The numerical analyses of queuing system with hyperexponential distribution of cooling time

The numerical analysis of QS with cooling is considered. Cooling is understood as the temporary delays coming after the period of continuous employment. In practice such delays arise when after service of the last request in system carrying out the actions connected with maintenance, carrying out maintenance, breaks in work, etc. is required. In this case pertinently to say that QS passes into the cooling mode. The beginning of service of again arrived request won't begin, all operations of cooling won't come to the end yet. Unlike a warming up, process of cooling of system doesn't depend on arrival of the first request of the period of employment. If the system manages to be cooled before arrival of this request, service will begin without additional delays. In this paper is presented the technique of the numerical analysis of multichannel QS with the Poisson arrival, exponential service and not-exponential distribution of cooling time. For approximation of cooling time it is offered to use the function R(x) which is formally coinciding with function of the hyperex-ponential distribution of the second order (H2), but allowing unlike hyper exhibitors complex type of parameters R( x) = yf'^ + y₂e^ ; y + y₂ = 1. Considering that prototype of function R(x) - the hyperexponential H2 distribution belongs to distributions of phase type, conditions of non-Markov QS and transitions between them can be presented in the form of discrete Markov process with continuous time. Possibilities of R-approximation in that case, when coefficient of a variation of initial distribution u<1 are shown in article. Function parameters thus accept complex values. Nevertheless, at calculation of QS with application of R-approximation in the field of complex values this pathology is shown only in intermediate results - probabilities of fictitious microstates of the chart of transitions on which physical conditions of QS are split. At a stage of summation of probabilities of microstates of circles their complex parts are annihilated also result of calculation - probability of number of requests in system - becomes material. Distribution of number of applications and Laplace transformation of a waiting time is received. In work the results of calculation of QS received by means of R-approximation and by means of imitating modeling are compared. Various initial distributions of cooling - determined uniform, scale with form parameter 0,5, exponential and Weibull with form parameter 0,46 are considered. It is shown that the accuracy of calculation of stationary distribution of number of requests and the first three moments waiting time is well. The maximum distance of Kolmogorov (for Weibull distribution) made 0,043. It is shown that when coefficient of a variation of cooling time distribution is increased, the average time of expectation is increased too. Scope of the offered calculation procedure is the analysis of functioning of organizational and technical systems with queues in which after end of the period of continuous employment there comes the temporary delay connected with the following reasons: - maintenance of system; - carrying out maintenance; - restoration of the material resources spent at service; - need of transfer to a higher body of information directorate about number of the requests served during continuous employment. Its expansion on a case of non-Markov service and the recurrent arrival, and also calculation of a queuing network which nodes is QS with "cooling" can be the further direction of development of the offered calculation procedure.

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