Section: New Results

Network Dynamics

Participants : François Baccelli, Abir Benabid, Florence Bénézit, Anne Bouillard, Ana Bušić, Emilie Coupechoux, Nadir Farhi, Furcy-Alexandre Pin, Aleksander Wieczorek.

This traditional research topic of TREC has several new threads like perfect simulation, active probing or Markov decision.

Network Calculus

Network calculus is a theory that aims at computing deterministic performance guarantees in communication networks. This theory is based on the (min,plus) algebra. Flows are modeled by an arrival curve that upper-bounds the amount of data that can arrive during any interval, and network elements are modeled by a service curve that gives a lower bound on the amount of service offered to the flows crossing that element. Worst-case performances are then derived by combining these curves.

Performance bounds networks with static priorities

In cooperation with Aurore Junier [INRIA/IRISA], we present in [29] algorithms to compute worst-case performance upper bounds when the service policy is static priorities, using linear programming. Linear programming does not lead to tight bounds, but when combining this method with (min,plus) methods, we obtain bounds that outperform the already known bounds. Also, we prove that in tandem networks, the the worst-case performance bound under arbitrary multiplexing can be obtain by a policy with static priorities, the “shortest-destination first” policy.

Feed-forward networks with wormhole routing discipline

In collaboration with Bruno Gaujal [INRIA Rhone Alpes] and Nadir Farhi [IFFSTAR] we are working on a model of performance bound calculus on feed-forward networks where data packets are routed under wormhole routing discipline. We are interested in determining maximum end-to-end delays and backlogs for packets going from a source node to a destination node, through a given virtual path in the network. Our objective is to give a “network calculus” approach to calculate the performance bounds. For this, we propose a new concept of curves that we call packet curves. The curves permit to model constraints on packet lengths for data flows, when the lengths are allowed to be different. We used this new concept to propose an approach for calculating residual services for data flows served under non preemptive service disciplines. This notion also enabled us to differentiate different classes of service policies: those that are based on a packet count (like round-robin and its generalized version), where the packet curve will be useful to tighten the bounds computed, and those that are based on the amount of data served (FIFO, priorities), where it won't be useful. These results can be found in [44] and have been presented in ILAS 2011.

Composition of service curves in Network Calculus

In envelope-based models for worst-case performance evaluation like Network Calculus or Real-Time Calculus, several types of service curves have been introduced to quantify some deterministic service guarantees. We compare those different classes of service curves regarding the composition (servers in tandem) and individual service curves (when several flows share a server, what service curve can be guaranteed to each of the flows?). In short, there are two main classes of service curves, simple and strict service curves. Individual service curve can not always be computed when simple service curves are considered, and strict service curves is not a stable class regarding the two operations described. We show that there can be no equivalence between the two main classes of service curves and that no notion of service curve in-between can be defined, that behaves well for the composition. We complete this study by studying other classes of service curves from this viewpoint. These results have been presented in [28]

Residuation in (max,plus) automata

With Éric Badouel, Philippe Darondeau [INRIA/IRISA] and Jan Komenda [Institute of Mathematics, Brnó], we study in [27] the decidability of existence and the rationality of delay controllers for systems with time weights in the tropical and interval semirings. Depending on the (max,+) or (min,+)-rationality of the series specifying the controlled system and the control objective, cases are identified where the controller series defined by residuation is rational, and when it is positive (i.e., when delay control is feasible). When the control objective is specified by a tolerance, i.e. by two bounding rational series, a nice case is identified in which the controller series is of the same rational type as the system specification series.

Queueing Theory and Active Probing

Inverse Problems

Active probing began by measuring end-to-end path metrics, such as delay and loss, in a direct measurement process which did not require inference of internal network parameters. The field has since progressed to measuring network metrics, from link capacities to available bandwidth and cross traffic itself, which reach deeper and deeper into the network and require increasingly complex inversion methodologies. The thesis of B. Kauffmann [6] investigates this line of thought as a set of inverse problems in queueing theory. Queueing theory is typically concerned with the solution of direct problems, where the trajectory of the queueing system, and laws thereof, are derived based on a complete specification of the system, its inputs and initial conditions. Inverse problems aim to deduce unknown parameters of the system based on partially observed trajectories. A general definition of the inverse problems in this class was provided and the key variants were mapped out: the analytical methods, the statistical methods and the design of experiments. We also show how this inverse problem viewpoint translates to the design of concrete Internet probing applications.

Inverse problems in bandwidth sharing networks theory were also investigated. A bandwidth sharing networks allocates the bandwidth to each flow in order to maximize a given utility function (typically an $\alpha$-fairness), with the constraints given by the capacity of the different servers. In particular, it has been shown that the equilibrium distribution of the bandwidth allocated by TCP to many competing connections is oscillating around an $\alpha$-fair allocation. As such, the theory of bandwidth sharing network is a high-level viewpoint of networks. The meaning of inverse problems in this theory, and their relation to the active probing paradigm are analyzed. In two simple examples of network, the capacity of the different servers and the flow population can estimated, and an algorithm to perform this estimation was proposed.

