Reliability Assessment for Wireless Mesh Networks Under Probabilistic

Region Failure Model

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Citation:

Jiajia Liu, Xiaohong jiang, Hiroki Nishiyama, and Nei Kato, “Reliability

Assessment for Wireless Mesh Networks Under Probabilistic Region Failure

Model,” IEEE Transactions on Vehicular Technology, vol. 60, no. 5, pp.

2253-2264, Jun. 2011.

URL:

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5713274

1

Reliability Assessment for Wireless Mesh NetworksUnder Probabilistic Region Failure Model

Jiajia Liu, Student-Member, IEEE,Xiaohong Jiang,Senior Member, IEEE,Hiroki Nishiyama,Member, IEEE,and Nei Kato,Senior Member, IEEE

Abstract—Wireless networks in open environment are exposedto various large region threats, like the natural disasters andmalicious attacks. Available work regarding region failures gen-erally adopt a kind of “deterministic” failure models, whichfailed to reflect some key features of a real region failure. Inthis paper, we provide a more general “probabilistic” regionfailure model to capture the key features of a region failure andapply it for the reliability assessment of wireless mesh networks.To facilitate such assessment, we develop a grid partition-basedscheme to estimate the expected flow capacity degradation from arandom region failure. We then establish a theoretical frameworkto determine a suitable grid partition such that a specifiedestimation error requirement is satisfied. The grid partitiontechnique is also useful for identifying the vulnerable zonesof a network, which can guide network designers to initiateproper network protection against such failures. The work inthis paper helps us understand the network reliability under aregion failure and facilitates the design and maintenance of futurehighly survivable wireless networks.

Index Terms—Wireless mesh networks, region failure, networkreliability.

I. I NTRODUCTION

In recent years, the wireless mesh networks have increas-ingly gained interests in both academia and industry. As apromising and flexible networking technology, the wirelessmesh network is expected to support data communicationsfor some important and mission critical applications, likethedisaster relief and battlefield headquarter construction.Dueto the nature of wireless communications, the nodes thereare exposed to various hazards [1]–[3], such as the naturaldisasters and malicious network attacks [4], [5]. Thus, thepre-active evaluation of network reliability and survivabilityagainst network failures becomes essential for the designand maintenance of future highly survivable wireless meshnetworks.

In the light of failure inevitability and its detrimentalconsequences, many studies have been dedicated to the designof failure-resilient networks. Stefanakoset al. in [6] examinedthe routing issue in networks that require guaranteed reliabilityagainst multiple link failures. Awerbuchet al. in [7] proposed

Copyright (c) 2011 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

J. Liu, H. Nishiyama and N. Kato are with the Graduate School ofInformation Sciences, Tohoku University, Aobayama 6-3-09, Sendai, 980-8579, JAPAN. E-mail:{liu-jia,kato}@it.ecei.tohoku.ac.jp.

X. Jiang is with the School of Systems Information Science, FutureUniversity Hakodate, Kamedanakano 116-2, Hakodate, Hokkaido, 041-8655,JAPAN. E-mail: jiang@fun.ac.jp.

an on-demand routing protocol for ad hoc wireless networks,which provides resilience to byzantine failures caused byindividual or colluding nodes. Bhandari and Vaidya [8] consid-ered the problem of reliable broadcast in a wireless network,where each node can fail independently. Yu and Zhang [9]proposed a novel scheduling algorithm for sensor networks tobound the service loss duration due to node failures and toprovide continuous surveillance coverage even when a subsetof sensors fail. The fast restoration and protection against linkand node failures have also been explored recently, see, forexample [10]–[13].

Most of available network survivability studies are basedon one common assumption that failures are random andindependent, which failed to reflect many real scenarios. Thereal-world disasters or attacks, like the earthquake, hurricane,physical bomb explosion or electromagnetic pulse (EMP)attack [14], [15], always happen in a particular geographicallocation and result in the so-calledregion failure( [16]–[26]).Under a region failure, multiple network components maysimultaneously corrupt but they are geographically correlatedand constrained within a specific region. Thus, it is importantto take into account the geographical information of networksin the study of such failures, and some research has beenconducted to understand the impact of region failures on wiredbackbone networks ( [19]–[26]). In this paper, we focus onthe reliability assessment of wireless mesh networks underarandom region failure.

There are few related works concerning region failures inwireless networks. Senet al. in [16], [18] explored the region-based connectivity issue in wireless networks and showed howto adjust the transmitting power to maintain a region-basedconnectivity in presence of region failures. This work wasfurther extended into multiple region failure model (MRFM)[17], where the failures are no longer confined within a singleregion. Xu et al. in [27] adopted the percolation theory tocharacterize the spread of correlated failures in large wire-less networks, and analyzed the condition under which aninitial node failure will/will not permeate the whole network.Azimi et al. in [28] addressed the problem of building dataredundancy with the minimum communication cost in a sensornetwork, where many nodes may simultaneously fail due to abomb attack or river overflow.

It is notable that the region failure models adopted inprevious region failure-related studies, like the single circularmodel in [16]–[18], [20], [28] and line cut model in [19]–[21], can be regarded as a kind of “deterministic” failuremodels in the sense that any network component intersecting

2

with the concerned failure region will always be destroyed(i.e., destroyed with probability 1). However, the real physicalattacks (such as physical bomb explosion and electromagneticpulse (EMP) attack) or natural disasters (such as earthquake,hurricane and flood) rarely have a deterministic nature. Theprobability that a network component is affected by an attackdepends on various factors, like the distance from the attackcenter to the component, the topography of the area aroundthe component, the component’s specifications, etc. In lightof the fact that such attacks have probabilistic rather thandeterministic effects on network components, a probabilisticfailure model would be more suitable for network reliabilitystudy under such attacks. In fact, the probabilistic regionfailure model has been recently employed for the reliabilitystudy of WDM backbone networks [25]. In [25], Agarwaletal. considered a simple probabilistic failure model, in whichnetwork components within the impact radius of the disasterfail with a fixed probability while other components furtheraway do not fail.

