Optimal Multiplicative Generalized Linear Search Plan for a Discrete Random Walker

Abstract

This paper formulates a search model that gives the optimal search plan for the problem of finding a discrete random walk target in minimum time. The target moves through one of n-disjoint real lines in ℝ^𝑛: we have n-searchers starting the searching process for the target from any point rather than the origin. We …find the conditions that make the expected value of the fi…rst meeting time between one of the searchers and the target fi…nite. Furthermore, we show the existence of the optimal search plan that minimizes the expected value of the fi…rst meeting time and fi…nd it. The effectiveness of this model is illustrated using numerical example.

1. Introduction

The search problem for a randomly moving target has a remarkable importance in our life due to its great applicability. This problem is very interesting because it may arise in many real world situations such as searching for randomly moving persons or targets on roads. That mathematical analysis collected all the researches derived from searching applications of the II World War. They solved with beauty two complementary objectives of the search: find the target with (1) the smallest cost and (2) in minimum time. Readers are referred to Koopman [1] for the early works and to Benkoski et al. [2] and Frost and Stone [3] for a more recent survey.

In the linear search problem, the target moves on the real line according to a known random process, and its initial position is given by the value of a random variable 𝑋₀ which has known probability distribution function. A searcher starts looking for the target at a point 𝐻₀(|𝐻₀|<∞). The searcher moves continuously along the line in both directions of the starting point 𝐻₀ until the target is met. The searcher would change its direction at suitable points many times before meeting its goal. Thus, we consider that the path length which we represented is the cost of the search. The aim of the searcher is to minimize 𝐸_𝜏𝜙,that is, the expected value of the first meeting time 𝜏_𝜙 between the searcher and the target. The problem is to find a search plan 𝜙(𝑡), such that 𝐸_𝜏𝜙<∞; in this case, we call that 𝜙(𝑡) is a finite search plan and if 𝐸_𝜏𝜙∗<𝐸_𝜏𝜙for all 𝜙(𝑡)∈Φ(𝑡), where Φ(𝑡) terms to the class of all search plans then we call that 𝜙^∗(𝑡) is an optimal search plan.

MC Cabe [4] found a finite search plan for a one-dimensional random walk target when the searcher starts the search from the origin, and the initial position of the target has a standard normal distribution. Mohamed [5] discussed the existence of a finite search plan for a one-dimensional random walk target in general case which means that the search may start from any point on the real line, and the initial position of the target has any distribution. El-Rayes and Mohamed [6] have shown the existence of a search plan which minimizes the expected value of the first meeting time between the searcher and the randomly moving target with imposed conditions. Fristedt and Heath [7] derived the conditions for optimal search path which minimizes the cost of effort of finding a randomly moving target on the real line. Ohsumi [8] presented an optimal search plan for a target moving with Markov process along one of 𝐾-nonintersecting arcs to a safe destination within a time limit, where the target starts at a safe base and tries to pass. El-Rayes et al. [9] illustrated this problem when the target moves on the real line with a Brownian motion, and the searcher starts the search from the origin. Recently, Mohamed et al. [10] discussed this problem for a Brownian target motion on one of 𝑛-intersected real lines in which any information of the target position is not available to the searchers all the time. Mohamed et al. formulate a search model and find the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite. Furthermore, they showed the existence of the optimal search plan that minimizes the expected value of the first meeting time and found it.

On the other hand, when the target is located somewhere on the real line according to a known probability distribution, the searcher searches for it with known velocity and tries to find it in minimal expected time. It is assumed that the searcher can change the direction of its motion without any loss of time. The target can be detected only if the searcher reaches the target. In an earlier work, this problem has been studied extensively in many variations, mostly by Beck et al. [11–17], Franck [18], Rousseeuw [19], Reyniers [20, 21], and Balkhi [22, 23].

Furthermore, Mohamed et al. [24, 25] have got more interesting results when they studied this problem to find a randomly located target in the plane. The target has symmetric or asymmetric distribution and with less information about this target available to the searchers. More recently, Mohamed and El-Hadidy [26] studied this problem when the target moves with parabolic spiral in the plane and starts its motion from a random point. Also, Mohamed and El-Hadidy [27] disscussed this problem in the plane when the target moves randomly with conditionally deterministic motion.

Some problems of search may impose using more than one searcher such as when we search for a valuable target (e.g., person lost on one of 𝑛-disjoint roads) or search for a serious target (e.g., a car filled with explosives which moves randomly in one of 𝑛-disjoint roads). Thus, the main contributions of this paper center around studying the problem of searching for a one-dimensional random walker target that is moving on one of a system of 𝑛-disjoint continuous real lines in ℝ^𝑛 (i.e., not intersected continuous real lines in the 𝑛-space). The problem that is studied here is very interesting where we argue to give the conditions on a strategy (or trajectories) of 𝑛-searchers, one on each line, that make the expected value of the first meeting time between one of the searchers and the target be finite and minimum. In this problem there exists a complication of such analysis. This complication is due to the fact that the searchers do not know the initial position of the target but only know its probability distribution. Otherwise, the problem would be reduced to determine the strategy of just one searcher on the same target’s line. This problem is already tackled in [6, 7]. This work focuses on the necessary conditions for the existence of finite and optimal search plan that finds a random walker target.

The optimal search plan that is proposed here shows that the special structure of the search problem can be exploited to obtain the efficient solution. For example, the search plan for a criminal drunk leaves its cache and walks up and down through one of 𝑛-disjoint streets, totally disoriented.

This paper is organized as follows. In Section 2 we formulate the problem. We display some properties that the search model should satisfy in Section 3. The search plan and the conditions that make the expected value of the first meeting time between one of the searchers and the target be finite is discussed in Section 4. The existence of optimal search plan that minimizes the expected value of the first meeting time is presented in Section 5. The optimal search plan is studied in Section 6. In Section 7 we illustrate the effectiveness of this model using numerical example. Finally, the paper concludes with a discussion of the results and directions for future research.

2. Problem Description

The problem under study can be formally described as follows. We have 𝑛-searchers 𝑆_𝑖,𝑖=1,2,…,𝑛, that start the searching process from any point rather than the origin of the line 𝐿_𝑖,𝑖=1,2,…,𝑛, respectively, as in Figure 1. Each of the searchers moves continuously along its line in both directions of the starting point. The searcher 𝑆_𝑖 would conduct its search in the following manner. Start at 𝛿_𝑖0 and go to the left (right) as far as 𝐻_𝑖1. Then, turn back to explore the right (left) part of 𝛿_𝑖0 as far as 𝐻_𝑖2. Retrace the steps again to explore the left (right) part of 𝐻_𝑖1 as far as 𝐻_𝑖3 and so forth. In this paper, we need to determine 𝐻_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…, that minimize the first meeting time between one of the searchers and the moving target.

Figure 1 

The search plan 𝜙(𝑡)=(𝜙1(𝑡),𝜙2(𝑡),…,𝜙𝑛(𝑡)) of the searchers 𝑆𝑖,𝑖=1,2,…,𝑛.

Let 𝐼 be a set of integer numbers and 𝐼⁺ a nonnegative part of 𝐼. We also assume that {𝑋_𝑖𝑗}_𝑗≥1 are sequences of independent identically distributed random variables in 𝐿_𝑖,𝑖=1,2,…,𝑛, respectively. In addition, we let a value of “−1” indicate a step to the left and a value “1” a step to the right, so that, for any 𝑗≥1, we have the 𝑛-dimensional probability vectors,[𝑃(𝑋_1𝑗=1)=𝑝₁,𝑃(𝑋_2𝑗=1)=𝑝₂,…,𝑃(𝑋_𝑛𝑗=1)=𝑝_𝑛]and  [𝑃(𝑋_1𝑗=−1)=𝑞₁,𝑃(𝑋_2𝑗=−1)=𝑞₂,…,𝑃(𝑋_𝑛𝑗=−1)=𝑞_𝑛], where [𝑝₁,𝑝₂,…,𝑝_𝑛]  +  [𝑞₁,𝑞₂,…,𝑞_𝑛]  =  [1,1,…,1].

Supposing that, for𝑡>0, 𝑡∈𝐼⁺, 𝑊(𝑡)=∑^𝑡_𝑗=1𝑋_𝑖𝑗,𝑊(0)=0 and 𝑋₀ is a random variable that represent the target initial position, valued in 2𝐼 or (2𝐼+1) and independent of 𝑊(𝑡),𝑡≥0; if 𝜉_𝑖>0 and 𝑋_𝑖𝑗 such that 0≤𝐾_𝑖1=(𝜉_𝑖+𝑥_𝑖)/2≤𝜉_𝑖, where 𝐾_𝑖1 is an integer, then 𝑃𝑊𝜉_𝑖=𝐾_𝑖1=𝜉_𝑖𝐾_𝑖1𝑝^𝐾𝑖1_𝑖𝑞^{𝜉𝑖−𝐾𝑖1}_𝑖,0,if𝐾_𝑖1doesnotexist. (1)

The target is assumed to move randomly on one of 𝑛-disjoint real lines according to the process {𝑊(𝑡),𝑡∈𝐼⁺}, where 𝐼⁺ is the set of positive integer numbers and 𝑊(𝑡) is a one-dimensional random walk motion. The initial position of the target is unknown but the searchers know its probability distribution (i.e., the probability distribution of the target is given at time 0) and the process {𝑊(𝑡),𝑡∈𝐼⁺} which controls the target’s motion.

