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., wormholerouting networks on chip). A worst-case end-to-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 untractable from an algorithmic standpoint: data structures tend to explode, making operations unfeasibly complex, 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 min-plus 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.

Nancy: an efficient parallel Network Calculus library

“…Others, such as the sub-additive closure, are exponential. This holds for both the number of elementary operations involved and the number of linear pieces of the result, and implies that chaining a number of NC operators may yield expressions that -while visually and algebraically neat -are algorithmically unfeasible, e.g., take days to compute, or may exhaust the hardware resources and not compute at all [3]. This constrains -sometimes severely -the practical uses of NC analysis, and calls for efficient software libraries that implement NC operations.…”

“…We remark that there are several common situations where UPP curves arise spontaneously: work [8] shows that this is the case whenever packetization and finite transmission speeds are factored in. Moreover, flow-controlled networks [3], currently being envisaged for data centers, have UPP service curves, and so do wormhole-routing networks [9] used in systems architectures.…”

“…This allows it to reap a considerable speedup over a single-thread implementation. Furthermore, it implements smart algorithms for NC operators working on particular sub-classes of UPP functions (e.g., sub-additive ones), as well as for minimizing the representation of functions, which concur to abate the computation times by orders of magnitude, as shown in [3]. Nancy can be used to implement existing methods of analysis (e.g., Total Flow Analysis or Single Flow Analysis, [14]) used in other tools, without the restrictive hypotheses on the type of input curves.…”

“…Nancy can be used to implement existing methods of analysis (e.g., Total Flow Analysis or Single Flow Analysis, [14]) used in other tools, without the restrictive hypotheses on the type of input curves. Moreover, it can be exploited to devise new methods of analysis, possibly for types of networks that were considered computationally untractable [3]. Last, but not least, it is designed so that a NC researcher will find it very easy to translate pen-and-paper computations to code: the names used for objects are those which are widespread in the research community, and the syntax is very intuitive.…”

“…As discussed in [3], the representation of the result of the above operations is often (unavoidably) much larger than necessary -i.e., it includes many more segments than required. This may lead to severe memory/speed inefficiency, especially with chained operations (recall that NC operators often involve algorithms that are superlinear in the number of segments).…”