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The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times

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  • George Luchak

    (Canadian Defence Research Board, Suffield Experimental Station, Ralston, Alberta, Canada)

Abstract

The single-channel queuing equations considered in this paper are characterized by a Poisson-distributed arrival rate and a general class of holding-time distributions. General solutions of the equations are obtained for the case in which the traffic intensity i is a continuous function of time and possesses continuous derivatives of all orders. The following particular cases are considered in detail (a) i constant and the holding-time distribution Pearson type-III (in this case the general solution is obtained in closed form in terms of a newly introduced function I n k ( z ), many of the properties of which are derived in the Appendix), (b) i directly proportional to time and the holding time exponentially distributed.

Suggested Citation

  • George Luchak, 1956. "The Solution of the Single-Channel Queuing Equations Characterized by a Time-Dependent Poisson-Distributed Arrival Rate and a General Class of Holding Times," Operations Research, INFORMS, vol. 4(6), pages 711-732, December.
    • Handle: RePEc:inm:oropre:v:4:y:1956:i:6:p:711-732
    DOI: 10.1287/opre.4.6.711
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    Cited by:

    1. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
    2. Ascione, Giacomo & Leonenko, Nikolai & Pirozzi, Enrica, 2020. "Fractional Erlang queues," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3249-3276.
    3. J. D. Griffiths & G. M. Leonenko & J. E. Williams, 2008. "Approximation to the Transient Solution of the M/E k /1 Queue," INFORMS Journal on Computing, INFORMS, vol. 20(4), pages 510-515, November.
    4. K. Murari, 1972. "A queueing problem with correlated arrivals and correlated phase-type service zur theorie des rangtests," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 18(1), pages 110-119, December.
    5. Leonenko, G.M., 2009. "A new formula for the transient solution of the Erlang queueing model," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 400-406, February.

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