Figure 1 gives an illustration of the search plan that the𝑛-searchers𝑆_𝑖,𝑖=1,2,…,𝑛, follow it. Moreover, it is to be noted that this search plan 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)) is a combination of continuous functions 𝜙_𝑖(𝑡) with speed 𝑣_𝑖 and given by 𝜙_𝑖(𝑡)∶𝐼⁺→𝐼,  𝑖=1,2,…,𝑛, such that 𝜙_𝑖𝑡₁−𝜙_𝑖𝑡₂<𝑣_𝑖𝑡₁−𝑡₂∀𝑡₁,𝑡₂∈𝐼⁺,𝜙_𝑖(0)=0,𝑖=1,2,…,𝑛. (2)

The first meeting time 𝜏_𝜙 is a random variable valued in 𝐼⁺, and it is defined by 𝜏_𝜙=inf𝑡∶eitheroneof𝜙_𝑖(𝑡)=𝑋₀(𝑡),},+𝑊𝑖=1,2,…,𝑛∞,ifthesetisempty, (3) where 𝑋₀∈𝐼, that is, a random variable independent of 𝑊(𝑡), and it represents the initial position of the target. We suppose that 𝑋₀=𝑋_𝑖 if the target moves on 𝐿_𝑖,𝑖=1,2,…,𝑛. Also assume that the set of all search plans of the searchers 𝑆_𝑖 with speeds 𝑣_𝑖,𝑖=1,2,…,𝑛, respectively, satisfying condition (2) be Φ_𝑣𝑖(𝑡),𝑖=1,2,…,𝑛. The problem is to find a search plan 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))∈Φ(𝑡) for the 𝑛-searchers such that 𝐸_𝜏𝜙<∞, where 𝐸_𝜏𝜙 is the expected value of the first meeting time between one of the searchers and the target, and Φ(𝑡)={(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡));𝜙_𝑖(𝑡)∈Φ_𝑣𝑖(𝑡),𝑖=1,2,…,𝑛}, is called a class of all sets of the search plans.

Assuming that 𝜗_𝑖, 𝜆_𝑖,𝑖=1,2,…,𝑛, are positive numbers such that 𝜗_𝑖>1,𝐶_𝑖=(𝜗_𝑖−1)/(𝜗_𝑖+1)>|𝑝_𝑖−𝑞_𝑖|,𝜆_𝑖=𝑏𝜛_𝑖,𝑏=1,2,…, and 𝜛_𝑖(1−𝐶_𝑖)/2 is positive number also. In this problem, we assume that 𝑣_𝑖=1,𝑖=1,2,…,𝑛. In addition, we define the sequences {𝐺_𝑖𝑗},{𝑑_𝑖𝑗}, and {𝐻_𝑖𝑗},𝑖=1,2,…,𝑛,𝑗=1,2,… by 𝐺_𝑖𝑗=𝜆_𝑖(𝜗^𝑗_𝑖−1),𝑑_𝑖𝑗=(−1)^𝑗+1𝐶_𝑖[𝐺_𝑖𝑗+1+(−1)^𝑗+1], and 𝐻_𝑖𝑗=𝑑_𝑖𝑗+𝛿_𝑖0. We also use the notations 𝜓_𝑖(𝐺_{𝑖(2𝑗+1)})=𝑊(𝐺_{𝑖(2𝑗+1)})−𝐾_𝑖1(𝐺_{𝑖(2𝑗+1)}), 𝜓_𝑖(𝐺_𝑖(2𝑗))=𝑊(𝐺_𝑖(2𝑗))+𝐾_𝑖2(𝐺_𝑖(2𝑗)), where 𝐾_𝑖1(𝐺_{𝑖(2𝑗+1)})=𝐶_𝑖𝐺_{𝑖(2𝑗+1)}, 𝐾_𝑖2(𝐺_𝑖(2𝑗))=𝐶_𝑖𝐺_𝑖(2𝑗) which are positive functions and the following remark.

Remark 1. If 0<𝜘_§<1,§=1,2,…,𝜅, then ∏^𝜅_§=1𝜘_§<∑^𝜅_§=1𝜘_§.

There are known probability measures 𝛾_𝑖,𝑖=1,2,…,𝑛, such that 𝛾₁+𝛾₂+⋯+𝛾_𝑛=1 on 𝐿₁∪𝐿₂∪⋯∪𝐿_𝑛. And, they describe the location of the target, where 𝛾_𝑖 is induced by the position of the target on 𝐿_𝑖.

The objective is to obtain the conditions that make the search plan 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)) be finite (i.e., 𝐸_𝜏𝜙<∞). We also need to show the existence of the optimal search plan 𝜙^∗(𝑡)=(𝜙^∗₁(𝑡),𝜙^∗₂(𝑡),…,𝜙^∗_𝑛(𝑡)) and find it, that is, to give the minimum expected value of the first meeting time.

3. Properties of the Search Model

In this section we start to discuss some properties that the search model for a one-dimensional random walk target should satisfy.

The searcher𝑆_𝑖would conduct its search as in the above manner that is detailed in Section 2. Consequently, for any line 𝐿_𝑖,𝑖=1,2,…,𝑛, 𝑡∈𝐼⁺, we have the following.

Case 1. If we consider that ℚ is a set of positive even numbers such that ℚ={2,4,6,…,𝑛}, for ℱ∈ℚ,𝑗∈𝐼⁺, then we have ⋯<𝐻_𝑖(𝑗+2)<0<𝐻_𝑖𝑗<𝐻_{𝑖(𝑗−2)}<⋯<𝐻_𝑖2<𝛿_𝑖0<𝐻_𝑖1<𝐻_𝑖3<⋯; (4) for any𝑡∈𝐼⁺, if 𝐺_{𝑖(2𝑗−1)}≤𝑡≤𝐺_𝑖(2𝑗),1≤𝑗≤ℱ/2, then 𝜙_𝑖(𝑡)=𝐻_{𝑖(2𝑗+1)}−𝛿_𝑖0−𝑡−𝐺_{𝑖(2𝑗−1)}; (5) if 𝐺_𝑖𝑗≤𝑡≤𝐺_𝑖(𝑗+1), 𝑗≥ℱ+1, then 𝜙_𝑖(𝑡)=𝐻_𝑖𝑗+(−1)^𝑗𝑡−𝐺_𝑖𝑗; (6) if 𝐺_𝑖(2𝑗)≤𝑡≤𝐺_{𝑖(2𝑗+1)}, 1≤𝑗≤ℱ/2, then 𝜙_𝑖(𝑡)=𝛿_𝑖0−𝐻_𝑖(2𝑗)+𝑡−𝐺_𝑖(2𝑗). (7)

Case 2. Also, if 𝑂 is a set of positive odd numbers such that 𝑂={1,3,5,…,𝑚}, for ℱ∈𝑂,𝑗∈𝐼⁺, then we have ⋯<𝐻_𝑖2<𝛿_𝑖0<𝐻_𝑖1<𝐻_𝑖3<⋯<𝐻_{𝑖(𝑗−2)}<𝐻_𝑖𝑗<0<𝐻_𝑖(𝑗+2)<⋯; (8) for any 𝑡∈𝐼⁺, if 𝐺_{𝑖(2𝑗−1)}≤𝑡≤𝐺_𝑖(2𝑗), 1≤𝑗≤(ℱ+1)/2, then 𝜙_𝑖(𝑡)=𝛿_𝑖0−𝐻_{𝑖(2𝑗−1)}−𝑡−𝐺_{𝑖(2𝑗−1)}; (9) if 𝐺_𝑖𝑗≤𝑡≤𝐺_𝑖(𝑗+1), 𝑗≥ℱ+1, then 𝜙_𝑖(𝑡)=𝐻_𝑖𝑗+(−1)^𝑗𝑡−𝐺_𝑖𝑗; (10) if 𝐺_𝑖(2𝑗)≤𝑡≤𝐺_{𝑖(2𝑗+1)}, 1≤𝑗≤(ℱ−1)/2, then 𝜙_𝑖(𝑡)=𝐻_𝑖(2𝑗)−𝛿_𝑖0+𝑡−𝐺_𝑖(2𝑗). (11)
It is clear that the search path of 𝑆_𝑖 depends on 𝜆_𝑖, 𝜗_𝑖 and 𝑗∈𝐼⁺. Since 𝑋₀∈𝐼 then the first meeting time is done at 𝑥_𝑖∈𝐼 where 𝑥_𝑖 is an integer number. For this reason we will prove the conditions (properties) that make 𝑆_𝑖 meet the target at 𝑥_𝑖 as in the following theorems.

Theorem 2. If 𝐶_𝑖 is a rational number different from 1 to −1 and, for all 𝜉_𝑖≥1,𝑉(𝜉_𝑖)=(𝑊(𝜉_𝑖𝜗_𝑖)−𝐶_𝑖𝜉_𝑖𝜗_𝑖)/2 then(i)there exists a sequence {𝑌_𝑖𝑗}_𝑗≥1 of independent identically distributed random variables such that 𝑉(𝜉_𝑖)=∑^Γ_𝑗=1𝑌_𝑖𝑗, and the distribution of 𝑌_𝑖𝑗 is concentrated on the integers 𝐸(𝑌_𝑖𝑗)  =  (𝜗_𝑖/2)[𝐸(𝑋_𝑖𝑗)−𝐶_𝑖] and the probability vector [𝑝(𝑌_1𝑗=ℱ)>0,𝑝(𝑌_2𝑗=ℱ)>0,…,𝑝(𝑌_𝑛𝑗=ℱ)>0], if and only if −𝜗_𝑖(1+𝐶_𝑖)/2≤ℱ≤𝜗_𝑖(1−𝐶_𝑖)/2.(ii)The probability vector, [𝑝(𝑉(𝜉₁)=𝑥₁)>0,𝑝(𝑉(𝜉₂)=𝑥₂)>0,…,𝑝(𝑉(𝜉_𝑛)=𝑥_𝑛)>0], holds if and only if 𝑥_𝑖 is an integer such that −𝜉_𝑖𝜗_𝑖(1+𝐶_𝑖)/2≤𝑥_𝑖≤𝜉_𝑖𝜗_𝑖(1−𝐶_𝑖)/2.(iii)If𝐶_𝑖≠𝑝_𝑖−𝑞_𝑖then there exist constants 𝑟_𝑖1 and 𝑟_𝑖2 depending on 𝐶_𝑖,𝑝_𝑖 such that, for any 𝑥_𝑖∈ℝ, where ℝ is the set of real numbers, if 𝜉_𝑖>𝑟_𝑖1𝑥_𝑖+𝑟_𝑖2 and if 𝑥_𝑖≥0 we have [𝑝(0≤𝑉(𝜉₁+1)≤𝑥₁),𝑝(0≤𝑉(𝜉₂+1)≤𝑥₂),…,𝑝(0≤𝑉(𝜉_𝑛+1)≤𝑥_𝑛)]  ≤  [𝑝(0≤𝑉(𝜉₁)≤𝑥₁),𝑝(0≤𝑉(𝜉₂)≤𝑥₂),…,𝑝(0≤𝑉(𝜉_𝑛)≤𝑥_𝑛)], and if 𝑥_𝑖<0 we have, [𝑝(𝑥₁≤𝑉(𝜉₁+1)<0),𝑝(𝑥₂≤𝑉(𝜉₂+1)<0),…,𝑝(𝑥_𝑛≤𝑉(𝜉_𝑛+1)<0)]  ≤  [𝑝(𝑥₁≤𝑉(𝜉₁)<0),𝑝(𝑥₂≤𝑉(𝜉₂)<0),…,𝑝(𝑥_𝑛≤𝑉(𝜉_𝑛)<0)].

