Application of negative binomial distribution to approximate the stationary distribution of the number of arrivals in a QS with an incoming MAP, the intensity of which depends on the state of the system

This paper considers a mathematical model of an infinite-linear queuing system with an incoming MAP with an intensity depending on the number of busy servers. The parameters of the incoming process, namely its conditional intensities, change every time the state of the system changes, that is, a new request appears or one of the requests completes servicing. The service discipline is determined by the fact that the request occupies any of the free devices in the system on which its service is performed for a random time distributed according to an exponential distribution. For this model, obtaining a stationary probability distribution of the number of applications in the system by analytical means is not possible, so this paper proposes a heuristic approach, namely, the use of a negative binomial distribution as an approximation for the desired distribution. Two approaches to such approximation are proposed, for which a numerical analysis of the accuracy is performed based on comparison with the results of simulation modeling. The first approach is based on calculating the parameters of the negative binomial distribution using the exact values of the expected value and dispersion of the number of applications in the system under consideration, and the second is based on the fact that the intensity of incoming applications is determined by the Markov chain controlling the incoming process. It was found that the first approximation method gives more accurate results, however, when the system is heavily loaded, both approximations have a large error.