Reliability Analysis of Wireless Sensor Networks Using Markovian Model

Abstract

This paper investigates reliability analysis of wireless sensor networks whose topology is switching among possible connections which are governed by a Markovian chain. We give the quantized relations between network topology, data acquisition rate, nodes' calculation ability, and network reliability. By applying Lyapunov method, sufficient conditions of network reliability are proposed for such topology switching networks with constant or varying data acquisition rate. With the conditions satisfied, the quantity of data transported over wireless network node will not exceed node capacity such that reliability is ensured. Our theoretical work helps to provide a deeper understanding of real-world wireless sensor networks, which may find its application in the fields of network design and topology control.

1. Introduction

The wireless sensor network (WSN), consisting of spatially distributed autonomous sensors to monitor physical or environmental conditions, is recently arousing lots of attention with flourishing results achieved. The development of wireless sensor network ascends to the 19th century and now can find its wide applications in many industrial and consumption fields, such as industrial process monitoring and control machine health monitoring and [1, 2]. The WSN is built of “nodes”—from a few to several hundred or even thousand which therefore compose a large-scaled complex network [3]. An important point can be drawn from above achievements that the complexity of network topology has great effect on the reliability or stability of WSN with many effective topology control methods proposed [4, 5]. Meanwhile, node’s calculation capacity as well as data acquisition rate is an important index for node ability and is also concerned with the network’s reliability. However in most cases, network topology is stochastic and it varies with random events occurring which are due to changes in sensor nodes’ position, reachability (due to jamming, noise, moving obstacles, etc.), available energy, malfunctioning, and task details. This variation alters not only the nodes’ ability of calculation and data acquisition rate, but also the network topology. Obviously, such changes are unpredictable which may cause network congestion and lead to network collapse in worst cases. How the holistic WSN can maintain reliable despite of all these stochastic perturbations, is of course, a big challenge for researchers.

For reliability of such WSN with topology switching, some feasible assumptions are necessary. As we know, network topology remains unchanged until next event breaks out and the occurrence of these events is usually governed by a Markov chain [6]. For this reason, such kind of special stochastic pattern is given the name “Markovian jump model” [7]. This model is now being used to solve many problems in network, such as approximation [8], synchronization [9], and stabilization [10]. By assuming the concerned complex network is a Markovian jump system with UNIQUE equilibrium, sufficient conditions for stability are proposed using Lyapunov method. However for practical WSN used to acquire sensory data from outside environment, failure of nodes or change of environment will cause topology switching and also change the data acquisition rate, for each node. Thus the considered WSN can be modelled as a Markovian jump complex network with switching equilibrium instead of UNIQUE, and its reliability analysis remains unsolved. In this paper, we focus on establishing quantized relations between WSN reliability and network parameters, especially network topology and data acquisition rate. From the point of system analysis, Lyapunov method is applied and the possible maximum date transport quantity over network node can be calculated for two cases: with constant data acquisition rate and with varying data acquisition rate. This quantity is certainly concerned with network topology, data acquisition rate, and nodes’ calculation capacity. Sufficient conditions for the reliability analysis of networks are proposed ensuring this calculated quantity will not exceed network capacity such that buffer zone is bounded and network congestion is avoided. For WSN which cannot satisfy the sufficient conditions, for example, to maintain network reliability, we should improve the nodes’ calculation capacity to a desired level, perform network topology control, or decrease the frequency of acquiring sensory data from outside environment. This work investigates WSN from the point of system analysis and will be of help for WSN topology design as well as traffic control.

The following of this paper is organized as follows: Section 2 begins with problem description. In Section 3, network model is given and reliability criteria are discussed. Section 4 presents a numerical example and a brief conclusion is drawn in Section 5.

Notation 1. Throughout the paper, unless otherwise specified, we denote by (Ω,ℱ,𝑃), a complete probability space. The superscript 𝑇 will denote transpose and matrix 𝑃>0(≥0) denotes 𝑃 is positive(nonnegative) definite matrix. Let |⋅| stand for the Euclidean norm for vectors and 𝜆_min(𝑃) denote the minimal eigenvalue of matrix 𝑃.

2. System Model Description

Consider the following WSN with 𝑁 nodes as shown in Figure 1 where WSN has 𝑀 possible topologies which switch randomly among set 𝑆={1,2,…,𝑀}. Each possible topology 𝑘∈𝑆 corresponds to one regime and this topology(regime) switching is governed by a Markovian chain 𝑟(𝑡) characterized by transition rate matrix Π=[𝜋_𝑘𝑙]_𝑀×𝑀, 𝑘,𝑙∈𝑆 as𝑃(𝑟(𝑡+𝑑𝑡)=𝑙∣𝑟(𝑡)=𝑘)=𝜋^𝑘𝑙1+𝑑𝜋𝑡+_𝑘𝑘𝑜(𝑑𝑡)if𝑘≠𝑙𝑑𝑡+𝑜(𝑑𝑡)if𝑘=𝑙,(2.1) where 𝑑𝑡>0 and 𝑜(𝑑𝑡) satisfie lim_𝑑𝑡→0(𝑜(𝑑𝑡)/𝑑𝑡)=0. Notice that the total probability axiom imposes 𝜋_𝑘𝑘 negative and ∑^𝑀𝑙=1𝜋_𝑘𝑙=0, for all 𝑘∈𝑆.

Figure 1 

Wireless sensor network.

