Congestion Control and Optimal Maintenance of Communication Networks with Stochastic Cost Functions: A Variational Formulation

We consider a game theory model of congestion control in communication networks, where each player is a user who wishes to maximize his/her flow over a path in the network. We allow for stochastic fluctuations of the cost function of each player, which consists of two parts: a pricing and a utility term. The solution concept we look for is the mean value of the (unique) variational Nash equilibrium of the game. Furthermore, we assume that it is possible to invest a certain amount of money to improve the network by enhancing the capacity of its links and, because of limited financial resources, an optimal choice of the links to improve has to be made. We model the investment problem as a nonlinear knapsack problem with generalized Nash equilibrium constraints in probabilistic Lebesgue spaces and solve it numerically for some examples.