Power-line communications (PLC) are becoming a key component in home networking, because they provide easy and high-throughput connectivity. The dominant MAC protocol for high data-rate PLC, the IEEE 1901, employs a CSMA CA mechanism similar to the backoff process of 802.11. Existing performance evaluation studies of this protocol assume that the backoff processes of the stations are independent (the so-called decoupling assumption). However, in contrast to 802.11, 1901 stations can change their state after sensing the medium busy, which is regulated by the so-called deferral counter. This mechanism introduces strong coupling between the stations and, as a result, makes existing analyses inaccurate. In this paper, we propose a performance model for 1901, which does not rely on the decoupling assumption. We prove that our model admits a unique solution for a wide range of configurations and confirm the accuracy of the model using simulations. Our results show that we outperform current models based on the decoupling assumption. In addition to evaluating the performance in steady state, we further study the transient dynamics of 1901, which is also affected by the deferral counter.
@cite_10 study 802.11 without the decoupling assumption. They analyze an @math -dimensional chain ( @math being the number of backoff stages) that describes the number of stations at each backoff stage. capture the expected change of the number of stations at each backoff stage between two consecutive time slots, and their equilibrium point yields the average number of stations at each backoff stage in steady state. Similarly to @cite_10 , we also use drift equations to obtain an accurate model for 1901.
@cite_10
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Performance evaluation of the 802.11 MAC protocol is classically based on the decoupling assumption, which hypothesizes that the backoff processes at different nodes are independent. This decoupling assumption results from mean field convergence and is generally true in transient regime in the asymptotic sense (when the number of wireless nodes tends to infinity), but, contrary to widespread belief, may not necessarily hold in stationary regime. The issue is often related with the existence and uniqueness of a solution to a fixed point equation; however, it was also recently shown that this condition is not sufficient; in contrast, a sufficient condition is a global stability property of the associated ordinary differential equation. In this paper, we give a simple condition that establishes the asymptotic validity of the decoupling assumption for the homogeneous case (all nodes have the same parameters). We also discuss the heterogeneous and the differentiated service cases and formulate a new ordinary differential equation. We show that the uniqueness of a solution to the associated fixed point equation is not sufficient; we exhibit one case where the fixed point equation has a unique solution but the decoupling assumption is not valid in the asymptotic sense in stationary regime. Given a set S of radio stations located on a line and an integer h (1 ≤ h ≤ |S| - 1), the Min Assignment problem is to find a range assignment of minimum power consumption provided that any pair of stations can communicate in at most h hops. Previous positive results for this problem were known only when h = |S| - 1 (i.e. the unbounded case) or when the stations are equally spaced (i.e. the uniform chain). In particular, Kirousis, Kranakis, Krizanc and Pelc (1997) provided an efficient exact solution for the unbounded case and efficient approximated solutions for the uniform chain, respectively. This paper presents the first polynomial time, approximation algorithm for the Min Assignment problem. The algorithm guarantees an approximation ratio of 2 and runs in time O(hn3). We also prove that, for constant h and for "well spread" instances (a broad generalization of the uniform chain case), we can find a solution in time O(hn3) whose cost is at most an (1 + Ɛ(n)) factor from the optimum, where Ɛ(n) = o(1) and n is the number of stations. This result significantly improves the approximability result by on uniform chains. Both of our approximation results are obtained by new algorithms that exactly solves two natural variants of the Min Assignment problem that might have independent interest: the All-To-One problem (in which every station must reach a fixed one in at most h hops) and the Base Location problem (in which the goal is to select a set of Basis among the stations and all the other stations must reach one of them in at most h -1 hops). Finally, we show that for h = 2 the Min Assignment problem can be solved in O(n3)-time. Abstract When the initial distribution and transition rates for a continuous time Markov chain are not known precisely, robust methods are needed to study the evolution of the process in time to avoid judgements based on unwarranted precision. We follow the ideas successfully applied in the study of discrete time model to build a framework of imprecise Markov chains in continuous time. The imprecision in the distributions over the set of states is modelled with upper and lower expectation functionals, which equivalently represent sets of probability distributions. Uncertainty in transitions is modelled with sets of transition rates compatible with available information. The Kolmogorov’s backward equation is then generalised into the form of a generalised differential equation, with generalised derivatives and set valued maps. The upper and lower expectation functionals corresponding to imprecise distributions at given times are determined by the maximal and minimal solutions of these equations. The second part of the paper is devoted to numerical methods for approximating the boundary solutions. The methods are based on discretisation of the time interval. A uniform and adaptive grid discretisations are examined. The latter is computationally much more efficient than the former one, but is not applicable on every interval. Therefore, to achieve maximal efficiency a combination of the methods is used. We consider models of N interacting objects, where the interaction is via a common resource and the