Proof. (i) Define 𝑌_𝑖𝑗=∑^𝜗𝑖_ℱ=1((𝑋_{𝑖(ℱ+(𝑗−1)𝜗𝑖)}−𝐶_𝑖)/2),𝑗≥1, and [𝑝(𝑌_1𝑗=𝑥₁),𝑝(𝑌_2𝑗=𝑥₂),…,𝑝(𝑌_𝑛𝑗=𝑥_𝑛)]=[𝑝(𝑊(𝜗₁))=2𝑥₁+𝜗₁𝐶₁,𝑝(𝑊(𝜗₂))=2𝑥₂+𝜗₂𝐶₂,…,𝑝(𝑊(𝜗_𝑛))=2𝑥_𝑛+𝜗_𝑛𝐶_𝑛].
Consequently, if [𝑝(𝑌_1𝑗=𝑥₁)>0,𝑝(𝑌_2𝑗=𝑥₂)>0,…,𝑝(𝑌_𝑛𝑗=𝑥_𝑛)>0], then from (1) we have 𝑥_𝑖+𝜗_𝑖(1+𝐶_𝑖)/2 that is an integer, and since 𝑥_𝑖 is an integer also then we have 0≤𝑥_𝑖+𝜗_𝑖(1+𝐶_𝑖)/2≤𝜗_𝑖. Therefore, −𝜗_𝑖(1+𝐶_𝑖)/2≤𝑥_𝑖≤𝜗_𝑖(1−𝐶_𝑖)/2. If 𝑥_𝑖=ℱ then we have [𝑝(𝑌_1𝑗=ℱ)>0,𝑝(𝑌_2𝑗=ℱ)>0,…,𝑝(𝑌_𝑛𝑗=ℱ)>0], if and only if −𝜗_𝑖(1+𝐶_𝑖)/2≤ℱ≤𝜗_𝑖(1−𝐶_𝑖)/2. In addition, by using (1), 𝐸(∑^𝜗𝑖_ℱ=1((𝑋_{𝑖(ℱ+(𝑗−1)𝜗𝑖)}−𝐶_𝑖)/2))=(𝜗_𝑖/2)[𝐸(𝑋_𝑖𝑗)−𝐶_𝑖] is proved.
(ii) Since 𝑉(𝜉_𝑖)=(𝑊(𝜉_𝑖𝜗_𝑖)−𝐶_𝑖𝜉_𝑖𝜗_𝑖)/2 then we have[𝑝(𝑉(𝜉₁)=𝑥₁),𝑝(𝑉(𝜉₂)=𝑥₂),…,𝑝(𝑉(𝜉_𝑛)=𝑥_𝑛)]=[𝑝((𝑊(𝜉₁𝜗₁)−𝐶₁𝜉₁𝜗₁)/2=𝑥₁),𝑝((𝑊(𝜉₂𝜗₂𝐶)−₂𝜉₂𝜗₂)/2=𝑥₂),…,𝑝((𝑊(𝜉_𝑛𝜗_𝑛)−𝐶_𝑛𝜉_𝑛𝜗_𝑛)/2=𝑥_𝑛)]=  [𝑝(𝑊(𝜉₁𝜗₁))=2𝑥₁+𝐶₁𝜉₁𝜗₁),𝑝(𝑊(𝜉₂𝜗₂))=2𝑥₂+𝐶₂𝜉₂𝜗₂),…,𝑝(𝑊(𝜉_𝑛𝜗_𝑛))=2𝑥_𝑛+𝐶_𝑛𝜉_𝑛𝜗_𝑛)], and using (1) the prove is completed.
(iii) By using (ii), if 𝑥_𝑖≥0 then we have [𝑝(0≤𝑉(𝜉₁)≤𝑥₁),𝑝(0≤𝑉(𝜉₂)≤𝑥₂),…,𝑝(0≤𝑉(𝜉_𝑛)≤𝑥_𝑛)]=[∑^[𝑥1]_𝑗=0𝑝(𝑉(𝜉₁)=𝑗),∑^[𝑥2]_𝑗=0𝑝(𝑉(𝜉₂)=𝑗),…,∑^[𝑥𝑛]_𝑗=0𝑝(𝑉(𝜉_𝑛)=𝑗)], and if 𝑥_𝑖<0 we get [𝑝(𝑥₁≤𝑉(𝜉₁+1)<0),𝑝(𝑥₂≤𝑉(𝜉₂+1)<0),…,𝑝(𝑥_𝑛≤𝑉(𝜉_𝑛+1)<0)]=[∑⁰_𝑗=[𝑥1]𝑝(𝑉(𝜉₁)=𝑗),∑⁰_𝑗=[𝑥2]𝑝(𝑉(𝜉₂)=𝑗),…,∑⁰_{𝑗=[𝑥𝑛]}𝑝(𝑉(𝜉_𝑛)=𝑗)], where [𝑥_𝑖] means the greatest integer less than or equal to𝑥_𝑖, 𝑖=1,2,…,𝑛. It is sufficient to show that 𝑝𝑉𝜉₁+1=ℱ,𝑝𝑉𝜉₂𝑝𝑉𝜉+1=ℱ,…,_𝑛≤𝑝𝑉𝜉+1=ℱ₁=ℱ,𝑝𝑉𝜉₂𝑝𝑉𝜉=ℱ,…,_𝑛=ℱif𝜉_𝑖>𝑟_𝑖1𝑥_𝑖+𝑟_𝑖2. (12)
We have the following cases.
Case (a). If 𝐶_𝑖<−1, then, from (ii), [𝑝(𝑉(𝜉₁)=ℱ)>0,𝑝(𝑉(𝜉₂)=ℱ)>0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)>0], if and only if 2ℱ/𝜗_𝑖(1−𝐶_𝑖)≤𝜉_𝑖≤−2ℱ/𝜗_𝑖(1+𝐶_𝑖). We take 𝑟_𝑖1=−2/𝜗_𝑖(1+𝐶_𝑖),𝑟_𝑖2=0  and then 𝜉_𝑖>𝑟_𝑖1|ℱ|⇒𝜉_𝑖>−2ℱ/𝜗_𝑖(1+𝐶_𝑖) that leads to [𝑝(𝑉(𝜉₁)=ℱ)=0,𝑝(𝑉(𝜉₂)=ℱ)=0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)=0]. Consequently, (12) holds.
Case (b). If 𝐶_𝑖>1, hence [𝑝(𝑉(𝜉₁)=ℱ)>0,𝑝(𝑉(𝜉₂)=ℱ)>0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)>0], if and only if −2ℱ/𝜗_𝑖(1+𝐶_𝑖)≤𝜉_𝑖≤2ℱ/𝜗_𝑖(1−𝐶_𝑖). From (ii) putting 𝑟_𝑖1=−2/𝜗_𝑖(1−𝐶_𝑖),𝑟_𝑖2=0 then 𝜉_𝑖>𝑟_𝑖1|ℱ|⇒𝜉_𝑖>2ℱ/𝜗_𝑖(1−𝐶_𝑖) that leads to[𝑝(𝑉(𝜉₁)=ℱ)=0,𝑝(𝑉(𝜉₂)=ℱ)=0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)=0].
Then, (12) holds.
Case (c). If  −1<𝐶_𝑖<1, then let 𝛼_𝑖=(1−𝐶_𝑖)/2,𝛽_𝑖=1−𝛼_𝑖 and Δ_𝑖=(𝛼_𝑖/𝑞_𝑖)^𝛼𝑖(𝛽_𝑖/𝑝_𝑖)^𝛽𝑖, where [𝑝₁,𝑝₂,…,𝑝_𝑛]+[𝑞₁,𝑞₂,…,𝑞_𝑛]=[1,1,…,1]. In addition, put 𝑟_𝑖2=1/(Δ_𝑖−1),  𝑟_𝑖1=Δ_𝑖max[(1/𝛼_𝑖,1/𝛽_𝑖)𝜗_𝑖(Δ_𝑖−1)]. Since 𝐸(𝑋_𝑖𝑗)≠𝐶_𝑖 and 0<𝛼_𝑖<1 then 𝛼_𝑖≠𝑞_𝑖 and then Δ_𝑖>1. Consequently, 𝑟_𝑖1 and 𝑟_𝑖2 are positive and well defined. Assuming that 𝜉_𝑖>𝑟_𝑖1|ℱ| then  −𝛽_𝑖𝜉_𝑖𝜗_𝑖<ℱ<𝛼_𝑖𝜉_𝑖𝜗_𝑖; therefore, from (ii), we have [𝑝(𝑉(𝜉₁)=ℱ)>0,𝑝(𝑉(𝜉₂)=ℱ)>0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)>0].
It remains to prove that if [𝑝(𝑉(𝜉₁)=ℱ)>0,𝑝(𝑉(𝜉₂)=ℱ)>0,…,𝑝(𝑉(𝜉_𝑛)=ℱ)>0] and 𝜉_𝑖>𝑟_𝑖1|ℱ|+𝑟_𝑖2 then (2) holds.
Considering that 𝑍_𝑖𝑗=(𝑋_𝑖𝑗−𝐶_𝑖)/2 then [𝑝(𝑍_1𝑗=𝛼₁)=𝑝₁,𝑝(𝑍_2𝑗=𝛼₂)=𝑝₂,…,𝑝(𝑍_𝑛𝑗=𝛼_𝑛)=𝑝_𝑛],  [𝑝(𝑍_1𝑗=−𝛽₁)=𝑞₁,𝑝(𝑍_2𝑗=−𝛽₂)=𝑞₂,…,𝑝(𝑍_𝑛𝑗=−𝛽_𝑛)=𝑞_𝑛], and [𝑝(𝑉(𝜉₁)=ℱ),𝑝(𝑉(𝜉₂)=ℱ),…,𝑝(𝑉(𝜉_𝑛)=ℱ)]=[𝑝(∑^𝜉1𝜗1_𝑗=1𝑍_1𝑗=ℱ),𝑝(∑^𝜉2𝜗2_𝑗=1𝑍_2𝑗=ℱ),…,𝑝(∑^{𝜉𝑛𝜗𝑛}_𝑗=1𝑍_𝑛𝑗=ℱ)]. By using (1) we have 𝑝𝑉𝜉₁+1=ℱ𝑝𝑉𝜉₁,𝑝𝑉𝜉=ℱ²+1=ℱ𝑝𝑉𝜉₂𝑝𝑉𝜉=ℱ,…,_𝑛+1=ℱ𝑝𝑉𝜉_𝑛==ℱ_{𝜗1𝛽1𝑗=1}𝜉₁𝜗₁+𝑗Δ₁𝜉₁𝜗₁+𝑗+𝑗𝛼₁+ℱ/𝛽₁×^𝜗1𝛼1_𝑗=1𝜉₁𝜗₁+𝑗+𝜗₁𝛽₁Δ₁𝜉₁𝜗₁+𝑗+𝑖𝛽₁−ℱ/𝛼_{1𝜗2𝛽2𝑗=1},𝜉₂𝜗₂+𝑗Δ₂𝜉₂𝜗₂+𝑗+𝑗𝛼₂+ℱ/𝛽₂×^𝜗2𝛼2_𝑗=1𝜉₂𝜗₂+𝑗+𝜗₂𝛽₂Δ₂𝜉₂𝜗₂+𝑗+𝑖𝛽₂−ℱ/𝛼₂,…,_{𝜗𝑛𝛽𝑛𝑗=1}𝜉_𝑛𝜗_𝑛+𝑗Δ_𝑛𝜉_𝑛𝜗_𝑛+𝑗+𝑗𝛼_𝑛+ℱ/𝛽_𝑛×^{𝜗𝑛𝛼𝑛}_𝑗=1𝜉_𝑛𝜗_𝑛+𝑗+𝜗_𝑛𝛽_𝑛Δ_𝑛𝜉_𝑛𝜗_𝑛+𝑗+𝑖𝛽_𝑛−ℱ/𝛼_𝑛. (13) Since 𝜉_𝑖>𝑟_𝑖1|ℱ|+𝑟_𝑖2 then every term is strictly less than 1. Consequently, [𝑝(𝑉(𝜉₁+1)=ℱ),𝑝(𝑉(𝜉₂+1)=ℱ),…,𝑝(𝑉(𝜉_𝑛+1)=ℱ)]<[𝑝(𝑉(𝜉₁)=ℱ),𝑝(𝑉(𝜉₂)=ℱ),…,𝑝(𝑉(𝜉_𝑛)=ℱ)].

