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.
Our works of @cite_14 , @cite_12 propose a simplification to the model of @cite_16 , reducing the Markov chain to a single fixed-point equation; by applying a similar theoretical technique to @cite_6 , these papers also prove that this fixed-point equation admits a unique solution. Being less accurate but simpler than the model introduced here, the model of @cite_12 enables us to optimize the performance of the protocol towards high throughput.
@cite_14 @cite_6 @cite_12 @cite_16
1968517161 2137298097 2166657206 2167223378 2134711723 2167877620 2514483739 2110398528 2112901936 2086672759 2147568833 2952684528 2107208179 2763281404 2051343255 2163932785 2133027741 2109528718 2136340918 2003095404
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. The Gilbert model (1-st order Markov chain model) and the single-multiplexer model are two frequently used models in the study of packet-loss processes in communication networks. In this paper we investigate the accuracy of the Gilbert model, and higher-order Markov chain extended Gilbert models, in characterizing the packet-loss process associated with a transport network modeled in terms of a single-multiplexer. More specifically, we quantitatively compare the packet-loss statistics predicted by the Gilbert models with those predicted by an exact queueing analysis of the single-multiplexer model. This topic is important since low-complexity Gilbert models are frequently used to characterize end-to-end network packet-loss behavior. On the other hand, network congestion behavior is often characterized in terms of a single bottleneck node modeled as a multiplexer. It is of some interest then to establish the relative accuracy of Gilbert models in predicting the packet-loss behavior on even such a simplified network model. We demonstrate that the Gilbert models have some serious deficiencies in accurately predicting the packet-loss statistics of the single-multiplexer model. The results are shown to have some serious consequences for the performance evaluation of forward error correction (FEC) coding schemes used to combat the effects of packet losses due to network buffer overflows. We study a fixed-point formalization of the well-known analysis of Bianchi. We provide a significant simplification and generalization of the analysis. In this more general framework, the fixed-point solution and performance measures resulting from it are studied. Uniqueness of the fixed point is established. Simple and general throughput formulas are provided. It is shown that the throughput of any flow will be bounded by the one with the smallest transmission rate. The aggregate throughput is bounded by the reciprocal of the harmonic mean of the transmission rates. In an asymptotic regime with a large number of nodes, explicit formulas for the collision probability, the aggregate attempt rate, and the aggregate throughput are provided. The results from the analysis are compared with ns2 simulations and also with an exact Markov model of the backoff process. It is shown how the saturated network analysis can be used to obtain TCP transfer throughputs in some cases. This paper has two parts. In the first part we consider a simple Markov chain for d-regular graphs on n vertices, where d = d(n) may grow with n. We show that the mixing time of this Markov chain is bounded above by a polynomial in n and d. In the second part of the paper, a related Markov chain for d-regular graphs on a varying number of vertices is introduced, for even constant d. This is a model for a certain peer-to-peer network. We prove that the related chain has mixing time which is bounded above by a polynomial in N, the expected number of vertices, provided certain assumptions are met about the rate of arrival and departure of vertices. We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest eigenvalue modulus (SLEM) of the transition probability matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the SLEM, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, @math ) using standard numerical methods for SDPs. Larger problems can be solved by exploiting various types of symmetry and structure in the problem, and far larger problems (say, 100,000 edges) can be solved using a subgradient method we describe. We compare the fastest mixing Markov chain to those obtained using two commonly used heuristics: the maximum-degree method, and the Metropolis--Hastings algorithm. For many of the examples considered, the fastest mixing Markov chain is substantially faster than those obtained using these heuristic methods. We derive the Lagrange dual of the fastest mixing Markov chain problem, which gives a sophisticated method for obtaining (arbitrarily good) bounds on the optimal mixing rate, as well as the optimality conditions. Finally, we describe various extensions of the method, including a solution of the problem of finding the fastest mixing reversible Markov chain, on a fixed graph, with a given equilibrium distribution. In this paper we consider a simple Markov chain for bipartite graphs with given degree sequence on @math vertices. We show that the mixing time of this Markov chain is bounded above by a polynomial in @math in case of half-regular degree sequence. The novelty of our approach lies in the construction of the multicommodity flow in Sinclair's method. This paper investigates the compress-and-forward scheme for an uplink cloud radio access network (C-RAN) model, where multi-antenna base stations (BSs) are connected to a cloud-computing-based central processor (CP) via capacity-limited fronthaul links. The BSs compress the received signals with Wyner-Ziv coding and send the representation bits to the CP; the CP performs the decoding of all the users’ messages. Under this setup, this paper makes progress toward the optimal structure of the fronthaul compression and CP decoding strategies for the compress-and-forward scheme in the C-RAN. On the CP decoding strategy design, this paper shows that under a sum fronthaul capacity constraint, a generalized successive decoding strategy of the quantization and user message codewords that allows arbitrary interleaved order at the CP achieves the same rate region as the optimal joint decoding. Furthermore, it is shown that a practical strategy of successively decoding the quantization codewords first, then the user messages, achieves the same maximum sum rate as joint decoding under individual fronthaul constraints. On the joint optimization of user transmission and BS quantization strategies, this paper shows that if the input distributions are