Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 760359 |

Jin Zhu, Liang Tang, Hongsheng Xi, Zhenghuan Zhang, "Reliability Analysis of Wireless Sensor Networks Using Markovian Model", Journal of Applied Mathematics, vol. 2012, Article ID 760359, 21 pages, 2012.

Reliability Analysis of Wireless Sensor Networks Using Markovian Model

Academic Editor: Chong Lin
Received11 Dec 2011
Accepted26 Mar 2012
Published03 Jun 2012


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,,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 𝜆min(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 𝑘𝑆 corresponds to one regime and this topology(regime) switching is governed by a Markovian chain 𝑟(𝑡) characterized by transition rate matrix Π=[𝜋𝑘𝑙]𝑀×𝑀, 𝑘,𝑙𝑆 as𝜋𝑃(𝑟(𝑡+𝑑𝑡)=𝑙𝑟(𝑡)=𝑘)=𝑘𝑙𝑑𝑡+𝑜(𝑑𝑡)if𝑘𝑙1+𝜋𝑘𝑘𝑑𝑡+𝑜(𝑑𝑡)if𝑘=𝑙,(2.1) where 𝑑𝑡>0 and 𝑜(𝑑𝑡) satisfie lim𝑑𝑡0(𝑜(𝑑𝑡)/𝑑𝑡)=0. Notice that the total probability axiom imposes 𝜋𝑘𝑘 negative and 𝑀𝑙=1𝜋𝑘𝑙=0, for all 𝑘𝑆.

𝑥𝑖(𝑡) is the data length, that is, quantity of data waiting to be transported in each nodes 𝑖; 𝑓𝑖(𝑥𝑖(𝑡),𝑡,𝑟(𝑡)) is the calculation capacity of node 𝑖. For each possible network topology 𝑟(𝑡), 𝑓𝑖(𝑥𝑖(𝑡),𝑡,𝑟(𝑡)) may be different. 𝐺𝑖𝑗(𝑟(𝑡)) is specified as follows: if there is a physical transport path or connection between node 𝑖 and node 𝑗(𝑖𝑗), 𝐺𝑗𝑖(𝑟(𝑡))=𝐺𝑖𝑗(𝑟(𝑡))=1; otherwise 𝐺𝑗𝑖(𝑟(𝑡))=𝐺𝑖𝑗(𝑟(𝑡))=0(𝑖𝑗). 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, 𝑥𝑗𝑥𝑖. 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 𝑑𝑖(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:̇𝑥𝑖(𝑡)=𝑓𝑖𝑥𝑖(𝑡),𝑟(𝑡)+𝑐(𝑟(𝑡))𝑁𝑗=1𝐺𝑖𝑗(𝑟(𝑡))𝑥𝑗(𝑡)+𝑑𝑖(𝑟(𝑡)),𝑖=1,2,𝑁𝑟(𝑡)𝑆={1,2,𝑀}.(2.2) In (2.2), 𝐺𝑖𝑖(𝑟(𝑡))=𝑁𝑗=1𝐺𝑖𝑗(𝑟(𝑡)), for all 𝑖𝑗. Noticing the fact that for all 𝐺𝑖𝑗(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 𝑥𝑖(𝑡)>𝑥𝑗(𝑡) and otherwise from 𝑗 to 𝑖 with 𝑥𝑖(𝑡)<𝑥𝑗(𝑡).

