Queueing Management for Reducing the Overlaps of Customers in Service Systems

During pandemic seasons, queueing management that prevents customers in service systems from interacting with each other can be effective in reducing their risk of infection. In this paper, we aim to understand how queueing topologies and flow control policies influence customers' infection risk in service systems. We define two novel metrics that serve to describe the customers' infection risk, namely the average overlapping time and the average number of overlapped customers. We prove that both metrics have a symmetry property, which allows us to derive closed-form expressions for these metrics. We investigate two widely-used queueing topologies, the serial topology and the parallel topology. For the serial topology systems, we design flow control schemes that are easy for service systems to apply in practice. We then consider the grocery store model as a special case of the serial system, where only a portion of customers can shop in store at the same time. We show that the threshold in the grocery model leads to a risk-efficiency tradeoff, and the waiting time by the risk-optimal threshold is always bounded by twice the minimum waiting time. In addition, we prove that the parallel topology system with even splitting results in a smaller number of overlapped customers, but it does not achieve a shorter overlapping time than the serial topology system when the traffic is heavy. We further prove that the parallel topology system under the join-the-shortest-queue scheme achieves smaller overlapping time and number than the serial topology system under the optimal flow control, with a slight loss of efficiency. Our findings reveal that reducing the waiting time of customers does not always reduce their overlaps. We also discuss the impact of risk preferences, mixed topologies, and service distributions on the policy design