The possibility of flexibly assigning spectrum resources with channels of different sizes greatly improves the spectral efficiency of optical networks, but can also lead to unwanted spectrum fragmentation. We study this problem in a scenario where traffic demands are categorized in two types (low or high bit-rate) by assessing the performance of three allocation policies. Our first contribution consists of exact Markov chain models for these allocation policies, which allow us to numerically compute the relevant performance measures. However, these exact models do not scale to large systems, in the sense that the computations required to determine the blocking probabilities—which measure the performance of the allocation policies—become intractable. In order to address this, we first extend an approximate reduced-state Markov chain model that is available in the literature to the three considered allocation policies. These reduced-state Markov chain models allow us to tractably compute approximations of the blocking probabilities, but the accuracy of these approximations cannot be easily verified. Our main contribution then is the introduction of reduced-state imprecise Markov chain models that allow us to derive guaranteed lower and upper bounds on blocking probabilities, for the three allocation policies separately or for all possible allocation policies simultaneously.
A general introduction to the theory of imprecise probabilities can be found in @cite_2 . This theory has been used to generalise both discrete-time and continuous-time MCs, the latter more recently than the former. A basic treatment of imprecise continuous-time MCs can be found in @cite_4 @cite_30 @cite_3 @cite_7 @cite_26 . For their discrete-time counterpart, see for example @cite_12 @cite_29 .
@cite_30 @cite_26 @cite_4 @cite_7 @cite_29 @cite_3 @cite_2 @cite_12
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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. This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities.In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research. We present a new Markov Chain Monte Carlo (MCMC) sampling algorithm for probabilistic programs. Our approach and tool, called R2, has the unique feature of employing program analysis in order to improve the efficiency of MCMC sampling. Given an input program P, R2 propagates observations in P backwards to obtain a semantically equivalent program P′ in which every probabilistic assignment is immediately followed by an observe statement. Inference is performed by a suitably modified version of the Metropolis-Hastings algorithm that exploits the structure of the program P′. This has the overall effect of preventing rejections due to program executions that fail to satisfy observations in P. We formalize the semantics of probabilistic programs and rigorously prove the correctness of R2. We also empirically demonstrate the effectiveness of R2--in particular, we show that R2 is able to produce results of similar quality as the CHURCH and STAN probabilistic programming tools with much shorter execution time. We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)