Optimal routing for end-to-end guarantees using Network Calculus

Abstract

In this paper we show how Network Calculus can be used to compute the optimal route for a flow (w.r.t. end-to-end guarantees on the delay or the backlog) in a network in the presence of cross-traffic. When cross-traffic is independent, the computation is shown to boild down to a functional shortest path problem. When cross-traffic perturbates the main flow over more than one node, then the "Pay Multiplexing Only Once" phenomenon makes the computation more involved. We provide an efficient algorithm to compute the service curve available for the main flow and show how to adapt the shortest path algorithm in this case.

Performance evaluation and design tradeoffs of on-chip interconnect architectures

“…This service curve is called the rate-latency service curve in which data is delayed by a fixed time T i and then routed out at a rate R i . These two curves are widely used in evaluating systems 31,32,33,34 . We use these curves to evaluate and compare the considered OCIs.…”

QoS multicast tree construction in IP/DWDM optical internet by bio-inspired algorithms

“…Since the all-optical wavelength converter is still in its early development stage and the optoelectronic conversion not only is very expensive but also has limited performance, we assume only partial nodes are equipped with full-range wavelength converter [33] QoS requirement is often flexible [1]. The lower bound and the upper bound of the delay interval are determined by the user and the application.…”

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

“…However at the next nodes, the interference due to the cross-traffic cannot be as severe since the competition for the resource has already been partially resolved at the first node (see [25,27] for a discussion about the PMOO appellation). The PMOO phenomenon can be quantified in the Network Calculus framework: a formula provided in [27] for a small example was generalized in [6,7] into an explicit formula of an end-to-end service curve offered by the whole path to the main flow. This formula is written under the form of a multi-dimensional (min, +) convolution.…”

“…It was claimed in [6] that this new operator, and consequently the end-toend service curve, could be computed in polynomial time. Unfortunately the algorithm had a flaw explained in [7]. The first part of our paper (Section 2) comes back to the properties and the computation of this multi-dimensional convolution under usual assumptions (concave arrival curves and convex service curves): we show that convexity is preserved (Theorem 3) and although the complexity of computing the whole output curve remains an open problem, one can compute end-to-end delay or backlog bounds in polynomial time thanks to linear programming (Theorem 4).…”

“…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.…”