The “deterministic” failure models in [16]–[20], [20], [21],[28] and the probabilistic model in [25], although simple andeasy to use, neglected two key facts of real-world regionfailures: network components can only be destroyed withcertain probability (not always probability 1), and more im-portantly, such failure probability of a network componenttends to monotonously decrease as it is farther away fromthe region center. Based on this observation, we believe a“probabilistic” model addressing these two key features will bemuch more suitable for network reliability study. In this paper,we consider such a probabilistic failure model and apply it toassess the reliability of wireless mesh networks. Actually, asimilar model has just been proposed recently in [26] for thereliability study of WDM optical backbone networks, whereeach optical fiber link is treated as a compound componentconsisting of consecutive amplifiers and the failure probabilityof an amplifier there is determined by its distance to attackcenter. The main contributions of this paper are as follows:

• We provide a general and more realistic probabilisticregion failure model to capture the key features of regionfailures, which covers the deterministic failure models in[16], [20] as special cases.

• Based on the new failure model, we formulate the ex-pected flow capacity degradation problem in wirelessmesh networks as a network zone partition problem,which is hard to solve for a large network. We thendevelop a grid partition scheme to efficiently estimate theexpected flow capacity degradation from a random regionfailure. The grid partition technique can also help us toidentify the vulnerable zones of a network.

• A theoretical framework is further established to analyzethe estimation error from using the grid partition tech-nique, which can guide us to determine a suitable gridpartition such that a specified estimation error require-ment is satisfied.

• We demonstrate through extensive theoretical and simu-lation studies that neglecting probabilistic behavior of aregion failure may significantly over-estimate or under-

Fig. 1. Probabilistic Region Failure Model

estimate its impact on network reliability.

The rest of this paper is outlined as follows. Section II in-troduces the general probabilistic region failure model and theproblem formulation of expected flow capacity degradation.In Section III, we develop a grid partition scheme to estimatethe average performance degradation caused by a randomregion failure, and also provide the theoretical analysis onthe estimation error from using such grid partition. Section IVpresents the numerical results to validate the new region failuremodel and the grid partition scheme. Finally we conclude thispaper in Section V.

II. M ODEL AND PROBLEM FORMULATION

In this section, we first define a general probabilistic regionfailure model, then formulate the expected flow capacitydegradation from such a failure as a network zone partitionproblem.

A. Probabilistic Region Failure Model

It is notable that one common feature of real-world attacks(like the physical bomb explosion, E-bomb or EMP attack)is that the power of such an attack gradually attenuates fromits center area to outer area. Due to this common feature, theregion failures caused by such attacks always share two com-mon behaviors, i.e., a network component near attack centerwill fail with high probability (may not always probability1),and such failure probability tends to monotonously decreaseas it is farther away from the attack center.

To emulate these common failure behaviors, we introducehere a general probabilistic region failure model.

Definition 1: (Probabilistic Region Failure Model) con-sists of a set ofM consecutive annulus, defined byMconcentric circles with radiusri, i = 1, . . . ,M , as illustratedin Fig. 1. A network component (like a network node) fallingwithin i-th annulus will fail with probabilitypi, where annulusare sequentially numbered from the failure center. To mimicthe above behaviors, the following properties hold forpi:

- The probabilitypi is monotonously decreasing, i,e,pi ≥pi+1, i = 1, . . . ,M − 1.

- The region failure is only confined within the circle areaof radius rM , beyond which the failure probability isregarded as 0.

3

It is noted our probabilistic region failure (PRF) model isdifferent from the previous “deterministic” failure models inthe sense that: 1) it is more general as it covers the formersingle circular model in [16], [20] as a special case ; 2) itis more realistic as it reflects the monotonously decreasingtrend of failure probability for real region failures; 3) itismore flexible and can be configured with different parametersettings to adapt to various realistic scenarios.

Remark 1:The PRF model aims at characterizing networkcomponent failures caused by large region physical attacks(such as physical bomb explosion and electromagnetic pulse(EMP) attack) or natural disasters (such as earthquake, hurri-cane and flood). Under such a region failure, multiple networkcomponents (nodes) may simultaneously corrupt but they aregeographically correlated and constrained within a specificregion. It is notable that, however, under some kind of net-work attacks, the induced network failures may be temporaryand not permanent, and the affected network nodes may begeographically independent and may not be constrained in aspecific region. Further, the attacker may just attack the datatraffic of the compromised node without physically destroyingthe node. This kind of network attacks are beyond the scopeof this paper, and some related work can be found in [29],[30].

Remark 2:For the simplicity of analysis, the PRF modelhere assumes that the failure probability of a component isonly determined by its distance to the failure center whileneglecting the effects from other issues, like the component’sprotection issue, component’s specification issue, the topogra-phy issue, etc.

Without loss of generality, in the following we focus onthe simple two-annulus PRF model withM = 2, p1 = 1 andp2 = p to simplify the presentation1.

