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We consider a queuing network that opens at a specified time, where customersare 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 waitingand service completion time. We restrict ourselves to a two class, two queuenetwork: this simplification is well motivated as the diversity in solutionstructure as a function of problem parameters is substantial even in thissimple setting (e.g., a specific routing structure involves eight differentregimes), suggesting a combinatorial blow up as the number of queues, routesand customer classes increase. We identify the unique Nash equilibrium customerarrival profile when the customer linear cost preferences are different. Thisprofile is a function of problem parameters including the size of each class,service rates at each queue, and customer cost preferences. When customer costpreferences match, under certain parametric settings, the equilibrium arrivalprofiles may not be unique and may lie in a convex set. We further make asurprising observation that in some parametric settings, customers in one classmay arrive in disjoint intervals. Further, the two classes may arrive incontiguous intervals or in overlapping intervals, and at varying rates withinan interval, depending upon the problem parameters.

Game of arrivals at a two queue network with heterogeneous customer routes