Quasi-Coordinate Search for a Randomly Moving Target

In this paper, we study the quasi-coordinated search technique for a lost target assumed to move randomly on one of two disjoint lines according to a random walk motion, where there are two searchers beginning their search from the origin on the first line and other two searchers begin their search from the origin on the second line. But the motion of the two searchers on the first line is independent from the motion of the other two searchers on the second line. Here we introduce a model of search plan and investigate the expected value of the first meeting time between one of the searchers and the lost target. Also, we prove the existence of a search plan which minimizes the expected value of the first meeting time between one of the searchers and the target.


Introduction
The searching for a lost target either located or moved is often a time-critical issue, that is, when the target is very important. The primary objective is to find and search for the lost target as soon as possible. The searching for lost targets has recently applications such as the search for a goldmine underground, the search for Landmines and navy mines, the search for the cancer cells in the human body, the search for missing black box of a plane crash in the depth of the sea of ocean, the search for a damaged unit in a large linear system such as telephone lines, and mining system, and so on [1] [2] [3]. Search problem when the We assume the searchers S 1 and S 2 begin their search path from O 1 on L 1 with speeds V 1 , and the searchers S 3 and S 4 begin their search path from O 2 on L 2 with speeds V 2 , following the search paths which are functions 1 and ( ) ( ) ( ) ( ) where V 1 and V 2 are constants in R + and ( ) ( ) ( ) ( ) Let the set of all search paths of the two searchers S 1 and S 2 , which satisfy condition (1), be respectively by The search plan of the four searchers be represented by is the set of all search plan.
We assume that 0 Z X = if the target moves on L 1 and 0 There is a known probabili- where v 1 is probability measure induced by the position of the target on L 1 , while v 2 on L 2 . The first meeting time valued in I + defined as where Z 0 is a random variable representing the initial position of the target and valued in 2I (or 2 1 I + ) and independent of ( ), 0 At the beginning of the search suppose that the lost target is existing on any integer point on L 1 but more than H 1 or less than In this case φ is said to be a finite search plan, and if ( ) ( ) where E terms to expectation value, then we call * φ is an optimal search plan.
Given 0 n > , if z is: We shall define the search path as follows: We define the notion on L 2 , the searchers S 1 and S 2 return to the origin of L 1 after searching successively common distances 1 2 3 , , , H H H  , and are finite.
Proof: Assume that X and Y are independent of ( ), 0 until the first meeting between S 1 and the target on L 1 , also if 0 X < , then until the first meeting between S 2 and the target on L 2 . We can apply this assumption on the second line by replacing X by Y and 1 1 , φ φ by 2 2 , φ φ respectively. Hence, for any to solve Equation (4) we shall find the value of ( ) We get substituting by (5), (6), (7) and (8) in (4) satisfies the condition of the renewal equation, for more details see [19].

Conclusions
We have described a new kind of search technique to find a lost moving target on one of two disjoint lines. The motion of the four searchers on the two lines in the quasi-coordinated search technique is independent, and this helps us to find the lost target without waste of time and cost, especially if this target is valuable as the search for lost children. Actually we calculated the finite search plan. Also; we proved the existence of an optimal search plan which minimizes the expected value of the first meeting time between one of the searchers and the target.
In the future work, we will introduce an important search problem, looking for a randomly moving target as a general case and the searchers will begin their mission from any point on the line.