Comparison and optimization of RESTART run time strategies

The RESTART method with multiple steps or thresholds is one of the most general methods for the evaluation of rare events by simulation. This method has had a revival due to its use in the evaluation of cell loss probabilities in Asynchronous Transfer Mode (ATM) networks. The main problem in applying this method is the choice of parameters: how many steps and at which points or which levels should the steps be placed? Previous work has given some formulas for these parameters, but, unfortunately, the formulas contain values that are to be determined by the simulation itself. This paper compares different approaches with their formulas and gives examples. Two different run time strategies are evaluated: the step-by-step and the global-step approaches. In the step-by-step approach, one step after the other is evaluated until the last step is completed. In this case, values that have been determined during the evaluation of a previous step can be used to determine parameters of the next step. The drawback of this method is that, in general, more system states have to be stored. In the global-step approach, all steps are fixed in advance so that every time a step is reached, this simulation run can be split into several runs. Here only one state per threshold has to be stored, but the thresholds and also the splitting factors have to be fixed in advance. This paper shows how the two methods can be combined determining the optimal thresholds and parameters in a short pre-run with the step-by-step approach and gaining the desired accuracy in a second longer run with the global-step approach. Results for actual simulation runs using both run time strategies with the RESTART/LRE algorithm and the combined approach are given for some important queueing systems: M/M/1/N, M/D/1/N, SSMP(Geo,Geo)/D/1/N with a correlated input stream, and a tandem network. The obtained result is the complete complementary distribution function of the arrival occupancy, which includes as a last value the loss probability of the system.