Controlled jump Markov processes with local transitions and their fluid approximation
WSEAS Transactions on Systems and Control
Convergence of trajectories and optimal buffer sizing for AIMD congestion control
Performance Evaluation
Mathematics of Operations Research
SIAM Journal on Control and Optimization
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Such processes are widely used in Engineering Sciences, Biology, Insurance, Inventory Theory and so on. We present a series of meaningful examples and describe rigorous mathematical models. One of the most powerful methods of obtaining an optimal control strategy is Dynamic Programming. Another modern method of attack is so called Convex Analytic Approach. In this report, the both methods will be briefly discussed. The second half of the article will be devoted to more specific models. First, we concentrate on the processes with local transitions, like birth-and-death processes. They are known to be successfully approximated by deterministic differential equations under the so called 'fluid scaling'. In this connection, a new look at the well known 'µC-rule' in the Queuing Theory will be discussed: we shall compare the stochastic and deterministic versions. Finally, we shall stop on applications of the theory to the optimal buffer sizing for Internet routers.