Upper and lower bounds for the multiplexing of multiclass Markovian on/off sources

On a Reduced Load Equivalence under Heavy Tail Assumptions

“…It was shown that source 1 dominates the behavior of the bu er if its on periods have an intermediate regular varying distribution 10 as long as the tail of the on periods of source 1 is heavier in some precise technical sense than that of source 2. A noteworthy byproduct of the results in 6, 18 is that the source 2 contributes to the asymptotic behavior of P W x only through its average uid generation rate 2 . In fact, the following rephrasing of these results was rst pointed out by Jelenkovic and Lazar 18, Thm 10 : Let A 1 and A 2 denote the on-o sources 1 and 2 with average rates 1 and 2 , and peak rates r 1 and r 2 , respectively.…”

“…The combined or multiplexed uid process A 1 +A 2 is o ered to a bu er which is drained at rate c uid units sec under the stability condition 1 + 2 c . Under the non-triviality condition c , 2 2 results from feeding source A 1 into a bu er drained at the reduced rate c , 2 . This asymptotic equivalence 1.2 re ects the following intuitive notion: The tail of Pareto distributed activity periodsof source 1 is considerably heavier than the exponential tails governing the exponential on-o source 2.…”

“…As a result, a single on period for source 1 is likely to correspond to a large numberof successive on and o periodsin source 2. Such a disparity in time scales is enough for the Law of Large Numbers to kick in for source 2, e ectively averaging out random uctuations about the mean 2 and replacing them by the average behavior of the source 2. It is now a small step to believe in the plausibility of 1.2.…”

“…The reduced load asymptotic equivalence 1.2 suggests a natural way of approximating the distribution of W A 1 +A 2 ;c with that of W A 1 ;c, 2 . As this latter quantity is associated with the single source A 1 , a reduction in computational e orts may result, at least in principle.…”

“…As this latter quantity is associated with the single source A 1 , a reduction in computational e orts may result, at least in principle. For instance, when applicable, 1.1 and 1.2 together imply for appropriate constants K 1 ; 1 0 determined by the statistics of A 1 and by the release rate c , 2 . Given its asymptotic basis, this approximation will become increasingly accurate with x large, a property that might make it well suited in various contexts for evaluating very small cell loss probabilities via bu er over ow probabilities.…”