On a Reduced Load Equivalence under Heavy Tail Assumptions

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Agrawal, Rakesh
Makowski, Armand M.
Nain, P.
We propose a general framework for obtaining asymptotic distributional bounds on the stationary backlog <I> W<SUP>A<SUB>1</SUB>+A<SUB>2</SUB>,c</SUP></I> in a buffer fed by a combined fluid process <I>A<SUB>1</SUB>+A<SUB>2</SUB></I> and drained at a constant rate <I>c</I>.<P>The fluid process <I>A<SUB>1</SUB></I> is an (independent) on-off source with average and peak rates <I> <FONT FACE="Symbol">r</FONT><SUB>1</SUB></I> and <I>r<SUB>1</SUB></I>, respectively, and with distribution <I>G</I> for the activity periods. The fluid process <I>A<SUB>2</SUB></I> of average rate <I><font face="Symbol">r</font><SUB>2</SUB></I> is arbitrary but independent of <I>A<SUB>1</SUB></I>.<P>These bounds are used to identify subexponential distributions <I>G</I> and fairly general fluid processes <I>A<SUB>2</SUB></I> such that the asymptotic equivalence <I><B>P</B>[W<SUP>A<SUB>1</SUB>+A<SUB>2</SUB>,c</SUP> > x]~<B>P</B>[W<SUP>A<SUB>1</SUB>,c-<font face="Symbol">r</font><SUB>2</SUB></SUP> > x](x<font face="Symbol">לּ/font><font face="Symbol">/font>)</I> holds under the stability condition <I><font face="Symbol">r</font><SUB>1</SUB>+<font face="Symbol">r</font><SUB>2</SUB> < c</I> and under the non-triviality condition <I>c-<font face="Symbol">r</font><SUB>2</SUB> < r<SUB>1</SUB></I>.<P>The stationary backlog <I>W<SUP>A<SUB>1</SUB>,c-<font face="Symbol">r</font><SUB>2</SUB></SUP></I>in these asymptotics results from feeding source <I>A<SUB>1</SUB></I> into a buffer drained at <I>reduced</I> rate <I>c-<font face="Symbol">r</font><SUB>2</SUB></I>. This reduced load asymptotic equivalence extends to a larger class of distributions <I>G</I> a result obtained by Jelenkovic and Lazar [18] in thecase when <I>G</I> belongs to the class of regular intermediatevarying distributions.<P><I>The equations in this abstract will not display properly unless you have the symbol font installed and your browser supports superscripts and subscripts. Otherwise, you will need to download the paper to see the equations properly.</I>