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：本文研究了一个多服务台排队系统，其到达率为周期函数，顾客的加入决策取决于其耐心阈值与延迟代理指标。具体而言，每位顾客的耐心阈值服从某个公共分布。顾客在到达时接收服务延迟估计值，仅当该估计值不超过其耐心阈值时才加入系统，否则选择犹豫离开。研究的主要目标是估计到达率与耐心分布的相关参数。其中的复杂因素在于：推断过程必须仅基于观测过程（即只包含实际加入系统的顾客），而犹豫离开的顾客始终未被观测到。我们构建了状态依赖有效到达过程（即对应实际加入顾客的到达过程）的似然函数，证明了极大似然估计量的强相合性，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，现有文献中的证明技术已不再适用。本文证明机制的新颖之处在于：通过将样本路径分解为独立同分布的再生循环，从而从非独立样本中构造独立同分布对象。通过一系列数值实验（采用多种延迟估计函数）探讨了极大似然估计法的可行性。实验结果表明，高服务容量情景下到达率的估计效果最优，而低服务容量情景下耐心分布的估计效果最佳。