Abstract
In the recent past, discrete-time queueing models of the DBMAP/G/1/N-type (D[B]MAP: Discrete [Batch] Markovian Arrival Process) were developed, which allow for computation of per-stream loss probabilities, one type of solution based on the method of the unfinished work, and the other using the M/G/1-paradigm. The method of the unfinished work (applicable for deterministic service time only) provides — besides formulae for computing per-stream loss probabilities — also expressions for obtaining per-stream waiting-time probability functions, and therefore arbitrary moments of the waiting-time of each stream [7]. What is missing up to now (see also [3]), is an algorithmic recipe to compute waiting-time moments for DBMAP/G/1/N queueing system. This paper derives the z-transform of the actual waiting-time and derives a fast algorithm to determine the waiting-time probability function for an arbitrary renewal service time. The framework presented allows dealing with a superposition DMAP+DBMAPas an input process. For both streams in the superposition, per-stream waiting-time probability functions are presented for the case of a deterministic service time, which is relevant for modelling of ATM networking (both wired and wireless). Together with findings for the continuous-time case ([2, 12]), this paper completes the insights into finite queueing systems of the M/G/1-type. In comparison with the method of the unfinished work, the M/G/1-paradigm provides much faster algorithms to compute loss probabilities and waiting-time moments, due to the smaller system matrix. Note that the D(B)MAP has proved a versatile stochastic process, which can also be tuned to represent periodic correlation functions [8] and not only geometrically decaying ones.
The author’s daily work on wireless ATM networks is supported by: The Federal Ministry of Education, Science, Research, and Technology, Germany
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35353-1_28
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© 2000 IFIP International Federation for Information Processing
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Herrmann, C. (2000). Efficient Computation of Waiting Time Moments for the DBMAP/G/1/N Queue. In: Kouvatsos, D. (eds) Performance Analysis of ATM Networks. ATM 1997. IFIP — The International Federation for Information Processing, vol 29. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35353-1_24
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DOI: https://doi.org/10.1007/978-0-387-35353-1_24
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