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2 results

  • Rate conservation for stationary processes

  • Josep M. Ferrandiz, Aurel A. Lazar
  • Journal:
    Journal of Applied Probability / Volume 28 / Issue 1 / March 1991
    Published online by Cambridge University Press:
    14 July 2016, pp. 146-158
    Print publication:
    March 1991

  • We derive a rate conservation law for distribution densities which extends a result of Brill and Posner. Based on this conservation law, we obtain a generalized Takács equation for the G/G/m/B queueing system that only requires the existence of a stochastic intensity for the arrival process and the residual service time distribution density for the G/GI/1/B queue. Finally, we solve Takács' equation for the N/GI/1/∞ queueing system.

  • On a single-server queue with group arrivals

  • J. W. Cohen
  • Journal:
    Journal of Applied Probability / Volume 13 / Issue 3 / September 1976
    Published online by Cambridge University Press:
    14 July 2016, pp. 619-622
    Print publication:
    September 1976

  • The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.