Rewrite (2.2) in matrix form aṡ𝑥(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),𝑥𝑁(𝑡)]𝑇𝑁 denotes the node data variables; 𝑓(𝑥(𝑡),𝑟(𝑡))=[𝑓1(𝑥1,𝑟(𝑡)),𝑓2(𝑥2,𝑟(𝑡)),,𝑓𝑁(𝑥𝑁,𝑟(𝑡))]𝑇, and 𝐷(𝑟(𝑡))=[𝑑1(𝑟(𝑡)),𝑑2(𝑟(𝑡)),𝑑𝑁(𝑟(𝑡))]𝑇. Matrix 𝐺(𝑟(𝑡))𝑁×𝑁 is coupling matrix and for each regime 𝑟(𝑡)𝑆, the elements 𝐺𝑖𝑗(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 𝑘𝑆, the function 𝑓(𝑥(𝑡),𝑘) in (2.2) satisfies the following sector condition:||||||||𝑓(𝑥(𝑡),𝑘)𝑓(𝑦(𝑡),𝑘)𝑚𝑥(𝑡)𝑦(𝑡),𝑥,𝑦𝑁,𝑘𝑆.(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 lim𝑡𝐸𝑥𝑇(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 𝑃lim𝑡𝑥(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=𝑄𝑇3, the following inequality holds: 𝑄𝑇1𝑄2+𝑄𝑇2𝑄1𝑄𝑇1𝑄3𝑄1+𝑄𝑇2𝑄31𝑄2.(2.7)

Lemma 2.4 (Schur complement). Let 𝑋=𝑋𝑇(𝑛+𝑚)×(𝑛+𝑚) be a symmetric matrix given by 𝑋=𝐴𝐵𝑇𝐵𝐶, where 𝐴𝑛×𝑛, 𝐵𝑚×𝑛, 𝐶𝑚×𝑚, and 𝐶 is nonsingular, then 𝑋 is positive definite if and only if 𝐶>0 and 𝐴𝐵𝑇𝐶1𝐵>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))=[𝑢1(𝑟(𝑡)),𝑢2(𝑟(𝑡)),𝑢𝑁(𝑟(𝑡))]𝑁, where 𝑢(𝑟(𝑡)) can be different or the same for each different regime 𝑟(𝑡), thus there is𝑓𝑢(𝑟(𝑡),𝑡)+𝑐(𝑟(𝑡))𝐺(𝑟(𝑡))𝑢(𝑟(𝑡))+𝐷(𝑟(𝑡))=0,𝑘𝑆.(3.1) In order to shift 𝑢(𝑟(𝑡)) to the origin, define 𝑧(𝑡,𝑟(𝑡))=𝑥(𝑡)𝑢𝑔𝑢(𝑟(𝑡)),(3.2)(𝑧(𝑡,𝑟(𝑡)),𝑟(𝑡))=𝑓(𝑥(𝑡),𝑟(𝑡))𝑓(𝑟(𝑡)),𝑟(𝑡).(3.3) Therefore 𝑔(𝑧(𝑡,𝑘),𝑘)=[𝑔1(𝑧1(𝑡,𝑘),𝑘),𝑔2(𝑧2(𝑡,𝑘),𝑘),,𝑔𝑁(𝑧𝑁(𝑡,𝑘),𝑘)]𝑇 and for each regime 𝑘𝑆, there is 𝑔(0,𝑘)=0. Substituting (3.1)–(3.3) into (2.3), we havė𝑧(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 𝑐𝑘, 𝐺𝑘, 𝐷𝑘.

