Surveillance problems: Poisson models with noise

The basic model starts with three stochastic processes satisfying y(t) = x(t) + z(t). Observations are made on the y‐process from which inferences are made about the production process, x(t). The z‐process constitutes noise. For each unit of time that x(t) = x the income is i(x). The producer, at a cost in both time and money, can repair the y‐process, i.e., bring its value back to zero. Continuous surveillance at no cost and intermittent surveillance with a fixed cost for each observation are considered, When the x‐ and z‐processes are independent Poisson processes, it is shown that the strategies which maximize the average income per unit of time depend only on the last observed value of the y‐process. Production is continued until the y‐process exceeds a specified integer which depends on the economic parameters. When i(x) is nonincreasing and costly inspections are being made, the time between inspections decreases as a function of the last observed value of the y‐process. The model is a generalization of the model used in Ref. [3] See References, page 13. .