GI|GI|1 tandem queue and to prove asymptotic efficiency. For the case of a single GI|GI|1 queue, this approach enables us to find an alternative proof for asymptotic efficiency of a state-independent change of measure.