We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d.\ regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities.

翻译：我们研究了一个具有周期性到达率的多服务台排队系统，其中顾客的加入决策取决于其耐心程度和延迟代理指标。具体而言，每位顾客的耐心水平服从某一共同分布。顾客到达时，会收到加入服务前的延迟估计，仅当该延迟不超过其耐心水平时才会加入系统，否则选择退缩。主要目标是估计到达率及耐心分布的相关参数。此处复杂之处在于，此类推断必须仅基于可观测过程（即仅通过实际加入的顾客数据）进行，而退缩顾客始终无法被观测到。我们基于状态依赖的有效到达过程（即对应于加入顾客的到达过程）构建了似然函数，证明了极大似然估计的强相合性，并推导了估计误差的渐近分布。由于泊松到达过程固有的非平稳性，先前工作中使用的证明技术不再适用。本文证明机制的新颖性在于：通过将样本路径分解为独立同分布的再生循环，从相依样本中构造独立同分布对象。通过一系列数值实验（针对多种提供延迟估计的函数）讨论了极大似然估计方法的可行性。特别地，观察到到达率在高服务容量下估计效果最佳，而耐心分布则在低服务容量下估计效果最优。