Almut Burchard

New strategies for assigning real-time tasks to multiprocessor systems

Optimal scheduling of real-time tasks on multiprocessor systems is known to be computationally intractable for large task sets. Any practical scheduling algorithm for assigning real-time tasks to a multiprocessor system presents a trade-off between its computational complexity and its performance. In this study, new schedulability conditions are presented for homogeneous multiprocessor systems where individual processors execute the rate-monotonic scheduling algorithm. The conditions are used to develop new strategies for assigning real-time tasks to processors. The performance of the new strategies is shown to be significantly better than suggested by the existing literature. Under the realistic assumption that the load of each real-time task is small compared to the processing speed of each processor, it is shown that the processors can be almost fully utilized.

A network calculus with effective bandwidth

This paper establishes a link between two principal tools for the analysis of network traffic, namely, effective bandwidth and network calculus. It is shown that a general version of effective bandwidth can be expressed within the framework of a probabilistic version of the network calculus, where both arrivals and service are specified in terms of probabilistic bounds. By formulating well-known effective bandwidth expressions in terms of probabilistic envelope functions, the developed network calculus can be applied to a wide range of traffic types, including traffic that has self-similar characteristics. As applications, probabilistic lower bounds are presented on the service given by three different scheduling algorithms: static priority, earliest deadline first, and generalized processor sharing. Numerical examples show the impact of specific traffic models and scheduling algorithms on the multiplexing gain in a network.

Statistical service assurances for traffic scheduling algorithms

Network services for the most demanding advanced networked applications which require absolute, per-flow service assurances can be deterministic or statistical. By exploiting the statistical properties of traffic, statistical assurances can extract more capacity from a network than deterministic assurances. We consider statistical service assurances for traffic scheduling algorithms. We present functions, so-called effective envelopes, which are, with high certainty, upper bounds of multiplexed traffic. Effective envelopes can be used to obtain bounds on the amount of traffic on a link that can be provisioned with statistical service assurances. We show that our bounds can be applied to a variety of traffic scheduling algorithms. In fact, one can reuse existing admission control functions for scheduling algorithms with deterministic assurances. We present numerical examples which compare the number of flows with statistical assurances that can be admitted with our effective envelope approach to those achieved with existing methods.

Holder Regularity and Dimension Bounds for Random Curves

Random systems of curves exhibiting fluctuating features on arbitrarily small scales (δ) are often encountered in critical models. For such systems it is shown that scale-invariant bounds on the probabilities of crossing events imply that typically all the realized curves admit Holder continuous parametrizations with a common exponent and a common random prefactor, which in the scaling limit (δ → 0) remains stochastically bounded. The regularity is used for the construction of scaling limits, formulated in terms of probability measures on the space of closed sets of curves. Under the hypotheses presented here the limiting measures are supported on sets of curves which are Holder continuous but not rectifiable, and have Hausdorff dimensions strictly greater than one. The hypotheses are known to be satisfied in certain two dimensional percolation models. Other potential applications are also mentioned.

A network service curve approach for the stochastic analysis of networks

The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical end-to-end delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that end-to-end performance measures computed with a network service curve are bounded by O(Hlog H), where H is the number of nodes traversed by a flow. Using currently available techniques that compute end-to-end bounds by adding single node results, the corresponding performance measures are bounded by O(H3).

A Min-Plus Calculus for End-to-End Statistical Service Guarantees

The network calculus offers an elegant framework for determining worst-case bounds on delay and backlog in a network. This paper extends the network calculus to a probabilistic framework with statistical service guarantees. The notion of a statistical service curve is presented as a probabilistic bound on the service received by an individual flow or an aggregate of flows. The problem of concatenating per-node statistical service curves to form an end-to-end (network) statistical service curve is explored. Two solution approaches are presented that can each yield statistical network service curves. The first approach requires the availability of time scale bounds at which arrivals and departures at each node are correlated. The second approach considers a service curve that describes service over time intervals. Although the latter description of service is less general, it is argued that many practically relevant service curves may be compliant to this description

Scaling properties of statistical end-to-end bounds in the network calculus

The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical end-to-end delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that end-to-end performance measures computed with a network service curve are bounded by /spl Oscr/(H log H), where H is the number of nodes traversed by a flow. Using currently available techniques, which compute end-to-end bounds by adding single node results, the corresponding performance measures are bounded by /spl Oscr/(H/sup 3/).