B. Problem Formulation

Based on the above PRF model, we will assess the reliabilityof a wireless mesh network under single random region failure.In this paper, we choose to explore the impact of region failureupon some specified key flows (like some mission criticalflows), and take the expected flow capacity degradation as thenetwork reliability metric2. Here, the expected flow capacitydegradation is measured over all concerned flows after regionfailure happens but before initiating the network recoverymechanism, so it indicates the possible worst case performancedegradation after a region failure. This problem can be definedas follows.

Expected Capacity Degradation Problem: For a givennetwork and the routing/capacity information of some specifiedflows in it, calculate the expected capacity degradation of theseflows under a random region failure.

To solve the Expected Capacity Degradation (ECD) prob-lem, one straightforward approach is to first apply the PRF

1For a small area around the region failure center, the failureprobabilitythere can be high enough to be approximated as 1.

2Some other metrics can also be adopted for network reliabilityevaluation,like the vertex based degree centrality [31], the operational O-D pairs or paths,the minimum shortest paths [32]–[34], the critical vertex/edge [35]–[37] andpairwise connectivity [38].

Fig. 2. RFL zones{Zi} and their impacts{wi} of a flow

model to partition the overall network area into some disjointand uniform region failure location (RFL) zones.

Definition 2: (RFL Zone) A RFL zone is a network sub-area that any PRF with center falling within it will alwaysinduce the same impact (i.e., the same flow capacity degrada-tion) to all the concerned flows.

For a simple scenario of having only one flowf withcapacityCf and 3 nodes, such RFL zone partition is illustratedin Fig. 2, where the network region is divided into differentRFL zones{Zi} with impacts{wi}, i = 1, ..., 10.

Based on the area of each RFL zone and its impact onflow capacity degradation, we can easily evaluate the overallECD of all concerned flows under a random region failure.For a network with coverage areaZ, suppose we have alreadydivided network region into different RFL zones{Zt} withdifferent areas{|Zt|} and impacts{wt}, then the overall ECDw can be determined as

w =∑

Zt

|Zt|

Z· wt (1)

Here |Zt|/Z is just the probability that the PRF’s center fallswithin the RFL zoneZt.

To apply (1) for the evaluation of ECD, we need to findout all the RFL zones. Such RFL zones depend on manyfactors, like the node topology distribution (e.g., distanceamong nodes, number of nodes), flow distribution (routingpath for each concerned flow, number of nodes per flow),and also parameter settings of the PRF model (M and ri,i = 1, . . . ,M ). Suppose the number of nodes of all concernedflows is N , then we can see that total number of RFL zonescan be as high asO((M+1)N ) in the worst case. Also, findingall these RFL zones and calculating their area involve a lot ofvery complicated geometric operations. In the next sectionwepresent an efficient scheme for the estimation ofw.

III. E STIMATION OF ECD

In this section, we first introduce a grid partition-basedscheme for the estimation of ECD, then provide a theoreticalanalysis on the estimation error from using such grid partitiontechnique.

4

A. Grid Partition-based Estimation for ECD

Without loss of generality, we assume that the networkcoverage area is anb × b square. We apply a grid to evenlydivide theb×b square intoc×c small cells{Sj , j = 1, ..., c2}with side lengtha = b/c each, as illustrated in Fig. 3. Basedon this grid partition, one simple way to estimate the ECD ofsome concerned flows is to regard each cell here as a “RFL”zone and take the impact of its center point as the impact ofthis cell. In this way, we can get an estimation of ECD basedon the Eq. (1).

Suppose that the set of concerned flows are{fk, k =1, ...,K}, and let(x∗

j , y∗j ) be the central point ofjth cell Sj ,

and letwfk(x, y) be the induced impact on flowfk when PRF

center is at point(x, y). Then the grid partition-based schemefor obtaining an estimationw of ECD can be summarized asthe following Algorithm 1.

Algorithm 1 ECD Estimation:Input: The network grid partition information, flow distribu-tion and failure model parameters;Output: ECD estimationw;

1. w ⇐ 0;2. for k = 1 to K do3. for j = 1 to c2 do4. calculatewfk

(x∗j , y

∗j );

5. w = w + (a2/b2) · wfk(x∗

j , y∗j );

6. end for7. end for8. return w;

In the Algorithm 1, we take the central point(x∗j , y

∗j )

of cell Sj as the sampling point and simply use its impactwfk

(x∗j , y

∗j ) as an approximation of the impact of all other

points in Sj . Since each cell here may not be a RFL zone,such approximation will cause an estimation error betweenw and w. For a flow with two nodes and capacityCf , apartition cell Sj that intersects with three RFL zones thereis illustrated in the Fig. 3. Notice that the three RFL zonesintersecting withSj have distinct impacts of 0,p · Cf and(1 − (1 − p)2) · Cf , respectively. Thus, talking the impactp ·Cf of the center(x∗

j , y∗j ) of Sj as an approximation of the

impacts of all other points (can be 0 or(1 − (1 − p)2) · Cf

here) will induce estimation error in the calculation of ECDw.

In the next subsection, we provide a theoretical model on thepossible estimation error that the Algorithm 1 may introducein the estimation ofw. Such a model can help us to determinea suitable grid partition (i.e., a suitable cell sizea) such thata specified estimation error requirement is satisfied.

B. ECD Estimation Error Modeling

Based on the grid partition introduced above, the ECDwand its estimationw for the set of concerned flows{fk, k =

Fig. 3. Illustration of network grid partition and ECD estimation error for aflow with only two nodesA andB. The cells of cases 2 and 3 will introduceECD estimation error while the cells of cases 1 and 4 will not.