𝑥_𝑖(𝑡) is the data length, that is, quantity of data waiting to be transported in each nodes 𝑖; 𝑓_𝑖(𝑥_𝑖(𝑡),𝑡,𝑟(𝑡)) is the calculation capacity of node 𝑖. For each possible network topology 𝑟(𝑡), 𝑓_𝑖(𝑥_𝑖(𝑡),𝑡,𝑟(𝑡)) may be different. 𝐺_𝑖𝑗(𝑟(𝑡)) is specified as follows: if there is a physical transport path or connection between node 𝑖 and node 𝑗(𝑖≠𝑗), 𝐺_𝑗𝑖(𝑟(𝑡))=𝐺_𝑖𝑗(𝑟(𝑡))=1; otherwise 𝐺_𝑗𝑖(𝑟(𝑡))=𝐺_𝑖𝑗(𝑟(𝑡))=0(𝑖≠𝑗). And data transported between node 𝑖 and 𝑗 satisfy that data will be transported from node 𝑗 to 𝑖 only if the quantity of data in node 𝑗 is larger than that of node 𝑖, that is, 𝑥_𝑗≥𝑥_𝑖. Weighted value 𝑐(𝑟(𝑡)) represents network status for data transport, if network status is good, 𝑐(𝑟(𝑡))=1 and all data transported can be received; otherwise 𝑐(𝑟(𝑡)) takes values in (0,1) and partial data will be lost during the transport process. Parameter 𝑑_𝑖(𝑟(𝑡)) represents the data acquired from outside environment for node 𝑖 under regime 𝑟(𝑡). Thus the data flow equation is described as follows:𝑥̇_𝑖(𝑡)=𝑓_𝑖𝑥_𝑖(𝑡),𝑟(𝑡)+𝑐(𝑟(𝑡))^𝑁_𝑗=1𝑖=1,2,…𝑁𝐺𝑟_𝑖𝑗(𝑡()𝑟∈(𝑡)𝑆)𝑥=_𝑗({𝑡)1+,2𝑑,…_𝑖(𝑟𝑀(𝑡)}),.(2.2) In (2.2), 𝐺_𝑖𝑖(𝑟(𝑡))=−∑^𝑁𝑗=1𝐺_𝑖𝑗(𝑟(𝑡)), for all 𝑖≠𝑗. Noticing the fact that for all 𝐺_𝑖𝑗(𝑟(𝑡))≠0, which means there is a transport path between nodes 𝑖 and 𝑗, data will be transported from 𝑖 to 𝑗 under the condition 𝑥_𝑖(𝑡)>𝑥_𝑗(𝑡) and otherwise from 𝑗 to 𝑖 with 𝑥_𝑖(𝑡)<𝑥_𝑗(𝑡).

Rewrite (2.2) in matrix form as𝑥̇(𝑡)=𝑓(𝑥(𝑡),𝑟(𝑡))+𝑐(𝑟(𝑡))𝐺(𝑟(𝑡))𝑥(𝑡)+𝐷(𝑟(𝑡)).(2.3) Here vector 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…𝑥_𝑁(𝑡)]^𝑇∈ℝ^𝑁 denotes the node data variables; 𝑓(𝑥(𝑡),𝑟(𝑡))=[𝑓₁(𝑥₁,𝑟(𝑡)),𝑓₂(𝑥₂,𝑟(𝑡)),…,𝑓_𝑁(𝑥_𝑁,𝑟(𝑡))]^𝑇, and 𝐷(𝑟(𝑡))=[𝑑₁(𝑟(𝑡)),𝑑₂(𝑟(𝑡)),…𝑑_𝑁(𝑟(𝑡))]^𝑇. Matrix 𝐺(𝑟(𝑡))∈ℝ^𝑁×𝑁 is coupling matrix and for each regime 𝑟(𝑡)∈𝑆, the elements 𝐺_𝑖𝑗(𝑟(𝑡)) are specified as above which means that the network is fully connected in the sense of having no isolated clusters. Obviously, zero is the largest eigenvalue of 𝐺 with multiplicity. For simplicity, we write 𝑓(𝑥(𝑡),𝑟(𝑡)=𝑘) as 𝑓(𝑥(𝑡),𝑘) and so on.

Because of calculation capacity limitation for nodes, assume that for each regime 𝑘∈𝑆, the function 𝑓(𝑥(𝑡),𝑘) in (2.2) satisfies the following sector condition:||𝑓(𝑥(𝑡),𝑘)−𝑓(𝑦(𝑡),𝑘)||≤𝑚||𝑥(𝑡)−𝑦(𝑡)||,∀𝑥,𝑦∈ℝ^𝑁,𝑘∈𝑆.(2.4) It is known by [11] that with inequality (2.4) established, there exists a unique solution 𝑥(𝑡,𝑟(𝑡)) for network (2.3).

For reliability analysis, the following definitions and lemmas are introduced.

Definition 2.1. Wireless sensor network (2.3) is stochastically reliable in mean-square sense if there exists a bounded positive constant 𝐶 such that for node data variable 𝑥(𝑡,𝑟(𝑡)), there is lim_𝑡→∞𝐸𝑥^𝑇(𝑡,𝑟(𝑡))𝑥(𝑡,𝑟(𝑡))<𝐶.(2.5)

Definition 2.2. Wireless sensor network (2.3) is asymptotically reliable almost surely if there exists a bounded positive constant 𝐶 such that 𝑃lim_𝑡→∞𝑥(𝑡,𝑟(𝑡))=𝐶=1.(2.6)

Lemma 2.3 (see [12]). Given any real matrices 𝑄₁, 𝑄₂, 𝑄₃ with appropriate dimensions such that 0<𝑄₃=𝑄^𝑇3, the following inequality holds: 𝑄^𝑇1𝑄₂+𝑄^𝑇2𝑄₁≤𝑄^𝑇1𝑄₃𝑄₁+𝑄^𝑇2𝑄³⁻¹𝑄₂.(2.7)

Lemma 2.4 (Schur complement). Let 𝑋=𝑋^𝑇∈ℝ^{(𝑛+𝑚)×(𝑛+𝑚)} be a symmetric matrix given by 𝑋=^𝐴𝐵𝑇_𝐵𝐶, where 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑚×𝑛, 𝐶∈ℝ^𝑚×𝑚, and 𝐶 is nonsingular, then 𝑋 is positive definite if and only if 𝐶>0 and 𝐴−𝐵^𝑇𝐶⁻¹𝐵>0.

3. Reliability Analysis

Considering WSN (2.3) with topology switching, since for each different topology, 𝐷(𝑟(𝑡)) may be different, thus the equilibrium for network (2.3) will not necessarily be the same. Assuming that for each regime 𝑟(𝑡), the equilibrium is given as 𝑢^∗(𝑟(𝑡)) with 𝑢^∗(𝑟(𝑡))=[𝑢^∗1(𝑟(𝑡)),𝑢^∗2(𝑟(𝑡)),…𝑢^∗𝑁(𝑟(𝑡))]∈ℝ^𝑁, where 𝑢^∗(𝑟(𝑡)) can be different or the same for each different regime 𝑟(𝑡), thus there is𝑓𝑢^∗(𝑟(𝑡),𝑡)+𝑐(𝑟(𝑡))𝐺(𝑟(𝑡))𝑢^∗(𝑟(𝑡))+𝐷(𝑟(𝑡))=0,∀𝑘∈𝑆.(3.1) In order to shift 𝑢^∗(𝑟(𝑡)) to the origin, define 𝑔(𝑧(𝑡,𝑟(𝑡)),𝑟𝑧(𝑡()𝑡),=𝑟(𝑡𝑓))(𝑥=(𝑥𝑡)(,𝑡𝑟)(−𝑡))𝑢^∗−(𝑟𝑓(𝑡)𝑢)_∗,(3.2)(𝑟(𝑡)),𝑟(𝑡).(3.3) Therefore 𝑔(𝑧(𝑡,𝑘),𝑘)=[𝑔₁(𝑧₁(𝑡,𝑘),𝑘),𝑔₂(𝑧₂(𝑡,𝑘),𝑘),…,𝑔_𝑁(𝑧_𝑁(𝑡,𝑘),𝑘)]^𝑇 and for each regime 𝑘∈𝑆, there is 𝑔(0,𝑘)=0. Substituting (3.1)–(3.3) into (2.3), we have𝑧̇(𝑡,𝑘)=𝑔(𝑧(𝑡,𝑘),𝑘)+𝑐(𝑘)𝐺(𝑘)𝑧(𝑡,𝑘).(3.4) Through this substituting, network (2.3) with switching equilibrium 𝑢^∗(𝑘) is transferred to network (3.4) with common equilibrium 0. Thus we study stability of network (2.3) around equilibria 𝑢^∗(𝑘) by investigating stability of network (3.4) around origin. For simplicity, we write 𝑧(𝑡,𝑘) as 𝑧(𝑘) and matrix 𝑐(𝑟(𝑡)=𝑘), 𝐺(𝑟(𝑡)=𝑘), 𝐷(𝑟(𝑡)=𝑘) as 𝑐_𝑘, 𝐺_𝑘, 𝐷_𝑘.

