Equilibria for Multiclass Routing in Multi-Agent Networks

We study optimal static routing problems in open multiclass networks with state-independent arrival and service rates. Our goal is to study the uniqueness of optimal routing under different scenarios. We consider first the overall optimal policy that is the routing policy whereby the overall mean cost of a job is minimized. We then consider an individually optimal policy whereby jobs are routed so that each job may feel that its own expected cost is minimized if it knows the mean cost for each path. This is related to the Wardrop equilibrium concept in a multiclass framework. We finally study the case of class optimization, in which each of several class of jobs tries to minimize the averaged cost per job within that class; this is related to the Nash equilibrium concept. For all three settings, we show that the routing decisions at optimum need not be unique, but that the utilizations in some large class of links are uniquely determined.