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The expected discounted reward from a Markov replacement process. (English) Zbl 0616.60089

Let \(\{X_ t\), \(t\geq 0\}\) be a homogeneous Markov process with state space \(I=\{1,...,r\}\). Its development can be influenced by the controller’s action called replacement. A stationary replacement policy f is the prescription to realize instantaneously the replacement \(j\to f(j)\) whenever the transition in state \(j\in I_ f\subset I\) occurs. No replacements are made in states \(j\not\in I_ f\). \(E^ f_ j\) denotes the mathematical expectation in a replacement process under the stationary replacement policy f and under the condition j being the initial state.
Let \(R_ T\) be the reward from the process up to the time T, \(R=\int^{\infty}_{0}e^{-\lambda T}dR(T)\), \(\lambda >0\), the discounting of the reward. The article derives a system of equations for establishing the expected discounted reward \(E^ f_ jR=\vartheta_ f(j)\), \(j=1,...,r\), and the uniqueness of its solution is proved.
R. A. Howard’s iteration procedure [Dynamic programming and Markov processes. (1960; Zbl 0091.16001)] for finding the maximal expected discounted reward \[ {\hat \vartheta}(j)=\max_{f}\{\vartheta_ f(j): f\text{ stationary replacement policy\}, }j\in I, \] and the responsive optimal stationary replacement policy is described.

MSC:

60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)

Citations:

Zbl 0091.16001
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References:

[1] Howard R. A.: Dynamic programming and Markov Processes. M. I. T. Press and John Wiley, New York-London (1960). · Zbl 0091.16001
[2] Mandl P.: An identity for Markovian replacement processes. J. Appl. Prob. 6, No. 2, 348-354 (1969). · Zbl 0192.55203
[3] Mandl P.: Řízené Markovovy řetězce. příloha časopisu Kybernetika, roč. 5, Academia Praha (1969).
[4] Sarymsakov T. A.: Osnovy teorii processov Markova. Moskva (1954).
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