Let 𝑃_𝑘, 𝑘∈𝑆 be a series of symmetric positive definite matrices and construct Lyapunov function as follows:𝑉(𝑧(𝑘),𝑡,𝑘)=𝑧(𝑘)^𝑇𝑃_𝑘𝑧(𝑘).(3.5) According to infinitesimal generator [11], there is ℒ𝑉(𝑧(𝑘),𝑡,𝑘)=𝑧̇^𝑇(𝑘)𝑃_𝑘𝑧(𝑘)+𝑧^𝑇(𝑘)𝑃_𝑘𝑧̇(𝑘)+^𝑀_𝑙=1=𝑔(𝑧(𝑘),𝑘)+𝑐_𝑘𝜋_𝑘𝑙𝑉(𝑧(𝑙),𝑡,𝑙)+𝐺_𝑘𝑧(𝑘)^𝑇𝑃_𝑘𝑧(𝑘)+𝑧^𝑇(𝑘)𝑃_𝑘𝑔(𝑧(𝑘),𝑘)+𝑐_𝑘𝐺_𝑘𝑧(𝑘)_𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑧^𝑇(𝑙)𝑃_𝑙𝑧(𝑙)+𝑧_𝑇𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘𝑧(𝑘)+𝑔^𝑇(𝑧(𝑘),𝑘)𝑃_𝑘𝑧(𝑘)(𝑘)𝑃_𝑘𝑔(𝑧(𝑘),𝑘)+^𝑀_𝑙=1≤𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑧^𝑇(𝑙)𝑃_𝑙𝑧(𝑙)+𝑔𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘𝑧(𝑘)_𝑇(𝑧(𝑘),𝑘)𝑄_3𝑘𝑔(𝑧(𝑘),𝑘)+𝑧^𝑇(𝑘)𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)+^𝑀_𝑙=1≤𝑧_𝑇𝜋_𝑘𝑙𝑧^𝑇(𝑙)𝑃_𝑙𝑧(𝑙)+𝑚(𝑘)𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘𝑧(𝑘)₂𝑧^𝑇(𝑘)𝑄_3𝑘𝑧(𝑘)+𝑧^𝑇(𝑘)𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)+^𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑧^𝑇(𝑙)𝑃_𝑙𝑧(𝑙)+𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑥(𝑡)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑥(𝑡)−𝑢^∗(𝑙)+𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑥(𝑡)−𝑢^∗(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑥(𝑡)−𝑢^∗(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙)+𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1𝜋_𝑘𝑙𝑧(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑧(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙).(3.6) The “derivative” of Lyapunov function is given by (3.6), and here 𝑄_3𝑘 are a series of symmetric positive definite matrices. For the stability analysis, we will discuss its property from two situations: 𝑢^∗(𝑘)≡𝑢^∗(𝑙), which means for each possible topology, the WSN acquires the same quantity of data from environment, and 𝑢^∗(𝑘)≠𝑢^∗(𝑙), 𝑘,𝑙∈𝑆 where the WSN acquires different quantity of data because of topology switching.

3.1. Constant Date Acquisition Rate

Consider the date acquisition rate is constant, which means 𝐷(𝑘)≡𝐷(𝑙) and 𝑢^∗(𝑘)≡𝑢^∗(𝑙), for all 𝑘,𝑙∈𝑆, thus (3.6) has the following form:ℒ𝑉(𝑧(𝑘),𝑡,𝑘)=≤𝑧𝑧(^𝑇𝑘()𝑘_𝑇)𝑃_𝑘𝑧(𝑘)𝑐^𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘+^𝑀_𝑙=1𝜋_𝑘𝑙𝑃_𝑙𝑧(𝑘).(3.7) In [13], Lasalle theorem of stochastic version was deduced for nonjump case and this theorem can be extended to Markovian jump case as follows.

Theorem 3.1 (Lasalle theorem). Consider Markovian jump system (3.4) with 𝑢^∗(𝑘)≡𝑢^∗(𝑙), for all 𝑘,𝑙∈𝑆, and Lyapunov function 𝑉(𝑥(𝑡,𝑘),𝑡,𝑘) satisfies that 𝑉(𝑥(𝑡,𝑘),𝑡,𝑘)∈𝐶^2,1(ℝ^𝑛×ℝ₊×𝑆;ℝ₊), if there exists 𝒦_∞ function 𝑊₁(𝑥(𝑡,𝑘)), 𝑊₂(𝑥(𝑡,𝑘)) and nonnegative continuous function 𝑊(𝑥(𝑡,𝑘)) such that 𝑉(ℒ0,𝑉𝑡,𝑘(𝑥)=(𝑡,0𝑘),𝑡,𝑊𝑘)₁≤(𝑥(−𝑡,𝑊𝑘))(≤𝑥(𝑉𝑡,𝑘(𝑥))(𝑡,𝑘∀)𝑘,∈𝑡,𝑘𝑆)≤𝑊𝑊(0₂)(𝑥=(0𝑡,,𝑘))(3.8) then the following equation stands: lim_𝑡→∞𝑊(𝑥(𝑡,𝑟(𝑡)))=0𝑎.𝑠.(3.9)

Proof. Please refer to the appendix.

By combining Theorem 3.1 and (3.7), we have the following theorem about the stability for network (2.3) with unique equilibrium.

