We develop an approximate generating function analysis (AGFA) technique which approximates the Laplace transform of the probability density function of customer response time in networks of queues with class-based priorities. From the approximated Laplace transform, we derive the first two moments of customer response time. This technique is applied to a model of a large hospitalÆs Accident and Emergency department for which we obtain the mean and standard deviation of total patient service time. We experiment with different patient-handling priority schemes and compare the AGFA moments with the results from a discrete event simulation.