Theorem 3. If 𝐸(𝑋_𝑖𝑗)<𝐶_𝑖,𝐶_𝑖∈ℝ (the set of real numbers), then there exists 𝜀_𝑖,0<𝜀_𝑖<1, such that [𝑝(𝑊(𝜉₁)),𝑝(𝑊(𝜉₂)),…,𝑝(𝑊(𝜉_𝑛))]≤[𝜀^𝜉1₁,𝜀^𝜉2₂,…,𝜀^𝜉𝑛_𝑛], for all 𝜉_𝑖,𝑖=1,2,…,𝑛.

Proof. For𝜁_𝑖>0,[𝑝(𝑊(𝜉₁)≥𝐶₁𝜉₁),𝑝(𝑊(𝜉₂)≥𝐶₂𝜉₂),…,𝑝(𝑊(𝜉_𝑛)≥𝐶_𝑛𝜉_𝑛)]=[𝑝(exp{𝜁₁(𝑊(𝜉₁𝐶)−₁𝜉₁)}≥1),𝑝(exp{𝜁₂(𝑊(𝜉₂)−𝐶₂𝜉₂)}≥1),…,𝑝(exp{𝜁_𝑛(𝑊(𝜉_𝑛)−𝐶_𝑛𝜉_𝑛)}≥1)]≤[𝐸(exp{𝜁₁(𝑊(𝜉₁)−𝐶₁𝜉₁)}),𝐸(exp{𝜁₂(𝑊(𝜉₂)−𝐶₂𝜉₂)}),…,𝐸(exp{𝜁_𝑛(𝑊(𝜉_𝑛)−𝐶_𝑛𝜉_𝑛)})]=[{𝑓₁(𝜁₁)}^𝜉1,{𝑓₂(𝜁₂)}^𝜉2,…,{𝑓_𝑛(𝜁_𝑛)}^𝜉𝑛], where[{𝑓₁(𝜁₁)},{𝑓₂(𝜁₂)},…,{𝑓_𝑛(𝜁_𝑛)}]=[𝐸(exp{𝜁₁(𝑋_1𝑗−𝐶₁)}),𝐸(exp{𝜁₂(𝑋_2𝑗−𝐶₂)}),…,𝐸(exp{𝜁_𝑛(𝑋_𝑛𝑗−𝐶_𝑛)})]=[𝑝₁exp{𝜁₁(1−𝐶₁)}+𝑞₁exp{𝜁₁(−1−𝐶₁)},𝑝₂exp{𝜁₂(1−𝐶₂)}+𝑞₂exp{𝜁₂𝐶(−1−₂)},…,𝑝_𝑛exp{𝜁_𝑛(1−𝐶_𝑛)}+𝑞_𝑛exp{𝜁_𝑛(−1−𝐶_𝑛)}]; if 𝐸(𝑋_𝑖𝑗)<𝐶_𝑖,𝑖=1,2,…,𝑛, then [{𝑓^′₁(0)},{𝑓^′₂(0)},…,{𝑓^′_𝑛(0)}]<[0,0,…,0], and since [{𝑓₁(0)},{𝑓₂(0)},…,{𝑓_𝑛(0)}]<[1,1,…,1], then [min_𝜁1>0{𝑓₁(𝜁₁)},min_𝜁2>0{𝑓₂(𝜁₂)},…,min_𝜁𝑛>0{𝑓_𝑛(𝜁_𝑛)}]=[𝜀₁,𝜀₂,…,𝜀_𝑛]<[1,1,…,1].
By similar arguments if 𝐸(𝑋_𝑖𝑗)>𝐶_𝑖,𝑖=1,2,…,𝑛, then there exist 𝜀_𝑖<1 such that [𝑝(𝑊(𝜉₁)),𝑝(𝑊(𝜉₂)),…,𝑝(𝑊(𝜉_𝑛))]≤[𝜀^𝜉1₁,𝜀^𝜉2₂,…,𝜀^𝜉𝑛_𝑛] for all 𝜉_𝑖,𝑖=1,2,…,𝑛.

4. Existence of a Finite Search Plan

In this section, we find the conditions that make the search plan be finite. In addition, we will discuss that under what these conditions are indeed finite. This is a crucial issue related to the existence of a finite search plan. So, we will provide useful theorems that help us to do it.

Theorem 4. Let 𝛾_𝑖 be the measure defined on ℝ by 𝑋_𝑖𝑗,𝑖=1,2,…,𝑛, and if 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)) is a search plan defined previously, the expectation 𝐸_𝜏𝜙 is finite if ^𝛿𝑖0_−∞^ℱ/2_𝑗=1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖+^∞_𝑗=ℱ/2+1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)≤−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (14)∫^∞_𝛿𝑖0∑^∞_𝑗=1𝜗_𝑖^2𝑗+1𝑝(𝜓_𝑖(𝐺_{𝑖(2𝑗+1)})>−𝑥_𝑖)𝛾_𝑖(𝑑𝑥_𝑖), are finite when 𝛿_𝑖0>0. And if 𝛿_𝑖0<0, then𝐸_𝜏𝜙is finite if ^∞_𝛿𝑖0^∞_{𝑗=(ℱ+1)/2}𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}>−𝑥_𝑖+^(ℱ−1)/2_𝑗=1𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}≤−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (15)∫^𝛿𝑖0_−∞∑^∞_𝑗=1𝜗_𝑖^2𝑗𝑝(𝜓_𝑖(𝐺_𝑖(2𝑗))<−𝑥_𝑖)𝛾_𝑖(𝑑𝑥_𝑖), are finite.