Theorem 3.2. Consider wireless sensor network (2.3) with constant date acquisition rate 𝐷, if there exist a series of symmetric positive definite matrices 𝑃_𝑘, 𝑄_3𝑘, 𝑘∈𝑆 such that 𝐽^𝑘𝑃_𝑘𝑃_𝑘𝑄_3𝑘>0,(3.10) the network (2.3) is asymptotically reliable almost surely, that is, there is lim_𝑡→∞𝑥(𝑡,𝑟(𝑡))=𝑢^∗𝑎.𝑠.,(3.11) where 𝐽_𝑘 is defined as 𝐽_𝑘=−𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+^𝑀_𝑙=1𝜋_𝑘𝑙𝑃_𝑙.(3.12)

Proof. Since 𝑢^∗(𝑘)≡𝑢^∗(𝑙)=𝑢^∗, for all 𝑘,𝑙∈𝑆, thus (3.6) is translated to ℒ𝑉(𝑧(𝑘),𝑡,𝑘)=≤𝑧𝑧^𝑇_𝑇(𝑘)𝑃_𝑘𝑧(𝑘)(𝑘)𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘+^𝑀_𝑙=1𝜋_𝑘𝑙𝑃_𝑙≜−𝑊(𝑧(𝑡,𝑘))≤0.𝑧(𝑘)(3.13) By checking the definition of 𝑉(𝑧(𝑘),𝑡,𝑘), ℒ𝑉(𝑧(𝑘),𝑡,𝑘) and applying Lemma 2.3 as well as Lemma 2.4, we have immediately: tlhima_𝑡→∞t𝑊is(𝑃𝑧(𝑡,𝑘li)m)_𝑡→∞=0a.s.𝑊(𝑧(𝑡,𝑘))=0=1∀𝑘∈𝑆.(3.14) And it is easily seen that 𝑊(𝑧(𝑡,𝑘)) is a classic 𝒦 function of 𝑧(𝑡,𝑘). According to the quality of 𝒦 function (seen in [14]), 𝑊(𝑧(𝑡,𝑘)) is strictly positive if 𝑧(𝑡,𝑘)≠0, thus 𝑊(𝑧(𝑡,𝑘))=0 implies that 𝑧(𝑡,𝑘)=0, which means sample set {𝜔∶𝑊(𝑧(𝑡,𝑘))=0}⊆{𝜔∶𝑧(𝑡,𝑘)=0} and we have 𝑃lim_𝑡→∞𝑧(𝑡,𝑘)=0≥𝑃lim_𝑡→∞𝑊(𝑧(𝑡,𝑘))=0.(3.15) Combined with (3.14), the following stands: 𝑃lim_𝑡→∞𝑧(𝑡,𝑘)=0≥𝑃lim_𝑡→∞𝑊(𝑧(𝑡,𝑘))=0=1(3.16) Immediately we have t𝑃hatilsim𝑃_𝑡→∞𝑧(𝑡l,i𝑘m_𝑡→∞)=0=1,𝑥(𝑡,𝑟(𝑡))=𝑢^∗=1.(3.17)

3.2. Varying Date Acquisition Rate

In most cases, the distribution of equilibria 𝑢^∗(𝑘) will differ with different network topology. Obviously this difference will bring effects on the trajectory of node state in network (2.3). For example, network regime is 𝑟(𝑡₁)=𝑘 at time point 𝑡₁ and behavior of state trajectory is determined by the corresponding dynamic 𝑥̇(𝑡)=𝑓(𝑥(𝑡,𝑘),𝑘)+𝑐_𝑘𝐺(𝑘)𝑥(𝑡,𝑘)+𝐷(𝑘), 𝑡≥𝑡₁ such that the trajectory 𝑥(𝑡) is going towards the desired equilibrium 𝑢^∗(𝑘) with reliability criteria satisfied. After a random time Δ𝑡, a sudden event occurs and now the regime is 𝑙 with new network topology described as 𝑥̇(𝑡)=𝑓(𝑥(𝑡,𝑙),𝑙)+𝑐_𝑙𝐺(𝑙)𝑥(𝑡,𝑙)+𝐷(𝑙), 𝑡≥𝑡₁+Δ𝑡; thus the trajectory 𝑥(𝑡) will going towards a new equilibrium 𝑢^∗(𝑙) following the new topology. Intuitively, node state 𝑥(𝑡) will keep going towards the corresponding equilibrium 𝑢^∗(𝑗) for each regime 𝑗 with reliability criteria ensured. Thus it will finally run into a region which is concerned with the distribution of all the equilibria. It is obvious that reliability criteria only guarantee the node trajectory goes towards the equilibrium and cannot explain how close it can be near the equilibrium. For quantitative analysis, we have the following theorem.

Theorem 3.3. Consider stochastic network (2.3) with switching equilibria 𝑢^∗(𝑘), if there exist a series of symmetric positive-definite matrices 𝑃_𝑘, 𝑄_3𝑘 and positive scalar 𝜖_𝑘 such that the following inequality is ensured: 𝐽^∗𝑘𝑃_𝑘𝑃_𝑘𝑄_3𝑘>0.(3.18) This network is stochastically reliable in mean-square sense, that is, there exists a positive constant 𝐶 such that lim_𝑡→∞𝐸𝑥^𝑇(𝑡)𝑥(𝑡)≤𝐶,(3.19) where 𝐽_𝑘 is defined as 𝐽^∗𝑘=−𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+^𝑀_𝑙=11+𝜖_𝑙𝜋_𝑘𝑙𝑃_𝑙.(3.20) Constant 𝐶 is determined by 𝑢^∗(𝑘) with network parameters 𝑓(𝑘), 𝐺_𝑘, 𝑐_𝑘 given and 𝑃_𝑘, 𝑄_3𝑘, 𝜖_𝑘 taken.

