Investigation of retrial queue system M|M|1 with phase-type retrial times

Retrial queueing models are used as stochastic modeling tool of many computer and communication systems such as call-centers, mobile phone systems, local computer networks with multiple random access. Consider a single-server queueing system. The customers form the Poisson input process with a rate L The customers' service times have an exponential distribution with a parameter If a customer finds server engaged, one joins the retrial queue (or orbit) in order to seek service again after a random delay that has a phase-type distribution. Let i(t) be the vector with components which are equal to a number of customers in each phase of the phase-type retrial at instant t. The process i(t) is a non-Markovian process. To extend i(t) to a Markov process, we introduce new random variables. Let k(t) be the indicator function of a server's state: k(t) = 0 if the server is vacant at moment t; k(t) = 1 if it is engaged at moment t. Therefore, the stochastic process {k(t), i(t)} is the Markovian process defined on the state space {(k,i)}, where one in N+1 components is finite and other N components are countable. Our purpose is to find the probability distribution of states for non-Markovian process N Z h (t). k=1 We use the asymptotic analysis method that allows us to find the probability distribution of process states in a marginal condition when the retrial time grows unlimited. It is shown that in the condition of unlimited growth, the probability distribution can be approximated by the multidimensional Gaussian distribution. The first and second orders of the asymptotic method provide as result the mean vector and covariate matrix correspondingly. To estimate the range of applicability of analytic results, we use simulations. Kolmogorov's distance as a comparison criteria of analytic and simulation results is used.

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