assumed to be Gaussian, then under joint decoding, the optimal quantization scheme for maximizing the achievable rate region is Gaussian. Moreover, Gaussian input and Gaussian quantization with joint decoding achieve to within a constant gap of the capacity region of the Gaussian multiple-input multiple-output (MIMO) uplink C-RAN model. Finally, this paper addresses the computational aspect of optimizing uplink MIMO C-RAN by showing that under fixed Gaussian input, the sum rate maximization problem over the Gaussian quantization noise covariance matrices can be formulated as convex optimization problems, thereby facilitating its efficient solution. This paper analyzes the optimum constant contention window (CW) for the HomePlug 1.0 and AV CSMA CA MAC. A discrete time, homogenous Markov chain, with the state specified by both the backoff counter (BC) and deferral counter (DC), is used to model a single node contending for transmission. The structure of the Markov chain admits a generalized expression for the stationary state probability mass function (pmf) associated with each state. The recursively defined state pmfs can be analytically reduced to a single expression relating the probability p, of the node finding the medium idle, the maximum window size W, the maximum deferral counter size, and the number of nodes n. Optimizing the MAC efficiency provides a target value for p, which can be attained with the proper selection of W and. It is shown that an optimal contention window size can be chosen based on a linear relationship with the number of nodes. Reasoning about the performance of models of software systems typically entails the derivation of metrics such as throughput, utilization, and response time. If the model is a Markov chain, these are expressed as real functions of the chain, called reward models. The computational complexity of reward-based metrics is of the same order as the solution of the Markov chain, making the analysis infeasible when evaluating large-scale systems. In the context of the stochastic process algebra PEPA, the underlying continuous-time Markov chain has been shown to admit a deterministic (fluid) approximation as a solution of an ordinary differential equation, which effectively circumvents state-space explosion. This paper is concerned with approximating Markovian reward models for PEPA with fluid rewards, i.e., functions of the solution of the differential equation problem. It shows that (1) the Markovian reward models for typical metrics of performance enjoy asymptotic convergence to their fluid analogues, and that (2) via numerical tests, the approximation yields satisfactory accuracy in practice. In this paper, we focus on the Markov model of IEEE 802.11 distributed coordination function with power saving mode. The throughput evaluation is based on the model, with the comparison with the simulation result for validation. Furthermore, we notice that after each beacon transmission, there're more stations in contention than usual. Excessive contention results in high collision probability and low throughput. To solve this problem, we propose a downlink access scheme which can be used in both the basic access scheme and the access scheme with RTS CTS. This novel scheme enables AP to constrain the number of stations in contention to be an optimal value. Simulation results prove the performance improvement declared. The proposed scheme can be extended for multi-rate or multi-service WLANs. We formulate some simple conditions under which a Markov chain may be approximated by the solution to a differential equation, with quantifiable error probabilities. The role of a choice of coordinate functions for the Markov chain is emphasised. The general theory is illustrated in three examples: the classical stochastic epidemic, a population process model with fast and slow variables, and core-finding algorithms for large random hypergraphs. We consider the problem of storing segments of encoded versions of content files in a set of base stations located in a communication cell. These base stations work in conjunction with the main base station of the cell. Users move randomly across the space based on a discrete-time Markov chain model. At each time slot each user accesses a single base station based on it's current position and it can download only a part of the content stored in it, depending on the time slot duration. We assume that file requests must be satisfied within a given time deadline in order to be successful. If the amount of the downloaded (encoded) data by the accessed base stations when the time deadline expires does not suffice to recover the requested file, the main base station of the cell serves the request. Our aim is to find the storage allocation that minimizes the probability of using the main base station for file delivery. This problem is intractable in general. However, we show that the optimal solution of the problem can be efficiently attained in case that the time deadline is small. To tackle the general case, we propose a distributed approximation algorithm based on large deviation inequalities. Systematic experiments on a real world data set demonstrate the effectiveness of our proposed algorithms. Index Terms: Mobility-aware Caching, Markov Chain, MDS Coding, Small-cell Networks A memory model specifies a correctness requirement for a distributed shared memory protocol. Sequential consistency (SC) is the most widely researched model; previous work citealur1996 has shown that, in general, the SC verification problem is undecidable. We identify two aspects of the formulation found in citealur1996 that we consider to be highly unnatural; we call these non-prefix-closedness and prophetic inheritance. We conjecture that preclusion of such behavior yields a decidable version of SC, which we call decisive sequential consistency (DSC). We also introduce a structure called a phview window (VW), which retains information about a protocol's history, and we define the notion of a phVW-bound, which essentially bounds the size of the VWs needed to maintain DSC. We prove that the class of DSC protocols with VW-bound k is decidable; left conjectured is the hypothesis that all DSC protocols have such a bound, and further that the bound is computable from the protocol description. This hypothesis is true for all real protocols known to us; we verify its truth for the Lazy Caching protocol citeafek1993. When solving consensus optimization problems over a graph, there is often an explicit characterization of the convergence rate of Gradient Descent (GD) using the spectrum of the graph Laplacian. The same type