## Quality control for Markov chains and free boundary problems

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- by Robert F. Anderson and Avner Friedman PDF
- 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.

## References

- Robert F. Anderson and Avner Friedman,
*A quality control problem and quasi-variational inequalities*, Arch. Rational Mech. Anal.**63**(1976/77), no. 3, 205–252. MR**456857**, DOI 10.1007/BF00251581 - Robert F. Anderson and Avner Friedman,
*Multidimensional quality control problems and quasivariational inequalities*, Trans. Amer. Math. Soc.**246**(1978), 31–76. MR**515529**, DOI 10.1090/S0002-9947-1978-0515529-8 - Daniel W. Stroock and S. R. S. Varadhan,
*Diffusion processes with continuous coefficients. I*, Comm. Pure Appl. Math.**22**(1969), 345–400. MR**253426**, DOI 10.1002/cpa.3160220304

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**246**(1978), 77-94 - MSC: Primary 93E20; Secondary 49A29, 62N10
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515530-4
- MathSciNet review: 515530