Game of arrivals to a two queue network withheterogeneous customer routes

We consider a queuing network that opens at a specified time, where users are non-atomic and belong to different classes. Each class has its own route, and as is typical in the literature, the costs are a linear function of waiting and service completion time. We restrict ourselves to a two class, two queue network: this simplification is well motivated as the diversity in solution structure as a function of problem parameters is substantial even in this simple setting (e.g., a specific routing structure involves eight different regimes), suggesting a combinatorial blow up as the number of queues, routes and user classes increase. We identify the unique Nash equilibrium user arrival profile when the user linear cost preferences are different. This profile is a function of problem parameters including the size of each class, service rates at each queue, and user cost preferences. When user cost preferences match, under certain parametric settings, the equilibrium arrival profiles may not be unique and may lie in a convex set. We further make a surprising observation that in some parametric settings, users in one class may arrive in disjoint intervals. Further, the two classes may arrive in contiguous intervals or in overlapping intervals, and at varying rates within an interval, depending upon the problem parameters.