This paper studies a service system in which arriving customers are provided with information about the delay they will experience. Based on this information they decide to wait for service or to leave the system. The main objective is to estimate the customers' patience-level distribution and the corresponding potential arrival rate, using knowledge of the actual queue-length process only. The main complication, and distinguishing feature of our setup, lies in the fact that customers who decide not to join are not observed, but, remarkably, we manage to devise a procedure to estimate the load they would generate. We express our system in terms of a multi-server queue with a Poisson stream of customers, which allows us to evaluate the corresponding likelihood function. Estimating the unknown parameters relying on a maximum likelihood procedure, we prove strong consistency and derive the asymptotic distribution of the estimation error. Several applications and extensions of the method are discussed. The performance of our approach is further assessed through a series of numerical experiments. By fitting parameters of hyperexponential and generalized-hyperexponential distributions our method provides a robust estimation framework for any continuous patience-level distribution.
翻译:本文研究一个服务系统,向抵达的客户提供有关他们将要经历的延误的信息。 根据这些信息,他们决定等待服务或离开系统。主要目标是评估客户的耐心水平分布和相应的潜在抵达率,仅使用实际排队长度过程的知识。我们设置的主要复杂之处和特点在于,决定不加入的客户没有被观察到,但显然,我们设法设计了一个程序来估计他们将产生的负荷。我们用一个多服务员队列来表达我们的系统,有波斯森的客户队伍,使我们能够评估相应的概率功能。根据最大可能性程序来估计未知的参数,我们证明我们非常一致,并得出估计错误的零点分布。讨论了方法的若干应用和扩展情况。我们的方法通过一系列数字实验进一步评估了我们的方法的绩效。我们的方法通过高耗度和普遍热度分布的适当参数,为任何连续的耐心水平分布提供了可靠的估计框架。