arXiv subjects

Jean-Yves Le Boudec

Publications and source records attributed to Jean-Yves Le Boudec.

At least 19 recordsLinked to original sources

arXiv2026

An Elementary Proof of the Dvoretzky--Kiefer--Wolfowitz--Massart Inequality

The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives an upper bound on the probability that the empirical distribution function of a finite sequence of independent random variables deviates from its theoretical value. It is widely used in statistics in tests and in the production of confidence intervals. The original proof by Massart, later reformulated by Dudley, is very long and technical. Recently, Reeve slightly extends Dvoretzky--Kiefer--Wolfowitz--Massart inequality and presents an alternative, much shorter proof, that uses a continuous time martingale. We present a simpler and less technical proof of the original Dvoretzky--Kiefer--Wolfowitz--Massart inequality, based on a discrete-time martingale and Sion's minimax theorem.

math.PR↗

arXiv2024

Saihu: A Common Interface of Worst-Case Delay Analysis Tools for Time-Sensitive Networks

Time-sensitive networks, as in the context of IEEE-TSN and IETF-Detnet, require bounds on worst-case delays. Various network analysis tools compute such bounds; these tools are based on different methods and provide delay bounds that are all valid but may differ; furthermore, it is generally not known which tool will provide the best bound. To obtain the best possible bound, users need to implement multiple pieces of code with a different syntax for every tool, which is impractical and error-prone. To address this issue, we present Saihu, a Python interface that integrates the three most frequently used worst-case network analysis tools: xTFA, DiscoDNC, and Panco. They altogether implement six analysis methods. Saihu provides a general interface that enables defining a network in a single file and executing all tools simultaneously without any modification. Saihu further exports analysis results as formatted reports automatically and allows quick generation of certain types of networks. With its simplified steps of execution, Saihu reduces the burden on users and makes it accessible for anyone working with time-sensitive networks.

cs.NI↗

arXiv2023

Computation of Ultra-Short-Term Prediction Intervals of the Power Prosumption in Active Distribution Networks

Microgrids and, in general, active distribution networks require ultra-short-term prediction, i.e., for sub-second time scales, for specific control decisions. Conventional forecasting methodologies are not effective at such time scales. To address this issue, we propose a non-parametric method for computing ultra short-term prediction intervals (PIs) of the power prosumption of generic electrical-distribution networks. The method groups historical observations into clusters according to the values of influential variables. It is applied either to the original or to the differentiated power-prosumption time series. The clusters are considered statistically representative pools of future realizations of power prosumption (or its derivative). They are used to determine empirical PDFs and, by extracting the quantiles, to deliver PIs for respective arbitrary confidence levels. The models are validated a posteriori by carrying out a performance analysis that uses experimentally observed power-prosumption for different building types, thus allowing the identification of the dominant model.

eess.SY↗

arXiv2023

Worst-case Delay Analysis of Time-Sensitive Networks with Deficit Round-Robin

In feed-forward time-sensitive networks with Deficit Round-Robin (DRR), worst-case delay bounds were obtained by combining Total Flow Analysis (TFA) with the strict service curve characterization of DRR by Tabatabaee et al. The latter is the best-known single server analysis of DRR, however the former is dominated by Polynomial-size Linear Programming (PLP), which improves the TFA bounds and stability region, but was never applied to DRR networks. We first perform the necessary adaptation of PLP to DRR by computing burstiness bounds per-class and per-output aggregate and by enabling PLP to support non-convex service curves. Second, we extend the methodology to support networks with cyclic dependencies: This raises further dependency loops, as, on one hand, DRR strict service curves rely on traffic characteristics inside the network, which comes as output of the network analysis, and on the other hand, TFA or PLP requires prior knowledge of the DRR service curves. This can be solved by iterative methods, however PLP itself requires making cuts, which imposes other levels of iteration, and it is not clear how to combine them. We propose a generic method, called PLP-DRR, for combining all the iterations sequentially or in parallel. We show that the obtained bounds are always valid even before convergence; furthermore, at convergence, the bounds are the same regardless of how the iterations are combined. This provides the best-known worst-case bounds for time-sensitive networks, with general topology, with DRR. We apply the method to an industrial network, where we find significant improvements compared to the state-of-the-art.

