A use of hyperexponential distribution in non-markovian queuing systems analyses

In this paper we consider the application of second order hyperexponential distribution (H₂) with complex-type parameters for analysis of non-markovian queuing systems. This distribution relates to the phase-type one and allows showing non-markovian queuing systems states and the transitions between them as discrete Markov process with continuous time. The complementary cumulative H2 distribution function is F (t) = y1e ^ + y ₂ e ^. Using of H2 distribution for queuing system calculations is reasoned by following reasons: - the possibility of saving the three initial moments of the original distribution that provides high accuracy in queuing system calculation; - much more compact (compared to the Erlang approximation) view transition diagrams and as a consequence necessity to calculate the probabilities of a smaller number of microstates for systems with low coefficients of variation of the service time or the intervals between customers; - simple calculating of complementary cumulative distribution function. Since the parameters of H₂-distributions {j}, J = 1, 2, are interpreted as the probabilities of random select of one of two phases, most specialists in queuing theory considered only the case when these parameters are defined on a real interval [0; 1], which corresponds to the approximation of the distribution with a coefficient of variation и > 1. In this article, it is showed the possibilities of the H₂-approximation in the case when the original distribution coefficient of variation и < 1. In this case parameters of H₂ distribution are complex. More detailed analysis of H₂-approximation of the original gamma distribution with a coefficient of variation и shows that: - if и > 1, then the parameters are real; - if 1 > и , then the parameters are real, but the paradox: one of the parameters {j}, J = 1, 2, is negative, and the other will exceed one; - equality и = is unacceptable (because corresponds to the Erlang distribution with consistent phase-change and, accordingly, can not be replaced by parallel); - when и <1lV2 , we have the complex parameters of H₂ distribution. However, when calculating the queuing system with H₂-approximation in the field of complex values of the parameters of its potential pathology manifests itself only in the intermediate results - in the probabilities of "fictitious" microstate-transition diagram, which split the "physical" state of queuing systems. At the summation of probabilities of microstates tiers of complex parts are annihilated and the result of the calculation - the probability of the number of customers in the system - becomes real. The paper compares the results of single-channel systems MIGI1 calculation invested by embedded Markov chain, which allows you to obtain an exact solution, with the results obtained by the fictitious phase using H₂-approximation of non-exponential service time. Various initial distribution services - deterministic, gamma with shape parameters of 1.5 or 0.5, and even in the interval [0, 2] are considered. It is shown that the accuracy of the above-mentioned result is high enough. The maximum Kolmogorov distance was 0.002, and the relative error of the probability of rare events (about 10) did not exceed 15%. At the same time the quality of approximation in the field of complex parameters H2 distribution is out of question because density of H₂-distribution in this case is negative and in general does not satisfy the requirements of the probability density function. Quality of approximations here should be assessed indirectly - through comparison of the distribution of the number of customers in the queuing system, obtained through H₂-approximation, with the results obtained by alternative methods.

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