Exponential bounds for the waiting time distribution in Markovian queues, with applications to TES / GI /1 systems

Abstract

Several services to be supported by emerging high-speed networks are expected to result in highly bursty (autocorrelated) traffic streams. A typical example is variable bit-rate (VBR) compressed video. Therefore, traffic modeling and performance evaluation techniques geared towards autocorrelated streams are extremely important for the design of practical networks.The TES (Transform - Expand - Sample) technique has emerged as a general methodology for modeling autocorrelated random processes with arbitrary marginal distributions. Because of their generality and practical applicability, TES models can be readily used to accurately characterize bursty traffic streams in ATM networks.Although TES models can be easily implemented for simulation studies, the need still exists for analytical results on the performance of queueing systems driven by autocorrelated traffic. Of particular interest are the tails of the waiting time distribution in queues driven by TES-modeled bursty traffic. Such tail probabilities, when they become exceedingly small, may be difficult to obtain via conventional simulation.In order to extend existing results, based on Large Deviations theory, to TES processes, the main difficulty is posed by the continuous state-space of the TES time-series. In this paper, we develop a general result concerning exponential bounds for the waiting time under continuous state-space Markov arrivals. We apply this result to TES/GI /1 queues, show numerical examples, and compare our bound with simulation results. Accurate estimates of extremely low probabilities are obtained by employing fast simulation techniques based on importance sampling.