cs.NI↗

arXiv2023

Quasi-Deterministic Burstiness Bound for Aggregate of Independent, Periodic Flows

Time-sensitive networks require timely and accurate monitoring of the status of the network. To achieve this, many devices send packets periodically, which are then aggregated and forwarded to the controller. Bounding the aggregate burstiness of the traffic is then crucial for effective resource management. In this paper, we are interested in bounding this aggregate burstiness for independent and periodic flows. A deterministic bound is tight only when flows are perfectly synchronized, which is highly unlikely in practice and would be overly pessimistic. We compute the probability that the aggregate burstiness exceeds some value. When all flows have the same period and packet size, we obtain a closed-form bound using the Dvoretzky-Kiefer-Wolfowitz inequality. In the heterogeneous case, we group flows and combine the bounds obtained for each group using the convolution bound. Our bounds are numerically close to simulations and thus fairly tight. The resulting aggregate burstiness estimated for a non-zero violation probability is considerably smaller than the deterministic one: it grows in $\sqrt{n\log{n}}$, instead of $n$, where $n$ is the number of flows.

cs.NI↗

arXiv2023

Network-Calculus Service Curves of the Interleaved Regulator

The interleaved regulator (implemented by IEEE TSN Asynchronous Traffic Shaping) is used in time-sensitive networks for reshaping the flows with per-flow contracts. When applied to an aggregate of flows that come from a FIFO system, an interleaved regulator that reshapes the flows with their initial contracts does not increase the worst-case delay of the aggregate. This shaping-for-free property supports the computation of end-to-end latency bounds and the validation of the network's timing requirements. A common method to establish the properties of a network element is to obtain a network-calculus service-curve model. The existence of such a model for the interleaved regulator remains an open question. If a service-curve model were found for the interleaved regulator, then the analysis of this mechanism would no longer be limited to the situations where the shaping-for-free holds, which would widen its use in time-sensitive networks. In this paper, we investigate if network-calculus service curves can capture the behavior of the interleaved regulator. We find that an interleaved regulator placed outside of the shaping-for-free requirements (after a non-FIFO system) can yield unbounded latencies. Consequently, we prove that no network-calculus service curve exists to explain the interleaved regulator's behavior. It is still possible to find non-trivial service curves for the interleaved regulator. However, their long-term rate cannot be large enough to provide any guarantee (specifically, we prove that for the regulators that process at least four flows with the same contract, the long-term rate of any service curve is upper bounded by three times the rate of the per-flow contract).

cs.NI↗

arXiv2023

An Analysis of the Completion Time of the BB84 Protocol

The BB84 QKD protocol is based on the idea that the sender and the receiver can reconcile a certain fraction of the teleported qubits to detect eavesdropping or noise and decode the rest to use as a private key. Under the present hardware infrastructure, decoherence of quantum states poses a significant challenge to performing perfect or efficient teleportation, meaning that a teleportation-based protocol must be run multiple times to observe success. Thus, performance analyses of such protocols usually consider the completion time, i.e., the time until success, rather than the duration of a single attempt. Moreover, due to decoherence, the success of an attempt is in general dependent on the duration of individual phases of that attempt, as quantum states must wait in memory while the success or failure of a generation phase is communicated to the relevant parties. In this work, we do a performance analysis of the completion time of the BB84 protocol in a setting where the sender and the receiver are connected via a single quantum repeater and the only quantum channel between them does not see any adversarial attack. Assuming certain distributional forms for the generation and communication phases of teleportation, we provide a method to compute the MGF of the completion time and subsequently derive an estimate of the CDF and a bound on the tail probability. This result helps us gauge the (tail) behaviour of the completion time in terms of the parameters characterising the elementary phases of teleportation, without having to run the protocol multiple times. We also provide an efficient simulation scheme to generate the completion time, which relies on expressing the completion time in terms of aggregated teleportation times. We numerically compare our approach with a full-scale simulation and observe good agreement between them.

