Modelling Complex Transport Network with Dynamic Routing: A Queueing Networks Approach

Abstract

In this paper we consider a Jackson type queueing network with unreliable nodes. The network consists of $m <\infty$ nodes, each node is a queueing system of M/G/1 type. The input flow is assumed to be the Poisson process with parameter $\varLambda (t)$. The routing matrix $\{r_{ij}\}$ is given, $i, j=0,1,...,m$, $\sum _{i = 1} ^ m r_ {0i} \le 1$. The new request is sent to the node i with the probability $r_{0i}$, where it is processed with the intensity rate $\mu _i(t,n_i(t))$. The intensity of service depends on both time t and the number of requests at the node $n_i(t)$. Nodes in a network may break down and repair with some intensity rates, depending on the number of already broken nodes. Failures and repairs may occur isolated or in groups simultaneously. In this paper we assumed if the node j is unavailable, the request from node i is send to the first available node with minimal distance to j, i.e. the dynamic routing protocol is considered in the case of failure of some nodes. We formulate some results on the bounds of convergence rate for such case.