Proof. According to inequality (3.6), there is ℒ𝑉(𝑧(𝑘),𝑡,𝑘)≤+𝑧(𝑘)^𝑇𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑧(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑧(𝑘)+𝑢^∗(𝑘)−𝑢^∗(𝑙)+𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1𝜋_𝑘𝑙𝑧(𝑘)^𝑇𝑃_𝑙𝑧(𝑘)+^𝑀_𝑙=1+𝜋_𝑘𝑙𝑧^𝑇(𝑘)𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)_𝑀_𝑙=1𝜋_𝑘𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑧(𝑘)+^𝑀_𝑙=1≤𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)+𝑐_𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘𝑧(𝑘)_𝑀_𝑙=1𝜋_𝑘𝑙𝑧(𝑘)^𝑇𝑃_𝑙𝑧(𝑘)+^𝑀_𝑙=1+𝜋_𝑘𝑙𝜖_𝑙𝑧^𝑇(𝑘)𝑃_𝑙𝑧^𝑇(𝑘)_𝑀_𝑙=1𝜋_𝑘𝑙1𝜖_𝑙𝑢^∗+(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)_𝑀_𝑙=1=𝑧(𝑘)_𝑇𝜋_𝑘𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)𝑐^𝑘𝐺_𝑘𝑃_𝑘+𝑐_𝑘𝑃_𝑘𝐺_𝑘+𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘+^𝑀_𝑙=11+𝜖_𝑙𝜋_𝑘𝑙𝑃_𝑙+𝑧(𝑘)_𝑀_𝑙=11+𝜖1_𝑙𝜋^𝑘𝑙𝑢^∗≤−𝛼𝑉(𝑧(𝑘),𝑡,𝑘)+𝛽(𝑘,)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)(3.21) where 𝛼≜min_𝑘∈𝑆𝜆^min−𝑐^𝑘𝐺_𝑘𝑃_𝑘−𝑐_𝑘𝑃_𝑘𝐺_𝑘−𝑚²𝑄_3𝑘+𝑃_𝑘𝑄^−13𝑘𝑃_𝑘−^𝑀_𝑙=11+𝜖_𝑙𝜋_𝑘𝑙𝑃_𝑙𝛽≜max_𝑘∈𝑆^𝑀_𝑙=11+𝜖1_𝑙𝜋^𝑘𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙)^𝑇𝑃_𝑙𝑢^∗(𝑘)−𝑢^∗(𝑙).(3.22) Similar to [15], apply generalized Itô formula and result that 𝐸𝑒^𝛼𝑡𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)=+𝑉𝛼(𝐸𝑧(0_𝑡0,𝑟(0)),𝑡,𝑟(0))+𝐸^𝑡0𝑒^𝛼𝑠ℒ𝑉(𝑧(𝑠,𝑟(𝑠)),𝑠,𝑟(𝑠))𝑑𝑠𝑉(𝑧(𝑠,𝑟(𝑠)),𝑠,𝑟(𝑠))𝑑𝑠.(3.23) Substitute (3.21) into (3.23) and we have 𝐸𝑒^𝛼𝑡𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)=≤𝑉𝑉((𝑧𝑧((00,,𝑟𝑟((00)))),,𝑡𝑡,,𝑟𝑟((00))))++𝛽𝛽^𝑡0𝑒^𝛼𝑠𝑑𝑠𝛼𝑒^𝛼𝑡−1(3.24) which immediately generates 𝐸{𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)}≤𝑒^−𝛼𝑡𝑉(𝑧(0,𝑟(0)),𝑡,𝑟(0))−𝛼𝛽+𝛼𝛽.(3.25) Let 𝑡→∞ in above inequality, there is lim_𝑡→∞𝐸{𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)}=lim_𝑡→∞𝐸𝑧^𝑇(𝑡,𝑘)𝑃_𝑘𝑧^𝑇(𝑡,𝑘)≤𝛽thatislim_𝑡→∞𝛼𝐸𝑥^𝑇(𝑡)𝑥(𝑡)≤𝛽𝛼𝜆_min𝑃_𝑘+𝑢^𝑘∗𝑇𝑢^∗𝑘≜𝐶.(3.26) The proof is complete.

Remark 3.4. In above analysis, we give the sufficient conditions for network (2.3) to be reliable in mean-square sense. It can be seen with (3.18) ensured, the trajectory of each regime 𝑘 is sure to enter an attractive region around the equilibrium 𝑢^∗(𝑘), and the radius of attractive region is an increasing function of the distance of all equilibria ∑^𝑀𝑙=1|𝑢^∗(𝑘)−𝑢^∗(𝑙)|², and this distance reflects the intrinsic characteristic of network which is determined by network topology. In order to decrease the radius, one possible way is to increase the value of 𝛼. From (3.25), we know that 𝛼 determines how fast the trajectory can converge to the attractive region, and 𝛼 is concerned with node calculation capacity 𝑓, network topology 𝐺, and network status 𝑐. The larger the parameter 𝛼 is, the faster the convergence speed is, and also the smaller radius is. For these reasons, we request a larger parameter 𝛼 for convergence speed and convergence precision.

Remark 3.5. Neither control variable nor decision action appears in the model of network (2.3), and the above analysis reflects the natural property of autonomous network. Note that parameter 𝐶 is a conservative result for the bound of node state, and the radius of attractive region for practical network will be less than 𝐶. Consider a network with performance demand that 𝐸{|𝑥(𝑡)|²}≤𝛾 after calculating we know 𝐶≤𝛾; thus we need to do nothing and just let the network work by itself. Otherwise, there may be a need taking control or decision to change either the network topology 𝐺_𝑘 or the transition rate matrix Π for the satisfaction of performance. This work will be of some help for network topology design and decision making.

4. Numerical Example

Consider the following 2-regime wireless sensor network as shown in Figure 2:𝑥̇(𝑡)=𝑓(𝑥(𝑡),𝑘)+𝑐_𝑘𝐺_𝑘𝑥(𝑡)+𝐷_𝑘𝑘=1,2,(4.1) where this network consists of 6 nodes (𝑁=6), and its topology switches between regime 1 and regime 2 with transition rate matrix as Π=^{−121220−20}. Initial node state is 𝑥(0)=[10,20,−10,−20,−7,33]^𝑇 and parameters are given asRegime1∶𝐺₁=⎡⎢⎢⎢⎢⎣−101002−110103−010113−100001−110002−001001⎤⎥⎥⎥⎥⎦,𝑐₁=0.05,(4.2)𝑓₁=−diag(6+1/20,6+2/20,6+3/20,6+4/20,6+5/20,6+6/20)𝑥(𝑡), Regime2∶𝐺₂=⎡⎢⎢⎢⎢⎣−100102−100012−000112−000101−011013−011002⎤⎥⎥⎥⎥⎦,𝑐₂=0.02,(4.3)𝑓₂=−diag(1+1/10,1+2/10,1+3/10,1+4/10,1+5/10,1+6/10)𝑥(𝑡), and 𝑚=6 for both regimes. For comparison of unique equilibrium case and multiple equilibria case, we adopt the same sample path, that is, Markovian jump is the same throughout numerical experiments as in Figure 3.

Figure 2 

Network topology switching.

Figure 3 

Simulation of Markovian jump.

