Computation of a (min,+) multi-dimensional convolution for end-to-end performance analysis.

Abstract

Network Calculus is an attractive theory to derive deterministic bounds on end-to-end performance measures. Nevertheless bounding tightly and quickly the worst-case delay or backlog of a flow over a path with cross-traffic remains a challenging problem. This paper carries on with the study of configurations where a main flow encounters some cross-traffic flows which interfere over connected sub-paths. We also assume that no information is available about scheduling policies at the nodes (blind multiplexing). Such configurations were first analyzed in [25,27] where a "Pay Multiplexing Only Once" (PMOO) phenomenon was identified, and then in [6,7] where a (min, +) multi-dimensional operator was introduced to compute a minimum service curve for the whole path. Under usual assumptions (concave arrival curves and convex service curves), we prove some properties of this new operator and we show how to use it to derive bounds on delays and backlogs in polynomial time.We also discuss the simpler case when there is no crosstraffic. Then the analysis is known to boil down to the (min, +) convolution of all the service curves over the path. For convex and piecewise affine service curves, a specific theorem enables to compute efficiently the convolution. This theorem has been used by several authors [6,8,17,21,22,25,27], but they all refer to a proof which is unfortunately incomplete [5]. To set definitely this theorem, we provide three different proofs. We also investigate the complexity of computing performances bounds in this case.

Fast weak–KAM integrators for separable Hamiltonian systems

“…A direct consequence of this lemma is Theorem 4.2, stated in [LBT01]. A complete proof is presented in [BJT08], but can also be deduced from previous works about the Legendre transform: the Legendre transform is an involution on the set of convex functions and can be computed in linear time (see [Luc97] for example). Moreover, the transform of the (min,plus)-convolution of two functions is the addition of the respective transforms of the two functions, inducing an alternative linear-time algorithm.…”

Quality and Cost of Deterministic Network Calculus - Design and Evaluation of an Accurate and Fast Analysis

“…4a. Other DNC tandem analyses implementing the PMOO principle for arbitrary multiplexing servers, namely (min,+) multi-dimensional convolution [11,12] and OBA [38,31], also require segregate bounding of cross-traffic and thus cause the same PSOO violation.…”

Container of (min, +)-linear systems

“…The exact computation of sum, inf-convolution and subadditive closure for ultimately pseudo-periodic and nondecreasing functions of F cp can be really time and memory consuming (see for instance Cottenceau et al 1998Cottenceau et al -2006Gaubert 1992;Bouillard and Thierry 2008). The main objective of this work is to get some efficient algorithms to handle these functions.…”

“…The main operations of the (min,+) algebra such as the sum and the inf-convolution 2 are available in MinMaxGD, COINC, DISCO and RTC, whereas the operation of subadditive closure 3 is only available with MinMaxGD and COINC. Moreover, for MinMaxGD and COINC, the algorithms of these operations are described in Gaubert (1992) and Cottenceau (1999) for the former, and in Bouillard and Thierry (2008) for the latter. In these toolboxes, the complexity of sum and inf-convolution operations is linear or quasi-linear, whereas the one for subadditive closure tends to be polynomial.…”