1, ...,K} can be expressed as

w =1

b2·

K∑

k=1

c2

∑

j=1

∫∫

(x,y)∈Sj

wfk(x, y)dxdy (2)

w =1

b2·

K∑

k=1

c2

∑

j=1

∫∫

(x,y)∈Sj

wfk(x∗

j , y∗j )dxdy (3)

If we use∆ to denote the estimation error ofw, then wehave

∆ = |w − w|

≤K

∑

k=1

∣

∣

∣

1

b2

c2

∑

j=1

∫∫

(x,y)∈Sj

(

wfk(x∗

j , y∗j ) − wfk

(x, y))

dxdy∣

∣

∣

(4)

The (4) indicates that the overall estimation error∆ is no morethan the sum of estimation error for the ECD of each flow.If we use∆fk

to denote the estimation error for the ECD offlow fk, then we have

∆fk=

∣

∣

∣

1

b2

c2

∑

j=1

∫∫

(x,y)∈Sj

(

wfk(x∗

j , y∗j ) − wfk

(x, y))

dxdy∣

∣

∣

(5)

≤

c2

∑

j=1

∣

∣

∣

1

b2

∫∫

(x,y)∈Sj

(

wfk(x∗

j , y∗j ) − wfk

(x, y))

dxdy∣

∣

∣

(6)

The (6) says that the estimation error∆fkfor flow fk is

upper bounded by the sum of corresponding estimation errorintroduced in each cell. If we use∆Sj

fkto denote the maximum

difference between the average impact (onfk) of any twopoints inSj , i.e.,

∆Sj

fk= max

(x,y)∈Sj

{wfk(x, y)} − min

(x,y)∈Sj

{wfk(x, y)} (7)

5

then we have∣

∣

∣

1

b2

∫∫

(x,y)∈Sj

(

wfk(x∗

j , y∗j ) − wfk

(x, y))

dxdy∣

∣

∣

≤1

b2

∫∫

(x,y)∈Sj

∣

∣

∣wfk

(x∗j , y

∗j ) − wfk

(x, y)∣

∣

∣dxdy

≤1

b2

∫∫

(x,y)∈Sj

∆Sj

fkdxdy

=1

c2∆

Sj

fk(8)

Combining (4), (5), (6) and (8), we have

∆ ≤K

∑

k=1

∆fk≤

K∑

k=1

c2

∑

j=1

1

c2∆

Sj

fk(9)

The (9) shows that we can control the overall estimationerror ∆ by properly selecting the number of cellsc (orequivalently the sizea = b/c of each cell) in the grid-partition based ECD estimation. LetCk denote the capacityof concerned flowfk, k = 1, ...,K, then we can define thefollowing ECD estimation error bounding problem.

ECD Estimation Error Bounding Problem: For an errorrequirementǫ > 0, to determine a low boundcǫ on the numberof cells c, such that whenc > cǫ we can always guaranteethat

∆ ≤

K∑

k=1

c2

∑

j=1

1

c2∆

Sj

fk≤ ǫ ·

K∑

k=1

Ck (10)

The (10) indicates that to determine the lower boundcǫ fora givenǫ, we need to identify each cell that has non-zero term∆

Sj

fk(i.e., the cell that introduces estimation error) and also to

determine the total number of such cells, as discussed in thefollowing subsections.

C. Identification of Cells with Estimation Error

Based on the simple two-annulus PRF model introducedin Section II-A, we can easily see that the area around anetwork node can also be divided into two same annulus (asshown in Fig. 3), where a PRF with center falling within theinner annulus (resp. the outer annulus) will cause the nodeto fail with probability 1 (resp. probabilityp). Thus, for agiven flow fk, whether a cell will introduce ECD estimationerror for this flow depends on how the cell intersects withthe boundaries of the outer annulus and inner annulus of allthe nodes of this flow (hereafter, we call these annulus asthe annulus of this flow). To characterize such intersectionbetween a cellS and the annulus of flowfk, we define a four-tuple (mk, nk, hk, gk), which indicates that the cell partiallyintersects with the boundaries ofmk outer annulus andhk

inner annulus of flowfk, but it completely falls within othernk outer annulus andgk inner annulus of the flow.

Based on the four-tuple(mk, nk, hk, gk) for flow fk and acell S, the ∆S

fkdefined by (7), i.e., the maximum difference

between the average impact (onfk) of any two points inS,

can be determined as

∆Sfk

=

0 case 1: ifgk ≥ 1,

Ck · qnk

case 2: ifgk = 0, hk ≥ 1,

Ck · qnk ·(

1 − qmk)

case 3: ifgk = 0, hk = 0,mk ≥ 1,

0 case 4: ifgk = 0, hk = 0,mk = 0.

(11)

where q = 1 − p is the non-failure probability of a nodefalling within the outer annulus defined by the simple two-annulus PRF model in Section II.A. The (11) indicates clearlythat only the cells of the cases 2 and 3 will introduce ECDestimation error for the flowfk, as illustrated in the Fig. 3.

