A fundamental problem in the delay and backlog analysis across multi-hop paths in wireless networks is how to account for the random properties of the wireless channel. Since the usual statistical models for radio signals in a propagation environment do not lend themselves easily to a description of the available service rate on a wireless link, the performance analysis of wireless networks has resorted to higher-layer abstractions, e.g., using Markov chain models. In this work, we propose a network calculus that can incorporate common statistical models of fading channels and obtain statistical bounds on delay and backlog across multiple nodes. We conduct the analysis in a transfer domain, which we refer to as the SNR domain', where the service process at a link is characterized by the instantaneous signal-to-noise ratio at the receiver. We discover that, in the transfer domain, the network model is governed by a dioid algebra, which we refer to as (min,x)-algebra. Using this algebra we derive the desired delay and backlog bounds. An application of the analysis is demonstrated for a simple multi-hop network with Rayleigh fading channels and for a network with cross traffic.
Analytical approaches for network-layer performance analysis of wireless networks include queueing theory, effective bandwidth and, more recently, network calculus. Since the service processes corresponding to the channel capacity of common fading channel models such as Rician, Rayleigh, or Nakagami- @math , require to take a logarithm of their distributions, researchers often turn to higher-layer abstractions to model fading channels, which lend themselves more easily to an analysis. A widely used abstraction is the two-state channel model developed by Gilbert @cite_29 and Elliott @cite_28 , and subsequent extensions to a finite-state Markov channel (FSMC) @cite_33 . Markov channel models are well suited to express the time correlation of fading channel samples. We refer to @cite_3 for a survey of the development and applications of FSMC models. @cite_37 evaluated the accuracy of first-order Markov channel models of fading channels, where the next channel sample depends only on the current state of the Markov process, and higher order processes that can capture memory extending further back in the process history. The authors found that a first-order Markov model is a good approximation of the fading channel, and that using higher order Markov processes does not significantly improve the accuracy of the model.
@cite_37 @cite_33 @cite_28 @cite_29 @cite_3
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The authors first study the behavior of a finite-state channel where a binary symmetric channel is associated with each state and Markov transitions between states are assumed. Such a channel is referred to as a finite-state Markov channel (FSMC). By partitioning the range of the received signal-to-noise ratio into a finite number of intervals, FSMC models can be constructed for Rayleigh fading channels. A theoretical approach is conducted to show the usefulness of FSMCs compared to that of two-state Gilbert-Elliott channels. The crossover probabilities of the binary symmetric channels associated with its states are calculated. The authors use the second-order statistics of the received SNR to approximate the Markov transition probabilities. The validity and accuracy of the model are confirmed by the state equilibrium equations and computer simulation. > The error structure on communication channels used for data transmission may be so complex as to preclude the feasibility of accurately predicting the performance of given codes when employed on these channels. Use of an approximate error rate as an estimate of performance allows the complex statistics of errors to be reduced to a manageable table of parameters and used in an economical evaluation of large collections of error detecting codes. Exemplary evaluations of error detecting codes on the switched telephone network are included in this paper. On channels which may be represented by Gilbert's model of a burst-noise channel, the probabilities of error or of retransmission may be calculated without approximations for both error correcting and error detecting codes A model of a burst-noise binary channel uses a Markov chain with two states G and B. In state G, transmission is error-free. In state B, the channel has only probability h of transmitting a digit correctly. For suitably small values of the probabilities, p, P of the B @math rA B transitions, the model simulates burst-noise channels. Probability formulas relate the parameters p, P, h to easily measured statistics and provide run distributions for comparison with experimental measurements. The capacity C of the model channel exceeds the capacity C(sym. bin.) of a memoryless symmetric binary channel with the same error probability. However, the difference is slight for some values of h, p, P; then, time-division encoding schemes may be fairly efficient. This article's goal is to provide an in-depth understanding of the principles of FSMC modeling of fading channels with its applications in wireless communication systems. While the emphasis is on frequency nonselective or flat-fading channels, this understanding will be useful for future generalizations of FSMC models for frequency-selective fading channels. The target audience of this article include both theory- and practice-oriented researchers who would like to design accurate channel models for evaluating the performance of wireless communication systems in the physical or media access control layers, or those who would like to develop more efficient and reliable transceivers that take advantage of the inherent memory in fading channels. Both FSMC models and flat-fading channels will be formally introduced. FSMC models are particulary suitable to represent and estimate the relatively fast flat-fading channel gain in each subcarrier.