Queueing Systems 11(1992)419-428 419 Short communication Sample path analysis of level crossings for the workload process* Michael A. Zazanis Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 60208, USA Received 6 March 1991; revised 28 September 1991 We examine level crossings of sample paths of queueing processes and investigate the conditions under which the limiting empirical distribution for the workload process exists and is absolutely continuous. The connection between the density of the workload distribution and the rate of downcrossings is established as a sample path result that does not depend on any stochastic assumptions. As a corollary, we obtain the sample path version of the Tak~cs formula connecting the time and customer stationary distributions in a queue. Defective limiting empirical distributions are considered and an expression for the mass at infinity is derived. Keywords: Level crossings, sample path analysis, empirical distributions, Tak~ics formula. 1. Introduction The investigation of relationships between time-stationary characteristics of the workload process and rates of downcrossings has a long history. We refer the reader to the monograph of Franken et al. [6, pp. 57 and 142]. Among the early results on level crossing methods for queues, we mention Brill and Posner [1,2], who examined queues with Poisson arrivals, Kt3ning et al. [6], Rolski [10], and Schmidt [11], who investigated level crossings in a stationary and ergodic context, and Cohen [4] and Shanthikumar [13], who examined regenerative queues using level crossing methods. Miyazawa [9] developed the general form of the Rate Conservation Law and used it to derive the connection between level crossings and the density of the time stationary workload. In the same vein is the paper by Ferrandiz and Lazar [5]. Besides their intrinsic interest, level crossings have been used in the analysis and control of priority and vacation queues and queues in a random environment. We refer the reader to Shanthikumar [13, 14], Miyazawa [9], and the references therein. *This research has been supported in part by NSF Grants ECS-8811003 and DDM-8905638. 9 J.C. Baltzer AG, Scientific Publishing Company