Internet Tomography

Most active probing techniques suffer of the “Bottleneck” limitation: all characteristics of the path after the bottleneck link are erased and unreachable. we are currently investigating a new tomography technique, based on the measurement of the fluctuations of point-to-point end-to-end delays, and allowing one to get insight on the residual available bandwidth along the whole path. For this, we combined classical queueing theory models with statistical analysis to obtain estimators of residual bandwidth on all links of the path. These estimators were proved to be tractable, consistent and efficient. In  [59] we evaluated their performance with simulation and trace-based experiments.

Lately this method has been generalized in [72] to a probing multicast tree instead of a single path. This work deals with the complexity of the combinatorials in trees, and gives an explicit formula for the iteration of the Expectation-Maximization (E-M) algorithm. The E-M algorithm is notoriously slow, and we provided three speed-up techniques which are effective in our case (up to a factor ${10}^3$ in the computation time). These techniques are general, and can be applied to other instances of E-M, or even several other iterative algorithms.

Perfect Sampling of Queueing Systems

Propp and Wilson introduced in 1996 a perfect sampling algorithm that uses coupling arguments to give an unbiased sample from the stationary distribution of a Markov chain on a finite state space $𝒳$. In the general case, the algorithm starts trajectories from all $x\in 𝒳$ at some time in the past until time $t=0$. If the final state is the same for all trajectories, then the chain has coupled and the final state has the stationary distribution of the Markov chain. Otherwise, the simulations are started further in the past. This technique is very efficient if all the events in the system have appropriate monotonicity properties. However, in the general (non-monotone) case, this technique requires that one consider the whole state space, which limits its application only to chains with a state space of small cardinality.

Piecewise Homogeneous Events

In collaboration with Bruno Gaujal [INRIA Grenoble - Rhone-Alpes], we proposed in [47] a new approach for the general case that only needs to consider two trajectories. Instead of the original chain, we used two bounding processes (envelopes) and we showed that, whenever they couple, one obtains a sample under the stationary distribution of the original chain. We showed that this new approach is particularly effective when the state space can be partitioned into pieces where envelopes can be easily computed. We further showed that most Markovian queueing networks have this property and we propose efficient algorithms for some of them.

The envelope technique has been implemented in a software tool PSI2 (see Section 5.2 ).

Acceleration of Perfect Sampling by Skipping Events

In collaboration with Bruno Gaujal [INRIA Grenoble - Rhone-Alpes], we proposed a new method to speed up perfect sampling of Markov chains by skipping passive events during the simulation [38] . We showed that this can be done without altering the distribution of the samples. This technique is particularly efficient for the simulation of Markov chains with different time scales such as queueing networks where certain servers are much faster than others. In such cases, the coupling time of the Markov chain can be arbitrarily large while the runtime of the skipping algorithm remains bounded. This was further illustrated by several experiments that also show the role played by the entropy of the system in the performance of our algorithm.

Aggregated Envelopes

When the cardinality of the state space is so huge that even storing the state of the Markov chain becomes challenging, we propose to combine the ideas of bounding processes and the aggregation of Markov chains [30] . We illustrate the proposed approach of aggregated envelope bounding chains on queueing models with joint arrivals and joint services, often referred to in the literature as assemble-to-order systems. Due to the finite capacity, and coupling in arrivals and services, the exact solving techniques are inefficient for larger problem instances. For instance, for the service tools model proposed by Vliegen and Van Houtum (2009), the aggregated envelope method reduces exponentially the dimension of the state space and allows effective perfect sampling algorithms. We also provide bounds for the coupling time, under the high service rate assumptions.

Markov Chains and Markov Decision Processes

Solving Markov chains is in general difficult if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and provide bounds for a reward function of interest. The bounds can be established by using different qualitative properties, such as stochastic monotonicity, convexity, submodularity, etc. In the case of Markov decision processes, similar properties can be used to show that the optimal policy has some desired structure (e.g. the critical level policies).

Stochastic Monotonicity

g In collaboration with Jean-Michel Fourneau [PRiSM, Université de Versailles Saint-Quentin] we consider two different applications of stochastic monotonicity in performance evaluation of networks [18] . In the first one, we assume that a Markov chain of the model depends on a parameter that can be estimated only up to a certain level and we have only an interval that contains the exact value of the parameter. Instead of taking an approximated value for the unknown parameter, we show how we can use the monotonicity properties of the Markov chain to take into account the error bound from the measurements. In the second application, we consider a well known approximation method: the decomposition into submodels. In such an approach, models of complex networks are decomposed into submodels whose results are then used as parameters for the next submodel in an iterative computation. One obtains a fixed point system which is solved numerically. In general, we have neither an existence proof of the solution of the fixed point system nor a convergence proof of the iterative algorithm. Here we show how stochastic monotonicity can be used to answer these questions. Furthermore, monotonicity properties can also help to derive more efficient algorithms to solve fixed point systems.

