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.
There are a few works analyzing the backoff mechanism of 1901 that rely on the decoupling assumption. First, @cite_16 introduce a model using a discrete-time Markov chain similar to Bianchi's model for 802.11 @cite_18 . The additional state required to capture the effect of the deferral counter @math significantly increases the complexity of the Markov chain.
@cite_18 @cite_16
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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. This paper proposes the analytical expressions for the IEEE 802.11 string-topology multi-hop networks using Markov-chain model. For achieving that, the proposed analysis procedure includes two proposals, which are: (i) Bianchi's Markov-chain model is modified for considering a relationship between the backoff timer and frame length, and (ii) the interferences, such as hidden node collision and carrier sensing, among network nodes are expressed by merging the proposed Markov-chain model and airtime expression. The analytical expressions are verified by comparisons with simulation results. We investigate the IEEE 1901 MAC protocol, the dominant protocol for high data rate power-line communications. 1901 employs a CSMA CA mechanism similar to - but much more complex than - the backoff mechanism of 802.11. Because of this extra complexity, and although this mechanism is the only widely used MAC layer for power-line networks, there are few analytical results on its performance. We propose a model for the 1901 MAC that comes in the form of a single fixed-point equation for the collision probability. We prove that this equation admits a unique solution, and we evaluate the accuracy of our model by using simulations. The IEEE 802.11n standard aims at providing a data transmission rate of up to 600 Mbps. The frame aggregation techniques of IEEE 802.11n MAC layer play an important role in the enhancement of channel utilization. In this paper, we investigate the effect of IEEE 802.11n frame aggregation for two aggregation mechanisms: MAC protocol data unit aggregation (A-MPDU) and MAC service data unit aggregation (A-MSDU). We propose an analytical model based on an enhanced discrete time Markov chain (DTMC) model in order to describe the post- backoff behavior due to frame aggregation, and then evaluate the throughput performance. Analytical results based on the mathematical model are verified through simulation. The throughput performance of the A-MSDU mechanism outperforms the A-MPDU mechanism as the frame aggregation size becomes larger under an error-free environment. In addition, the throughput performance is significantly improved as the frame aggregation size increases. This letter presents a new approach to evaluate the throughput delay performance of the 802.11 distributed coordination function (DCF). Our approach relies on elementary conditional probability arguments rather than bidimensional Markov chains (as proposed in previous models) and can be easily extended to account for backoff operation more general than DCF's one. We prove that under mild positivity assumptions, the entropy rate of a continuous-state hidden Markov chain, observed when passing a finite-state Markov chain through a discrete-time continuous-output channel, is analytic as a function of the transition probabilities of the underlying Markov chain. We further prove that the entropy rate of a continuous-state hidden Markov chain, observed when passing a mixing finite-type constrained Markov chain through a discrete-time Gaussian channel, is smooth as a function of the transition probabilities of the underlying Markov chain. 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. Power-line communications are employed in home networking to provide easy and high-throughput connectivity. IEEE 1901, the MAC protocol for power-line networks, employs a CSMA CA protocol similar to that of 802.11, but is substantially more complex, which probably explains why little is known about its performance. One of the key differences between the two protocols is that whereas 802.11 only reacts upon collisions, 1901 also reacts upon several consecutive transmissions and thus can potentially achieve better performance by avoiding unnecessary collisions. In this paper, we propose a model for the 1901 MAC. Our analysis reveals that the default configuration of 1901 does not fully exploit its potential and that its performance degrades with the number of stations. We derive analytically the optimal configuration parameters for 1901; this drastically improves throughput and achieves optimal performance without requiring the knowledge of the number of stations in the network. In contrast, to provide a similar performance, 802.11 requires knowing the number of contending stations, which is unfeasible for realistic traffic patterns. Our solution can be readily implemented by vendors, as it only consists in modifying existing MAC parameters. We corroborate our results with testbed measurements, unveiling a significant signaling overhead in 1901 implementations. Let @math . We show that that the mixing time of a continuous-time Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of the state space with stationary measure @math . Suitably modified results hold in discrete time and or without the reversibility assumption. The key technical tool in the proof is the construction of random set @math such that the hitting time of @math is a light-tailed stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi. We analyze and solve a game in which a player chooses which of several Markov chains to advance, with the object of minimizing the expected time (or cost) for one of the chains to reach a target state. The solution entails computing (in polynomial time) a function @math ---a variety of "Gittins index"---on the states of the individual chains, the minimization of which produces an optimal strategy. It turns out that @math is a useful cousin of the expected hitting time of a Markov chain but is defined, for example, even for random walks on infinite graphs. We derive the basic properties of @math and consider its values in some natural situations. In this paper, we develop a method for analyzing time-varying wireless channels in the context of the modern theory of the stochastic network calculus. In particular, our technique is applicable to channels that can be modeled as Markov chains, which is the case of channels subject to Rayleigh fading. Our approach relies on theoretical results on the convergence time of reversible Markov processes and is applicable to chains with an arbitrary number of states. We provide two expressions for the delay tail distribution of traffic transmitted over a fading channel fed by a Markov source. The first expression is tighter and only requires a simple numerical minimization, the second expression is looser, but is in closed form. Binary Exponential Backoff (BEB) is a key component of the IEEE 802.11 DCF protocol. It has been shown that BEB can achieve the theoretical limit of throughput as long as the initial backoff window size is properly selected. It, however, suffers from significant delay degradation when the network becomes saturated. It is thus of special interest for us to further design backoff schemes for IEEE 802.11 DCF networks that can achieve comparable throughput as BEB, but provide better delay performance. This paper presents a systematic study on the