Heavy Traffic Limits for Unobservable Queues with Clearing Times

In many service systems, customers may choose to abandon the system if their wait is deemed excessive. In an observable queue, the server is always aware that an abandoning customer has left the queue. However, in unobservable queues, the fact that the customer has abandoned the system may be unbeknownst to the server. Not only are the abandoning customers unobservable to the server, the server may also spend unnecessary time attempting to locate the customer to preserve their position in line. We call this additional time that the server has to spend to confirm whether a customer has abandoned the queue a clearing time. In this paper, we investigate the impact of clearing times on the dynamics of single server unobservable queues with abandonment. Our analysis involves the derivation of heavy traffic limit theorems for the queue length and workload processes. These heavy traffic limits illustrate that not only the number of customers in the system at any time has a steady state distribution given by a truncated Gaussian, but also that the clearing times serve to reduce the impact or strength of the state dependent drift towards the origin. Moreover, we also develop approximations for the fraction of wasted time that the server spends trying to locate customers, the fraction of balking customers, and the fraction of reneging customers. Comparisons between simulation and our explicit heavy traffic formulas confirm that the approximations are accurate at quantifying the impact of the clearing times and unobservable dynamics.