Componentwise Bounds

In collaboration with Jean-Michel Fourneau [PRiSM, Université de Versailles Saint-Quentin] we proposed an iterative algorithm to compute component-wise bounds of the steady-state distribution of an irreducible and aperiodic Markov chain [17] . These bounds are based on very simple properties of $(max,+)$ and $(min,+)$ sequences. We showed that, under some assumptions on the Markov chain, these bounds converge to the exact solution. In that case we have a clear tradeoff between computation and the tightness of bounds. Furthermore, at each step we know that the exact solution is within an interval, which provides a more effective convergence test than usual iterative methods.

Markov Reward Processes and Aggregation

In a joint work with I.M. H. Vliegen [University of Twente, The Netherlands] and A. Scheller-Wolf [Carnegie Mellon University, USA] [19] , we presented a new bounding method for Markov chains inspired by Markov reward theory: Our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications of the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for an initial chain by analyzing a much smaller chain. We illustrated our method by using it to prove monotonicity results and bounds for assemble-to-order systems.

Critical Level Policies in Controlled Queuing Systems

In a joint work with Emmanuel Hyon [University of Paris Ouest Nanterre La Defense and LIP6] [39] , we consider a single-item lost sales inventory model with different classes of customers. Each customer class may have different lost sale penalty costs. We assume that the demands follow a Poisson process and we consider a single replenishment hypoexponential server. We give a Markov decision process associated with this optimal control problem and prove some structural properties of its dynamic programming operator. This allows us to show that the optimal policy is a critical level policy. We also discuss some possible extensions to other replenishment distributions and give some numerical results for the hyperexponential server case.

Dynamic Systems with Local Interactions

Dynamic systems with local interactions can be used to model problems in distributed computing: gathering a global information by exchanging only local information. The challenge is two-fold: first, it is impossible to centralize the information (cells are indistinguishable); second, the cells contain only a limited information (represented by a finite alphabet $𝒜$; $𝒜=\{0,1\}$ in our case). Two natural instantiations of dynamical systems are considered, one with synchronous updates of the cells, and one with asynchronous updates. In the first case, time is discrete, all cells are updated at each time step, and the model is known as a Probabilistic Cellular Automaton (PCA) (e.g. Dobrushin, R., Kryukov, V., Toom, A.: Stochastic cellular systems: ergodicity, memory, morphogenesis, 1990). In the second case, time is continuous, cells are updated at random instants, at most one cell is updated at any given time, and the model is known as a (finite range) Interacting Particle System (IPS) (e.g. Liggett, T.M.: Interacting particle systems, 2005).

Density Classification on Infinite Lattices and Trees

In a joint work with N. Fatès [INRIA Nancy – Grand-Est], J. Mairesse and I. Marcovici [LIAFA, CNRS and Université Paris 7] [46] we consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter $p$. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that evolves on this graph and decides whether $p$ is smaller or larger than $1/2$. Precisely, the trajectories should converge (weakly) to the uniform configuration with only $0^{\text{'}}s$ if $p<1/2$, and only $1^{\text{'}}s$ if $p>1/2$. We present solutions to that problem on ${\mathbb{Z}}^d$, for any $d\ge 2$, and on the regular infinite trees. For $\mathbb{Z}$, we propose some candidates that we back up with numerical simulations.

Stochastic Stability

Ergodicity of Probabilistic Cellular Automata

In a joint work with J. Mairesse and I. Marcovici [LIAFA, CNRS and Université Paris 7] [31] , we considered ergodicity properties of probabilistic cellular automata (PCA). A classical cellular automaton (CA) is a particular case of PCA. For a 1-dimensional CA, we proved that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then proposed an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity properties of the local rule. It is based on a bounding process which is shown to be also a PCA. We then focused on the PCA Majority, whose asymptotic behavior is unknown, and performed numerical experiments using the perfect sampling procedure.

Spatial Queues

In a joint work with S. Foss [Heriot–Watt University, UK]  [13] , we considered a queue where the server is the Euclidean space, the customers are random closed sets of the Euclidean space arriving according to a Poisson rain and where the discipline is a hard exclusion rule: no two intersecting random closed sets can be served at the same time. We use the max plus algebra and Lyapunov exponents to show that under first come first serve assumptions, this queue is stable for a sufficiently small arrival intensity. We also discuss the percolation properties of the stationary regime of the random closed sets in the queue.