Searching for Tight Performance Bounds in Feed-Forward Networks

Computing tight performance bounds in feed-forward networks under general assumptions about arrival and server models has turned out to be a challenging problem. Recently it was even shown to be NP-hard [1]. We now address this problem in a heuristic fashion, building on a procedure for computing provably tight bounds under simple traffic and server models. We use a decomposition of a complex problem with more general traffic and server models into a set of simpler problems with simple traffic and server models. This set of problems can become prohibitively large, and we therefore resort to heuristic methods such as Monte Carlo. This shows interesting tradeoffs between performance bound quality and computational effort. 1 Motivation and Related Work When designing or analyzing a network, one of the most important aspects is its performance under various load conditions. A number of methods for that kind of analysis have been devised, among them network calculus, which describes methods for calculating performance bounds, i.e., describing worst-case behavior. Network calculus is a (min,+) system theory for deterministic queuing systems which builds on the calculus for network delay in [2, 3]. The important service curve concept was introduced in [4–8] to perform efficient analysis of tandem queues. Scaling properties in the number of traversed network nodes are linear, as is shown in [9], a phenomenon also known as pay bursts only once phenomenon [10]. Detailed descriptions of the (min,+) algebra and of network calculus can be found in [11] and [10, 12]. Network calculus has found numerous applications, most prominently in the Internet’s Quality of Service (QoS) proposals IntServ and DiffServ [13, 14], but it has also become a valuable method in other fields, such as wireless sensor networks [15, 16], switched Ethernets [17], Systems-on-Chip (SoC) [18], or even to speed-up simulations [19]. However, as a relatively young theory, compared to, e.g., traditional queueing theory, there is also a number of challenges network calculus still has to master. A very tough challenge is found in the treatment of non-tandem topologies with aggregate multiplexing of multiple flows. While this has been addressed from the beginning [3], there are still many open issues. For aggregate multiplexing in general network topologies there is a very fundamental issue about the circumstances under which a finite delay bound exists at all [20, 21]. In [22] a sufficient condition for stability in general network topologies and an explicit delay bound are given. Extensions of this approach are provided in [23, 24]. Yet, for larger networks this severely limits the utilization of the network since the maximum allowable utilization is inversely proportional to the network diameter. The problems in the analysis of general topologies arise from cyclic dependencies between flows and the resulting difficulties in bounding their network-internal burstiness. A special class of topologies which avoids those problems are feedforward networks, which are known to be stable for all utilizations ≤ 1 [3]. In this paper, we focus on this class of networks. While many networks are obviously not feed-forward, many important instances like switched networks, wireless sensor networks, or MPLS networks with multipoint-to-point label switched paths are, or can be made, feed-forward by using, e.g., the turn-prohibition algorithm [25]. In feed-forward networks, there has been some work on aggregate multiplexing recently: [26] treats the case of feed-forward networks under FIFO multiplexing for token-bucket constrained flows and rate-latency servers, showing that the derived left-over service curve for a flow of interest is again of the rate-latency type with minimally possible latency. [27] shows that this does not result in a tight delay bound, and derives tight delay bounds under knowledge about the arrival curve of the flow of interest for the special case of sink-trees and, again, under token bucket constrained flows and rate-latency servers. Another work [28] also investigates sink-tree networks, but now under dual token-bucket constrained flows and constant rate servers, for which delay bounds are derived by summing per-node bounds, which unsurprisingly does not yield tight bounds but is still reported as being close under practical conditions. Besides being very specific with respect to traffic and server models, all of the above work assumes FIFO aggregate multiplexing. However in practice, as argued in [29], many devices cannot be accurately described by FIFO because packets arriving at the output queue from different input ports may experience different delays when traversing a node. This is due to the fact that many networking devices like routers are implemented using input-output buffered crossbars and/or multistage interconnections between input and output ports. Hence, packet reordering on the aggregate level is a frequent event (unlike on the flow level) and should not be neglected in modelling. Therefore, in this work we drop the FIFO multiplexing assumption and make essentially no assumptions on the way aggregates are multiplexed at servers, i.e. we assume arbitrary multiplexing also known as general or blind multiplexing [2, 10]. On the level of a single flow, however, we still assume FIFO. This