We analyze a call center with multiple customer types and dynamic priority service discipline, in which a low-priority customer becomes high priority when its waiting time exceeds a given deterministic service level threshold. Within each priority queue, the service discipline is first come, first served. Based on a fluid approximation of the system, we apply infinitesimal perturbation analysis (IPA) to derive estimators for the derivatives of the queue lengths with respect to the threshold parameter. Numerical examples illustrate the validity of the fluid model approximation and the accuracy of the IPA estimators.