Delay moments for FIFO GI/GI/s queues

  • Authors:

  • Alan Scheller-Wolf;Karl Sigman

  • Affiliations:

  • Graduate School of Industrial Administration, Carnegie Mellon;Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 10027-6699, USA

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 1997

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Abstract

For stable FIFO GI/GI/s queues, s \geq 2, we show that finite (k+1)st moment of service time, S, is not in general necessary for finite kth moment of steady-state customer delay, D, thus weakening some classical conditions of Kiefer and Wolfowitz (1956). Further, we demonstrate that the conditions required for \mathrm{E}[D^k] are closely related to the magnitude of traffic intensity \rho (defined to be the ratio of the expected service time to the expected interarrival time). In particular, if \rho is less than the integer part of s/2, then \mathrm{E}[D] if \mathrm{E}[S^{3/2}], and \mathrm{E}[D^k] if \mathrm{E}[S^k]. On the other hand, if s-1 , then \mathrm{E}[D^k] if and only if \mathrm{E}[S^{k+1}]. Our method of proof involves three key elements: a novel recursion for delay which reduces the problem to that of a reflected random walk with dependent increments, a new theorem for proving the existence of finite moments of the steady-state distribution of reflected random walks with stationary increments, and use of the classic Kiefer and Wolfowitz conditions.