🤖 AI Summary
This paper investigates the Nash equilibrium arrival strategies of heterogeneous customers in a two-class, two-queue queuing network with an open-starting time epoch, where customer cost is a linear function of waiting time and service completion time. Using nonatomic game modeling and differential game equilibrium analysis, we derive piecewise-continuous arrival rate functions. We establish, for the first time, that equilibrium arrival intervals for the two customer classes may be non-connected. We rigorously prove uniqueness of the Nash equilibrium arrival distribution when customer cost preferences differ, and convexity of the equilibrium set when preferences coincide. The system identifies and classifies eight parameter-driven structured equilibrium regimes—including overlapping, separated, and piecewise-varying arrival patterns—and provides a complete parametric sensitivity analysis along with precise characterizations of existence and uniqueness phase transitions.
📝 Abstract
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.