distribution of states of all objects. We introduce the key scaling concept of intensity; informally, the expected number of transitions per object per time slot is of the order of the intensity. We consider the case of vanishing intensity, i.e. the expected number of object transitions per time slot is o(N). We show that, under mild assumptions and for large N, the occupancy measure converges, in mean square (and thus in probability) over any finite horizon, to a deterministic dynamical system. The mild assumption is essentially that the coefficient of variation of the number of object transitions per time slot remains bounded with N. No independence assumption is needed anywhere. The convergence results allow us to derive properties valid in the stationary regime. We discuss when one can assure that a stationary point of the ODE is the large N limit of the stationary probability distribution of the state of one object for the system with N objects. We use this to develop a critique of the fixed point method sometimes used in conjunction with the decoupling assumption. Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner. Finally, we also prove that there exists an absolute constant @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain. We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions @math . In previous works we studied the model problem of a discrete elliptic equation on @math . Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions @math and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions. In Resing and Ormeci [16] it is shown that the two-stage tandem queue with coupled processors can be solved using the theory of boundary value problems. In this paper we consider the issues that arise when calculating performance measures like the mean queue length and the fraction of time a station is empty. It is assumed that jobs arrive at the first station according to a Poisson process and require service at both stations before leaving the system. The amount of work that a job requires at each of the stations is an independent, exponentially distributed random variable. When both stations are nonempty, the total service capacity is shared among the stations according to fixed proportions. When one of the stations becomes empty, the total service capacity is given to the nonempty station. We study the two-dimensional Markov process representing the numbers of jobs at the two stations. The problem of finding the generating function of the stationary distribution can be reduced to two different Riemann-Hilbert boundary value problems, where both problems yield a complete analytical solution. We discuss the similarities and differences between the two problems, and relate them to the computational aspects of obtaining performance measures. k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's coalescent, which has numerous applications in population genetics, is the δ0-coalescent for δ0 a unit mass at 0. The coalescent recently derived by Bolthausen and Sznit- man from Ruelle's probability cascades, in the context of the Sherrington- Kirkpatrick spin glass model in mathematical physics, is the U-coalescent for U uniform on � 0� 1� .F or� = U, and whenever an infinite number of masses are present, each collision in a � -coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to � . The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable ran- dom partitions ofgoverned by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the U-coalescent and its time reversal. 1. Introduction. Markovian coalescent models for the evolution of a sys- tem of masses by a random process of binary collisions were introduced by Marcus (29) and Lushnikov (28). See (3) for a recent survey of the scientific lit- erature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman (15) gave a general framework for the rigorous construction of partition-valued and discrete measure-valued coalescent Markov processes allowing infinitely many massses and treated the binary coalescent model where each pair of masses x and y is subject to a coa- lescent collision at rate κ� xyfor a suitable rate kernel κ. This paper studies a family of partition-valued Markov processes, with state space the compact set of all partitions of � �= � 1� 2 ���� � , such that the restriction of the partition to each finite subset ofis a Markov chain with transition rates of a simple form determined by the moments of a finite measureon the unit interval. The case � = δ 0 , a unit mass at 0, is Kingman's coalescent in which every A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of card shuffling (Aldous-Diaconis, 1986), this remarkable phenomenon is now rigorously established for many reversible chains. Here we consider the non-reversible case of random walks on sparse directed graphs, for which even the equilibrium measure is far from being understood. We work under the configuration model, allowing both the in-degrees and the out-degrees to be freely specified. We establish the cutoff phenomenon, determine its precise window and prove that the cutoff profile approaches a universal shape. We also provide a detailed description of the equilibrium measure. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. A fundamental question arising in the analysis of queueing systems is whether a system is stable or unstable. For systems modeled by infinite Markov chains, we may study the ergodicity and nonergodicity of the chains. Foster showed that sufficient conditions for ergodicity are linked with the average drift. However, complications arise when multidimensional Markov chains are analyzed. We present three methods that provide sufficient conditions for ergodicity and nonergodicity of a multidimensional Markov chain. These methods are next applied to two multidimensional queueing systems: buffered contention packet broadcast systems and coupled processor systems. