Research of Non-Markovian Queuing Networks

The article presents models of mathematical multiplexing and demultiplexing of flows, as well as the expressions for determining the numerical characteristics of the intervals distribution in the output flow from the queuing system (QS). Together they allow us to write the equations of equilibrium with respect to the mean values and variance in the time intervals distributions between neighboring states in queuing networks in case of random receiving flows patterns and service time. QS M-/M-/1 with a delay in time, H2/H2/1 and H2/M/1 are considered to be the nodes of the network. With general assumptions about the probability of time distribution between neighboring states in the input flows and service time at the nodes, the proposed approach allows to determine average values and variances of time intervals between all the states of queuing network flows, as well as all the probabilistic properties of such networks. The approach can also be extended, if necessary, to third moments. From the perspective of probability theory, this will be more accurate than the calculation at the level of two moments of the distributions. Having incomplete information on the patterns of time intervals distribution, the proposed approach may be acceptable for the analysis of the performance of non-Markovian queuing networks.