cs.PF↗

arXiv2022

Worst-case Delay Bounds in Time-Sensitive Networks with Packet Replication and Elimination

Packet replication and elimination functions are used by time-sensitive networks (as in the context of IEEE TSN and IETF DetNet) to increase the reliability of the network. Packets are replicated onto redundant paths by a replication function. Later the paths merge again and an elimination function removes the duplicates. This redundancy scheme has an effect on the timing behavior of time-sensitive networks and many challenges arise from conducting timing analyses. The replication can induce a burstiness increase along the paths of replicates, as well as packet mis-ordering that could increase the delays in the crossed bridges or routers. The induced packet mis-ordering could also negatively affect the interactions between the redundancy and scheduling mechanisms such as traffic regulators (as with per-flow regulators and interleaved regulators, implemented by TSN asynchronous traffic shaping). Using the network calculus framework, we provide a method of worst-case timing analysis for time-sensitive networks that implement redundancy mechanisms in the general use case, i.e., at end-devices and/or intermediate nodes. We first provide a network calculus toolbox for bounding the burstiness increase and the amount of reordering caused by the elimination function of duplicate packets. We then analyze the interactions with traffic regulators and show that their shaping-for-free property does not hold when placed after a packet elimination function. We provide a bound for the delay penalty when using per-flow regulators and prove that the penalty is not bounded with interleaved regulators. Finally, we use an industrial use-case to show the applicability and the benefits of our findings.

cs.NI↗

arXiv2022

The Stationary Behaviour of Fluid Limits of Reversible Processes is Concentrated on Stationary Points

Assume that a stochastic processes can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.

math.DS↗

arXiv2022

Equivalent Versions of Total Flow Analysis

Total Flow Analysis (TFA) is a method for conducting the worst-case analysis of time sensitive networks without cyclic dependencies. In networks with cyclic dependencies, Fixed-Point TFA introduces artificial cuts, analyses the resulting cycle-free network with TFA, and iterates. If it converges, it does provide valid performance bounds. We show that the choice of the specific cuts used by Fixed-Point TFA does not affect its convergence nor the obtained performance bounds, and that it can be replaced by an alternative algorithm that does not use any cut at all, while still applying to cyclic dependencies.

cs.NI↗

arXiv2022

Improved Network Calculus Delay Bounds in Time-Sensitive Networks

In time-sensitive networks, bounds on worst-case delays are typically obtained by using network calculus and assuming that flows are constrained by bit-level arrival curves. However, in IEEE TSN or IETF DetNet, source flows are constrained on the number of packets rather than bits. A common approach to obtain a delay bound is to derive a bit-level arrival curve from a packet-level arrival curve. However, such a method is not tight: we show that better bounds can be obtained by directly exploiting the arrival curves expressed at the packet level. Our analysis method also obtains better bounds when flows are constrained with g-regulation, such as the recently proposed Length-Rate Quotient rule. It can also be used to generalize some recently proposed network-calculus delay-bounds for a service curve element with known transmission rate.

cs.NI↗

arXiv2022

A Palm Calculus Approach to the Distribution of the Age of Information

A key metric to express the timeliness of status updates in latency-sensitive networked systems is the age of information (AoI), i.e., the time elapsed since the generation of the last received informative status message. This metric allows studying a number of applications including updates of sensory and control information in cyber-physical systems and vehicular networks as well as, job and resource allocation in cloud clusters. State-of-the-art approaches to analyzing the AoI rely on queueing models that are composed of one or many queuing systems endowed with service order, e.g., FIFO, LIFO, or last-generated-first-out order. A major difficulty arising in these analysis methods is capturing the AoI under message reordering when the delivery is non-preemptive and non-FIFO, i.e., when messages can overtake each other and the reception of informative messages may obsolete some messages that are underway. In this paper, we derive an exact formulation for the distribution of AoI in non-preemptive, non-FIFO systems where the main ingredients of our analysis are Palm calculus and time inversion. Owing to the rationality of the Laplace-Stieltjes transforms that are used in our approach, we obtain computable exact expressions for the distribution of AoI.

