Some Markov queueing networks with absolute priorities (Q789826)

An open network of N service systems (vertices) is considered. There are \(n_ i\) service channels in each vertex. M classes of customers are served in each vertex. The service time in the i-th vertex is independent of the customer class and has an exponential distribution with the parameter \(\mu_ i\). The queue size in each vertex is unbounded. To each vertex M independent customer flows are coming. The intensity of the k-th flow into the i-th vertex is \(\lambda_{ik}\). The customers of each class after being served in the i-th vertex go to the j-th vertex with probability r(i,j) or leave the network with probability \(r(i,N+1)\). In each vertex there is the same system of priorities. Priority of the customer is constant. Let \(\xi_{ik}(t)\) denote the number of customers of class k in the vertex i at the moment t. In the paper explicit formulas for stationary expected values \(\xi_{ik}=\lim M \xi_{ik}(t)\) are given. For the loss function \(F=\sum^{N}_{i=1}\sum^{M}_{k=1}c_ kM_{ik}\) an optimal rule for designing priorities to the classes of customers is given. The analogous results are derived for so called closed networks. In this case, for each i \(r(i,N+1)=0\). It is also assumed that \(\lambda_{ik}=0\) for each i and k.