Modification of the asymptotic analysis method under heavy load condition on the example of the study of the retrial queueing system M|M|1

In the paper, three methods of the asymptotic analysis modification are offered for retrial queueing systems under heavy load condition as mathematical models of telecommunication systems. The characteristic feature of these models is existence of repeated attempts of calls to get service after a random time. There are a large number of papers devoted to retrial queues researching, but mainly problems are solved by numerical methods and simulation. Analytical results are obtained only for systems with Poisson input process. Earlier we proposed the method of asymptotic analysis (first and second order) for retrial queueing systems researching under heavy load condition [13, 14]. It has been shown that the first order asymptotics has the form of gamma distributions for all types of the system, but its application range is quite narrow. The asymptotics of second order has allowed to extend the application range of the results by 4 times, but the asymptotic formula has not unified form and it needs time-taking rather laborious formula derivations for each system. Therefore, we suggest ways to modify the method of the asymptotic analysis under heavy load condition to increase the method accuracy. The method is described on the example of the simplest retrial queue M|M|1 in order to compare the results with the known exact distribution. The system of differential Kolmogorov equations for Markov process {k(t), i(t)} is composed, then it is rewritten for characteristic functions in steady state. X The parameter p = - is the load rate of the system. The principle of asymptotic analysis method under heavy load condition (where p 11) is briefly described. Asymptotic characteristic functions of the first and second order are presented. Note that during the derivation of the first order asymptotic formula the function F₀(w) was obtained. We propose to study "intermediate" asymptotics h (u), which has the form of a weighted sum of two gamma distribution characteristic functions: h (u) = q^(u) + (1 - q)Г₀(и) , { \-a / \-a+1 i where Г₁ (u) = f 1 - , Г₀(u) = |l- j , q = - , « = ^ +1 и P = 1 -p . We suggest to call this distribution as hypergamma distribution (like a hyperexponential distribution). The numerical analysis shows that hypergamma asymptotics is more accurately than the first order asymptotics, but it is worth than the second order asymptotics. However, the derivation of the second order asymptotic formula for more complex retrial queues is difficult, whereas "intermediate" asymptotics does not require additional calculations, because its shape is the same for all retrial queues and hypergamma distribution parameters are obtained in the first order asymptotic formula theorem. Then a modification of hypergamma asymptotics of shifting the argument to the parameter 5 is proposed: F(x) = qr ₁ (x + 8) + (1 - q)r₀(x + 8), where 5 is found from the condition Г ₁(8) = (1 - p) . The numerical analysis shows that "modified" distribution significantly improves results of hypergamma asymptotics. In the paper conclusion a combination of "modified" distribution and the second asymptotics Pm = 2(Pm + P). is considered. As a result of numerical analysis, it was observed that this modification is more accurately than the second order asymptotics and can be applied for load rate p>0.7 and even p>0.5 for large values of delay parameter a. In summary, it can be concluded that for practical tasks which does not require high accuracy of approximation, it is useful to apply the "intermediate" asymptotics results because of its simplicity of construction and rather broad area of applicability (p> 0.8), and for applications requiring higher accuracy of approximation, it is advisable to apply the proposed modifications 1 and 2. In the future, the modifications can be applied to the study of various retrial queueing systems including with non Poisson incoming process.

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