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Sample-path analysis of processes with imbedded point processes

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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 callY=λX. We then applyY=λX 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.

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Authors and Affiliations

  1. Department of Operations Research, CB 3180, Smith Building, University of North Carolina, 27599-3180, Chapel Hill, NC, USA

    Shaler Stidham Jr.

  2. Department of Mathematics and Statistics, University of Southern Maine, 96 Falmouth Street, 04103, Portland, ME, USA

    Muhammad El Taha

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  1. Shaler Stidham Jr.
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  2. Muhammad El Taha
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Additional information

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 Carolina, 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.

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Stidham, S., Taha, M.E. Sample-path analysis of processes with imbedded point processes. Queueing Syst 5, 131–165 (1989). https://doi.org/10.1007/BF01149190

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