Proof. The hypotheses 𝑋_𝑖 and 𝛿_𝑖0 are valued in 2𝐼 or (2𝐼+1) then 𝑋_𝑖+𝑊(𝑡) is greater than 𝜙_𝑖(𝑡) until the first meeting; also if 𝑋_𝑖 is smaller than 𝛿_𝑖0 then 𝑋_𝑖+𝑊(𝑡) is smaller than 𝜙_𝑖(𝑡) until the first meeting. Since 𝜏_𝜙𝑖>𝑡,𝑖=1,2,…,𝑛, are mutually exclusive events then 𝑝(𝜏_𝜙𝑖>𝑡)=𝑝(𝜏_𝜙1>𝑡 or 𝜏_𝜙2>𝑡 or ⋯ or 𝜏_𝜙𝑛>𝑡)=∑^𝑛_𝑖=1𝑝(𝜏_𝜙𝑖>𝑡), and, for any 𝑗≥0 we have 𝑝𝜏_𝜙𝑖>𝐺_{𝑖(2𝑗+1)}≤^𝛿𝑖0_−∞𝑝𝑋_𝑖+𝑊𝐺_𝑖(2𝑗)<𝐻_𝑖(2𝑗)∣𝑋_𝑖=𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0𝑝𝑋_𝑖+𝑊𝐺_{𝑖(2𝑗+1)}>𝐻_{𝑖(2𝑗+1)}∣𝑋_𝑖=𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖. (16)
Using the notation 𝜓_𝑖(𝐺_𝑖(2𝑗))=𝑊(𝐺_𝑖(2𝑗))+𝐶_𝑖𝐺_𝑖(2𝑗), we obtain 𝑊(𝐺_𝑖(2𝑗))−𝐻_𝑖(2𝑗)<−𝑋_𝑖=𝑥_𝑖; then 𝑊(𝐺_𝑖(2𝑗))−(−1)^2𝑗+1𝐶_𝑖[𝐺_𝑖(2𝑗)+1+(−1)^2𝑗+1]<−𝑥_𝑖 leads to 𝑊(𝐺_𝑖(2𝑗))−(−1)𝐶_𝑖[𝐺_𝑖(2𝑗)+1+(−1)]=𝑊(𝐺_𝑖(2𝑗))+𝐶_𝑖[𝐺_𝑖(2𝑗)]=𝜓_𝑖(𝐺_𝑖(2𝑗))<−𝑥_𝑖. Similarly, by using the notation 𝜓_𝑖(𝐺_{𝑖(2𝑗+1)})=𝑊(𝐺_{𝑖(2𝑗+1)})−𝐶_𝑖𝐺_{𝑖(2𝑗+1)}, we get 𝜓_𝑖(𝐺_{𝑖(2𝑗+1)})>−𝑥_𝑖. Consequently, 𝑝𝜏_𝜙𝑖>𝐺_{𝑖(2𝑗+1)}≤^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (17) also, 𝑝𝜏_𝜙𝑖>𝐺_𝑖(2𝑗)≤^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_{𝑖(2𝑗−1)}>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖. (18)
From Remark 1, we obtain 𝐸_𝜏𝜙=^∞₀𝑝𝜏_𝜙≤>𝑡𝑑𝑡_{∞𝑗=0𝐺1(𝑗+1)𝐺1𝑗}𝑝𝜏_𝜙1+>𝑡𝑑𝑡+⋯_{∞𝑗=0𝐺𝑛(𝑗+1)𝐺𝑛𝑗}𝑝𝜏_𝜙𝑛>𝑡𝑑𝑡, (19) where 𝐺_𝑖0=0; then 𝐸_𝜏𝜙≤^∞_𝑗=0^{𝐺1(𝑗+1)}_𝐺1𝑗𝑝𝜏_𝜙1>𝐺_1𝑗+𝑑𝑡+⋯_{∞𝑗=0𝐺𝑛(𝑗+1)𝐺𝑛𝑗}𝑝𝜏_𝜙𝑛=>𝑡𝑑𝑡_{𝑛∞𝑖=1𝑗=0}𝐺_𝑖(𝑗+1)−𝐺_𝑖𝑗𝑝𝜏_𝜙𝑖>𝐺_𝑖𝑗=^𝑛∞_𝑖=1𝑗=0𝜆_𝑖𝜗_𝑖^𝑗+1−1−𝜆_𝑖𝜗^𝑗_𝑖−1𝑝𝜏_𝜙𝑖>𝐺_𝑖𝑗=^𝑛∞_𝑖=1𝑗=0𝜆_𝑖𝜗^𝑗_𝑖𝜗_𝑖−1𝑝𝜏_𝜙𝑖>𝐺_𝑖𝑗=^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖−1^∞_𝑗=0𝜗^𝑗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖𝑗=^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖×𝑝𝜏−1_𝜙𝑖>𝐷+𝜗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖1+𝜗²_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖2+𝜗³_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖3+⋯. (20)
If 𝛿_𝑖0<0, then we get𝐸_𝜏𝜙≤^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖×𝑝𝜏−1_𝜙𝑖>0+𝜗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖1+𝜗²_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖1>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖2<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗³_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖3>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖2<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗⁴_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖3>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖4<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯+𝜗^ℱ_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_{𝑖(ℱ−1)}>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖ℱ<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗_𝑖^ℱ+1∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖(ℱ+1)>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖ℱ<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗_𝑖^ℱ+2∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖(ℱ+1)>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖(ℱ+2)<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗_𝑖^ℱ+3∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖(ℱ+3)>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖0_−∞𝑝𝜓_𝑖𝐺_𝑖(ℱ+2)<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯; (21)thus,𝐸_𝜏𝜙≤^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖×𝑝𝜏−1_𝜙𝑖>0+𝜗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖1+𝜗²_𝑖𝜗_𝑖×+1_{𝛿𝑖0−∞}𝑝𝜓_𝑖𝐺_𝑖2<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯+𝜗^ℱ_𝑖𝜗_𝑖×+1_{𝛿𝑖0−∞}𝑝𝜓_𝑖𝐺_𝑖ℱ<−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯+𝜗³_𝑖𝜗_𝑖+1^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖3>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+𝜗⁵_𝑖𝜗_𝑖+1×^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖5>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯+𝜗_𝑖^ℱ+3𝜗_𝑖×+1_∞𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖(ℱ+3)>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+⋯. (22)
Leads to 𝐸_𝜏𝜙≤^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖×−1₁𝑔_𝑖+𝜗_𝑖×+1_{𝛿𝑖01−∞}𝑤_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0¹𝑞_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0¹ℎ_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖=^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖−1¹𝑔_𝑖+^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖−1𝜗_𝑖×+1_{𝛿𝑖01−∞}𝑤_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0¹𝑞_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0¹ℎ_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (23) where ¹𝑔_𝑖=𝑝𝜏_𝜙𝑖>0+𝜗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖1+𝜗²_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖1>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖,₁𝑤_𝑖𝑥_𝑖=^∞_𝑗=1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖,₁𝑞_𝑖𝑥_𝑖=^(ℱ−1)/2_𝑗=1𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}≤−𝑥_𝑖,₁ℎ_𝑖𝑥_𝑖=^∞_{𝑗=(ℱ+1)/2}𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}>−𝑥_𝑖. (24)
Then 𝐸_𝜏𝜙 is finite if ^∞_𝛿𝑖0^∞_{𝑗=(ℱ+1)/2}𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}>−𝑥_𝑖+^(ℱ−1)/2_𝑗=1𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}≤−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (25)∫^𝛿𝑖0_−∞∑^∞_𝑗=1𝜗_𝑖^2𝑗𝑝(𝜓_𝑖(𝐺_𝑖(2𝑗))<−𝑥_𝑖)𝛾_𝑖(𝑑𝑥_𝑖), are finite.
By similar way if 𝛿_𝑖0>0, then 𝐸_𝜏𝜙 is finite if 𝐸_𝜏𝜙≤^𝑛_𝑖=1𝜆_𝑖𝜗_𝑖−1²𝑔_𝑖+𝜆_𝑖𝜗_𝑖−1𝜗_𝑖×+1_{𝛿𝑖02−∞}𝑤_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^𝛿𝑖02_−∞𝑞_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖+^∞_𝛿𝑖0²ℎ_𝑖𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (26) where ²𝑔_𝑖=𝑝𝜏_𝜙1>0+𝜗_𝑖𝑝𝜏_𝜙𝑖>𝐺_𝑖1+𝜗²_𝑖^∞_𝛿𝑖0𝑝𝜓_𝑖𝐺_𝑖1>−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖,₂𝑤_𝑖𝑥_𝑖=^ℱ/2_𝑗=1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖,₂𝑞_𝑖𝑥_𝑖=^∞_𝑗=ℱ/2+1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)≤−𝑥_𝑖,₂ℎ_𝑖𝑥_𝑖=^∞_𝑗=1𝜗_𝑖^2𝑗+1𝑝𝜓_𝑖𝐺_{𝑖(2𝑗+1)}>−𝑥_𝑖. (27)
And 𝐸_𝜏𝜙 is finite if ^𝛿𝑖0_−∞^ℱ/2_𝑗=1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)<−𝑥_𝑖+^∞_𝑗=ℱ/2+1𝜗_𝑖^2𝑗𝑝𝜓_𝑖𝐺_𝑖(2𝑗)≤−𝑥_𝑖𝛾_𝑖𝑑𝑥_𝑖, (28)∫^∞_𝛿𝑖0∑^∞_𝑗=1𝜗_𝑖^2𝑗+1𝑝(𝜓_𝑖(𝐺_{𝑖(2𝑗+1)})>−𝑥_𝑖), are finite.

Lemma 5 (see El-Rayes et al. [9]). If 𝑎𝑚≥0,𝑎𝑚₊₁≤𝑎𝑚,𝑚=1,2,…, and {𝑑𝑚},𝑚=1,2,…, are a strictly increasing sequence of integer numbers with𝑑₀=0, then, for any 𝑘=1,2,…, 𝑚^∞₌𝑘𝑑𝑚₊₁−𝑑𝑚𝑎_𝑑𝑚₊₁≤^∞𝑚_=𝑑𝑘𝑎𝑚≤^∞𝑚₌𝑘𝑑𝑚₊₁−𝑑𝑚𝑎_𝑑𝑚, (29) where ∑^∞𝑚₌𝑘[𝑑𝑚₊₁−𝑑𝑚]𝑎_𝑑𝑚₊₁,∑^∞𝑚_=𝑑𝑘𝑎𝑚, and ∑^∞𝑚⁼𝑘[𝑑𝑚₊₁−𝑑𝑚]𝑎_𝑑𝑚 are vectors in formulas and they are taken to be row vectors. These vectors are defined as follows: 𝑚^∞₌𝑘𝑑𝑚₊₁−𝑑𝑚𝑎_𝑑𝑚₊₁=^∞𝑚₌𝑘𝑑₁₍𝑚₊₁₎−𝑑₁𝑚𝑎_1𝑑1(𝑚_+1)∞,𝑚₌𝑘𝑑₂₍𝑚₊₁₎−𝑑₂𝑚𝑎_2𝑑2(𝑚₊₁₎,…,_∞𝑚₌𝑘𝑑_𝑛(𝑚₊₁₎−𝑑_𝑛𝑚𝑎_{𝑛𝑑𝑛(}𝑚_+1)∞,𝑚_=𝑑𝑘𝑎𝑚=^∞𝑚_=𝑑1𝑘𝑎₁𝑚,^∞𝑚_=𝑑2𝑘𝑎₂𝑚,…,^∞𝑚_=𝑑𝑛𝑘𝑎_𝑛𝑚,_∞𝑚₌𝑘𝑑𝑚₊₁−𝑑𝑚𝑎_𝑑𝑚=^∞𝑚₌𝑘𝑑₁₍𝑚₊₁₎−𝑑₁𝑚𝑎_1𝑑1𝑚_∞,𝑚₌𝑘𝑑₂₍𝑚₊₁₎−𝑑₂𝑚𝑎_2𝑑2𝑚,…,_∞𝑚₌𝑘𝑑_𝑛(𝑚₊₁₎−𝑑_𝑛𝑚𝑎_𝑛𝑑𝑛𝑚. (30)

Now, we will discuss under what conditions of the chosen search plan should be satisfied to make the previous integrals in Theorem 4 are indeed finite. This is a crucial issue related to the existence of a finite search plan.

