Optimal Coordinated Search for a Discrete Random Walker

This paper presents the search technique for a lost target. A lost target is random walker on one of two intersected real lines, and the purpose is to detect the target as fast as possible. We have four searchers start from the point of intersection, they follow the so called Quasi-Coordinated search plan. The expected value of the first meeting time between one of the searchers and the target is investigated, also we show the existence of the optimal search strategy which minimizes this first meeting time.


Introduction
The search problem for a randomly moving target is very interesting because it may arise in many real world situations such as searching for lost persons on roads, the cancer cells in the human body and missing black box of a plane crash in the depth of the sea or ocean, also searching for a gold mine underground, Landmines and navy mines, a faulty unit in a large linear system such as electrical power lines, telephone lines, and mining system, and so on (see [1], [2], [3], [4] and [5]).
The aim of search, in many cases (see [6], and [7]) is to calculate the expected cost of detecting the target and is to obtain the search plan, which minimizes this expected cost. In the case of linear search for stationary or randomly moving targets many studies are made (see [8]- [26]).
The coordinated search method is one of the famous search methods which consider the searchers starting together from the origin and moving, seeking for a random walk target. Therefore, coordinated search technique is one of many techniques which studied previously on the line where the located targets have symmetric and unsymmetric distributions (see [27], [28], [29] and [30]), this technique has been illustrated on the circle with a known radius and the target equally likely to be anywhere on its circumference (see [31]), also this technique has been discussed in the plane when the located target has symmetric and asymmetric distribution (see [32] and [33]). There is obviously some similarity between this problem and the well known linear search problem.
In the present paper, we introduce the search problem for a random walk target motion on one of two intersected lines. This will happen by coordinating search between four searchers, all the searchers will start together at the same point of intersected their lines with zero as the starting and meeting point of the searchers. So that we may assume that two searchers always search to the right part and the other searchers search to the left part of intersected point. They return to zero after searching successively common distances until the target is found, we call this search as Quasi-Coordinated Linear Search Problem. We aim to minimize the expected value of the first meeting time between one of the searchers and the target. This paper is organized as follows. In Section 2 we formulate the problem and we give the conditions that make the expected value of the first meeting time between one of the searchers and the target which is finite.
In Section 3 the existence of optimal search plan that minimizes the expected value of the first meeting time is presented. Finally, the paper concludes with a discussion of the results and directions for future research.

Problem Formulation
A target is assumed to move randomly on one of two intersected line according to a stochastic process Let the search plan be represented by We assume that 0 Z is a random variable represented the initial position of the target and valued in 2I (or 2I + 1) and independent with ( ), 0 There is a known probability measures by the position of the target on 1 L , while 2 v on 2 L . The first meeting time is a random variable valued in I + defined as: At the beginning of the search suppose that the lost target is existing on any integer point on 1 L but more than 11 H or less than 11 -H or the lost target is existing on any integer point on 2 L but more than The main objective is to find the search plan such that where E terms to expectation value, then we call 0 φ * is an optimal search plan. Given 0 n > , if x is: Figure 1. The searchers S 1 and S 2 start from the origin of L 1 after searching successively distances H 11 and −H 11 , respectively, they return to the origin (note the black arrow) and then they search the distances H 12 and −H 12 , respectively, they return to the origin (note the blue arrow) and so on also the same procedure for the searchers S 3 and S 4 on L 2 .

Existence of a Finite Search Plan
Assuming that , λ ζ be positive integers such that:  and θ are positive integer numbers greater than one and 1 V = . We will shall define the following sequences { } 1 , for all the searchers , 1, 2,3, 4 r S r = on the line j L , to obtain the distances which the searcher should do them as the functions of λ and ζ . In Figure 2 we can de- Also, we shall define the search paths as follows: for any t I + ∈ , If We define the notations Proof: The hypothesis 1 Z and 2 Z are valued in 2I (or 2I + 1) and independent of ( ), 0 until the first meeting between 2 S and the target on 1 L .
The same thing for the second line by replacing 1 Z by 2 Z in the second line respectively. Hence, for any 0 i > : To solve this equation we shall find the value of ( ) Hence, we can get, For any ≥ 0, if 0 n a ≥ for 0 n > , and  2) but for 1 0 z > , we have from theorem 2 see [13], we obtain: Let us defined the following: B is a sequence of independent identically distributed random variable.
( ) ( ) is a sequence of independent identically distributed random variable. + , m 2 is an integer such that