Let 𝑃𝑘, 𝑘𝑆 be a series of symmetric positive definite matrices and construct Lyapunov function as follows:𝑉(𝑧(𝑘),𝑡,𝑘)=𝑧(𝑘)𝑇𝑃𝑘𝑧(𝑘).(3.5) According to infinitesimal generator [11], there is 𝑉(𝑧(𝑘),𝑡,𝑘)=̇𝑧𝑇(𝑘)𝑃𝑘𝑧(𝑘)+𝑧𝑇(𝑘)𝑃𝑘̇𝑧(𝑘)+𝑀𝑙=1𝜋𝑘𝑙=𝑉(𝑧(𝑙),𝑡,𝑙)𝑔(𝑧(𝑘),𝑘)+𝑐𝑘𝐺𝑘𝑧(𝑘)𝑇𝑃𝑘𝑧(𝑘)+𝑧𝑇(𝑘)𝑃𝑘𝑔(𝑧(𝑘),𝑘)+𝑐𝑘𝐺𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧𝑇(𝑙)𝑃𝑙𝑧(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘𝑧(𝑘)+𝑔𝑇(𝑧(𝑘),𝑘)𝑃𝑘𝑧(𝑘)+𝑧𝑇(𝑘)𝑃𝑘𝑔(𝑧(𝑘),𝑘)+𝑀𝑙=1𝜋𝑘𝑙𝑧𝑇(𝑙)𝑃𝑙𝑧(𝑙)𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑔𝑧(𝑘)𝑇(𝑧(𝑘),𝑘)𝑄3𝑘𝑔(𝑧(𝑘),𝑘)+𝑧𝑇(𝑘)𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧𝑇(𝑙)𝑃𝑙𝑧(𝑙)𝑧𝑇𝑐(𝑘)𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘𝑧+𝑚(𝑘)2𝑧𝑇(𝑘)𝑄3𝑘𝑧(𝑘)+𝑧𝑇(𝑘)𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧𝑇(𝑙)𝑃𝑙𝑧(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑥(𝑡)𝑢(𝑙)𝑇𝑃𝑙𝑥(𝑡)𝑢(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑥(𝑡)𝑢(𝑘)+𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑥(𝑡)𝑢(𝑘)+𝑢(𝑘)𝑢(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘𝑧+(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧(𝑘)+𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑧(𝑘)+𝑢(𝑘)𝑢.(𝑙)(3.6) The “derivative” of Lyapunov function is given by (3.6), and here 𝑄3𝑘 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,𝑙𝑆 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,𝑙𝑆, thus (3.6) has the following form:𝑉(𝑧(𝑘),𝑡,𝑘)=𝑧𝑇(𝑘)𝑃𝑘𝑧(𝑘)𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑀𝑙=1𝜋𝑘𝑙𝑃𝑙𝑧(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,𝑙𝑆, and Lyapunov function 𝑉(𝑥(𝑡,𝑘),𝑡,𝑘) satisfies that 𝑉(𝑥(𝑡,𝑘),𝑡,𝑘)𝐶2,1(𝑛×+×𝑆;+), if there exists 𝒦 function 𝑊1(𝑥(𝑡,𝑘)), 𝑊2(𝑥(𝑡,𝑘)) and nonnegative continuous function 𝑊(𝑥(𝑡,𝑘)) such that 𝑉(0,𝑡,𝑘)=0𝑊1(𝑥(𝑡,𝑘))𝑉(𝑥(𝑡,𝑘),𝑡,𝑘)𝑊2(𝑥(𝑡,𝑘))𝑉(𝑥(𝑡,𝑘),𝑡,𝑘)𝑊(𝑥(𝑡,𝑘))𝑘𝑆𝑊(0)=0,(3.8) then the following equation stands: lim𝑡𝑊(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 𝑃𝑘, 𝑄3𝑘, 𝑘𝑆 such that 𝐽𝑘𝑃𝑘𝑃𝑘𝑄3𝑘>0,(3.10) the network (2.3) is asymptotically reliable almost surely, that is, there is lim𝑡𝑥(𝑡,𝑟(𝑡))=𝑢𝑎.𝑠.,(3.11) where 𝐽𝑘 is defined as 𝐽𝑘𝑐=𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑀𝑙=1𝜋𝑘𝑙𝑃𝑙.(3.12)