Theorem 6. The chosen search plan should satisfy ²ℎ(𝑥)≤𝐿(|𝑥|), ²𝑞(𝑥)≤𝐿(|𝑥|), if 𝛿_𝑖0>0 and ¹𝑤(𝑥)≤𝑀(|𝑥|), ¹ℎ(𝑥)≤𝑀(|𝑥|) if 𝛿_𝑖0<0, where 𝐿(|𝑥|),𝐿(|𝑥|),𝑀(|𝑥|), and 𝑀(|𝑥|) are vectors of linear functions given by ¹𝑤(𝑥)=[¹𝑤₁(𝑥₁),¹𝑤₂(𝑥₂),…,¹𝑤_𝑛(𝑥_𝑛)],¹ℎ(𝑥)=[¹ℎ₁(𝑥₁),¹ℎ₂(𝑥₂),…,¹ℎ_𝑛(𝑥_𝑛)],²[ℎ(𝑥)=₂ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)],²𝑞(𝑥)=[²𝑞₁(𝑥₁),²𝑞₂(𝑥₂),…,²𝑞_𝑛(𝑥_𝑛)],𝐿(|𝑥|)=[𝐿₁(|𝑥₁|),𝐿₂(|𝑥₂|),…,𝐿_𝑛(|𝑥_𝑛|)],𝐿(|𝑥|)=[𝐿₁(|𝑥₁|),𝐿₂(|𝑥₂|),…,𝐿_𝑛(|𝑥_𝑛|)],𝑀(|𝑥|)=[𝑀₁(|𝑥₁|),𝑀₂(|𝑥₂|),…,𝑀_𝑛(|𝑥_𝑛|)] and 𝑀(|𝑥|)=[𝑀₁(|𝑥₁|),𝑀₂(|𝑥₂|),…,𝑀_𝑛(|𝑥_𝑛|)].

Proof. We will prove this theorem for ²ℎ(𝑥) when 𝛿_𝑖0>0, where ²ℎ(𝑥)=[²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(𝜓₁(𝐺_1(2𝑗+1))>−𝑥₁),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(𝜓₂(𝐺_2(2𝑗+1))>−𝑥₂),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})>−𝑥_𝑛)], and we obtain the following cases.
(I) If 𝑥_𝑖>𝛿_𝑖0, then we have [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[²ℎ₁(𝛿₁₀),²ℎ₂(𝛿₂₀),…,²ℎ_𝑛(𝛿_𝑛0)]+[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(−𝑥₁<𝜓₁(𝐺_1(2𝑗+1)𝛿)≤−₁₀),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(−𝑥₂<𝜓₂(𝐺_2(2𝑗+1))≤−𝛿₂₀),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(−𝑥_𝑛<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})≤−𝛿_𝑛0)].
(II) If 0≤𝑥_𝑖≤𝛿_𝑖0, then we have [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)]+[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(−𝑥₁<𝜓₁(𝐺_1(2𝑗+1))≤0),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(−𝑥₂<𝜓₂(𝐺_2(2𝑗+1))≤0),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(−𝑥_𝑛<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})≤0)].
(III) If 𝑥_𝑖≤𝛿_𝑖0, then we get [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)]−[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(0<𝜓₁(𝐺_1(2𝑗+1)𝑥)≤−₁),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(0<𝜓₂(𝐺_2(2𝑗+1))≤−𝑥₂),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(0<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})≤−𝑥_𝑛)].
From (II) we have [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]≤[²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)], and for 𝑥_𝑖≥𝛿_𝑖0[²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[²ℎ₁(𝛿₁₀),²ℎ₂(𝛿₂₀),…,²ℎ_𝑛(𝛿_𝑛0)]+[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(−𝑥₁<𝜓₁(𝐺_1(2𝑗+1))≤−𝛿₁₀),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(−𝑥₂<𝜓₂(𝐺_2(2𝑗+1))≤−𝛿₂₀),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(−𝑥_𝑛<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})≤−𝛿_𝑛0)], but [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]=[²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)]+[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(−𝑥₁<𝜓₁(𝐺_1(2𝑗+1))≤0),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(−𝑥₂<𝜓₂(𝐺_2(2𝑗+1))≤0),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(−𝑥_𝑛<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})≤0)]. Consequently, from Theorem 3, we get [²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)]=[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(𝜓₁(𝐺_1(2𝑗+1))>0),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(𝜓₂(𝐺_2(2𝑗+1))>0),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(𝜓_𝑛(𝐺_{𝑛(2𝑗+1)})>0)]≤[∑^∞_𝑗=1𝜗₁^2𝑗+1𝜀^{𝐺1(2𝑗+1)}₁,∑^∞_𝑗=1𝜗₂^2𝑗+1𝜀^{𝐺2(2𝑗+1)}₂,…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝜀^{𝐺𝑛(2𝑗+1)}_𝑛]≤[𝜗³₁/(𝜗²₁−1),𝜗³₂/(𝜗²₂−1),…,𝜗³_𝑛/(𝜗²_𝑛−1)],0<𝜀_𝑖<1.
Suppose the following holds:(i)𝑑_𝑖𝜉𝑖=𝐺_{𝑖(2𝜉𝑖+1)}/𝜎_𝑖=𝜇_𝑖(𝜗^2𝜉𝑖_𝑖⁺¹−1), (ii)𝜇_𝑖(𝜉_𝑖)=𝜓_𝑖(𝜉_𝑖𝜎_𝑖)/2=∑^𝜉𝑖_𝑗=1𝑋_𝑖𝑗, where {𝑋_𝑖𝑗},𝑗=1,2,…, is a sequence of independent identically distributed random variables,(iii)𝑎_𝑖(𝜉_𝑖)=𝑝(−𝑥_𝑖/2<𝜇_𝑖(𝜉_𝑖)≤0)=∑^|𝑥𝑖|/2_ℱ=0𝑝(−(ℱ+1)<𝜇_𝑖(𝜉_𝑖)≤−ℱ), (iv)𝑚 is an integer such that 𝑑_𝑖𝑚=𝑟_𝑖1|𝑥_𝑖|+𝑟_𝑖2,(v)𝛼_𝑖=𝜗²_𝑖/𝜇_𝑖(𝜗_𝑖−1),(vi)𝑈_𝑖(ℱ,ℱ+1)=∑^∞_𝜉𝑖=0𝑝(−(ℱ+1)<𝜇_𝑖(𝜉_𝑖)≤−ℱ).
Thus, from Theorem 2, we have 𝑎_𝑖(𝜉_𝑖) is nonincreasing if 𝜉_𝑖>𝑑_𝑖𝜉𝑖. By applying Lemma 5, we have [²ℎ₁(𝑥₁),²ℎ₂(𝑥₂),…,²ℎ_𝑛(𝑥_𝑛)]−[²ℎ₁(0),²ℎ₂(0),…,²ℎ_𝑛(0)]=[∑^∞_𝑗=1𝜗₁^2𝑗+1𝑝(−𝑥₁<𝜓₁(𝐺_1(2𝑗+1))≤0),∑^∞_𝑗=1𝜗₂^2𝑗+1𝑝(−𝑥₂<𝜓₂(𝐺_2(2𝑗+1))≤0),…,∑^∞_𝑗=1𝜗_𝑛^2𝑗+1𝑝(−𝑥_𝑛<𝜓_𝑛(𝐺_{𝑛(2𝑗+1)}𝑚)≤0)]=[∑_𝜉1=1𝜗^2𝜉1₁⁺¹𝑎₁(𝑑_1𝜉1),∑𝑚_𝜉2=1𝜗^2𝜉2₂⁺¹𝑎₂(𝑑_2𝜉2),…,∑𝑚_𝜉𝑛=1𝜗^2𝜉𝑛_𝑛⁺¹𝑎_𝑛(𝑑_𝑛𝜉𝑛)]  +  [∑^∞_𝜉1=𝑚₊₁𝜗^2𝜉1₁⁺¹𝑎₁(𝑑_1𝜉1),∑^∞_𝜉2=𝑚₊₁𝜗^2𝜉2₂⁺¹𝑎₂(𝑑_2𝜉2),…,∑^∞_𝜉𝑛=𝑚₊₁𝜗^2𝜉𝑛_𝑛⁺¹𝑎_𝑛(𝑑_𝑛𝜉𝑛)]  ≤  [∑𝑚_𝜉1=1𝜗^2𝜉1₁⁺¹,∑𝑚_𝜉2=1𝜗^2𝜉2₂⁺¹,…,∑𝑚_𝜉𝑛=1𝜗^2𝜉𝑛_𝑛⁺¹]  +  [𝛼₁∑^∞_𝜉1=𝑚⁺¹(𝑑_1𝜉1−𝑑_1(𝜉1−1))𝑎₁(𝑑_1𝜉1),𝛼₂∑^∞_𝜉2=𝑚⁺¹(𝑑_2𝜉2−𝑑_2(𝜉2−1))𝑎₂(𝑑_2𝜉2),…,𝛼_𝑛∑^∞_𝜉𝑛=𝑚⁺¹(𝑑_𝑛𝜉𝑛−𝑑_{𝑛(𝜉𝑛−1)})𝑎_𝑛(𝑑_𝑛𝜉𝑛)]   ≤  [∑𝑚_𝜉1=1𝜗^2𝜉1₁⁺¹,∑𝑚_𝜉2=1𝜗^2𝜉2₂⁺¹,…,∑𝑚_𝜉𝑛=1𝜗^2𝜉𝑛_𝑛⁺¹]  +  [𝛼₁∑^∞_𝜉1=𝑑1𝑚𝑎₁(𝜉₁),𝛼₂∑^∞_𝜉2=𝑑2𝑚𝑎₂(𝜉₂),…,𝛼_𝑛∑^∞_{𝜉𝑛=𝑑𝑛}𝑚𝑎_𝑛(𝜉_𝑛)]   ≤  [∑𝑚_𝜉1=1𝜗^2𝜉1₁⁺¹,∑𝑚_𝜉2=1𝜗^2𝜉2₂⁺¹,…,∑𝑚_𝜉𝑛=1𝜗^2𝜉𝑛_𝑛⁺¹]  +  [𝛼₁∑^|𝑥1|/2_ℱ=0𝑈₁(ℱ,ℱ+1),𝛼₂∑^|𝑥2|/2_ℱ=0𝑈₂(ℱ,ℱ+1),…,𝛼_𝑛∑^|𝑥𝑛|/2_ℱ=0𝑈_𝑛(ℱ,ℱ+1)].
Since 𝑈_𝑖(ℱ,ℱ+1) satisfies the conditions of renewal theorem as in Feller [28], then 𝑈_𝑖(ℱ,ℱ+1) is bounded for all ℱ,𝑖=1,2,…,𝑛, by a constant so ²ℎ(𝑥)=[𝑞₁(𝑥₁),𝑞₂(𝑥₂),…,𝑞_𝑛(𝑥_𝑛)]≤[²ℎ₁(𝛿₁₀),²ℎ₂(𝛿₂₀),…,²ℎ_𝑛(𝛿_𝑛0)]+[²Λ₁₁,²Λ₂₁,…,²Λ_𝑛1[]+₂Λ₁₂|𝑥₁|,²Λ₂₂|𝑥₂|,…,²Λ_𝑛2|𝑥_𝑛|]=[𝐿₁(|𝑥₁|),𝐿₂(|𝑥₂|),…,𝐿_𝑛(|𝑥_𝑛|)]=𝐿(|𝑥|), then, ²ℎ(𝑥)≤𝐿(|𝑥|). Similar to the previous analysis way we can prove that ²𝑞(𝑥)≤𝐿(|𝑥|) and also ¹𝑤(𝑥)≤𝑀(|𝑥|), ¹ℎ(𝑥)≤𝑀(|𝑥|)  if 𝛿_𝑖0<0. Therefore, the proof is completed.

