On a reduced load equivalence for fluid queues under subexponentiality

  • Authors:

  • Rajeev Agrawal;Armand M. Makowski;Philippe Nain

  • Affiliations:

  • Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA E-mail: agrawal@engr.wisc.edu;Electrical Engineering Department and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA E-mail: armand@isr.umd.edu;INRIA, B.P. 93, 06902 Sophia Antipolis Cedex, France E-mail: nain@sophia.inria.fr

  • Venue:
  • Queueing Systems: Theory and Applications
  • Year:
  • 1999

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Abstract

We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog W^{A_1+A_2,c} in a buffer fed by a combined fluid process A_1+A_2 and drained at a constant rate c. The fluid process A_1 is an (independent) on–off source with average and peak rates \rho_1 and r_1, respectively, and with distribution G for the activity periods. The fluid process A_2 of average rate \rho_2 is arbitrary but independent of A_1. These bounds are used to identify subexponential distributions G and fairly general fluid processes A_2 such that the asymptotic equivalence \mathbf{P}l[W^{A_1+A_2,c}x \sim \mathbf{P}l[W^{A_1,c-\rho_2}x]\quad (x\to\infty) holds under the stability condition \rho_1+\rho_2 and the non-triviality condition c-\rho_2. In these asymptotics the stationary backlog W^{A_1,c-\rho_2} results from feeding source A_1 into a buffer drained at reduced rate c-\rho_2. This reduced load asymptotic equivalence extends to a larger class of distributions G a result obtained by Jelenkovic and Lazar [19] in the case when G belongs to the class of regular intermediate varying distributions.