Proof. Since 𝑢(𝑘)𝑢(𝑙)=𝑢, for all 𝑘,𝑙𝑆, thus (3.6) is translated to 𝑉(𝑧(𝑘),𝑡,𝑘)=𝑧𝑇(𝑘)𝑃𝑘𝑧(𝑘)𝑧𝑇𝑐(𝑘)𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑀𝑙=1𝜋𝑘𝑙𝑃𝑙𝑧(𝑘)𝑊(𝑧(𝑡,𝑘))0.(3.13) By checking the definition of 𝑉(𝑧(𝑘),𝑡,𝑘), 𝑉(𝑧(𝑘),𝑡,𝑘) and applying Lemma 2.3 as well as Lemma 2.4, we have immediately: lim𝑡𝑊(𝑧(𝑡,𝑘))=0a.s.thatis𝑃lim𝑡𝑊(𝑧(𝑡,𝑘))=0=1𝑘𝑆.(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 {𝜔𝑊(𝑧(𝑡,𝑘))=0}{𝜔𝑧(𝑡,𝑘)=0} and we have 𝑃lim𝑡𝑧(𝑡,𝑘)=0𝑃lim𝑡𝑊(𝑧(𝑡,𝑘))=0.(3.15) Combined with (3.14), the following stands: 𝑃lim𝑡𝑧(𝑡,𝑘)=0𝑃lim𝑡𝑊(𝑧(𝑡,𝑘))=0=1(3.16) Immediately we have 𝑃lim𝑡𝑧(𝑡,𝑘)=0=1,thatis𝑃lim𝑡𝑥(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 ̇𝑥(𝑡)=𝑓(𝑥(𝑡,𝑘),𝑘)+𝑐𝑘𝐺(𝑘)𝑥(𝑡,𝑘)+𝐷(𝑘), 𝑡𝑡1 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 ̇𝑥(𝑡)=𝑓(𝑥(𝑡,𝑙),𝑙)+𝑐𝑙𝐺(𝑙)𝑥(𝑡,𝑙)+𝐷(𝑙), 𝑡𝑡1+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 𝑃𝑘, 𝑄3𝑘 and positive scalar 𝜖𝑘 such that the following inequality is ensured: 𝐽𝑘𝑃𝑘𝑃𝑘𝑄3𝑘>0.(3.18) This network is stochastically reliable in mean-square sense, that is, there exists a positive constant 𝐶 such that lim𝑡𝐸𝑥𝑇(𝑡)𝑥(𝑡)𝐶,(3.19) where 𝐽𝑘 is defined as 𝐽𝑘𝑐=𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑀𝑙=11+𝜖𝑙𝜋𝑘𝑙𝑃𝑙.(3.20) Constant 𝐶 is determined by 𝑢(𝑘) with network parameters 𝑓(𝑘), 𝐺𝑘, 𝑐𝑘 given and 𝑃𝑘, 𝑄3𝑘, 𝜖𝑘 taken.