of problems under the Alternating Direction Method of Multipliers (ADMM) are, however, poorly understood. For instance, simple but important non-strongly-convex consensus problems have not yet being analyzed, especially concerning the dependency of the convergence rate on the graph topology. Recently, for a non-strongly-convex consensus problem, a connection between distributed ADMM and lifted Markov chains was proposed, followed by a conjecture that ADMM is faster than GD by a square root factor in its convergence time, in close analogy to the mixing speedup achieved by lifting several Markov chains. Nevertheless, a proof of such a claim is is still lacking. Here we provide a full characterization of the convergence of distributed over-relaxed ADMM for the same type of consensus problem in terms of the topology of the underlying graph. Our results provide explicit formulas for optimal parameter selection in terms of the second largest eigenvalue of the transition matrix of the graph's random walk. Another consequence of our results is a proof of the aforementioned conjecture, which interestingly, we show it is valid for any graph, even the ones whose random walks cannot be accelerated via Markov chain lifting. This paper introduces a novel technique for access by a cognitive Secondary User (SU) to a spectrum with an incumbent Primary User (PU), which uses Type-I Hybrid ARQ. The technique allows the SU to perform selective retransmissions of previously corrupted SU data packets. The temporal redundancy introduced by the primary ARQ protocol and by the selective SU retransmission process can be exploited by the SU receiver to perform Interference Cancellation (IC) over the entire interference pattern, thus creating a "clean" channel for the decoding of the concurrent message. The chain decoding technique, initiated by a successful decoding operation of a SU or PU message, consists in the iterative application of IC, as previously corrupted messages become decodable. Based on this scheme, we design an optimal policy that maximizes the SU throughput under a constraint on the average long-term PU throughput degradation. We show that the optimal policy can be found by first optimizing the SU access policy using a Markov Decision Process formulation, and then applying a chain decoding protocol defined by five basic rules. Such an approach enables a compact state representation of the protocol, and its efficient numerical optimization. Finally, we show by numerical results the throughput benefit of the proposed technique. Received 28 May 2002; revised 18 September 2002; accepted 24 September 2002; published 14 February 2003. [1] When an inverse problem is solved to estimate an unknown function such as the hydraulic conductivity in an aquifer or the contamination history at a site, one constraint is that the unknown function is known to be everywhere nonnegative. In this work, we develop a statistically rigorous method for enforcing function nonnegativity in Bayesian inverse problems. The proposed method behaves similarly to a Gaussian process with a linear variogram (i.e., unrestricted Brownian motion) for parameter values significantly greater than zero. The method uses the method of images to define a prior probability density function based on reflected Brownian motion that implicitly enforces nonnegativity. This work focuses on problems where the unknown is a function of a single variable (e.g., time). A Markov chain Monte Carlo (MCMC) method, specifically, a highly efficient Gibbs sampler, is implemented to generate conditional realizations of the unknown function. The new method is applied to the estimation of the trichloroethylene (TCE) and perchloroethylene (PCE) contamination history in an aquifer at Dover Air Force Base, Delaware, based on concentration profiles obtained from an underlying aquitard. INDEX TERMS: 1831 Hydrology: Groundwater quality; 1869 Hydrology: Stochastic processes; 3260 Mathematical Geophysics: Inverse theory; KEYWORDS: stochastic inverse modeling, contaminant source identification, inference under constraints, Markov chain Monte Carlo (MCMC), Gibbs sampling, Bayesian inference 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. In this paper, we develop a rigorous, unified framework based on ordinary differential equations (ODEs) to study epidemic routing and its variations. These ODEs can be derived as limits of Markovian models under a natural scaling as the number of nodes increases. While an analytical study of Markovian models is quite complex and numerical solution impractical for large networks, the corresponding ODE models yield closed-form expressions for several performance metrics of interest, and a numerical solution complexity that does not increase with the number of nodes. Using this ODE approach, we investigate how resources such as buffer space and the number of copies made for a packet can be traded for faster delivery, illustrating the differences among various forwarding and recovery schemes considered. We perform model validations through simulation studies. Finally we consider the effect of buffer management by complementing the forwarding models with Markovian and fluid buffer models. We consider the problem of resource allocation in downlink OFDMA systems for multi service and unknown environment. Due to users' mobility and intercell interference, the base station cannot predict neither the Signal to Noise Ratio (SNR) of each user in future time slots nor their probability distribution functions. In addition, the traffic is bursty in general with unknown arrival. The probability distribution functions of the SNR, channel state and traffic arrival density are then unknown. Achieving a multi service Quality of Service (QoS) while optimizing the performance of the system (e.g. total throughput) is a hard and interesting task since it depends on the unknown future traffic and SNR values. In this paper we solve this problem by modeling the multiuser queuing system as a discrete time linear dynamic system. We develop a robust H∞ controller to regulate the queues of different users. The queues and Packet Drop Rates (PDR) are controlled by proposing a minimum data rate according to the demanded service type of each user. The data rate vector proposed by the controller is then fed as a constraint to an instantaneous resource allocation framework. This instantaneous problem is formulated as a convex optimization problem for instantaneous subcarrier and power allocation decisions. Simulation results show small delays and better fairness among users. 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.