Confocal FRET Microscopy to Measure Clustering of Ligand-Receptor Complexes in Endocytic Membranes

The dynamics of protein distribution in endocytic membranes are relevant for many cellular processes, such as protein sorting, organelle and membrane microdomain biogenesis, protein-protein interactions, receptor function, and signal transduction. We have developed an assay based on Fluorescence Resonance Energy Microscopy (FRET) and novel mathematical models to differentiate between clustered and random distributions of fluorophore-bound molecules on the basis of the dependence of FRET intensity on donor and acceptor concentrations. The models are tailored to extended clusters, which may be tightly packed, and account for geometric exclusion effects between membrane-bound proteins. Two main criteria are used to show that labeled polymeric IgA-ligand-receptor complexes are organized in clusters within apical endocytic membranes of polarized MDCK cells: 1), energy transfer efficiency (E%) levels are independent of acceptor levels; and 2), with increasing unquenched donor: acceptor ratio, E% decreases. A quantitative analysis of cluster density indicates that a donor-labeled ligand-receptor complex should have 2.5-3 labeled complexes in its immediate neighborhood and that clustering may occur at a limited number of discrete membrane locations and/or require a specific protein that can be saturated. Here, we present a new sensitive FRET-based method to quantify the co-localization and distribution of ligand-receptor complexes in apical endocytic membranes of polarized cells.

Cases of equality in the riesz rearrangement inequality

We determine the cases of equality in the Riesz rearrangement inequality ZZ f (y)g(x ? y)h(x) dydx ZZ f (y)g (x ? y)h (x) dydx where f , g , and h are the spherically decreasing rearrangements of the functions f , g, and h on R n. We apply our results to the weak Young inequality. The Riesz rearrangement inequality states that the functional I(f; g; h) := Z f gh dx = ZZ f(y)g(x?y)h(x) dydx (1:1) never decreases under spherical rearrangement, that is, 1 for any triple (f; g; h) of nonnegative measurable functions on R n for which the right hand side is deened. The spherically decreasing rearrangement, f , of a nonnegative measurable function f is the spherically decreasing function equimeasurable to f. We will deene it by f (x) = sup n s > 0 j (N s (f)) ! n jxj n o ; where N s (f) := n x 2 R n j f(x) > s o is the level set of f at height s, and ! n denotes the measure of the unit ball in R n. That is, the level sets of f are the centered balls of equal measure as the corresponding level sets of f. This deenition makes sense if all level sets corresponding to positive values of f have nite measure, for example, if f is in L p for some p < 1. In this paper, we determine the cases of equality in (1.2). A triple of functions that satisses (1.2) with equality will be called an optimizing triple, or optimizer, of the inequality. There are many optimizers of (1.2). One reason is that I is invariant under a large group of aane transformations: For any linear map, L, of determinant 1, and vectors a, b, and c = a + b in R n , we have where g ? denotes the function deened by g ? (x) := g(?x). Clearly, any triple of functions that is equivalent to a triple of spherically decreasing functions under these symmetries is an optimizer. There is a second reason to expect many optimizers. Consider the case when f and g have compact support. Then also the convolution f g has compact support. If h is the characteristic function of a set that contains the support of f g, then f; g; h produce equality in (1.2) regardless …

Delay Bounds in Communication Networks With Heavy-Tailed and Self-Similar Traffic

Traffic with self-similar and heavy-tailed characteristics has been widely reported in communication networks, yet, the state-of-the-art of analytically predicting the delay performance of such networks is lacking. This work addresses heavy-tailed traffic that has a finite first moment, but no second moment, and presents end-to-end delay bounds for such traffic. The derived performance bounds are non-asymptotic in that they do not assume a steady state, large buffer, or many sources regime. The analysis follows a network calculus approach where traffic is characterized by envelope functions and service is described by service curves. The system model is a multi-hop path of fixed-capacity links with heavy-tailed self-similar cross traffic at each node. A key contribution of the paper is a probabilistic sample-path bound for heavy-tailed arrival and service processes, which is based on a scale-free sampling method. The paper explores how delay bounds scale as a function of the length of the path, and compares them with lower bounds. A comparison with simulations illustrates pitfalls when simulating self-similar heavy-tailed traffic, providing further evidence for the need of analytical bounds.