A Fluid-Diffusion-Hybrid Limiting Approximation for Priority Systems with Fast and Slow Customers

We consider a large service system with two customer classes that are distinguished by their urgency and service requirements. In particular, one of the customer classes is considered urgent, and is therefore prioritized over the other class; further, the average service time of customers from the urgent class is significantly larger than that of the nonurgent class. We therefore refer to the urgent class as “slow,” and to the nonurgent class as “fast.” Due to the complexity and intractability of the system’s dynamics, our goal is to develop and analyze an asymptotic approximation, which captures the prevalent fact that, in practice, customers from both classes are likely to experience delays before entering service. However, under existing many-server limiting regimes, only two of the following options can be captured in the limit: (i) either the customers from the prioritized (slow) customer class do not wait at all, or (ii) the fast-class customers do not receive any service. We therefore propose a novel Fluid-Diffusion Hybrid (FDH) many-server asymptotic regime, under which the queue of the slow class behaves like a diffusion limit, whereas the queue of the fast class evolves as a (random) fluid limit that is driven by the diffusion process. That FDH limit is achieved by assuming that the service rate of the fast class scales with the system’s size, whereas the service rate of the slow class is kept fixed. Numerical examples demonstrate that our FDH limit is accurate when the difference between the service rates of the two classes is sufficiently large. We then employ the FDH approximation to study the costs and benefits of de-pooling the service pool, by reserving a small number of servers for the fast class. We prove that, in some cases, a two-pool structure is the asymptotically optimal system design.