4.1. Constant Date Acquisition Rate

We investigate network (2.3) with unique equilibrium 𝑢^∗𝑖=0, 𝑖=1,…,6, where 𝐷₁=𝐷₂=[0,0,0,0,0,0]^𝑇. It is easy to see that positive definite matric 𝑃₁=𝑃₂=𝑄₃₁=𝑄₃₂=𝐼 can satisfy−𝑐₁𝐺₁𝑃₁−𝑐₁𝑃₁𝐺₁−𝑚²𝑄₃₁+𝑃₁𝑄⁻¹³¹𝑃₁−²_𝑙=1𝜋_1𝑙𝑃_𝑙−𝑐₂=⎡⎢⎢⎢⎢⎣−−110000...59..2330009000010000700−−111800....5525..33000900000100000700−11810....5262.30009000010000700−1010.9..73000090010070−−111000..2.2..6330009000010000700−−11100..2..93300090001000700⎤⎥⎥⎥⎥⎦>0𝐺₂𝑃₂−𝑐₂𝑃₂𝐺₂−𝑚²𝑄₃₂+𝑃₂𝑄⁻²³²𝑃₂−²_𝑙=1𝜋_2𝑙𝑃_𝑙=⎡⎢⎢⎢⎢⎣−−−100000...446...111780222600000700000−−00100...446..110887220060000700−−−010000...684...111078222060000070000−−2000..54..1100582260007000−−000100.....6465.110805220006000007000−−00200...444..110878220600007000⎤⎥⎥⎥⎥⎦>0.(4.4) Thus all the parameters satisfy the reliability criteria in Theorem 3.2, and node state trajectory is shown as follows in Figure 4.

Figure 4 

Node state trajectory of constant data acquisition rate.

4.2. Varying Date Acquisition Rate

In this subsection, we simulate the case that Markovian jumps change not only the topology structure of network, but also date acquisition rate. And this network has two regimes with two different equilibria: 𝑢^∗𝑖(1)=0, 𝑢^∗𝑖(2)=5, 𝑖=1,2,…,6 with 𝐷₁=0,𝐷₂=[5.4861,5.9917,6.4861,6.9972,7.5194,8.0194]^𝑇. Positive definite matric 𝑃₁=𝑃₂=𝑄₃₁=𝑄₃₂=𝐼 and scalar 𝜖₁=0.2, 𝜖₂=0.5 can satisfy−𝑐₁𝐺₁𝑃₁−𝑐₁𝑃₁𝐺₁−𝑚²𝑄₃₁+𝑃₁𝑄⁻¹³¹𝑃₁−²_𝑙=11+𝜖_𝑙𝜋_1𝑙𝑃_𝑙−𝑐₂=⎡⎢⎢⎢⎢⎣−−61000...659..330900001000070000−−111400....5259..33000900000100000700−11510....5202.30009000010000700−080..91.3000090010070−−17100...202..330090000100007000−−8100..32..3300900010007000⎤⎥⎥⎥⎥⎦>0𝐺₂𝑃₂−𝑐₂𝑃₂𝐺₂−𝑚²𝑄₃₂+𝑃₂𝑄⁻¹³²𝑃₂−²_𝑙=11+𝜖₂𝜋_2𝑙𝑃_𝑙=⎡⎢⎢⎢⎢⎣−−−700000...446...111780222600000700000−−00700...446..110887220060000700−−−070000...684...111078222060000070000−−8000..54..1100582260007000−−000700.....6465.110805220006000007000−−00800...444..110878220600007000⎤⎥⎥⎥⎥⎦>0.(4.5) Thus all the parameters satisfy the reliability criteria in Theorem 3.3, and node sate trajectory is shown in Figure 5

Figure 5 

Node state trajectory of multiple equilibria 𝑢∗𝑖(1)=0, 𝑢∗𝑖(2)=5.

Consider network (2.3) has multiple equilibria 𝑢^∗𝑖(1)=0, 𝑢^∗𝑖(2)=10, 𝑖=1,…,6, thus 𝐷₁=[0,0,0,0,0,0]^𝑇, 𝐷₂=[10.9722,11.9833,12.9722,13,9944,15.0389,16.0389]^𝑇 while other parameters keep unchanged as well as the Markovian jump sample. Noticing that the reliability criteria are the same as the case of 𝑢^∗𝑖(1)=0, 𝑢^∗𝑖(2)=5, 𝑖=1,2,…,6. According to Theorem 3.3, such network is also stable in mean-square sense, and its node state trajectory is shown in Figure 6.

Figure 6 

Node state trajectory of multiple equilibria 𝑢∗𝑖(1)=0, 𝑢∗𝑖(2)=10.

It can be seen from Figures 5 and 6 that with the same parameters 𝑃₁, 𝑃₂, 𝑄₃₁, 𝑄₃₂,𝜀₁, 𝜀₂ satisfying the reliability criteria in Theorem 3.2, the node state trajectory can converge to the attractive region with the same convergence speed, which is dependent on parameter 𝑓(𝑘), 𝐺_𝑘, 𝑐_𝑘, 𝑁, 𝑚, Π with the same 𝑃_𝑘, 𝑄_3𝑘, 𝜀_𝑘, that is, this speed is determined by the natural property of network. However, the radius of this region is different: for the former case 𝑢^∗𝑖(1)=0, 𝑢^∗𝑖(2)=5, the radius is smaller while for the latter case 𝑢^∗𝑖(1)=0, 𝑢^∗𝑖(2)=10, the radius is larger, which means the radius of attractive region is an increasing function of the distance between the equilibria.

5. Conclusion

Reliability problems of stochastically switching wireless sensor network are studied in this paper. This switching, governed by Markovian chain, changes not only network topology, but also the data acquisition rate from outside environment. Our work reveals that reliability is dependent on three elements: network topology, network parameters and Markovian chain. When data acquisition rate keeps unchanged so that all network topologies share a common equilibrium despite of regime jump, network node state can be asymptotically reliable almost surely if the above three elements satisfy sufficient conditions. For varying data acquisition rate case, node state can converge to an attractive region similarly with sufficient conditions ensured, while its radius is concerned with the distribution of all the equilibria. Numerical simulations present an intuitive understanding of these relations and all the reliability criteria in this paper can be feasible for a general network.

Appendix

Lasalle Theorem in Markovian Jump Systems

Lemma A.1 (Supermartingale inequality [16]). Let 𝜉_𝑡, 𝑡∈ℝ₊ be a right-continuous supermartingale, there is for all 𝑠₀<𝑡₀∈ℝ₊, 𝜆>0, one has 𝑃𝜔∶sup_{𝑠0≤𝑡≤𝑡0}𝜉_𝑡(𝜔)≥𝜆≤1𝜆𝐸𝜉^𝑠0≤1∣𝜔∶sup_{𝑠0≤𝑡≤𝑡0}𝜉_𝑡(𝜔)≥𝜆𝜆𝐸𝜉^𝑠0.(A.1)

Lemma A.2 (Fatou's lemma). Let {𝑓_𝑘(𝑥)} be a series of measurable nonnegative functions defined on ℝ^𝑛, one has _ℝ𝑛lim_𝑘→∞𝑓_𝑘(𝑥)𝑑𝑥≤lim_𝑘→∞_ℝ𝑛𝑓_𝑘(𝑥)𝑑𝑥.(A.2)

Now we introduce the Lasalle theorem in Markovian jump systems as follows.

