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ANALYSIS OF THE BLOCKING PROBABILITY FOR THE M/G/1 QUEUING SYSTEM WITH THE CORRELATED SERVICE TIME
Igor V. Kartashevskiy, Povovlzhskiy state university of telecommunications and informatics, Samara, Russia, ivk@psuti.ru
Abstract
The traffic of modern networks has certain properties determined by the fact that one-dimensional probability density functions of inter-arrival time and service time for requests are characterized by heavy tailed distributions and, in addition, these sequences of time intervals have clearly expressed correlation properties. And if positive values prevail among correlation coefficients, it leads to formation of packs of requests that block the system operation. The blocking probability together with the average waiting time in queue are the most important characteristics of the traffic service quality. For the M/G/1/m queuing system, the analysis of blocking probability is rather complicated even if there are no correlation properties of the sequence of time intervals for servicing requests. The presence of correlation further complicates the problem, which, however, can be solved by methods developed for the case of lack of correlation. The key point of this approach is the use of approximation of arbitrary probability density function of service time intervals by the hyper-exponential distribution with parameters taking in account the correlation properties of the initial sequence of time intervals through the renewal process. In fact, the M/G/1/m queuing system with a correlated sequence of service intervals and arbitrary density is replaced with a system with non-correlated sequence and a hyper-exponential distribution of service time intervals with parameters considering the correlation properties of the original sequence. The conversion of the parameters of the correlated sequence of service time intervals into the parameters of the renewal process with the hyper-exponential distribution is based on equating the index of dispersion of the initial sequence to the squared coefficient of variation of the renewal process. The paper shows that such a replacement allows to get a quantitative assessment of the blocking probability for the queuing system with a correlated service time for any size of the buffer.
Keywords:queuing system, blocking probability, correlated traffic, renewal process, hyper-exponential distribution.
References
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Information about author:
Igor V. Kartashevskiy, assistant professor, Software department, Povovlzhskiy state university of telecommunications and informatics, Samara, Russia