Calculation of moments in retrial queueing system MMPP|M|1 | Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika – Tomsk State University Journal of Control and Computer Science. 2014. № 4(29).

Calculation of moments in retrial queueing system MMPP|M|1

In the paper, we investigate a retrial queueing system with the input Markovian Modulated Poisson Process (MMPP) which is defined by matrixes pX and Q, and the service time of each call is distributed by the exponential law with rate We introduce the following denotations: i(t) is the process defining the call number in the orbit, n(t) is the underlying Markov chain of the input ММР-process, and k(t) describes states of the server. Stochastic multidimensional process {k(t), i(t), n(t)} is a Markov process. To obtain the probability distribution P(k,i,n,t), the system of the Kolmogorov differentiate equations is composed. This system is written in matrix form at stationary regime. Then, it is rewritten with making use of characteristic functions and the obtained system is studied by the method of moments. Using mathematical transforms, the following systems for components of the mean and the second order initial moment of probability distribution of calls number in the orbit are obtained: jam 0 = R Q(Q -рЛ) + [am _q E = ц - R _q (|aE + pЛE), mjQ = am ₀ - m ₀Q - R ₁pЛ, mj(pE -pAE) = 2 R^E + -j m ₀(2рЛБ + aE), ad о = m q(Q -рЛ) + цш _1;ad ₀E = ц-mjE - pm ₀ ЛE, djQ = -d ₀Q + 2ad ₀ - 2m ₁pЛ - RjpX -am ₀, d ₁(pЛE + |aE) = d _Q(pЛE + aE) - m ₁pЛE - 3 R ₁pЛE - 3 am ₀ E. However, the form of vectors {R ₀, R ₁}, which is two-dimensional joint stationary probability distribution of the MMP-process and the server states, is unknown. So, we assume the independence of the MMP-process and the server states distributions, and the vectors {R0, R1} have the following form: [R 0 = (1 -p) • R, IR =p-R. The analysis of the results shows that moments calculated through obtained formulas are sufficiently close to exact ones obtained by numerical methods. Thus, the moments calculated by proposed way can be called «quasiexact». The formulas for moments calculation can be used in future researching by composing approximate distributions and also for other practical problems.

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Keywords

слова: RQ-система, источник повторных вызовов, метод моментов, retrial queueing system, orbit, method of moments

Authors

Name	Organization	E-mail
Fedorova Ekaterina А.	Tomsk State University	moiskate@mail.ru

Всего: 1

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