Scheduling homogeneous impatient customers

Customer impatience has become an integral component of analyzing services, especially in the context of call centers. Typically, when customers arrive to such systems, they seem identical or homogeneous; however, from the system's perspective, as they wait in the queue, their residual willingness to wait changes. For instance, a customer who has already waited for 10 minutes may have a different residual willingness to wait compared with a customer who has only waited for 1 minute. In this manner, as time progresses, customers become differentiated on their estimated patience levels. We exploit this dimension of customer heterogeneity to construct scheduling policies in overloaded systems that dynamically prioritize customers based on their time in queue to optimize any given system performance metric. Interestingly, the optimal policy has a very simple structure, and we find that implementing it can lead to significant improvements over the first-come, first-served policy.