Stochastic comparisons for rooted butterfly networks and tree networks, with random environments

Consider a rooted tree network, where the items enter at the system and they proceed away from the root until they reach their destination and exit the system, and they are served by a FIFO policy at each arc (server) of the network. The routing is defined by a discrete probability distribution with a given probability for each destination. For such systems, stochastic modelling of the departure times and the delay times is proposed, by the incorporation of random parameters of the inter-arrival times and of the service times, describing dynamic environments. A mixture model for the departure times is introduced. This mixture has an arbitrary mixing distribution defined by the environmental parameter distributions and the routing distribution. The main results provide conditions to compare stochastically the departure times (delay times) for two rooted tree networks characterized by different routing disciplines or by environmental and correlated random vectors of parameters. Furthermore, bounds for these measures are obtained from some well-known dependence concepts, as the PQD property, and ageing properties of the random environment. Similar results for butterfly networks, tree networks with possible failure during the service and other networks are provided. Within the computer networks, our framework and our results provide explorative tools to assess the design, the performance and the security of communication systems.