Nash Equilibrium Flow in a Routing Game with Random Queues

In this paper, we present a new dynamic flow routing model on networks of parallel paths with capacities on edges and a regime of random queues at vertices. According to the introduced rules of waiting at vertices, we estimate the routing cost functions expressed by a mathematical expectation of a routing duration. For the corresponding routing game with nonatomic agents we prove the existence of Nash equilibrium flows and provide a method for their computation. On a basis of the investigated properties of cost functions and Nash equilibrium flows, we show that the routing costs can be changed in such a way that one of the resulting equilibrium flows minimizes with some accuracy the time a network is used by agents.