Quickest Detection of a Minimum of Two Poisson Disorder Times

A multi-source quickest detection problem is considered. Assume there are two independent Poisson processes X and X with disorder times θ1 and θ2, respectively; that is, the intensities of X and X change at random unobservable times θ1 and θ2, respectively. θ1 and θ2 are independent of each other and are exponentially distributed. Define θ , θ1 ∧ θ2 = min{θ1, θ2} . For any stopping time τ that is measurable with respect to the filtration generated by the observations define a penalty function of the form Rτ = P(τ < θ) + cE [ (τ − θ) ] , where c > 0 and (τ − θ) is the positive part of τ − θ. It is of interest to find a stopping time τ that minimizes the above performance index. This performance criterion can be useful for example 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 θ1, · · · , θn. The router is said to be under attack at θ = θ1 ∧ · · · ∧ θn. The administrator of the router is interested in detecting θ as quickly as possible. Since both observations X and X reveal information about the disorder time θ, even this simple problem is more involved than solving the disorder problems for X and X 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.