Theorem 7. If there exists a finite search plan 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))∈Φ(𝑡), then 𝐸(|𝑋₀|) is finite.

Proof. For 𝐸_𝜏𝜙𝑖<∞,𝑖=1,2,…,𝑛, we have 𝑝  (𝜏_𝜙 is finite) = 1 and so 𝑝 (one of 𝜏_𝜙𝑖 is finite, 𝑖=1,2,…,𝑛) = 1. Therefore, 𝑝^𝑛_𝑖=1𝜏_𝜙𝑖<∞=^𝑛_𝑖=1𝑝𝜏_𝜙𝑖<∞−_𝑖,𝛽𝑝𝜏_𝜙𝑖<∞∩𝜏_𝜙𝛽+()<∞+⋯−1^𝑛𝑝^𝑛_𝑖=1𝜏_𝜙𝑖<∞. (31)
Since 𝜏_𝜙𝑖<∞, 𝑖=1,2,…,𝑛, are mutually exclusive events, then ^𝑛_𝑖=1𝑝𝜏_𝜙𝑖<∞=1. (32)
Consequently, 𝑝  (𝜏_𝜙𝑖 is finite) = 1 or 𝑝  (𝜏_𝜙𝜃 is finite) = 1  for all 𝑖≠𝜃,𝑖,𝜃=1,2,…,𝑛. If 𝑝  (𝜏_𝜙𝑖 is finite) = 1 then 𝑋₀=𝜙_𝑖(𝜏_𝜙𝑖)−𝑊(𝜏_𝜙𝑖) with probability one and |𝑋₀|≤|𝜙_𝑖(𝜏_𝜙𝑖)|+|𝑊(𝜏_𝜙𝑖)|≤𝜏_𝜙𝑖+|𝑊(𝜏_𝜙𝑖)| that leads to 𝐸(|𝑋₀|)≤𝐸_𝜏𝜙𝑖+𝐸(|𝑊(𝜏_𝜙𝑖)|). But𝑊(𝜏_𝜙𝑖)<𝜏_𝜙𝑖, then 𝐸(|𝑊(𝜏_𝜙𝑖)|)<𝐸_𝜏𝜙𝑖, and 𝐸(|𝑋₀|) is finite.
On the other hand, if 𝑝  (𝜏_𝜙𝜃 is finite) = 1 then 𝑋₀=𝜙_𝜃(𝜏_𝜙𝜃)−𝑊(𝜏_𝜙𝜃) with probability one, and by the same manner we can get 𝐸(|𝑋₀|) is finite.

Remark 8. A direct consequence of Theorems 4, 6, and 7 satisfies the existence of a finite search plan if and only if 𝐸(|𝑋₀|) is finite.

5. Existence of an Optimal Search Plan

The goal of the searching strategy could minimize the expected value of the first meeting time between one of the searchers and the target. Therefore, the main problem here is to find a search paths 𝜙_𝑖(𝑡),𝑖=1,2,…,𝑚. If such a search paths exists, we call it optimal search paths.

Definition 9. Let {𝜙_𝑖ℎ(𝑡)}_ℎ≥1∈Φ₁(𝑡),𝑖=1,2,…,𝑚, be sequences of search plans; we say that 𝜙_𝑖ℎ(𝑡) converges to 𝜙_𝑖(𝑡) as ℎ tends to ∞ if and only if, for any 𝑡∈𝐼⁺,𝜙_𝑖ℎ(𝑡) converges to 𝜙_𝑖(𝑡) uniformly on every compact space (El-Rayes et al. [9]).

Theorem 10. Let for any 𝑡∈𝐼⁺ and 𝑊(𝑡) be a one-dimensional random walk. Then, the mapping, 𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)⟶𝐸_𝜏𝜙∈𝐼⁺, (33) is lower semicontinuous on Φ₁(𝑡).

Proof. Let 𝐵(𝜙_𝑖,𝑡) be the indicator function of the set {𝜏_𝜙𝑖>𝑡,𝑖=1,2,…,𝑛}; by the Fatou-Lesbesque theorem, we get 𝐸_𝜏𝜙=𝐸^∞_𝑡=0𝐵𝜙_𝑖,𝑡=𝐸^∞_𝑡=0lim_ℎ→∞inf𝐵𝜙_𝑖ℎ(𝑡)≤,𝑡_∞𝑡=0lim_ℎ→∞inf𝐸1_{{𝜏𝜙ℎ>𝑡}}, (34) for any sequences 𝜙_𝑖ℎ→𝜙_𝑖 in Φ₁(𝑡), where Φ₁(𝑡) is sequentially compact; see El-Rayes et al. [9]. Thus the mapping (𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))→𝐸_𝜏𝜙 is lower semicontinuous mapping on Φ₁(𝑡) and this completed the proof.

6. Optimal Search Plan

In this section, we will find the optimal search plan that minimize the expected value of the first meeting time.

Since 𝜏_𝜙 depends on infimum either one of the functions 𝜙_𝑖(𝑡),𝑖=1,2,…,𝑛, then we need to minimize 𝐸_𝜏𝜙. That happens when we obtain the optimal search plan 𝜙^∗(𝑡)={𝜙^∗₁(𝑡),𝜙^∗₂(𝑡),…,𝜙^∗_𝑛(𝑡)}, where 𝜙_𝑖(𝑡) is a function on a distance 𝐻_𝑖𝑗, 𝑖=1,2,…,𝑛, 1≤𝑗≤ℱ/2. Therefore, it can be seen that the vector (𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)) are𝑛-distinct objective functions. Since the speed of the 𝑛-searchers (search team) is 𝑣_𝑖=1,𝑖=1,2,…,𝑛, then the objective functions 𝜙_𝑖(𝑡),𝑖=1,2,…,𝑛, are distinct set of constrains given from Cases 1 and 2. In this problem, we consider the constraint 𝐺_{𝑖(2𝑗−1)}≤𝑡≤𝐺_𝑖(2𝑗),𝑖=1,2,…,𝑛,1≤𝑗≤ℱ/2. Then, the problem takes the form min𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)subjectto𝑡∈𝑇=𝑡∈𝐼⁺∣𝐺_{𝑖(2𝑗−1)}≤𝑡≤𝐺_𝑖(2𝑗),ℱ𝑖=1,2,…,𝑛,1≤𝑗≤2, (35) where 𝑡 is an 𝑛-dimensional vector of the decision variables, 𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡) are𝑛-distinct objective functions of the decision vector of inequality constraints, and 𝑇 is the feasible set of constrained decisions.

But 𝑣_𝑖=1,𝑖=1,2,…,𝑛, that leads to 𝑡=𝐻_𝑖𝑗, 𝑖=1,2,…,𝑛,1≤𝑗≤ℱ/2; then the above problem is represented as the following multiobjective nonlinear programming problem (MONLP): ΞMONLP:min₁𝐻_1𝑗,Ξ₂𝐻_2𝑗,…,Ξ_𝑛𝐻_𝑛𝑗,subjectto𝐺_{𝑖(2𝑗−1)}≤𝐻_𝑖𝑗≤𝐺_𝑖(2𝑗),𝐻_𝑖𝑗∈𝐼⁺ℱ,𝑖=1,2,…,𝑛,1≤𝑗≤2,whereΞ_𝑖𝐻_𝑖𝑗=𝐻_{𝑖(2𝑗+1)}−𝛿_𝑖0−𝑡−𝐺_{𝑖(2𝑗−1)}. (36)

However, on formulating the MONLP which closely describes and represents the real decision situation, various factors of the real system should be reflected in the description of the objective functions and the constraints. Naturally, these objective functions and the constraints involve many variables and parameters whose possible values may be assigned by the experts. In the conventional approach, such parameters are changed according to the variables in an experimental and/or subjective manner through the experts understanding the nature of the variables.

