Queueing Systems, 5 (1989) 131-166 131 SAMPLE-PATH ANALYSIS OF PROCESSES WITH IMBEDDED POINT PROCESSES Shaler STIDHAM, Jr. * Department of Operations Research, CB 3180, Smith Building, University of North Carolina, Chapel Hill, NC 27599-3180, U.S.A. and Muhammad El TAHA * * Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, Portland, ME 04103, U.S.A. Received 24 June 1988; revised 21 February 1989 Abstract The purpose of this paper is to review, unify, and extend previous work on sample-path analysis of queues. Our main interest is in the asymptotic behavior of a discrete-state, continuous-time process with an imbedded point process. We present a sample-path analogue of the renewal-reward theorem, which we call Y = XX. We then apply Y = XX to derive several relations involving the transition rates and the asymptotic (long-run) state frequencies at an arbitrary point in time and at the points of the imbedded point process. Included are sample-path versions of the rate-conservation principle, the global-balance conditions, and the insensitivity of the asymptotic frequency distribution to the distribution of processing time in a LCFS-PR service facility. We also provide a natural sample-path characterization of the PASTA property. Keywords: Sample-path analysis, queues, point processes, time averages, customer averages, rate conservation, global balance, insensitivity, LCFS-PR discipline. 1. Introduction Sample-path analysis has been recognized as a rigorous approach to proye various results in queueing theory. It has been used to derive relations between * The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University, and by the National Science Foundation, Grant No. ECS-8719825, at the University of North Carofina, Chapel Hill. * * The research of this author was partially supported by the U.S. Army Research Office, Contract DAAG29-82-K-0151 at N.C. State University. 9 J.C. Baltzer A.G. Scientific Pubhshing Company