Quickest Detection of a Minimum of Two Poisson Disorder Times

A multisource quickest detection problem is considered. Assume there are two independent Poisson processes X¹ and X² with disorder times θ₁ and θ₂, respectively; i.e., the intensities of X¹ and X² 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)+c E[(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, 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 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 X¹ and X² reveal information about the disorder time θ, even this simple problem is more involved than solving the disorder problems for X¹ and X² 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.