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.

翻译：考虑一个在指定时间开放的排队网络，其中顾客是非原子的且属于不同类别。每类顾客有其既定路径，与文献惯例一致，成本是等待时间与服务完成时间的线性函数。我们限定于两类顾客、两个队列的网络：这一简化具有充分动机，因为即使在此简单场景中，问题参数变化导致的解结构多样性也相当可观（例如，特定路由结构涉及八种不同模式），暗示随着队列、路径及顾客类别数量的增加将出现组合爆炸。当顾客线性成本偏好不同时，我们确定了唯一的纳什均衡顾客到达分布。该分布是问题参数的函数，包括各类别规模、各队列服务速率及顾客成本偏好。若顾客成本偏好一致，在某些参数设定下，均衡到达分布可能不唯一且位于凸集中。我们进一步发现一个令人惊异的规律：在某些参数设定下，某一类顾客的到达时间可能呈现不连续区间。此外，两类顾客的到达时间可能表现为连续区间或重叠区间，且区间内到达速率随问题参数变化。