Quickest Detection of a Minimum of Two Poisson Disorder Times

Abstract

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 $\theta \triangleq \theta_1 \wedge \theta_2=\min\{\theta_{1},\theta_{2}\}$. For any stopping time τ that is measurable with respect to the filtration generated by the observations, define a penalty function of the form $R_{\tau}=\mathbb{P}(\tau<\theta)+c \mathbb{E}\left[(\tau-\theta)^{+}\right],$ where $c>0$ and $(\tau-\theta)^{+}$ is the positive part of $\tau-\theta$. It is of interest to find a stopping time τ 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 $\theta_1, \ldots, \theta_n$. The router is said to be under attack at $\theta=\theta_1 \wedge \cdots \wedge \theta_n$. The administrator of the router is interested in detecting $\theta$ as quickly as possible. Since both observations $X^{1}$ and $X^{2}$ reveal information about the disorder time θ, 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.

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