The amount of overtaking in a network of queues

Abstract

Abstract Customer B overtakes customer A in a queueing system if A arrives before B but B departs first. To understand the phenomenon of overtaking in a network of queues and its impact on sojourn times, it is important to develop concepts of the amount of overtaking. In this paper, two stochastic measures of the amount overtaking are defined: the random number of customers overtaken by an arbitrary customer and the random permutation of the original order of n customers upon departure. Describing these measures, however, often presents rather difficult combinatorial and probabilistic problems. In this paper, we make several conjectures about the amount of overtaking in open Jackson networks and its impact on sojourn times. Then we describe the amount of overtaking in several smaller networks, for example, a single node with the first‐come first‐served discipline and instantaneous Bernoulli feedback.

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