cs.PF↗

arXiv2021

Analysis of Dampers in Time-Sensitive Networks with Non-ideal Clocks

Dampers are devices that reduce delay jitter in the context of time-sensitive networks, by delaying packets for the amount written in packet headers. Jitter reduction is required by some real-time applications; beyond this, dampers have the potential to solve the burstiness cascade problem of deterministic networks in a scalable way, as they can be stateless. Dampers exist in several variants: some apply only to earliest-deadline-first schedulers, whereas others can be associated with any packet schedulers; some enforce FIFO ordering whereas some others do not. Existing analyses of dampers are specific to some implementations and some network configurations; also, they assume ideal, non-realistic clocks. In this paper, we provide a taxonomy of all existing dampers in general network settings and analyze their timing properties in presence of non-ideal clocks. In particular, we give formulas for computing residual jitter bounds of networks with dampers of any kind. We show that non-FIFO dampers may cause reordering due to clock non-idealities and that the combination of FIFO dampers with non-FIFO network elements may very negatively affect the performance bounds. Our results can be used to analyze timing properties and burstiness increase in any time-sensitive network, as we illustrate on an industrial case-study.

cs.NI↗

arXiv2021

On Packet Reordering in Time-Sensitive Networks

Time-sensitive networks (IEEE TSN or IETF DetNet) may tolerate some packet reordering. Re-sequencing buffers are then used to provide in-order delivery, the parameters of which (timeout, buffer size) may affect worst-case delay and delay jitter. There is so far no precise understanding of per-flow reordering metrics nor of the dimensioning of re-sequencing buffers in order to provide worst-case guarantees, as required in such networks. First, we show that a previously proposed per-flow metric, reordering late time offset (RTO), determines the timeout value. If the network is lossless, another previously defined metric, the reordering byte offset (RBO), determines the required buffer. If packet losses cannot be ignored, the required buffer may be larger than RBO, and depends on jitter, an arrival curve of the flow at its source, and the timeout. Then we develop a calculus to compute the RTO for a flow path; the method uses a novel relation with jitter and arrival curve, together with a decomposition of the path into non order-preserving and order-preserving elements. We also analyse the effect of re-sequencing buffers on worst-case delay, jitter and propagation of arrival curves. We show in particular that, in a lossless (but non order-preserving) network, re-sequencing is "for free", namely, it does not increase worst-case delay nor jitter, whereas in a lossy network, re-sequencing increases the worst-case delay and jitter. We apply the analysis to evaluate the performance impact of placing re-sequencing buffers at intermediate points and illustrate the results on two industrial test cases.

cs.NI↗

arXiv2021

Deficit Round-Robin: A Second Network Calculus Analysis

Deficit Round-Robin (DRR) is a widespread scheduling algorithm that provides fair queueing with variable-length packets. Bounds on worst-case delays for DRR were found by Boyer et al., who used a rigorous network calculus approach and characterized the service obtained by one flow of interest by means of a convex strict service curve. These bounds do not make any assumptions on the interfering traffic hence are pessimistic when the interfering traffic is constrained by some arrival curves. For such cases, two improvements were proposed. The former, by Soni et al., uses a correction term derived from a semi-rigorous heuristic; unfortunately, these bounds are incorrect, as we show by exhibiting a counter-example. The latter, by Bouillard, rigorously derive convex strict service curves for DRR that account for the arrival curve constraints of the interfering traffic. In this paper, we improve on these results in two ways. First, we derive a non-convex strict service curve for DRR that improves on Boyer et al. when there is no arrival constraint on the interfering traffic. Second, we provide an iterative method to improve any strict service curve (including Bouillard's) when there are arrival constraints for the interfering traffic. As of today, our results provide the best-known worst-case delay bounds for DRR. They are obtained by using the method of the pseudo-inverse.

