Discrete-Time Markov Decision Processes in .NET

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Discrete-Time Markov Decision Processes
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60 INTRODUCTION
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In the previous chapters we saw that in the analysis of many operational systems the concepts of a state of a system and a state transition are of basic importance For dynamic systems with a given probabilistic law of motion, the simple Markov model is often appropriate However, in many situations with uncertainty and dynamism, the state transitions can be controlled by taking a sequence of actions The Markov decision model is a versatile and powerful tool for analysing probabilistic sequential decision processes with an in nite planning horizon This model is an outgrowth of the Markov model and dynamic programming The latter concept, being developed by Bellman in the early 1950s, is a computational approach for analysing sequential decision processes with a nite planning horizon The basic ideas of dynamic programming are states, the principle of optimality and functional equations In fact dynamic programming is a recursion procedure for calculating optimal value functions from a functional equation This functional equation re ects the principle of optimality, stating that an optimal policy has the property that whatever the initial state and the initial decision, the remaining decisions must constitute an optimal policy with regard to the state resulting from the rst transition This principle is always valid when the number of states and the number of actions are nite At much the same time as Bellman (1957) popularized dynamic programming, Howard (1960) used basic principles from Markov chain theory and dynamic programming to develop a policy-iteration algorithm for solving probabilistic sequential decision processes with an in nite planning horizon In the two decades following the pioneering work of Bellman and Howard, the theory of Markov decision processes has expanded at a fast rate and a powerful technology has developed However, in that period relatively little effort was put into applying the quite useful Markov decision model to practical problems
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A First Course in Stochastic Models HC Tijms c 2003 John Wiley & Sons, Ltd ISBNs: 0-471-49880-7 (HB); 0-471-49881-5 (PB)
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DISCRETE-TIME MARKOV DECISION PROCESSES
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The Markov decision model has many potential applications in inventory control, maintenance, manufacturing and telecommunication among others Perhaps this versatile model will see many more signi cant applications when it becomes more familiar to engineers, operations research analysts, computer science people and others To that end, s 6 and 7 focus on the algorithmic aspects of Markov decision theory and illustrate the wide applicability of the Markov decision model to a variety of realistic problems The presentation is con ned to the optimality criterion of the long-run average cost (reward) per time unit For many applications of Markov decision theory this criterion is the most appropriate optimality criterion The average cost criterion is particularly appropriate when many state transitions occur in a relatively short time, as is typically the case for stochastic control problems in computer systems and telecommunication networks Other criteria are the expected total cost and the expected total discounted cost These criteria are discussed in length in Puterman (1994) and will not be addressed in this book This chapter deals with the discrete-time Markov decision model in which decisions can be made only at xed equidistant points in time The semi-Markov decision model in which the times between the decision epochs are random will be the subject of the next chapter In Section 61 we present the basic elements of the discrete-time Markov decision model A policy-improvement procedure is discussed in Section 62 This procedure is the key to various algorithms for computing an average cost optimal polity The so-called relative values of a given policy play an important role in the improvement procedure The relative values and their interpretation are the subject of Section 63 In Section 64 we present the policy-iteration algorithm which generates a sequence of improved policies Section 65 discusses the linear programming formulation for the Markov decision model, including a formulation to handle probabilistic constraints on the state-action frequencies The policy-iteration algorithm and the linear programming formulation both require the solving of a system of linear equations in each iteration step In Section 66 we discuss the alternative method of value iteration which avoids the computationally burdensome solving of systems of linear equations but involves only recursive computations The value-iteration algorithm endowed with quickly converging lower and upper bounds on the minimal average cost is usually the most effective method for solving Markov decision problems with a large number of states Section 67 gives convergence proofs for the policy-iteration algorithm and the value-iteration algorithm
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