Let uk(λ) denote the total number of cells of the case2with nk = λ, and letvk(β, γ) denote the total number of cellsof the case3 with mk = β andnk = γ, then the overall ECDestimation error for the flowfk is given by

c2

∑

j=1

1

c2∆

Sj

fk

=Ck

c2·(

∑

λ≥1

uk(λ)qλ +∑

β≥1,γ≥0

vk(β, γ)(

1 − qβ)

qγ)

(12)

D. Counting the Cells with Estimation Error

The (10) and (12) indicate that to solve the overall esti-mation error bounding problem, we need to determine thevalues ofuk(λ) and vk(β, γ) for each flowfk with λ ≥ 1,β ≥ 1 and γ ≥ 0. However, determining the exact value ofuk(λ) and vk(β, γ) for each flowfk is still a very difficulttask, which involves the complicated geometric operation toidentify the relationship (intersecting or containing) betweencells and annulus boundaries of a flow. We instead providehere a tractable upper bound to efficiently approximate theECD estimation error in (12).

Notice from the Fig. 3 that the effect of a PRF upon anetwork node is defined by the two annulus around the node,and our basic idea here is to first derive a general “node-level estimation error” for each node of flowfk based on theintersection between its two annulus and cells around the node,then apply the node-level estimation error of each node to geta general bound on the ECD estimation error of this flow.

To get a general node-level estimation error for each node,we need to identify all the cells around a node that will“contribute” to the ECD estimation error. For this purpose,weconsider a tagged nodeA and one its neighbor nodeB that isdmax away, as illustrated in Fig. 4. Here thedmax is definedas the maximum distance between any two neighbor nodes ofany flow, which is controlled by the maximum communicationrange (or power) of the network. We useC

Ain and C

Aout, C

Bin

andCBout to denote the inner annulus and outer annulus ofA

and B, respectively. Then the cells that may introduce ECDestimation error to the nodeA can be defined by the followingvariables:

6

Fig. 4. Illustration for the cell counting, where the distance between thenodesA and B is fixed asdmax, and the network is partitioned with cellsof sizea each.

• v1: the number of cells partially intersecting with theboundary of C

Aout but not completely falling within

CBin ∪ C

Bout.

• v2: the number of cells partially intersecting with theboundary ofCA

out and completely falling withinCBin ∪

CBout.

• u1: the number of cells partially intersecting with theboundary ofCA

in but not completely falling within theC

Bin ∪ C

Bout.

• u2: if u1 > 0, u2 is defined as the total number ofcells that partially intersect with the boundary ofC

Ain

and completely fall within theCBin ∪ C

Bout. In the case

u1 = 0, u2 is the number of cells that partially intersectwith the boundary ofCA

in and not completely fall withinthe C

Bin.

For the example shown in Fig. 4, we can easily prove thatthe u1, u2, v1 andv2 there are given by the formulas in Table Iand II.

Remark 3:The values ofu1, u2, v1 and v2 in Tables I andII are only determined by the network parametersdmax, r1

andr2 and thus independent of flows.Remark 4:For the tagged nodeA and its neighbor node

B in Fig. 4, the boundaries of their outer annulus are notintersecting with the network boundary. Thus, the results inTables I and II represent the maximum values ofu1, u2, v1

and v2.Based on theu1, u2, v1 and v2 in the Tables I and II, we

can defineNEEk (i.e., the node-level estimation error foreach node of flowfk) as

NEEk =Ck

c2·(

u1q + u2q2 + v1p + v2pq

)

(13)

We now show that theNEEk can be used to establish anupper bound for the overall ECD estimation error (12) of theflow fk, as summarized in the following lemma (See Appendixfor the proof).

Lemma 1:Given the PRF model parametersr1 andr2, cellside lengtha, then for any flowfk with Nk nodes,1 ≤ k ≤ K,

TABLE Iu1 AND u2

TABLE IIv1 AND v2

dmax v1 v2

dmax ≤ 2r2 4⌈ 2r2

a⌉ − v2 4⌈ 2r2

a⌉ 1

πarccos

dmax

2r2

dmax > 2r2 4⌈ 2r2

a⌉ 0

we have

Ck

c2·(

∑

λ≥1

uk(λ)qλ +∑

β≥1,γ≥0

vk(β, γ)(

1 − qβ)

qγ)

≤ Nk · NEEk (14)

E. A Lower Bound for Estimation Error Guarantee

By combining the (10), (12) and (14), we can easily provethe following theorem regarding a lower boundcǫ of c for aspecified error requirementǫ.

Theorem 1:For a specified error requirementǫ > 0, we candetermine a lower boundcǫ for c as follows such that whenc ≥ cǫ, the (10) always holds.

1) whendmax ≤ r2 − r1 & r2 ≤ 3r1, or dmax ≤ 2r1 &r2 > 3r1,

cǫ =8

ǫπb∑K

k=1 Ck

×

K∑

k=1

NkCk

(

r1(π − arccosdmax

2r1)(1 − p)2

+ r2(π − p arccosdmax

2r2) · p

)

(15)

2) when2r1 < dmax ≤ r2 − r1 & r2 > 3r1,

cǫ =8

ǫπb∑K

k=1 Ck

K∑

k=1

NkCk

(

r1π(1 − p)2

+ r2(π − p arccosdmax

2r2) · p

)

(16)

3) whenr2 − r1 < dmax ≤ r2 + r1,

cǫ =8

ǫπb∑K

k=1 Ck

×

K∑

k=1

NkCk

(

r1(π − arccosd2

max + r21 − r2

2

2dmaxr1)(1 − p)

+ r1 arccosd2

max + r21 − r2

2

2dmaxr1(1 − p)2

+ r2(π − p arccosdmax

2r2) · p

)

(17)

7

4) whenr2 + r1 < dmax ≤ 2r2,

cǫ =8

ǫπb∑K

k=1 Ck

K∑

k=1

NkCk

(

r1π(1 − p)

+ r2(π − p arccosdmax

2r2) · p

)

(18)

5) whendmax > 2r2,

cǫ =8

ǫb∑K

k=1 Ck

K∑

k=1

NkCk

(

r1(1 − p) + r2p)

(19)

IV. N UMERICAL RESULTS

In this section, we first verify the efficiency of the ECDestimation scheme through simulation, then apply it to assessthe network reliability under the new PRF model.