Proof. According to inequality (3.6), there is 𝑉(𝑧(𝑘),𝑡,𝑘)𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧(𝑘)+𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑧(𝑘)+𝑢(𝑘)𝑢(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧(𝑘)𝑇𝑃𝑙𝑧(𝑘)+𝑀𝑙=1𝜋𝑘𝑙𝑧𝑇(𝑘)𝑃𝑙𝑢(𝑘)𝑢+(𝑙)𝑀𝑙=1𝜋𝑘𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑧(𝑘)+𝑀𝑙=1𝜋𝑘𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑢(𝑘)𝑢(𝑙)𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑧(𝑘)𝑀𝑙=1𝜋𝑘𝑙𝑧(𝑘)𝑇𝑃𝑙𝑧(𝑘)+𝑀𝑙=1𝜋𝑘𝑙𝜖𝑙𝑧𝑇(𝑘)𝑃𝑙𝑧𝑇(+𝑘)𝑀𝑙=1𝜋𝑘𝑙1𝜖𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑢(𝑘)𝑢+(𝑙)𝑀𝑙=1𝜋𝑘𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑢(𝑘)𝑢(𝑙)=𝑧(𝑘)𝑇𝑐𝑘𝐺𝑘𝑃𝑘+𝑐𝑘𝑃𝑘𝐺𝑘+𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘+𝑀𝑙=11+𝜖𝑙𝜋𝑘𝑙𝑃𝑙+𝑧(𝑘)𝑀𝑙=111+𝜖𝑙𝜋𝑘𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑢(𝑘)𝑢(𝑙)𝛼𝑉(𝑧(𝑘),𝑡,𝑘)+𝛽,(3.21) where 𝛼min𝑘𝑆𝜆min𝑐𝑘𝐺𝑘𝑃𝑘𝑐𝑘𝑃𝑘𝐺𝑘𝑚2𝑄3𝑘+𝑃𝑘𝑄13𝑘𝑃𝑘𝑀𝑙=11+𝜖𝑙𝜋𝑘𝑙𝑃𝑙𝛽max𝑘𝑆𝑀𝑙=111+𝜖𝑙𝜋𝑘𝑙𝑢(𝑘)𝑢(𝑙)𝑇𝑃𝑙𝑢(𝑘)𝑢.(𝑙)(3.22) Similar to [15], apply generalized Itô formula and result that 𝐸𝑒𝛼𝑡𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)=𝑉(𝑧(0,𝑟(0)),𝑡,𝑟(0))+𝐸𝑡0𝑒𝛼𝑠𝑉(𝑧(𝑠,𝑟(𝑠)),𝑠,𝑟(𝑠))𝑑𝑠+𝛼𝐸𝑡0𝑉(𝑧(𝑠,𝑟(𝑠)),𝑠,𝑟(𝑠))𝑑𝑠.(3.23) Substitute (3.21) into (3.23) and we have 𝐸𝑒𝛼𝑡𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)𝑉(𝑧(0,𝑟(0)),𝑡,𝑟(0))+𝛽𝑡0𝑒𝛼𝑠𝛽𝑑𝑠=𝑉(𝑧(0,𝑟(0)),𝑡,𝑟(0))+𝛼𝑒𝛼𝑡1(3.24) which immediately generates 𝐸{𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)}𝑒𝛼𝑡𝛽𝑉(𝑧(0,𝑟(0)),𝑡,𝑟(0))𝛼+𝛽𝛼.(3.25) Let 𝑡 in above inequality, there is lim𝑡𝐸{𝑉(𝑧(𝑡,𝑘),𝑡,𝑘)}=lim𝑡𝐸𝑧𝑇(𝑡,𝑘)𝑃𝑘𝑧𝑇𝛽(𝑡,𝑘)𝛼thatislim𝑡𝐸𝑥𝑇(𝛽𝑡)𝑥(𝑡)𝛼𝜆min𝑃𝑘+𝑢𝑘𝑇𝑢𝑘𝐶.(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 𝐶𝛾; 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 𝐺𝑘 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:̇𝑥(𝑡)=𝑓(𝑥(𝑡),𝑘)+𝑐𝑘𝐺𝑘𝑥(𝑡)+𝐷𝑘𝑘=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=12122020. Initial node state is 𝑥(0)=[10,20,10,20,7,33]𝑇 and parameters are given asRegime1𝐺1=210100131010013011100100011020001001,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), Regime2𝐺2=210010120001002011000110101130011002,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.

4.1. Constant Date Acquisition Rate

We investigate network (2.3) with unique equilibrium 𝑢𝑖=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 satisfy𝑐1𝐺1𝑃1𝑐1𝑃1𝐺1𝑚2𝑄31+𝑃1𝑄131𝑃12𝑙=1𝜋1𝑙𝑃𝑙=10.29171.50000.30000.90000.300001.50008.59171.50000.30001.20000.30000.30001.50008.691701.20001.20000.90000.3000011.7917000.30001.20001.2000010.69170.300000.30001.200000.300011.9917>0𝑐2𝐺2𝑃2𝑐2𝑃2𝐺2𝑚2𝑄32+𝑃2𝑄232𝑃22𝑙=1𝜋2𝑙𝑃𝑙=1.47670.48000.12000.12000.60000.12000.48001.67670.120000.120000.48000.12000.12001.87670.12000.60000.48000.120000.12002.55670.480000.60000.12000.60000.48001.55670.12000.12000.48000.480000.12002.4767>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.

