Capacity Rationing in Multiserver, Nonpreemptive Priority Queues

Problem definition: Many service and manufacturing systems use both capacity rationing (CR) and priority to differentiate among their customers. We model these as a two-class nonpreemptive priority $M/M/c$ queueing model and the practice of CR; an arriving low-priority customer can directly enter service only when the number of idle servers is higher than the CR level, k. For these systems, we separately discuss two important features that are common in practice but ignored in the literature; supply is narrowly matched with demand, and service rates are heterogeneous, reflecting different customer types. Methodology and results: When the service times of both classes are identical, our asymptotic results indicate that for a system with a large number of servers, the nondegenerative CR level does not exceed $\mathcal{O}(\sqrt{c})$. When the service times of classes differ, we derive exact solutions for different performance measures of interest using queueing and Markov chain decomposition. We numerically demonstrate the impact of system parameters on these performance measures and provide insights on the CR level. Management implications: We show that as predicted by the asymptotic analysis, an $\mathcal{O}(\sqrt{c})$ CR level can significantly reduce the waits of high-priority customers with little effect on low-priority customers’ waiting. We establish that this insight is robust to heterogeneous service times across classes and other system parameters, such as the number of servers and the arrival rates of the classes.

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2021.0106.