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]1 .