Section: New Results

Network Dynamics

Participants : Abir Benabid, Julieta Bollati, Anne Bouillard, Ana Bušić, Emilie Coupechoux, Nadir Farhi.

Queueing network, stability, inversion formula, probing, estimator, product-form, insensitivity, markov decision, max-plus algebra, network calculus.

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 in FIFO tandem networks

In cooperation with Giovanni Stea [University of Pisa, Italy], we present in [31] algorithms to compute worst-case performance upper bounds when the service policy is FIFO, using linear programming. Linear programming leads to tight bounds; however, the computation corst is too high for reasonable-size networks. We then develop approximate solution schemes to find both upper and lower delay bounds on the worst-case delay. Both of them only require to solve just one LP problem, and they produce bounds which are generally more accurate than those found in the literature. Finally, we have a conjecture on what sould be the worst-case trajectory under usual assumptions.

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 have been presented at Valuetools (invited paper, [29] ).

Using arrival curves for detecting anomalies in a network

In cooperation with Aurore Junier [Inria/IRISA] and Benoît Ronot [Alcatel-Lucent], we present an on-line algorithm that performs a flow of messages analysis. More precisely, it is able to highlight hidden abnormal behaviors that existing network management methods would not detect. Our algorithm uses the notion of constraint curves, introduced in the Network Calculus theory, defining successive time windows that bound the flow. The advantage of this algorithm is that it can be performed online, and in a second version has different levels of precision. This work has been presented in [30] and a patent [57] has been submitted.

Min,plus algorithms for fast weak-KAM integrators

In cooperation with Erwan Faou [IPSO-Inria Rennes, DMA-ENS] and Maxime Zavidovique [Paris 6]. We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way. The results can be found in [49] .

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 [15] 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 ).

Perfect Sampling of Networks with Finite and Infinite Capacity Queues

In [33] , we consider open Jackson queueing networks with mixed finite and infinite buffers and analyze the efficiency of sampling from their exact stationary distribution. We show that perfect sampling is possible, although the underlying Markov chain has a large or even infinite state space. The main idea is to use a Jackson network with infinite buffers (that has a product form stationary distribution) to bound the number of initial conditions to be considered in the coupling from the past scheme. We also provide bounds on the sampling time of this new perfect sampling algorithm under hyper-stability conditions (to be defined in the paper) for each queue. These bounds show that the new algorithm is considerably more efficient than existing perfect samplers even in the case where all queues are finite. We illustrate this efficiency through numerical experiments.

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

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 [14] . 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.

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] [16] , 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.

Bounded State Space Truncation

Markov chain modeling often suffers from the curse of dimensionality problems and many approximation schemes have been proposed in the literature that include state-space truncation. Estimating the accuracy of such methods is difficult and the resulting approximations can be far from the exact solution. Censored Markov chains (CMC) allow to represent the conditional behavior of a system within a subset of observed states and provide a theoretical framework to study state-space truncation. However, the transition matrix of a CMC is in general hard to compute. Dayar et al. (2006) proposed DPY algorithm, that computes a stochastic bound for a CMC, using only partial knowledge of the original chain. In [32] , we prove that DPY is optimal for the information they take into account. We also show how some additional knowledge on the chain can improve stochastic bounds for CMC.

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] [43] 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.