General hazard-type scaling of abandonment time distribution for a G/Ph+GI queue in the Halfin–Whitt heavy-traffic regime

摘要

We consider the diffusion approximation of a G/Ph queue with customer abandonment in the Halfin–Whitt heavy-traffic regime and extend the conventional locally Lipschitz hazard-type scaling of abandonment time distribution to a more general scaling in the study. Under that new scaling of abandonment, not only the non-locally Lipschitz hazard-type case such as the non-locally bounded hazard-rate scaling which has never been solved to date, but also a wider range of abandonment time distributions including the non-absolutely continuous ones becomes subject to the analysis of diffusion approximation. Due to the general character of our scaling scheme of abandonment time, the stochastic equation for the limit of C-tight, scaled, and centered customer-count processes contains a nonlinear drift term as the limit of abandonment-count process, which does not satisfy the local Lipschitz condition so that the continuous mapping method in the literature of this area is invalid in our case. Instead, applying the Girsanov transformation to the localized multidimensional equation satisfied by the limit of customer-count process in multiphase of service time, we show the uniqueness in law of the solution to establish the desired diffusion approximation.