Staffing to Stabilize the Tail Probability of Delay in Service Systems with Time-Varying Demand

Analytic formulas are developed to set the time-dependent number of servers to stabilize the tail probability of customer waiting times for the G_t/GI/s_t + GI queueing model, which has a nonstationary non-Poisson arrival process (the G_t), nonexponential service times (the first GI), and allows customer abandonment according to a nonexponential patience distribution (the +GI). Specifically, for any delay target w > 0 and probability target α ∈ (0, 1), we determine appropriate staffing levels (the s_t) so that the time-varying probability that the waiting time exceeds a maximum acceptable value w is stabilized at α at all times. In addition, effective approximating formulas are provided for other important performance functions such as the probabilities of delay and abandonment, and the means of delay and queue length. Many-server heavy-traffic limit theorems in the efficiency-driven regime are developed to show that (i) the proposed staffing function achieves the goal asymptotically as the scale increases, and (ii) the proposed approximating formulas for other performance measures are asymptotically accurate as the scale increases. Extensive simulations show that both the staffing functions and the performance approximations are effective, even for smaller systems having an average of three servers.

The e-companion is available at https://doi.org/10.1287/opre.2017.1678.