Abstract
We consider a single-server queue with unlimited waiting space, the first-come, first-served discipline, a periodic arrival-rate function, and independent and identically distributed service requirements, where the service-rate function is subject to control. We previously showed that a rate-matching control, whereby the service rate is made proportional to the arrival rate, stabilizes the queue-length process but not the (virtual) waiting-time process. To minimize the maximum expected waiting time (and stabilize the expected waiting time), we now consider a modification of the service-rate control involving two parameters: a time lag and a damping factor. We develop an efficient simulation search algorithm to find the best time lag and damping factor. That simulation algorithm is an extension of our recent rare-event simulation algorithm for the GIt/GI/1 queue to the GIt/GIt/1 queue, allowing the time-varying service rate. To gain insight into these controls, we establish a heavy-traffic limit with periodicity in the fluid scale. This produces a diffusion control problem for the stabilization, which we solve numerically by the simulation search in the scaled family of systems with ρ ↑ 1. The state space collapse in that theorem shows that there is a time-varying Little’s law in heavy traffic, implying that the queue length and waiting time cannot be simultaneously stabilized in this limit. We conduct simulation experiments showing that the new control is effective for stabilizing the expected waiting time for a wide range of model parameters, but we also show that it cannot stabilize the expected waiting time perfectly.
Highlights
The state space collapse in that theorem shows that there is a time-varying Little’s law in heavy traffic, implying that the queue length and waiting time cannot be simultaneously stabilized in this limit
This is a stochastic design problem instead of a real-time stochastic control problem; that is, the service-rate control is to be determined in advance, assuming full knowledge of the model, but without knowledge of the system state that will prevail at any time
We extend the rare-event simulation algorithm for the time-varying workload process in the periodic GIt/GI/1 model in Ma and Whitt (2018) to the GIt/GIt/1 model, where the service rate is time varying as well. (The notation GIt means that the process is a deterministic time transformation of a renewal process; see Section 4.) The workload L(t) represents the amount of work in service time in the system at time t, whereas the waiting time can be represented as the first-passage time
Summary
The stabilization is to be achieved with a deterministic service-rate function, under the assumption that the customer service requirements are specified independently of the service-rate control This is a stochastic design problem instead of a real-time stochastic control problem; that is, the service-rate control is to be determined in advance, assuming full knowledge of the model, but without knowledge of the system state (e.g., the value of the stochastic queue-length process) that will prevail at any time. Related Literature There is a large literature on similar stochastic design problems involving setting staffing levels (the number of servers) in a multiserver queue to stabilize performance in the face of time-varying demand (e.g., Jennings et al 1996, Feldman et al 2008, Stolletz 2008, Liu and Whitt 2012b, Defraeye and van Nieuwenhuyse 2013, He et al 2016, Pender and Massey 2017, Liu 2018, Whitt 2018). In Whitt (2015), theorem 4.2 shows that the rate-matching control stabilizes the queue-length process, theorem 5.1 gives an expression for the waiting time with the ratematching control, whereas theorems 5.2 and 5.3 establish heavy-traffic limits showing that the queue length is asymptotically stable, but the waiting time is not, being asymptotically inversely proportional to the arrivalrate function
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