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: 𝑢𝑖(1)=0, 𝑢𝑖(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 satisfy𝑐1𝐺1𝑃1𝑐1𝑃1𝐺1𝑚2𝑄31+𝑃1𝑄131𝑃12𝑙=11+𝜖𝑙𝜋1𝑙𝑃𝑙=6.69171.50000.30000.90000.300001.50004.99171.50000.30001.20000.30000.30001.50005.091701.20001.20000.90000.300008.1917000.30001.20001.200007.09170.300000.30001.200000.30008.3917>0𝑐2𝐺2𝑃2𝑐2𝑃2𝐺2𝑚2𝑄32+𝑃2𝑄132𝑃22𝑙=11+𝜖2𝜋2𝑙𝑃𝑙=7.47670.48000.12000.12000.60000.12000.48007.67670.120000.120000.48000.12000.12007.87670.12000.60000.48000.120000.12008.55670.480000.60000.12000.60000.48007.55670.12000.12000.48000.480000.12008.4767>0.(4.5) Thus all the parameters satisfy the reliability criteria in Theorem 3.3, and node sate trajectory is shown in Figure 5

Consider network (2.3) has multiple equilibria 𝑢𝑖(1)=0, 𝑢𝑖(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 𝑢𝑖(1)=0, 𝑢𝑖(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.

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), 𝐺𝑘, 𝑐𝑘, 𝑁, 𝑚, Π with the same 𝑃𝑘, 𝑄3𝑘, 𝜀𝑘, that is, this speed is determined by the natural property of network. However, the radius of this region is different: for the former case 𝑢𝑖(1)=0, 𝑢𝑖(2)=5, the radius is smaller while for the latter case 𝑢𝑖(1)=0, 𝑢𝑖(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.


Lasalle Theorem in Markovian Jump Systems

Lemma A.1 (Supermartingale inequality [16]). Let 𝜉𝑡, 𝑡+ be a right-continuous supermartingale, there is for all 𝑠0<𝑡0+, 𝜆>0, one has 𝑃𝜔sup𝑠0𝑡𝑡0𝜉𝑡1(𝜔)𝜆𝜆𝐸𝜉𝑠0𝜔sup𝑠0𝑡𝑡0𝜉𝑡1(𝜔)𝜆𝜆𝐸𝜉𝑠0.(A.1)

