### Abstract

A multisource quickest detection problem is considered. Assume there are two independent Poisson processes X^{1} and X^{2} with disorder times θ_{1} and θ_{2}, respectively; i.e., the intensities of X^{1} and X^{2} change at random unobservable times θ_{1} and θ_{2}, respectively. θ_{1} and θ_{2} are independent of each other and are exponentially distributed. Define θ delta equal to sign θ_{1} Λ θ_{2} = min{θ_{1}, θ_{2}}- For any stopping time T that is measurable with respect to the filtration generated by the observations, define a penalty function of the form R_{τ} = 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 θ_{1},..., θ_{n}. The router is said to be under attack at θ = θ_{1} Λ⋯Λ θ_{n}. The administrator of the router is interested in detecting θ 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.

Original language | English (US) |
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Pages (from-to) | 308-331 |

Number of pages | 24 |

Journal | SIAM Journal on Control and Optimization |

Volume | 46 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2007 |

### All Science Journal Classification (ASJC) codes

- Control and Optimization
- Applied Mathematics

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## Cite this

*SIAM Journal on Control and Optimization*,

*46*(1), 308-331. https://doi.org/10.1137/050630933