  We consider a queuing network that opens at a specified time, where customers
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 customer classes increase. We identify the unique Nash equilibrium customer
arrival profile when the customer 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 customer cost preferences. When customer 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, customers 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.
