Computationally Efficient Worst-Case Analysis of Flow-Controlled Networks With Network Calculus

Abstract

Networks with hop-by-hop flow control occur in several contexts, from data centers to systems architectures (e.g., wormhole-routing networks on chip). A worst-case endto-end delay in such networks can be computed using Network Calculus (NC), an algebraic theory where traffic and service guarantees are represented as curves in a Cartesian plane. NC uses transformation operations, e.g., the min-plus convolution, to model how the traffic profile changes with the traversal of network nodes. NC allows one to model flow-controlled systems, hence one can compute the end-to-end service curve describing the minimum service guaranteed to a flow traversing a tandem of flow-controlled nodes. However, while the algebraic expression of such an end-to-end service curve is quite compact, its computation is often intractable from an algorithmic standpoint: data structures tend to grow quickly to unfeasibly large sizes, making operations intractable, even with as few as three hops.In this paper, we propose computational and algebraic techniques to mitigate the above problem. We show that existing techniques (such as reduction to compact domains) cannot be used in this case, and propose an arsenal of solutions, which include methods to mitigate the data representation space explosion as well as computationally efficient algorithms for the minplus convolution operation. We show that our solutions allow a significant speedup, enable analysis of previously unfeasible case studies, and -since they do not rely on any approximation -still provide exact results.

Extending the Network Calculus Algorithmic Toolbox for Ultimately Pseudo-Periodic Functions: Pseudo-Inverse and Composition

“…This is an instance of a general issue encountered with algorithms for UPP curves -also discussed in [30]. Generic algorithms, that make no assumptions on the shape of the operands (such as the ones presented here for the pseudo-inverse), may in general yield non-minimal representations of the result.…”

“…Note that, given R f , one can compute f (t) for all t ≥ 0, and also S D f for any interval D. Furthermore, being finite, R f can be used as a data structure to represent f in code. As is discussed in depth in [30], R f is not unique, and using a non-minimal representation of f can affect the efficiency of the computations. Given a sequence S, let n(S) be its cardinality.…”

“…Unlike UPP, the class of UA functions is not closed with respect to NC operations. For instance, [30] shows that flow-controlled networks with ratelatency (hence UA) service curves yield closed-loop service curves that are UPP, but not necessarily UA again. Moreover, in many cases, the service curves of individual flows served by Round-Robin schedulers are UPP, but not UA either (see, e.g., [24,25,31]).…”

“…However, addressing the issue of representation minimization a priori when implementing NC operators is too hard (if doable at all), since one would need to make a comprehensive list of subcases, and, of course, as many formal proofs of correctness. It is instead considerably more efficient to devise a generic algorithm for an operator, neglecting minimization, and then use a simple algorithm a posteriori that minimizes the representation of the result -see [30].…”

“…Nancy is a C# library implementing the UPP model and its operators, as described in [19,20]. Moreover, it implements state-of-theart algorithms that improve the efficiency of NC operators, described in [30], and lower pseudo-inverse, upper pseudo-inverse and composition operators, described in this paper. Nancy makes extensive use of parallelism.…”