Profile curves for the m/g/1 queue with group arrivals

Abstract

The purpose of profile curves is to serve as an “upper (or lower) stochastic envelope” of the path functions which are deemed to be statistically compatible with the proposed model. In defining these curves, we consider the joint stationary probability density of the queue length, given that the queue exceeds a given threshold K, and of the duration of such excursions. The profile curves are then the loci of given percentiles of the conditional probability densities of the queue length after successive numbers of services completed with the queue above threshold. Excessive excursions are either those that exceed the profile curve rapidly or those that stay above threshold for longer than can be expected. The occurrence of such excursions may require control actions or is an indication that the actual queue is no longer adequately described by the posited model. The profile curves are proposed as means to identify excessive excursions as soon as possible and serve in formalizing monitoring schemes for queu...