Lemma A.2 (Fatou's lemma). Let {𝑓𝑘(𝑥)} be a series of measurable nonnegative functions defined on 𝑛, one has 𝑛lim𝑘𝑓𝑘(𝑥)𝑑𝑥lim𝑘𝑛𝑓𝑘(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)𝐶2,1(𝑛×+×𝑆;+) and class 𝒦 functions 𝑊1, 𝑊2, such that 𝑊1(|𝑥|)𝑉(𝑥,𝑡,𝑖)𝑊2𝑉(|𝑥|),(A.4)𝑉(𝑥,𝑡,𝑖)𝑊(𝑥),(0,𝑡,𝑖)=0,(𝑥,𝑡,𝑖)𝑛×𝑅+×𝑆,(A.5) where 𝑊()𝐶(𝑛;+). Then the equilibrium 𝑥=0 is globally stable in probability and there is 𝑃lim𝑡𝑊𝑥𝑥0,𝑖0,𝑡=0=1,𝑥0𝑛,𝑟0𝑆.(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 𝑉0 and 𝑉0,which means 𝑉 is a supermartingale on probability space (Ω,,{𝑡}𝑡0,𝑃). For any class 𝒦 function 𝛾(), with supermartingale inequality applied, there is 𝑃sup0𝑠𝑡𝑉(𝑥,𝑠,𝑖)𝑊1𝛾||𝑥0||𝑉𝑥0,0,𝑖0𝑊1𝛾||𝑥0||.(A.7) According to the quality of 𝒦 function, we have 𝑃sup0𝑠𝑡||𝑥|𝑥|𝛾0||𝑊𝑃1sup0𝑠𝑡|𝑥|𝑊1𝛾||𝑥0||.(A.8) Connect the above inequality with (A.4) and (A.7) 𝑃sup0𝑠𝑡||𝑥|𝑥|𝛾0||𝑊𝑃1sup0𝑠𝑡|𝑥|𝑊1𝛾||𝑥0||𝑃sup0𝑠𝑡𝑉(𝑥,𝑠,𝑖)𝑊1𝛾||𝑥0||𝑉𝑥0,0,𝑖0𝑊1𝛾||𝑥0||.(A.9) Thus there is 𝑃sup0𝑠𝑡||𝑥|𝑥|<𝛾0||𝑉𝑥10,0,𝑖0𝑊1𝛾||𝑥0||.(A.10) For any given 𝜁>0, choose 𝛾() such that 𝛾||𝑥0||𝑊11𝑉𝑥0,0,𝑖0𝜁,𝑊1𝛾||𝑥0||𝑉𝑥0,0,𝑖0𝜁.(A.11) Then we have 𝑃||𝑥𝑥0,𝑟0||||𝑥,𝑡<𝛾0||1𝜁𝑡0,𝑥0,𝑖0𝑛×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={𝜔limsup𝑡𝑊(𝑥(𝑡,𝜔))=0},(2)𝐴2={𝜔liminf𝑡𝑊(𝑥(𝑡,𝜔))>0},(3)𝐴3={𝜔liminf𝑡𝑊(𝑥(𝑡,𝜔))=0andlimsup𝑡𝑊(𝑥(𝑡,𝜔))>0}.and our aim is to prove that 𝑃{𝐴2}=𝑃{𝐴3}=0.
Let =1,2, be a positive integer. Define the stopping time as 𝜏||𝑥||=inf𝑡>0(𝑡)(A.13) and it could been easily seen that as , 𝜏(a.s.). According to (A.10) we have 𝑃Ω1=𝜔sup0𝑡<||||𝑥(𝑡)<=1.(A.14) By generalized Itô formula and (A.5) 𝐸𝑉𝑡𝑥=𝑉0,0,𝑖0+𝐸𝑡0𝑥𝑉(𝑥,𝑠,𝑟(𝑠))𝑑𝑠𝑉0,0,𝑖0𝐸𝑡0.𝑊(𝑥(𝑠))𝑑𝑠(A.15) Here 𝑡 is defined as 𝑡=𝑡𝜏, for all 𝑡0. Since 𝐸𝑉𝑡0, therefore, 𝐸𝑡0𝑥𝑊(𝑥(𝑠))𝑑𝑠𝑉0,0,𝑖0.(A.16) Let 𝑡,, by applying Fatou’s lemma, there is 𝐸0𝑥𝑊(𝑥(𝑠))𝑑𝑠𝑉0,0,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))crossesfrombelow𝜖toabove2𝜖andbackinnitelymanytimes}2𝜖.(A.19) It is easily seen that 𝑃{Ω1𝐴3}2𝜖. We now define a sequence of stopping times 𝜎1𝜎=inf{𝑡0𝑊(𝑥(𝑡))2𝜖},2𝑘=inf𝑡𝜎2𝑘1,𝜎𝑊(𝑥(𝑡))𝜖2𝑘+1=inf𝑡𝜎2𝑘.𝑊(𝑥(𝑡))2𝜖(A.20) By hypothesis (𝐻), for any |𝑥|, there exists a constant 𝐾>0 such that ||||||||𝑓(𝑥,𝑡,𝑟(𝑡))𝑔(𝑥,𝑡,𝑟(𝑡))𝐾|𝑥|.