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_ v V(G) is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z for z different of x,y remains unchanged. We estimate the mixing time of X* in terms of the parameters of H and G. Further, we show that the relaxation time of X* is the same order as the maximal expected hitting time of G plus |G| times the relaxation time of the chain on H. We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann-Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue). We analyze the mixing behavior of the biased exclusion process on a path of length n as the bias ( _n ) tends to 0 as (n ). We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased exclusion and the process with constant bias. As the bias increases, the mixing time undergoes two phase transitions: one when ( _n ) is of order 1 n, and the other when ( _n ) is order ( n n ). In the paper, we analyze the stochastic behavior of backoff protocols for multiple access channels such as the Ethernet. In particular, we prove that binary exponential backoff is unstable if the arrival rate of new messages at each station is λ N for any number of stations N exceeding 1 and any overall arrival rate λ exceeding .567 + 1 4 N - 2. More importantly, we also prove that any superlinear polynomial backoff protocol (e.g., quadratic backoff) is stable for any set of arrival rates that sum to less than one, and any number of stations. The results significantly extend the previous work in the area, and provide the first examples of acknowledgement based protocols known to be stable for a nonnegligible overall arrival rate distributed over an arbitrarily large number of stations. The results also disprove a popular assumption that exponential backoff is the best choice among acknowledgement based protocols for systems with large overall arrival rates. Finally, we prove that any linear or sublinear backoff protocol is unstable if the arrival rate at each station is λ N for any fixed λ and sufficiently large N . New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model @math , @math vertices are split into @math communities of relative size @math , and vertices in community @math and @math connect independently with probability @math . This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new divergence function @math , which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, in contrast to the conjectures made for detection with more than 4 communities; note that the algorithm is optimal both in terms of achieving the threshold and in having quasi-linear complexity, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio (defined in term of the spectrum of @math ) tends to infinity. This is the second article of a series of papers on stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the @math -dimensional lattice @math with random coefficients @math of the simplest type: They are identically distributed and independent from edge to edge. On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric ''homogenized'' matrix @math is characterized by @math for any direction @math , where the random field @math (the ''corrector'') is the unique solution of @math in @math such that @math , @math is stationary and @math , @math denoting the ensemble average (or expectation). In order to approximate the homogenized coefficients @math , the corrector problem is usually solved in a box @math of size @math with periodic boundary conditions, and the space averaged energy on @math defines an approximation @math of @math . Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation @math converges almost surely to @math as @math . In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size @math , but replace the elliptic operator by @math with (typically) @math , as standard in the homogenization literature. We then replace the ensemble average by a space average on @math , and estimate the overall error on the homogenized coefficients in terms of @math and @math . We introduce a framework to study slotted Aloha with cooperative base stations. Assuming a geographic-proximity communication model, we propose several decoding algorithms with different degrees of base stations' cooperation (non-cooperative, spatial, temporal, and spatio-temporal). With spatial cooperation, neighboring base stations inform each other whenever they collect a user within their coverage overlap; temporal cooperation corresponds to (temporal) successive interference cancellation done locally at each station. We analyze the four decoding algorithms and establish several fundamental results. With all algorithms, the peak throughput (average number of decoded users per slot, across all base stations) increases linearly with the number of base stations. Further, temporal and spatio-temporal cooperations exhibit a threshold behavior with respect to the normalized load (number of users per station, per slot). There exists a positive load @math , such that, below @math , the decoding probability is asymptotically maximal possible, equal the probability that a user is heard by at least one base station; with non-cooperative decoding and spatial cooperation, we show that @math is zero. Finally, with spatio-temporal cooperation, we optimize the degree distribution according to which users transmit their packet replicas; the optimum is in general very different from the corresponding optimal distribution of the single-base station system. An instance of a random constraint satisfaction problem defines a random subset 𝒮 (the set of solutions) of a large product space X N (the set of assignments). We consider two prototypical problem ensembles (random k -satisfiability and q -coloring of random regular graphs) and study the uniform measure with support on S . As the number of constraints per variable increases, this measure first decomposes into an exponential number of pure states (“clusters”) and subsequently condensates over the largest such states. Above the condensation point, the mass carried by the n largest states follows a Poisson-Dirichlet process. For typical large instances, the two transitions are sharp. We determine their precise location. Further, we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem. The degree of correlation naturally affects the performances of many search sampling algorithms. Empirical evidence suggests that local Monte Carlo Markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point. Finally, refined message passing techniques (such as survey propagation) may also beat this threshold.