On the Computation of Dynamic User Equilibrium in the Multiclass Transient Fluid Queue

We consider the arrival timing problem faced by multiclass strategic customers to a single queue. The customers sensitivities to delay as well as service completion time preferences may be heterogeneous and the latter may vary non linearly with time. This captures many realistic settings where customers have preferences on when to arrive at a queue. We consider a fluid setup, so each customer is a point in a continuum and service rate is deterministic. This problem has been well studied in the transportation literature as the bottleneck model and the equilibrium customer arrival profile is shown to uniquely exist using intricate fixed point arguments. We develop a simple, elegant and geometrically insightful iterative method to arrive at this equilibrium profile, and provide an equally simple uniqueness proof. Further, under somewhat stringent assumptions, we arrive at the rate of convergence of the proposed algorithm. The simple geometric proof allows easy incorporation of useful extensions - to illustrate, we consider time varying service rates where the equilibrium profile is easily computed. Further, our results easily extend to the case of customers balking when their costs are above a class dependent threshold.