A. Simulation Setting

We developed a simulator in C++ to simulate the impactof a random PRF upon on some specified flows. Similar tothe settings used in [16], we consider a random network with32 nodes, in which the coordinates (x, y) of each node areuniformly generated in a2000× 2000 m2 field. We randomlygenerate eight flows, where the number of nodes per flow isdrawn randomly in[3, 5], each link distance is drawn randomlyin [100, 400] m, and the flow capacity is drawn randomly in[3, 10] Mbps. The final network graph for simulation is shownin Fig. 8a, in which the maximum distance between any twoneighbor nodes of any flow is determined asdmax = 360.555.The metric adopted for performance evaluation is the averageimpact ratio, defined as

AIR =w

∑Kk=1 Ck

(20)

The simulated average impact ratio was calculated as theaverage value of ten batches of simulation results, whereeach batch consists of one million random and independentsimulations.

B. PRF Model and “Deterministic” Failure Models

To illustrate how the general PRF model is different fromthe “Deterministic” failure models, we first conducted a sim-ulation under the general parameter setting for(p1, p2). TheFig. 5 illustrates the variations of average impact ratio with theparameters(r1, r2, p1, p2), where the settings(p1 = 1, p2 = 1)and(p1 = 1, p2 = 0) correspond to the “deterministic” modelscenarios.

The results here indicate clearly that the “deterministic”models, although simple and easy to use, may result ina significant overestimation or underestimation of networkreliability. For example, when we setr2/r1 = 3.6 in Fig. 5a,we get an average impact ratio of0.102 with the setting of(p1 = 0.95, p2 = 0.25), while this ratio decreases to0.032with the setting of (p1 = 1, p2 = 0). Regarding the resultsof fixed r2/r1 in Fig. 5b, we can see that whenr1 = 80,the average impact ratio is estimated as0.116 for the case

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 1.5 2 2.5 3 3.5 4 4.5 5

aver

age

impa

ct r

atio

outer circle radius / inner circle radius ( r2 / r1)

r1 = 100p1 = 1, p2 = 1

p1 = 0.95, p2 = 0.25

p1 = 1, p2 = 0

p1 = 0.95, p2 = 0.75

(a) Average Impact Ratio vs.r2/r1

0

0.05

0.1

0.15

0.2

0.25

0.3

60 80 100 120 140

aver

age

impa

ct r

atio

inner circle radius r1

r2 / r1 = 3p1 = 1, p2 = 1

p1 = 0.95, p2 = 0.25

p1 = 1, p2 = 0

p1 = 0.95, p2 = 0.75

(b) Average Impact Ratio vs.r1

Fig. 5. PRF Model and “Deterministic” Failure Models.

(p1 = 0.95, p2 = 0.75), and this estimated ratio increases to0.141 when bothp1 andp2 are regarded as 1 there. It is alsointeresting to note that asr1 (or r2/r1) increases (and thusfailure region becomes bigger), the estimation gap betweenthe probabilistic model and the corresponding “deterministic”ones tends to increase sharply, and such gaps can be verysignificant if the probabilistic feature of region failure is notproperly “rounded-off”.

C. ECD Estimation Scheme Validation

To verify the ECD estimation scheme, further simulationwas conducted under the simple PRF model of (p1 = 1, p2 =p). The parameters used in the simulation are summarizedin Table III, where each case of parameter setting is corre-sponding to one individual case discussed in the Theorem 1.We verified the ECD estimation scheme under two errorrequirements ofǫ = 0.01 and ǫ = 0.005. The correspondingsimulation results and estimation results from our scheme aresummarized in the Table IV.

The Table IV indicates clearly that when we setc ≥ cǫ, ourscheme could provide an efficient estimation for the averageimpact ratio, and the induced overall estimation error is alwaysless than the specifiedǫ. It is also notable that for eachtest case here, the actual error between the simulation andestimation results is several orders smaller than the specifiedǫ. This very small overall estimation error (and thus a very

8

TABLE IIIFAILURE MODEL PARAMETER SETTINGS

p r1 r2

case 1 0.50 50 100case 2 0.35 80 200case 3 0.25 100 500case 4 0.75 180 200case 5 0.15 200 600case 6 0.10 200 700

TABLE IVCOMPARISON BETWEEN SIMULATION AND ESTIMATION RESULTS FOR

MODEL VALIDATION , C =∑

8

k=1Ck

w/C w/C ∆/C cǫ

case 1 ǫ = 0.01 0.0201332 0.0200634 6.98e-005 126ǫ = 0.005 0.0201248 0.0201419 1.70e-005 252

case 2 ǫ = 0.01 0.05511 0.0550707 3.93e-005 199ǫ = 0.005 0.0551315 0.0551305 1.03e-006 398

case 3 ǫ = 0.01 0.160276 0.160349 7.28e-005 284ǫ = 0.005 0.160279 0.160315 3.62e-005 569

case 4 ǫ = 0.01 0.106664 0.106649 1.47e-005 295ǫ = 0.005 0.106683 0.106636 4.70e-005 590