Theorem A.3 (Lasalle theorem). Considering Markovian jump system of the form: 𝑑𝑥=𝑓(𝑥,𝑡,𝑟(𝑡))𝑑𝑡+𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡),(A.3) where 𝐵(𝑡) is a standard Wiener process which is independent of Markov process 𝑟(𝑡), suppose there exist a function 𝑉(𝑥,𝑡,𝑖)∈𝐶^2,1(ℝ^𝑛×ℝ₊×𝑆;ℝ₊) and class 𝒦_∞ functions 𝑊₁, 𝑊₂, such that 𝑉(0,𝑊𝑡,₁𝑖)ℒ(|=𝑥𝑉|)0(,≤𝑥,𝑉∀𝑡,((𝑖𝑥𝑥),,≤𝑡𝑡,,𝑖𝑖−))≤∈𝑊𝑊ℝ(_𝑛𝑥₂)(|,𝑥|),(A.4)×𝑅₊×𝑆,(A.5) where 𝑊(⋅)∈𝐶(ℝ^𝑛;ℝ₊). Then the equilibrium 𝑥=0 is globally stable in probability and there is 𝑃lim_𝑡→∞𝑊𝑥𝑥₀,𝑖₀,𝑡=0=1,∀𝑥₀∈ℝ^𝑛,𝑟₀∈𝑆.(A.6)

