Heterogeneous queueing system with Markov renewal arrivals and service times dependent on states of arrival process

Abstract

In the proposed work, we consider a heterogeneous queueing system with a Markov renewal process and an unlimited number of servers. The service time for requests on the servers is a positive random variable with an exponential probability distribution. The service parameters depend on the state of the Markov chain nested over the renewal moments. It should be noted that these parameters do not change their values until the end of maintenance. Thus, the devices in the system under consideration are heterogeneous. The object of the study is a multidimensional random process - the number of servers of each type being served with different intensities in the stationary regime. The method of asymptotic analysis under the condition of equivalent growing of service times in the units of servers is applied for the study. The method of asymptotic analysis is implemented in the construction of a sequence of asymptotic of increasing order, in which the asymptotic of the first order determines the asymptotic mean value of the number of occupied servers. The second-order asymptotic allows one to construct a Gaussian approximation of the probability distribution of the number of occupied servers in the system. It is shown that this approximation coincides with the Gaussian distribution.