case 5 ǫ = 0.01 0.193783 0.193646 1.36e-004 350ǫ = 0.005 0.193819 0.19378 3.96e-005 701

case 6 ǫ = 0.01 0.189801 0.18987 6.90e-005 346ǫ = 0.005 0.189838 0.189847 9.57e-006 693

safe cǫ) are due to the following factors. The first factor isthat the maximum possible estimation error (rather than thereal estimation error) of each cell is adopted in the evaluationof the overall ECD estimation error (Eq. 12). The second factorlies in the estimation of the number of error-inducing cellsin(13), in which only the maximum values foru1 and v1 areconsidered, while the errors of other cells are approximatedthroughu2 andv2. The last factor is that the distance betweenany two neighbor nodes is always regarded asdmax in theTheorem 1, which helps us to derive an unified and closedform formula for cǫ but leads to an overestimation for theparameter.

The above results indicate that our ECD estimation schemeand the related theoretical framework for estimation errorbounding, although may lead to a “conservative” estimationfor the overall ECD, are simple and efficient. To apply suchscheme, we just need to divide the network area intoc × ccells and simply use the central point of each cell to calculatethe ECD metric, which avoids the complicated geometricoperations for the identification and area evaluation of allRFL zones. As long as the cell sizea is small enough (orequivalently the number of cellsc is big enough such thatc ≥ cǫ), our scheme can always result in a very efficientestimation for the ECD with an error upper bounded byǫ.

Hereafter, the numerical results in performance evaluationare obtained based on our ECD estimation scheme, wherethe simple PRF model of (p1 = 1, p2 = p) and an errorrequirement ofǫ = 0.005 are assumed.

D. Average Impact Ratio vs. Failure Model Parameters

The Fig. 6 shows the relationship between the averageimpact ratio and the failure probabilityp under differentsettings ofr2/r1, where the settingr2/r1 = 1 corresponds toa “deterministic” model. We can see from the Fig. 6 that with

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

aver

age

impa

ct r

atio

failure probability p

r2 / r1 = 5

r2 / r1 = 4

r2 / r1 = 2

r2 / r1 = 1

r1 = 100 r2 / r1 = 7

Fig. 6. Average Impact Ratio vs. Failure Probabilityp

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 1.5 2 2.5 3 3.5 4 4.5 5

aver

age

impa

ct r

atio

outer circle radius / inner circle radius ( r2 / r1)

p = 0.5

r1 = 100

r1 = 200

r1 = 360

r1 = 450

Fig. 7. Average Impact Ratio vs.r2/r1.

r1 = 100, a non-negligible difference between the averageimpact ratio of r2/r1 > 1 (r2/r1 =2,4,5,7) and that ofr2/r1 = 1 can be observed even for a very small value of thefailure probabilityp. For example, whenp = 0.2, the averageimpact ratio of the scenarior2/r1 = 2 is 0.050, which is nearly1.56 times as that of the scenarior2/r1 = 1 (0.032 there); forthe case thatp = 0.35, the estimated ratio of the first scenario(0.062) is nearly1.94 times as that of the later case (0.032).The Fig. 6 also shows clearly that asr2 or p increase, thedifference between the average impact ratio ofr2/r1 > 1 andthat of r2/r1 = 1 increases sharply. This results indicates thateven for a very small failure probabilityp, the probabilisticouter annulus part of a PRF may significantly affect the overallnetwork capacity.

When failure probability is set asp = 0.5, the relationshipbetween the average impact ratio andr2/r1 is illustrated inthe Fig. 7. We can see from the figure that in general, asr1

increases, the average impact ratio becomes more sensitiveto the variation of the ratior2/r1. For example, whenr2/r1

varies from 1.6 to 3.6, the estimated ratio for the scenario ofr1 = 100 varies from 0.056 to 0.159 and that for the scenarioof r1 = 200 varies from 0.162 to 0.404, respectively. Theresults here also show clearly that for the caser1 = 450, theaverage impact ratio is not sensitive to the variation ofr2/r1

anymore as it increases beyond the pointr2/r1 = 4.4. This isbecause that whenr1 andr2 are large enough, the PRF starts

9

(a) Network topology(K = 8, dmax = 360.555)

(b) Vulnerable zone distribution(r1 = 200, r2 = 600, a = 40)

Fig. 8. Network topology and vulnerable network zone distribution estimatedby our scheme.

to cover the whole network area and thus all the flows. Amore careful observation of the Fig. 7 indicates that even witha PRF ofr1 = 360 (roughly same as thedmax = 360.555)and r2/r1 = 2.4, we may achieve an over50% reduction(i.e.,AIR > 0.5) to the overall capacity of the concernedflows.

E. Vulnerable Network Zone Identification

In our grid partition-based ECD estimation scheme, wedivide the whole network area equally intoc × c cells, thentake the central point of each cell as the sampling point anduse its impact to approximate the impact of the cell. Thus,one attractive application of the estimation scheme is thatithelps us to identify the geographical distribution and sizeofthe vulnerable network zones.

For the network adopted in our study (Fig. 8a) and thesetting of (r1 = 200, r2 = 600, a = 40), the vulnerablenetwork zone distribution estimated by our scheme is illus-trated in Fig. 8b. Based on such vulnerable network zonedistribution, one can also easily identify the most vulnerablenetwork zone(s), i.e., the zones in which each cell there hasthebiggest impact to the network flow capacity. Such vulnerablenetwork zone distribution and the most vulnerable networkzone information will be helpful for network designers to ini-tiate proper network protection strategy against region failures.

