Quickest detection of a minimum of two poisson disorder times

Erhan Bayraktar, H. Vincent Poor

Research output: Contribution to journalArticle

13 Scopus citations

Abstract

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 delta equal to sign θ1 Λ θ2 = min{θ1, θ2}- For any stopping time T that is measurable with respect to the filtration generated by the observations, define a penalty function of the form Rτ = double-strock P sign (τ < θ) + c double-strock E sign [(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 Λ⋯Λ θn. The administrator of the router is interested in detecting θ as quickly as possible. Since both observations X1 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 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.

Original languageEnglish (US)
Pages (from-to)308-331
Number of pages24
JournalSIAM Journal on Control and Optimization
Volume46
Issue number1
DOIs
StatePublished - Dec 1 2007

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics

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