Capacity Sizing in Queueing Models with a Random Number of Servers

Abstract

One prevalent assumption in queueing theory is that the number of servers in a queueing model is deterministic. However, randomness in the number of servers often arises in practice, particularly when the servers themselves may be viewed as strategic decision makers, e.g., as in virtual call centers or ride-sharing services where agents are allowed to set their own schedules, often at very short time notice. In this paper, we study the problem of staffing many-server queues with general abandonment and a random number of servers. We model the number of agents available to work in a given period as a binomial random variable, and rely on a fluid model to determine cost-minimizing staffing levels. We provide theoretical support for the usage of the fluid model by demonstrating the asymptotic accuracy of fluid-based performance measures and staffing prescriptions. As an application, we characterize the optimal staffing policy with self-scheduling servers, and show that it is not straightforward: It may be optimal to either understaff or overstaff the system, depending on (i) self-scheduling agent behavior and (ii) the abandonment-time distribution.