Quickest detection of a minimum of two Poisson disorder times

A multisource quickest detection problem is considered. Assume there are two independent Poisson processes X-1 and X-2 with disorder times theta(1) and theta(2), respectively; i.e., the intensities of X-1 and X-2 change at random unobservable times theta(1) and theta(2), respectively theta(1) and theta(2) are independent of each other and are exponentially distributed. Define theta Delta(=) theta 1 boolean AND theta(2) = min{theta(1),theta(2)}. For any stopping time tau that is measurable with respect to the filtration generated by the observations, define a penalty function of the form R tau = P(tau < theta) + cE [(tau-theta)(+)], where c > 0 and (tau -theta)(+) is the positive part of tau - theta. It is of interest to find a stopping time tau that minimizes the above performance index. This performance criterion can be useful, e.g., in the following scenario: There are two assembly lines that produce products A and B, respectively. Assume that the malfunctioning (disorder) of the machines producing A and B are independent events. Later, the products A and B are to be put together to obtain another product C. A product manager who is worried about the quality of C will want to detect the minimum of the disorder times (as accurately as possible) in the assembly lines producing A and B. Another problem to which we can apply our framework is the Internet surveillance problem: A router receives data from, say, n channels. The channels are independent, and the disorder times of channels are theta(1),...,theta(n). The router is said to be under attack at theta = theta(1)boolean AND...boolean AND theta(n). The administrator of the router is interested in detecting as quickly as possible. Since both observations X-1 and X-2 reveal information about the disorder time theta, even this simple problem is more involved than solving the disorder problems for X-1 and X-2 separately. This problem is formulated in terms of a three-dimensional sufficient statistic, and the corresponding optimal stopping problem is examined. The solution is characterized by iterating a suitable functional operator.