assumption is sometimes called FIFO-per-microflow [30] or locally FCFS multiplexing [2]. Work on bounds for networks with arbitrary multiplexing has become frequent only recently, but there are already several important results. Some older results are reported in [10] (see Section 2), and there is some work on the burstiness increase due to arbitrary multiplexing at a single node [31]. Adversarial queueing theory [32] provides results for general networks, however it is more concerned with network stability than with the determination of performance bounds. In previous work related to network calculus tool support, we have proposed and implemented a number of network calculus analysis methods for arbitrary multiplexing in feed-forward networks [33], but as will be demonstrated here, they were not the ultimate solution. A similar approach has been taken in [34], regarding a wider class of traffic and service specifications. The goal of our work is to search for tight delay bounds in feed-forward networks of arbitrary multiplexers. With respect to traffic and server models we address a more general case than previous work on FIFO multiplexing, in particular we assume piecewise linear concave arrival curves and convex service curves, which encompass the majority of practical traffic and server models. Compared to our previous work in [35], we now try to solve an issue that arises from the algebra used in network calculus, which, while allowing for an easy analysis, hides certain properties, and may lead to pessimistic bounds. After a short introduction to network calculus, we present an approach to network analysis based on an optimization problem, and show how a solution to that problem can be approximated by heuristics. We show how the quality of the performance bounds obtained by that new method compares with traditional results. 2 Network Calculus Background As network calculus is built around the notion of cumulative functions for input and output flows of data, the set of real-valued, non-negative, and wide-sense increasing functions passing through the origin plays a major role: F = {f : R → R |∀t ≥ s : f (t) ≥ f (s) , f (0) = 0} In particular, the input function F (t) and the output function F (t), which cumulatively count the number of bits that are input to, respectively output from, a system S, are in F . Throughout the paper, we assume inand output functions to be continuous in time and space. Note that this is not a general limitation as there exist transformations between discrete and continuous time models [10]. Definition 1. (Min-plus Convolution and Deconvolution) The min-plus convolution ⊗ and deconvolution ⊘ of two functions f, g ∈ F are defined as (f ⊗ g) (t) = inf 0≤s≤t {f(t− s) + g(s)} (f ⊘ g) (t) = sup u≥0 {f(t+ u)− g(u)} It can be shown that the triple (F ,∧,⊗), where ∧ denotes the pointwise minimum operator, constitutes a dioid [10]. Also, the min-plus convolution is a linear operator on the dioid (R ∪ {+∞},∧,+), whereas the min-plus deconvolution is not. These algebraic characteristics result in a number of rules that apply to those operators, many of which can be found in [10, 12]. Let us now turn to the performance characteristics of flows which can be bounded by network calculus means: Definition 2. (Backlog and Delay) Assume a flow with input function F that traverses a system S resulting in the output function F . The backlog of the flow at time t is defined as x(t) = F (t)− F (t) Assuming FIFO delivery, the virtual delay for a bit input at time t is defined as d(t) = inf {τ ≥ 0 : F (t) ≤ F (t+ τ)} Next, the arrival and departure processes specified by input and output functions are bounded based on the central network calculus concepts of arrival and service curves: Definition 3. (Arrival Curve) Given a flow with input function F a function α ∈ F is an arrival curve for F iff ∀t, s ≥ 0, s ≤ t : F (t)− F (t− s) ≤ α(s) ⇔ F ≤ F ⊗ α A typical example of an arrival curve is given by an affine arrival curve γr,b (t) = b + rt, t > 0 and γr,b (t) = 0, t ≤ 0 which corresponds to token-bucket traffic regulation. Definition 4. (Service Curve) If the service provided by a system S for a given input function F results in an output function F ′ we say that S offers a service curve β iff F ′ ≥ F ⊗ β A typical example of a service curve is given by a so-called rate-latency function βR,T (t) = R [t− T ] + , where [x] + := x∨0, and ∨ denotes the maximum operator. A number of systems fulfill, however, a stricter definition of the service curve [10], which is particularly useful as it permits certain derivations that are not feasible under the more general minimum service curve model. Definition 5. (Strict Service Curve) Let β ∈ F . System S offers a strict service curve β to a flow if during any backlogged period of duration u, the output of the flow is at least equal to β(u). Note that any strict service curve is also a service curve, but not the other way around. Many schedulers offer strict service curves, for example most of the generalized processor sharing-emulating schedulers offer a strict service curve of the rate-latency type. Strict service curves will play a crucial role in this paper, since they, in contrast to service curves, allow to bound the maximum backlogged period of a system. More specifically, that bound d̄ is given as the non-zero intersection point between arrival and service curve, i.e. α (