Substituting 𝐺_𝑖𝑗=𝑏𝜛_𝑖(𝜗^𝑗_𝑖−1) in the above MONLP we have (I):ΞMONLPmin₁𝜗₁;𝜛₁,Ξ₂𝜗₂;𝜛₂,…,Ξ_𝑛𝜗_𝑛;𝜛_𝑛,subjectto𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1−1≤𝐻_𝑖𝑗,𝐻_𝑖𝑗≤𝑏𝜛_𝑖𝜗_𝑖^2𝑗,𝐻−1_𝑖𝑗∈𝐼⁺ℱ,𝑖=1,2,…,𝑛,1≤𝑗≤2,whereΞ_𝑖𝐻_𝑖𝑗=𝐻_{𝑖(2𝑗+1)}−𝛿_𝑖0−𝑡−𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1−1, (37) where the variable 𝜗_𝑖 takes +ve numbers greater than 1 and the parameter 𝜛_𝑖 is the least positive integer such that 𝜛_𝑖(1−(𝜗_𝑖−1)/(𝜗_𝑖+1))/2 is positive number; 𝐻_𝑖𝑗≥0for all 𝑖=1,2,…,𝑛,1≤𝑗≤ℱ/2. Furthermore, 𝐻_𝑖𝑗 is a function of the variable 𝜗_𝑖 and the parameter 𝜛_𝑖, where 𝐻_𝑖𝑗=(−1)^𝑗+1𝐶_𝑖𝐺_𝑖𝑗+1+(−1)^𝑗+1+𝛿_𝑖0=(−1)^𝑗+1𝜗_𝑖−1𝜗_𝑖+1𝑏𝜛_𝑖𝜃^𝑗_𝑖−1+1+(−1)^𝑗+1+𝛿_𝑖0. (38)

Then, the previous MONLP(I) takes the form ():ΞMONLPIImin₁𝜗₁;𝜛₁,Ξ₂𝜗₂;𝜛₂,…,Ξ_𝑛𝜗_𝑛;𝜛_𝑛,subjectto𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1≤ℎ−1₁𝜗−1^𝑖−1𝜗_𝑖×+1𝑏𝜛_𝑖𝜗^𝑗_𝑖−1+ℎ₁+𝛿_𝑖0,𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1≥ℎ₁𝜗−1^𝑖−1𝜗_𝑖×+1𝑏𝜛_𝑖𝜗^𝑗_𝑖−1+ℎ₁+𝛿_𝑖0,whereΞ_𝑖𝜗_𝑖;𝜛_𝑖=ℎ₃𝜗_𝑖−1𝜗_𝑖×+1𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1−1+ℎ₄−2𝛿_𝑖0+ℎ₂𝜗_𝑖−1𝜗_𝑖×+1𝑏𝜛_𝑖𝜗^𝑗_𝑖−1+ℎ₅+𝑏𝜛_𝑖𝜗_𝑖^2𝑗−1,ℎ−1₁=1+(−1)^𝑗+1,ℎ₂=(−1)^𝑗+2,ℎ₃=(−1)^2𝑗+1,ℎ₄=1+(−1)^2𝑗,ℎ₅=1+(−1)^𝑗,𝜗_𝑖𝜛>1,_𝑖ℱ>0∀𝑖=1,2,…,𝑛,𝑏=1,2,…,1≤𝑗≤2. (39)

Choosing a certain weights ℧_𝑖,𝑖=1,2,…,𝑛, such that ∑^𝑛_𝑖=1℧_𝑖=1 and the certain parameters 𝜛_𝑖=1,𝑖=1,2,…,𝑛, then the previous MONLP(II) takes the formMONLP(III):min℧_𝑖ℎ₃𝜗_𝑖−1𝜗_𝑖×𝑏𝜗+1_{𝑖2𝑗−1}−1+ℎ₄−2𝛿_𝑖0+ℎ₂𝜗_𝑖−1𝜗_𝑖×𝑏𝜗+1_𝑗𝑖−1+ℎ₅+𝑏𝜗_𝑖^2𝑗−1,𝜗−1subjectto𝑏_𝑖^2𝑗−1≤ℎ−1₁𝜗−1^𝑖−1𝜗_𝑖×𝑏𝜗+1_𝑗𝑖−1+ℎ₁+𝛿_𝑖0,𝑏𝜗_𝑖^2𝑗≥ℎ−1₁𝜗−1^𝑖−1𝜗_𝑖×𝑏𝜗+1_𝑗𝑖−1+ℎ₁+𝛿_𝑖0,1−𝜗_𝑖ℱ<0,∀𝑏=1,2,…,1≤𝑗≤2. (40)

From the Kuhn-Tucker conditions we have 122𝜗_𝑖+1²ℎ₃−1𝑏𝜗_𝑖^2𝑗−1−1+ℎ₄+ℎ₂−1𝑏𝜗^𝑗_𝑖−1+ℎ₅+𝜗_𝑖−1𝜗_𝑖𝑏ℎ+1³−12𝑗−1𝜗_𝑖^2𝑗−2+ℎ₂−1𝑗𝜗_𝑖^𝑗−1+𝑏2𝑗−1𝜗_𝑖^2𝑗−2+𝑢₁𝑏2𝑗−1𝜗_𝑖^2𝑗−2−ℎ₁−1𝜗_𝑖×2+1𝜗_𝑖𝑏𝜗+1^𝑗𝑖−1+ℎ₁+𝜗_𝑖−1𝑗𝑏𝜗_𝑖^𝑗−1+𝑢₂−𝑏2𝑗−1𝜗_𝑖^2𝑗−2+ℎ₁−1𝜗_𝑖×2+1𝜗_𝑖𝑏𝜗+1^𝑗𝑖−1+ℎ₁+𝜗_𝑖−1𝑗𝑏𝜗_𝑖^𝑗−1−𝑢₃𝑢=0,₁𝑏𝜗_𝑖^2𝑗−1−1𝜗_𝑖+1−ℎ₁−1𝜗_𝑖×𝑏𝜗−1_𝑖𝑗−1−1+ℎ₁𝑢=0,₂ℎ₁−1𝜗_𝑖−1𝑏𝜗^𝑗_𝑖−1+ℎ₁−𝑏𝜗_𝑖^2𝑗−1−1𝜗_𝑖𝑢+1=0,₃1−𝜗_𝑖=0. (41)(42)(43)(44)

Solving (41)–(44), we found the optimal values of 𝜗_𝑖 that give the optimal distances 𝐻_𝑖𝑗for all 𝑏=1,2,…,1≤𝑗≤ℱ/2, and they are given by 𝜗_𝑖=Υ+Λ𝑏1−2𝑗𝜗_𝑖^2𝑗−2−1,𝜗_𝑖>1, (45) where Υ=2ℎ₃−1𝑏𝜗_𝑖^2𝑗−1−1+ℎ₄+ℎ₂−1𝑏𝜗^𝑗_𝑖−1+ℎ₅,Λ=𝑏ℎ₃−12𝑗−1𝜗_𝑖^2𝑗−𝑏ℎ₃−12𝑗−1𝜗_𝑖^2𝑗−2+𝑏ℎ₂−1𝑗𝜗_𝑖^𝑗+1−𝑏ℎ₂−1𝑗𝜗_𝑖^𝑗−1. (46)

7. An Illustrative Example

In this section, we clarify the effectiveness of this model by considering the following example.

Example 1. Suppose that a criminal drunk (target) leaves its cache and walks up and down through one of 𝑛-disjoint streets, totally disoriented. We model the streets as real lines with cache at the point 𝑋₀. In addition, assume that the criminal drunk takes unit steps, so we may record his position with an integer. Thus, for example, if he takes 5 steps to the left, he will be at a position 𝑋₀−5. If we consider the previous conditions in Theorems 2 and 3 hold, then the optimal expected value of the first meeting time depends on the optimal values of the distances 𝐻_𝑖𝑗 that the searcher 𝑆_𝑖 should do it and the optimal values of 𝐺_𝑖𝑗,𝑗=2,4,6. Using the constraint 𝐺_{𝑖(2𝑗−1)}≤𝑡≤𝐺_𝑖(2𝑗),𝑖=1,2,…,𝑛,1≤𝑗≤ℱ/2,from Case 2, such that 𝑗=2,4,6 and for 𝑏=1,2, in MONLP(III), and also using (38), we obtain the optimal values of 𝜗_𝑖 that give the optimal values of 𝐻_𝑖𝑗,𝐺_𝑖𝑗 in Table 1.

Table 1 

The optimal values of 𝜗_𝑖,𝐻_𝑖𝑗 and 𝐺_𝑖𝑗,𝑗=2,4,6,𝑏=1,2.

It is clear that the optimal distances 𝐻_𝑖𝑗 that 𝑆_𝑖 should do it depend on 𝑏 where 𝜆_𝑖=𝑏𝜛_𝑖 and 𝐺_𝑖𝑗=𝜆_𝑖(𝜗^𝑗_𝑖−1).

From the previous numerical calculations and by considering 𝛿_𝑖0=10, we get Figure 2 that shows the projection of the target’s motion (one-dimensional random walk) on the 𝑆_𝑖 path with 𝑏=1,2 where the current position on the line represented in the vertical axis versus the time steps at horizontal axis.

(a) 𝑏=1

(b) 𝑏=2

Figure 2 

(a) and (b) give the optimal expected value of the first meeting time between 𝑆𝑖 and the target that moves with discrete random walk in one dimension.

8. Discussion and Conclusions

A multiplicative generalized linear search plan for a one-dimensional random walk target on one of𝑛-disjoint real lines has been presented, where the target initial position is given by a random variable 𝑋₀. Therefore, the target will be meet if 𝐸_𝜏𝜙<∞, where 𝜙(𝑡)=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡)) and 𝐸_𝜏𝜙 is the expected value of the first meeting time between one of the searchers and the target. We discuss some properties that the search model should satisfy in Theorems 2 and 3. We introduce the proof of conditions that make a search plan finite in Theorem 4, based on the continuity of the search plan and the conditions in Theorems 2 and 3 to show that 𝐸_𝜏𝜙<∞. We provide more analysis by using Lemma 5, Theorems 2 and 3 in Theorem 6 to show that the search plan 𝜙(𝑡) is finite if the conditions ²ℎ(𝑥)≤𝐿(|𝑥|),²𝑞(𝑥)≤𝐿(|𝑥|), if 𝛿_𝑖0>0 and ¹𝑤(𝑥)≤𝑀(|𝑥|), ¹ℎ(𝑥)≤𝑀(|𝑥|) if 𝛿_𝑖0<0, where 𝐿(|𝑥|), 𝐿(|𝑥|), 𝑀(|𝑥|), and 𝑀(|𝑥|) are vectors of linear functions. We use Theorem 7 to show that if there exist a finite search plan then the expected value of the target initial position 𝐸(|𝑋₀|) is finite. It will also be interesting to see a direct consequence of Theorems 4, 6, and 7 satisfying the existence of a finite search plan if and only if 𝐸(|𝑋₀|) is finite. The existence of an optimal search plan has been proved in Theorem 10.

To find the optimal distances 𝐻_𝑖𝑗,𝑖=1,…,𝑛,  𝑗=1,2,…, that give the optimal search plan 𝜙^∗(𝑡)=(𝜙^∗₁(𝑡),𝜙^∗₂(𝑡),…,𝜙^∗_𝑛(𝑡)) we solve the multiobjective nonlinear programming problem (MONLP) which contains the variables 𝜗_𝑖 and the parameters 𝜆_𝑖,𝑖=1,…,𝑛. We use the Kuhn-Tucker conditions to obtain the optimal values of𝜗_𝑖that can give the optimal distances, after substituting (38), and then minimize the expected value of the first meeting time. The effectiveness of this model is illustrated using numerical example.

In future research, it seems that the proposed model will be extendible to the multiple searchers case by considering the combinations of movement of multiple targets.