Stabilizing Customer Abandonment in Many-Server Queues with Time-Varying Arrivals

An algorithm is developed to determine time-dependent staffing levels to stabilize the time-dependent abandonment probabilities and expected delays at positive target values in the M-1/GI/s(1) + GI many-server queueing model, which has a nonhomogeneous Poisson arrival process (the M-1), has general service times (the first GI), and allows customer abandonment according to a general patience distribution (the +GI). New offered-load and modified-offered-load approximations involving infinite-server models are developed for that purpose. Simulations show that the approximations are effective. A many-server heavy-traffic limit in the efficiency-driven regime shows that (i) the proposed approximations achieve the goal asymptotically as the scale increases, and (ii) it is not possible to simultaneously stabilize the mean queue length in the same asymptotic regime.