# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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## Quality control for Markov chains and free boundary problemsHTML articles powered by AMS MathViewer

by Robert F. Anderson and Avner Friedman
Trans. Amer. Math. Soc. 246 (1978), 77-94 Request permission

## Abstract:

A machine can manufacture any one of n Markov chains $P_x^{{\lambda _j}} \left ( {1 \leq j \leq n} \right )$; the $P_x^{{\lambda _j}}$ are defined on the space of all sequences $x = \left \{ {x\left ( m \right )} \right \} \left ( {1 \leq m \leq \infty } \right )$ and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains $P_x^{{\lambda _j}}$ evolve in a random way, dictated by a Markov chain $\theta \left ( m \right )$ with n states, so that when $\theta \left ( m \right ) = j$ the machine is producing $P_x^{{\lambda _j}}$. One observes the $\sigma$-fields of $x\left ( m \right )$ in order to determine when to inspect $\theta \left ( m \right )$. With each product $P_x^{{\lambda _j}}$ there is associated a cost ${c_j}$. One inspects $\theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta = n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced, in this paper, to solving a certain free boundary problem. In case $n = 2$ the latter problem is solved.
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