cs.PF↗

arXiv2020

Latency and Backlog Bounds in Time-Sensitive Networking with Credit Based Shapers and Asynchronous Traffic Shaping

We compute bounds on end-to-end worst-case latency and on nodal backlog size for a per-class deterministic network that implements Credit Based Shaper (CBS) and Asynchronous Traffic Shaping (ATS), as proposed by the Time-Sensitive Networking (TSN) standardization group. ATS is an implementation of the Interleaved Regulator, which reshapes traffic in the network before admitting it into a CBS buffer, thus avoiding burstiness cascades. Due to the interleaved regulator, traffic is reshaped at every switch, which allows for the computation of explicit delay and backlog bounds. Furthermore, we obtain a novel, tight per-flow bound for the response time of CBS, when the input is regulated, which is smaller than existing network calculus bounds. We also compute a per-flow bound on the response time of the interleaved regulator. Based on all the above results, we compute bounds on the per-class backlogs. Then, we use the newly computed delay bounds along with recent results on interleaved regulators from literature to derive tight end-to-end latency bounds and show that these are less than the sums of per-switch delay bounds.

cs.NI↗

arXiv2020

Security Measures for Grids against Rank-1 Undetectable Time-Synchronization Attacks

Time-synchronization attacks on phasor measurement units (PMU) pose a real threat to smart grids; it was shown that they are feasible in practice and that they can have a non-negligible negative impact on the state estimation, without triggering the bad-data detection mechanisms. Previous works identified vulnerability conditions when targeted PMUs measure a single phasor. Yet, PMUs are capable of measuring several quantities. We present novel vulnerability conditions in the general case where PMUs measure any number of phasors and can share the same time reference. One is a sufficient condition that does not depend on the measurement values. We propose a security requirement that prevents it and provide a greedy offline algorithm that enforces it. If this security requirement is satisfied, there is still a possibility that the grid can be attacked, although we conjecture that it is very unlikely. We identify two sufficient and necessary vulnerability conditions which depend on the measurement values. For each, we provide a metric that shows the distance between the observed and vulnerability conditions. We recommend their monitoring for security. Numerical results, on the IEEE-39 bus benchmark with real load profiles, show that the measurements of a grid satisfying our security requirement are far from vulnerable.

eess.SY↗

arXiv2020

On Time Synchronization Issues in Time-Sensitive Networks with Regulators and Nonideal Clocks

Flow reshaping is used in time-sensitive networks (as in the context of IEEE TSN and IETF Detnet) in order to reduce burstiness inside the network and to support the computation of guaranteed latency bounds. This is performed using per-flow regulators (such as the Token Bucket Filter) or interleaved regulators (as with IEEE TSN Asynchronous Traffic Shaping). Both types of regulators are beneficial as they cancel the increase of burstiness due to multiplexing inside the network. It was demonstrated, by using network calculus, that they do not increase the worst-case latency. However, the properties of regulators were established assuming that time is perfect in all network nodes. In reality, nodes use local, imperfect clocks. Time-sensitive networks exist in two flavours: (1) in non-synchronized networks, local clocks run independently at every node and their deviations are not controlled and (2) in synchronized networks, the deviations of local clocks are kept within very small bounds using for example a synchronization protocol (such as PTP) or a satellite based geo-positioning system (such as GPS). We revisit the properties of regulators in both cases. In non-synchronized networks, we show that ignoring the timing inaccuracies can lead to network instability due to unbounded delay in per-flow or interleaved regulators. We propose and analyze two methods (rate and burst cascade, and asynchronous dual arrival-curve method) for avoiding this problem. In synchronized networks, we show that there is no instability with per-flow regulators but, surprisingly, interleaved regulators can lead to instability. To establish these results, we develop a new framework that captures industrial requirements on clocks in both non-synchronized and synchronized networks, and we develop a toolbox that extends network calculus to account for clock imperfections.

cs.NI↗