The problem of evaluating compound probability distributions where one of the involved distributions is normal, frequently occurring when modelling communication channels in indoor industrial environments, is considered. Three different methods are investigated. They will here be named Gauss Newton Raphson (GNR), based on the Laplace approximation, Gauss- Hermite Quadratures (GHQ), and a Discrete Convolutional Sum (DCS). These three methods are investigated and compared for a one point problem assuming data that are continuous in amplitude, and a problem where data are assumed to be received in quantized bins. The relative integral approximation error, resulting from computing the compound distribution, is evaluated for the different methods and their complexities are compared. Simulations are provided to illustrate the advantages and disadvantages of the different methods.

The reliability and throughput in an industrial wireless sensor network can be improved by incorporating the predictions of channel gains when forming routing tables. Necessary conditions for such predictions to be useful are that statistical dependences exist between the channel gains and that those dependences extend over a long enough time to accomplish a rerouting. In this paper, we have studied such long-term dependences in channel gains for fixed wireless links in three factories. Long-term fading properties were modeled using a switched regime model, and Bayesian change point detection was used to split the channel gain measurements into segments. In this way, we translated the study of long-term dependences in channel gains into the study of dependences between fading distribution parameters describing the segments. We measured the strengths of the dependences using mutual information and found that the dependences exist in a majority of the examined links. The strongest dependence appeared between mean received power in adjacent segments, but we also found significant dependences between segment lengths. In addition to the study of statistical dependences, we present the summaries of the distribution of the fading parameters extracted from the segments, as well as the lengths of these segments.

Wireless sensor networks (WSNs) are being increasingly used for monitoring and control in modern process industries, which until a few years ago were almost exclusively done by cables. In order to successfully integrate WSNs in process industries, proper understanding of the radio channel characteristics is necessary. Predominantly, in control applications, the sensors are static, and variations in the channel are caused by the objects in the region, which move to different locations every now and then. Hence, over a period of several hours, the radio channel will have completely different properties, which will have a significant impact on network performance. In this paper, we conduct and analyze channel measurements in a fully functional factory over a much longer time horizon of 20 hours, which, to the best of our knowledge, have not been undertaken before. A key observation from our analysis is that the radio channel, over a longer time scale, shows a more complicated behavior than over a relatively shorter time period. We also show that if the long-term channel variations are not effectively characterized, then we run the risk of transmitting at a rate, as low as, 94% less than what the channel could actually support.

One of the challenges for a successful use of wireless sensor networks in process industries is to design networks with energy efficient transmission, to increase the lifetime of the deployed network while maintaining the required latency and bit-error rate. The design of such transmission schemes depend on the radio channel characteristics of the region. This paper presents an investigation of the statistical properties of the radio channel in a typical process industry, particularly when the network is meant to be deployed for a long time duration, e.g., days, weeks, and even months. Using 17-20-h-long extensive measurement campaigns in a rolling mill and a paper mill, we highlight the non-stationarity in the environment and quantify the ability of various distributions, given in the literature, to describe the variations on the links. Finally, we analyze the design of an optimal received signal-to-noise ratio (SNR) for the deployed nodes and show that improper selection of the distribution for modeling of the variations in the channel can lead to an overuse of energy by a factor of four or even higher.

Model selection based on observed data sequences is used to decide between different model structures within the class of multinomial, Markov, and hidden Markov models. In a unified Bayesian treatment, we derive posterior probabilities for different model structures without assuming prior knowledge of transition probabilities. We emphasize the following tests: 1) Given a particular data sequence of n outcomes, is each state equally likely? 2) Do the data support an independent model, or is a Markov model a more plausible description? 3) Are two data sequences generated from a) the same Markov model? b) the same hidden Markov model? For Markov models and independent multinomial models, all results are exact. For hidden Markov models, the exact solution is computationally prohibitive, and instead, an approximate solution is proposed