effect of backoff schemes on throughput and delay performance of saturated IEEE 802.11 DCF networks. In particular, a backoff scheme is defined as a sequence of backoff window sizes @math . The analysis shows that a saturated IEEE 802.11 DCF network has a single steady-state operating point as long as @math is a monotonic increasing sequence. The maximum throughput is found to be independent of @math , yet the growth rate of @math determines a fundamental tradeoff between throughput and delay performance. For illustration, Polynomial Backoff is proposed, and the effect of polynomial power @math on the network performance is characterized. It is demonstrated that Polynomial Backoff with a larger @math is more robust against the fluctuation of the network size, but in the meanwhile suffers from a larger second moment of access delay. Quadratic Backoff (QB), i.e., Polynomial Backoff with @math , stands out to be a favorable option as it strikes a good balance between throughput and delay performance. The comparative study between QB and BEB confirms that QB well preserves the robust nature of BEB and achieves much better queueing performance than BEB. 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. Perturbation theory for Markov chains addresses the question how small differences in the transitions of Markov chains are reflected in differences between their distributions. We prove powerful and flexible bounds on the distance of the @math th step distributions of two Markov chains when one of them satisfies a Wasserstein ergodicity condition. Our work is motivated by the recent interest in approximate Markov chain Monte Carlo (MCMC) methods in the analysis of big data sets. By using an approach based on Lyapunov functions, we provide estimates for geometrically ergodic Markov chains under weak assumptions. In an autoregressive model, our bounds cannot be improved in general. We illustrate our theory by showing quantitative estimates for approximate versions of two prominent MCMC algorithms, the Metropolis-Hastings and stochastic Langevin algorithms. IEEE 802.11 is the standard for wireless local area networks (WLANs) promoted by the Institute of Electrical and Electronics Engineers. Wireless technologies in the LAN environment are becoming increasingly important and the IEEE 802.11 is the most mature technology to date. Previous works have pointed out that the standard protocol can be very inefficient and that an appropriate tuning of its congestion control mechanism (i.e., the backoff algorithm) can drive the IEEE 802.11 protocol close to its optimal behavior. To perform this tuning, a station must have exact knowledge of the network contention level; unfortunately, in a real case, a station cannot have exact knowledge of the network contention level (i.e., number of active stations and length of the message transmitted on the channel), but it, at most, can estimate it. We present and evaluate a distributed mechanism for contention control in IEEE 802.11 wireless LANs. Our mechanism, named asymptotically optimal backoff (AOB), dynamically adapts the backoff window size to the current network contention level and guarantees that an IEEE 802.11 WLAN asymptotically achieves its optimal channel utilization. The AOB mechanism measures the network contention level by using two simple estimates: the slot utilization and the average size of transmitted frames. These estimates are simple and can be obtained by exploiting information that is already available in the standard protocol. AOB can be used to extend the standard 802.11 access mechanism without requiring any additional hardware. The performance of the IEEE 802.11 protocol, with and without the AOB mechanism, is investigated through simulation. Simulation results indicate that our mechanism is very effective, robust, and has traffic differentiation potentialities. Formal verification is used to establish the compliance of software and hardware systems with important classes of requirements. System compliance with functional requirements is frequently analyzed using techniques such as model checking, and theorem proving. In addition, a technique called quantitative verification supports the analysis of the reliability, performance, and other quality-of-service (QoS) properties of systems that exhibit stochastic behavior. In this paper, we extend the applicability of quantitative verification to the common scenario when the probabilities of transition between some or all states of the Markov models analyzed by the technique are unknown, but observations of these transitions are available. To this end, we introduce a theoretical framework, and a tool chain that establish confidence intervals for the QoS properties of a software system modelled as a Markov chain with uncertain transition probabilities. We use two case studies from different application domains to assess the effectiveness of the new quantitative verification technique. Our experiments show that disregarding the above source of uncertainty may significantly affect the accuracy of the verification results, leading to wrong decisions, and low-quality software systems. We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of σ-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365―393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space. Analytical modeling of the 802.11e enhanced distributed channel access (EDCA) mechanism is today a fairly mature research area, considering the very large number of papers that have appeared in the literature. However, most work in this area models the EDCA operation through per-slot statistics, namely probability of transmission and collisions referred to "slots." In so doing, they still share a methodology originally proposed for the 802.11 Distributed Coordination Function (DCF), although they do extend it by considering differentiated transmission collision probabilities over different slots.We aim to show that it is possible to devise 802.11e models that do not rely on per-slot statistics. To this purpose, we introduce and describe a novel modeling methodology that does not use per-slot transmission collision probabilities, but relies on the fixed-point computation of the whole (residual) backoff counter distribution occurring after a generic transmission attempt. The proposed approach achieves high accuracy in describing the channel access operations, not only in terms of throughput and delay performance, but also in terms of low-level performance metrics. Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case. We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods. The IEEE 802.11 standard for ad hoc networks uses a distributed access mechanism in the shared medium that attempts to avoid collisions by performing carrier sensing, inter-frame spaces, and a backoff mechanism. This paper aims at deriving an analytical expression for the maximum throughput of a communication between two nodes. The communication is achieved in a multi-hop scenario and the path from the source to the destination consists of a chain of nodes. We analyze the multi-path communication and show that the maximum throughput can be increased with the simultaneous use of two paths: the shortest path and an appropriate alternative path that takes into account the interference problem. We also derive the constraints for this alternative path. The use of multiple alternative paths can be considered in order to achieve a load and energy balancing.