We study the optimal provision of information for two natural performance measures of queuing systems: throughput and makespan. A set of parallel links is equipped with deterministic capacities and stochastic travel times where the latter depend on a realized scenario. A continuum of flow particles arrives at the system at a constant rate. A system operator knows the realization of the scenario and may (partially) reveal this information via a public signaling scheme to the flow particles. Upon arrival, the flow particles observe the signal issued by the system operator, form an updated belief about the realized scenario, and decide on a link to use. Inflow into a link exceeding the link's capacity builds up in a queue that increases the travel time on the link. Dynamic inflow rates are in a Bayesian dynamic equilibrium when the expected travel time along all links with positive inflow is equal at every point in time. We provide an additive polynomial time approximation scheme (PTAS) that approximates the optimal throughput by an arbitrary additive constant $\epsilon>0$. The algorithm solves a Langrangian dual of the signaling problem with the Ellipsoid method whose separation oracle is implemented by a cell decomposition technique. We also provide a multiplicative fully polynomial time approximation scheme (FPTAS) that does not rely on strong duality and, thus, allows to compute also the optimal signals. It uses a different cell decomposition technique together with a piece-wise convex under-estimator of the optimal value function. Finally, we consider the makespan of a Bayesian dynamic equilibrium which is defined as the last point in time when a total given value of flow leaves the system. Using a variational inequality argument, we show that full information revelation is a public signaling scheme that minimizes the makespan.

翻译：我们研究了排队系统中两种自然性能指标——吞吐量和完工时间——的最优信息提供问题。一组平行链路具有确定性容量和随机旅行时间，后者依赖于已实现的场景。连续流体粒子以恒定速率到达系统。系统操作员知晓场景的实现情况，并可能通过公开信号方案向流体粒子（部分）揭示该信息。到达时，流体粒子观察系统操作员发出的信号，形成关于已实现场景的更新信念，并决定使用哪条链路。流入链路超出其容量的流量会形成队列，从而增加该链路的旅行时间。当所有具有正流入的链路的预期旅行时间在每个时间点均相等时，动态流入率处于贝叶斯动态均衡状态。我们提出了一个加性多项式时间近似方案（PTAS），该方案能以任意加性常数$\epsilon>0$逼近最优吞吐量。该算法通过椭球法求解信号问题的拉格朗日对偶，其分离预言机通过单元分解技术实现。我们还提供了一个不依赖于强对偶性的乘性完全多项式时间近似方案（FPTAS），从而也能计算最优信号。该方案采用不同的单元分解技术，并结合最优值函数的分段凸下估计。最后，我们考虑了贝叶斯动态均衡的完工时间，定义为给定总流量离开系统的最后一个时间点。通过变分不等式论证，我们证明完全信息揭示是使完工时间最小化的公开信号方案。