Proof. First we will prove the global stability in probability of the Markovian jump system (A.3).
By (A.4) and (A.5), we have ℒ𝑉≤0 and 𝑉≥0,which means 𝑉 is a supermartingale on probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). For any class 𝒦_∞ function 𝛾(⋅), with supermartingale inequality applied, there is 𝑃sup_{0≤𝑠≤𝑡}𝑉(𝑥,𝑠,𝑖)≥𝑊₁𝛾||𝑥₀||≤𝑉𝑥⁰,0,𝑖₀𝑊₁𝛾||𝑥₀||.(A.7) According to the quality of 𝒦_∞ function, we have 𝑃sup_{0≤𝑠≤𝑡}|𝑥|≥𝛾||𝑥₀||≤𝑃𝑊₁sup_{0≤𝑠≤𝑡}|𝑥|≥𝑊₁𝛾||𝑥₀||.(A.8) Connect the above inequality with (A.4) and (A.7) 𝑃sup_{0≤𝑠≤𝑡}|𝑥|≥𝛾||𝑥₀||≤𝑃𝑊₁sup_{0≤𝑠≤𝑡}≤𝑃sup_{0≤𝑠≤𝑡}|𝑥|≥𝑊₁𝛾||𝑥₀||≤𝑉𝑥𝑉(𝑥,𝑠,𝑖)≥𝑊₁𝛾||𝑥₀||₀,0,𝑖₀𝑊₁𝛾||𝑥₀||.(A.9) Thus there is 𝑃sup_{0≤𝑠≤𝑡}|𝑥|<𝛾||𝑥₀||≥1−𝑉𝑥⁰,0,𝑖₀𝑊₁𝛾||𝑥₀||.(A.10) For any given 𝜁>0, choose 𝛾(⋅) such that 𝛾||𝑥₀||≥𝑊¹⁻¹𝑉𝑥⁰,0,𝑖₀𝑊₁𝜁,𝛾||𝑥₀||≥𝑉𝑥⁰,0,𝑖₀𝜁.(A.11) Then we have 𝑃||𝑥𝑥₀,𝑟₀,𝑡||<𝛾||𝑥₀||≥1−𝜁∀𝑡≥0,∀𝑥₀,𝑖₀∈ℝ^𝑛×𝑆(A.12) and the global stability in probability is proved.
Next we would prove the establishment of (A.6).
We decompose the sample space into three mutually exclusive events:(1)𝐴₁={𝜔∶limsup_𝑡→∞𝑊(𝑥(𝑡,𝜔))=0},(2) 𝐴₂={𝜔∶liminf_𝑡→∞𝑊(𝑥(𝑡,𝜔))>0},(3) 𝐴₃={𝜔∶liminf_𝑡→∞𝑊(𝑥(𝑡,𝜔))=0andlimsup_𝑡→∞𝑊(𝑥(𝑡,𝜔))>0}.and our aim is to prove that 𝑃{𝐴₂}=𝑃{𝐴₃}=0.
Let ℎ=1,2,… be a positive integer. Define the stopping time as 𝜏_ℎ=inf𝑡>0∶||𝑥(𝑡)||≥ℎ(A.13) and it could been easily seen that as ℎ→∞, 𝜏_ℎ→∞(a.s.). According to (A.10) we have 𝑃Ω₁=𝜔∶sup_0≤𝑡<∞||𝑥(𝑡)||<∞=1.(A.14) By generalized Itô formula and (A.5) 𝐸𝑉_𝑡ℎ=≤𝑉𝑉𝑥𝑥₀₀,0,𝑖₀+𝐸^𝑡ℎ₀ℒ𝑉(𝑥,𝑠,𝑟(𝑠))𝑑𝑠,0,𝑖₀−𝐸^𝑡ℎ₀𝑊(𝑥(𝑠))𝑑𝑠.(A.15) Here 𝑡_ℎ is defined as 𝑡_ℎ=𝑡∧𝜏_ℎ, for all 𝑡≥0. Since 𝐸𝑉_𝑡ℎ≥0, therefore, 𝐸^𝑡ℎ₀𝑊(𝑥(𝑠))𝑑𝑠≤𝑉𝑥₀,0,𝑖₀.(A.16) Let 𝑡→∞,ℎ→∞, by applying Fatou’s lemma, there is 𝐸^∞0𝑊(𝑥(𝑠))𝑑𝑠≤𝑉𝑥₀,0,𝑖₀.(A.17) Hence ^∞0𝑊(𝑥(𝑠))𝑑𝑠<∞a.s.(A.18) which follows immediately that 𝑃{𝐴₂}=0.
Now we proceed to show that 𝑃{𝐴₃}=0 by contradiction. Suppose 𝑃{𝐴₃}>0, then there exists 𝜖>0 such that 𝑃{𝑊(𝑥(𝑡))crossesfrombelow𝜖toabove2𝜖andbackinfinitelymanytimes}≥2𝜖.(A.19) It is easily seen that 𝑃{Ω₁∩𝐴₃}≥2𝜖. We now define a sequence of stopping times 𝜎𝜎₁=inf{𝑡≥0∶𝑊(𝑥(𝑡))≥2𝜖},_2𝑘𝜎=inf𝑡≥𝜎_2𝑘−1∶𝑊(𝑥(𝑡))≤𝜖_2𝑘+1,=inf𝑡≥𝜎_2𝑘∶𝑊(𝑥(𝑡))≥2𝜖.(A.20) By hypothesis (𝐻), for any |𝑥|≤ℎ, there exists a constant 𝐾_ℎ>0 such that ||𝑓(𝑥,𝑡,𝑟(𝑡))||∨||𝑔(𝑥,𝑡,𝑟(𝑡))||≤𝐾_ℎ|𝑥|.(A.21) From (A.3), we compute 𝐸sup_{0≤𝑡≤𝑇}||𝑥𝜏_ℎ∧𝜎_2𝑘−1+𝑡−𝑥𝜏_ℎ∧𝜎_2𝑘−1||²=𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑡)_{ℎ∧𝜎2𝑘−1}𝑓(𝑥,𝑡,𝑟(𝑡))𝑑𝑡+𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²≤2𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑡)_{ℎ∧𝜎2𝑘−1}𝑓(𝑥,𝑡,𝑟(𝑡))𝑑𝑡||||²+2𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑡)_{ℎ∧𝜎2𝑘−1}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²≤2𝐾_2ℎ𝑇²+2𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑡)_{ℎ∧𝜎2𝑘−1}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²=2𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑇)_{ℎ∧𝜎2𝑘−1}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)−^{𝜏ℎ∧(𝜎2𝑘−1𝜏+𝑇)}_{ℎ∧𝜎2𝑘−1+𝑡}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²+2𝐾^2ℎ𝑇²≤4𝐸sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑇)_{ℎ∧𝜎2𝑘−1}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²+sup_{0≤𝑡≤𝑇}||||^𝜏_ℎ^∧(𝜎_2𝑘−1^𝜏+𝑇)_{ℎ∧𝜎2𝑘−1+𝑡}𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)||||²+2𝐾^2ℎ𝑇²≤8𝐸||||^{𝜏ℎ∧(𝜎2𝑘−1𝜏+𝑇)}_{ℎ∧𝜎2𝑘−1}sup_{0≤𝑡≤𝑇}||||𝑔(𝑥,𝑡,𝑟(𝑡))||𝑑𝐵(𝑡)||₂+2𝐾^2ℎ𝑇²=8𝐸^{𝜏ℎ∧(𝜎2𝑘−1𝜏+𝑇)}_{ℎ∧𝜎2𝑘−1}sup_{0≤𝑡≤𝑇}||𝑔(𝑥,𝑡,𝑟(𝑡))||²𝑑𝑡+2𝐾^2ℎ𝑇²≤2𝐾^2ℎ𝑇²+8𝐾^2ℎ𝑇.(A.22) Since 𝑊(⋅) is continuous in ℝ^𝑛, it must be uniformly continuous in the closed ball 𝐵_ℎ={𝑥∈ℝ^𝑛∶|𝑥|≤ℎ}. We can therefore choose 𝛿=𝛿(𝜖)>0 small enough such that ||𝑊(𝑥)−𝑊(𝑦)||<𝜖,∀𝑥,𝑦∈𝐵_ℎ,||𝑥−𝑦||<𝛿.(A.23) We furthermore choose 𝑇=𝑇(𝜖,𝛿,ℎ) sufficiently small for 2𝐾^2ℎ𝑇²+8𝐾^2ℎ𝑇𝛿²<𝜖.(A.24) By Chebyshev’s inequality, it can be deduced that 𝑃≤su𝐸p_{0≤𝑡≤𝑇}||𝑥su𝜎p_{2𝑘−10≤𝑡≤𝑇}+𝑡−𝑥𝜎_2𝑘−1||≥𝛿||𝑥𝜎_2𝑘−1+𝑡−𝑥𝜎_2𝑘−1||²𝛿²≤2𝐾^2ℎ𝑇²+8𝐾^2ℎ𝑇𝛿²<𝜖.(A.25) According to operation principle of sets, we have 1≥=𝑃𝑃Ω₁Ω₁∩𝐴₃∪sup_{0≤𝑡≤𝑇}||𝑥𝜎_2𝑘−1+𝑡−𝑥𝜎_2𝑘−1||<𝛿−𝑃∩Ω𝐴₃₁+𝑃sup_{0≤𝑡≤𝑇}||𝑥𝜎_2𝑘−1+𝑡−𝑥𝜎_2𝑘−1||<𝛿≥2𝜖+(1−𝜖∩)𝐴−₃𝑃∩Ωs₁up_{0≤𝑡≤𝑇}||𝑥𝜎_2𝑘−1+𝑡𝑥𝜎_2𝑘−1||<𝛿∩𝐴₃∩sup_{0≤𝑡≤𝑇}||𝑥𝜎_2𝑘−1+𝑡−𝑥𝜎_2𝑘−1||<𝛿.(A.26) Thus 𝑃Ω₁∩𝐴₃∩sup_{0≤𝑡≤𝑇}||𝑥𝜎_2𝑘−1+𝑡−𝑥𝜎_2𝑘−1||<𝛿≥𝜖.(A.27) According to (A.23), we have 𝑃Ω∩𝐴₃∩sup_{0≤𝑡≤𝑇}||𝑊𝑥𝜎_2𝑘−1+𝑡−𝑊𝑥𝜎_2𝑘−1||<𝜖≥𝜖.(A.28) Define probability sample set as Ω_𝑘=sup_{0≤𝑡≤𝑇}||𝑊𝑥𝜎_2𝑘−1+𝑡−𝑊𝑥𝜎_2𝑘−1||<𝜖.(A.29) Let 𝟏(⋅) denote the indicator function of set, and noticed that 𝜎_2𝑘−𝜎_2𝑘−1≥𝑇, we derive from (A.18) and (A.26) that ∞>≥𝐸^∞0𝑊(𝑥(𝑡))𝑑𝑡_∞_𝑘=1𝐸𝟏_{{Ω1∩𝐴3}}^𝜎2𝑘_{𝜎2𝑘−1}≥𝜖_∞𝑊(𝑥(𝑡))𝑑𝑡_𝑘=1≥𝜖_∞𝐸𝟏_{{Ω1∩𝐴3}}𝜎_2𝑘−𝜎_2𝑘−1_𝑘=1≥𝜖𝑇_∞𝐸𝟏_{{Ω1∩𝐴3}∩Ω𝑘}𝜎_2𝑘−𝜎_2𝑘−1_𝑘=1≥𝜖𝑇_∞𝑃Ω₁∩𝐴₃∩Ω_𝑘_𝑘=1𝜀=∞(A.30) which is a contradiction. Thus 𝑃{𝐴₃}=0. Therefore 𝑃{𝐴₁}=1 and the proof is therefore completed.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 60904021, the Fundamental Research Funds for the Central Universities under Grant WK2100060004, and National High-Tech Research and Development Program of China 863 Program under Grant 2008AA01A317.