Bounded-Variance Network Calculus: Computation of Tight Approximations of End-to-End Delay

Abstract

 Currently, the most advanced framework for stochastic network calculus is the min-plus algebra, providing bounds for the end-to-end delay in networks. The bounds calculated with the min-plus algebra are tight, if compared with previous methods, but we still observe a significant degradation of the tightness of bounds as the number of nodes crossed by flows increases. Moreover, even if the calculations are greatly simplified relatively to previous methods, they are still complicated as numerical optimizations are necessary. In this paper, we propose a novel framework for the approximated calculation of end-to-end delay: the bounded-variance network calculus, by which we provide two important results. Firstly, we obtain an evaluation of end-to-end delay significantly tighter than that offered by the min-plus algebra. Secondly, the calculations needed to compute our approximations of delay are much simpler and we show that in a typical application scenario used to test the accuracy of the frameworks for network calculus, our approximations are obtained in a closed analytical form, as opposed to the numerical bounds of the other methods. These two advantages constitute an important progress in the direction of evolving statistical network calculus into a practical tool for network analysis.

Network-Layer Performance Analysis of Multihop Fading Channels

“…This is used for the analysis of an FSMC model of a Rayleigh fading channel. For computing the bounds for Markovian arrivals, they apply the bounded-variance network calculus introduced in [14], which is an extension of the central limit theorem methods by Choe and Shroff [8] to multihop paths. Verticale [34] has applied the same methodology to constant bit rate traffic.…”

End-to-End Delay Approximation in Cascades of Generalized Processor Sharing Schedulers

“…For each value of H the Figure shows results obtained with min-plus algebra, our analysis, and simulation. The comparison of the numerical values of delays shows that the bounded-variance network calculus [13] (in all cases (a), (b), and (c)) provides a significantly better estimation of the end-to-end delay than the min-plus algebra. For example, in (b) with H=10 and a total of 300 flows per node, the simulation results provide a measure of the end-to-end delay in the range 2.54 ms +/-0.1 ms, the min-plus algebra provides a delay threshold of 31.6 ms (with a percentage deviation of +1,144.1% from the simulation result), while our delay approximation with our analysis is equal to 5.1 ms (with a percentage deviation of +100.8% from the simulation result).…”

“…In order to extend the two-moment analysis to end-to-end paths it is necessary to introduce the concept of bounded variance network calculus, firstly discussed in [13]. One of the hardest problems to be faced by multi-node network analysis is the characterization of the output traffic of a scheduler, an unavoidable task, as the traffic output by a scheduler is the input traffic of the next scheduler.…”

“…The result is obtained by applying the bounded variance network calculus [13] whose basic principles have been outlined in Section II.B. The traffic envelope of the input cross flow at the scheduler h is referred to as ( ) (17) and (19): …”

“…Our approach is alternative, as we construct our method starting from the two-moments analysis of isolated nodes, as developed and formalized by Choe and Shroff [12] and, in particular, we use the Maximum Variance Asymptotic upper bound to calculate the distribution of delay at a single node. We iterate our single-node formulas by adopting the methods developed in the Bounded Variance Network Calculus [13]. In this way, starting from the single-node expression of delay, we are able to calculate an approximated analytical expression of the end-to-end delay.…”