(A.21) From (A.3), we compute 𝐸sup0𝑡𝑇||𝑥𝜏𝜎2𝑘1𝜏+𝑡𝑥𝜎2𝑘1||2=𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑡)𝜎2𝑘1||||𝑓(𝑥,𝑡,𝑟(𝑡))𝑑𝑡+𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)22𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑡)𝜎2𝑘1𝑓||||(𝑥,𝑡,𝑟(𝑡))𝑑𝑡2+2𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑡)𝜎2𝑘1𝑔||||(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)22𝐾2𝑇2+2𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑡)𝜎2𝑘1||||𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)2=2𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1+𝑡||||𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)2+2𝐾2𝑇24𝐸sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1||||𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)2+sup0𝑡𝑇||||𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1+𝑡||||𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)2+2𝐾2𝑇2||||8𝐸𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1sup0𝑡𝑇||||||||𝑔(𝑥,𝑡,𝑟(𝑡))𝑑𝐵(𝑡)2+2𝐾2𝑇2=8𝐸𝜏(𝜎2𝑘1𝜏+𝑇)𝜎2𝑘1sup0𝑡𝑇||||𝑔(𝑥,𝑡,𝑟(𝑡))2𝑑𝑡+2𝐾2𝑇22𝐾2𝑇2+8𝐾2𝑇.(A.22) Since 𝑊() is continuous in 𝑛, it must be uniformly continuous in the closed ball 𝐵={𝑥𝑛|𝑥|}. We can therefore choose 𝛿=𝛿(𝜖)>0 small enough such that ||||𝑊(𝑥)𝑊(𝑦)<𝜖,𝑥,𝑦𝐵,||||𝑥𝑦<𝛿.(A.23) We furthermore choose 𝑇=𝑇(𝜖,𝛿,) sufficiently small for 2𝐾2𝑇2+8𝐾2𝑇𝛿2<𝜖.(A.24) By Chebyshev’s inequality, it can be deduced that 𝑃sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||𝐸𝛿sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||2𝛿22𝐾2𝑇2+8𝐾2𝑇𝛿2<𝜖.(A.25) According to operation principle of sets, we have Ω1𝑃1𝐴3sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||Ω<𝛿=𝑃1𝐴3+𝑃sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||Ω<𝛿𝑃1𝐴3sup0𝑡𝑇||𝑥𝜎2𝑘1𝑥𝜎+𝑡2𝑘1||Ω<𝛿2𝜖+(1𝜖)𝑃1𝐴3sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||.<𝛿(A.26) Thus 𝑃Ω1𝐴3sup0𝑡𝑇||𝑥𝜎2𝑘1𝜎+𝑡𝑥2𝑘1||<𝛿𝜖.(A.27) According to (A.23), we have 𝑃Ω𝐴3sup0𝑡𝑇||𝑊𝑥𝜎2𝑘1𝑥𝜎+𝑡𝑊2𝑘1||<𝜖𝜖.(A.28) Define probability sample set as Ω𝑘=sup0𝑡𝑇||𝑊𝑥𝜎2𝑘1𝑥𝜎+𝑡𝑊2𝑘1||<𝜖.(A.29) Let 𝟏() denote the indicator function of set, and noticed that 𝜎2𝑘𝜎2𝑘1𝑇, we derive from (A.18) and (A.26) that >𝐸0𝑊(𝑥(𝑡))𝑑𝑡𝑘=1𝐸𝟏{Ω1𝐴3}𝜎2𝑘𝜎2𝑘1𝑊(𝑥(𝑡))𝑑𝑡𝜖𝑘=1𝐸𝟏{Ω1𝐴3}𝜎2𝑘𝜎2𝑘1𝜖𝑘=1𝐸𝟏{Ω1𝐴3}Ω𝑘𝜎2𝑘𝜎2𝑘1𝜖𝑇𝑘=1𝑃Ω1𝐴3Ω𝑘𝜖𝑇𝑘=1𝜀=(A.30) which is a contradiction. Thus 𝑃{𝐴3}=0. Therefore 𝑃{𝐴1}=1 and the proof is therefore completed.


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.


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