V. CONCLUSION AND FUTURE WORK

In this paper, we proposed a more realistic probabilisticregion failure (PRF) model to capture some main features ofgeographically correlated region failures, and then developeda framework to apply the PRF model for the reliabilityassessment of wireless mesh networks. Our framework canbe applied to estimate the worst case network performancedegradation from a PRF and also to identify the geographicaldistribution and size of vulnerable network zones. Such pre-assessment and evaluation can help network designers toselect suitable routing and protection strategies againstregionfailures and thus achieve a failure-resilient network design.Our results indicate that neglecting some key properties ofreal region failures can result in a significant overestimationor underestimation of network reliability, which may misleadnetwork designers in initiating proper and cost-efficient net-work protection against such failures. It is expected that our

work in this paper will contribute to the future network designand planning against possible region failures.

Some possible extensions of this work are:• Routing issue: Notice that our framework in this paper

can be applied to evaluate a network with a given routingstrategy. How to apply this work and the correspondingresults (like the vulnerable network zone distributioninformation) to find efficient and region failure-tolerantrouting algorithm to alleviate the impacts of region failurecan be an interesting work.

• Recovery issue: We only considered the PRF impactbefore network recovery (like topology reconfigure andflow rerouting), so it only indicates the worst case net-work performance degradation. How to extend this workto estimate the network performance degradation (likethe flow capacity degradation) after network recoverydeserves further study.

• Reliability under other metrics: To have a more deeperunderstanding of network reliability under PRF, anotherfuture work is to extend the framework established in thispaper to further evaluate the network performance degra-dation under other metrics, like the pairwise connectivity[38], critical vertex/edge [35]–[37], etc.

APPENDIX

PROOF OF THELEMMA 1

Based on the (12) and (13) we know that to prove the (14),we just need to show that for anyfk the following conditionholds

∑

λ≥1

uk(λ)qλ +∑

β≥1,γ≥0

vk(β, γ)(

1 − qβ)

qγ

≤ Nk

(

u1q + u2q2 + v1p + v2pq

)

(21)

From the discussion in Section III-C we know that only thecells of cases 2 and 3 in (11) will introduce the ECD estimationerror. We first consider the first term regarding the case 2 cellsin the left side of (21), i.e., the term

∑

λ≥1 uk(λ)qλ. Sincethedmax is the maximum distance between any two neighbornodes of any flow, it is trivial to see thatuk(1) ≤ Nk · u1.Based on the definitions ofu1 and u2, and also notice thatqλ

monotonically decreases asλ increases, we have∑

λ≥1

uk(λ)qλ ≤ Nk

(

u1q + u2q2)

(22)

Similarly, for the cells belonging to the case 3 in (11), wehave the following inequalities for anyfk

∑

β≥1

β · vk(β, 0) ≤ Nk · v1 (23)

and∑

β≥1,γ≥1

β · vk(β, γ) ≤ Nk · v2 (24)

Now we consider the second term in the left side of (21),i.e.,

∑

β≥1,γ≥0 vk(β, γ)(

1−qβ)

qγ , where the component(1−qβ)qγ increases withβ but decreases withγ. Since some cellsof the case 3 are associated with a big value ofβ but a small

10

value ofγ, we can not directly derive a similar inequality likethe (22) for the case 3 cells. For a case3 cell Sj countedasvk(β, γ), its maximum estimation error, i.e., the maximumdifference between the average impact (onfk) of any twopoints in it, is given by

∆Sj

fk= Ckqγ

(

1 − qβ)

= Ckqγ

β∑

i=1

(−1)

(

β

i

)

(−p)i

≤ Ckqγ · p · β

= Ckqγ(

1 − q)

· β (25)

Based on the (25), we first establish the following resultsregarding the second term in the left side of (21) under thespecial case thatβ ≥ 1 andγ = 0,

∑

β≥1

vk(β, 0)(

1 − qβ)

≤∑

β≥1

vk(β, 0) · p · β (26)

=∑

β≥1

β · vk(β, 0) · p

≤ Nk · v1 · p (27)

where inequalities (26) and (27) are due to the (25) and (23),respectively.

We now show that for the general caseβ ≥ 1, γ ≥ 1, wehave

∑

β≥1,γ≥1

vk(β, γ)(

1 − qβ)

qγ

≤∑

β≥1,γ≥1

vk(β, γ) · qγ · p · β (28)

≤∑

β≥1,γ≥1

β · vk(β, γ) · q · p

≤ Nk · v2 · q · p (29)

where inequalities (28) and (29) are due to the (25) and (24),respectively.

Combining the (27) and (29), the following inequality forthe cells of case 3 follows,

∑

β≥1,γ≥0

vk(β, γ)(

1 − qβ)

qγ ≤ Nk

(

v1p + v2pq)

(30)

Finally, the (21) comes after the (22) and (30).

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Jiajia Liu Jiajia Liu received his B.S. and M.S.Degrees both in Computer Science from HarbinInstitute of Technology in 2004 and from XidianUniversity in 2009, respectively. He is currently aPhD candidate at the Graduate School of Informa-tion Sciences at Tohoku University. His researchinterests include performance modeling and evalu-ation, scaling laws of wireless networks, stochasticnetwork optimization, and optimal control.

Xiaohong Jiang Xiaohong Jiang received his B. S,M. S and Ph D degrees from Xidian University,Xian, China, in 1989, 1992 and 1999, respectively.Dr. Jiang is currently a full professor of FutureUniversity Hakodate, Japan. Before joining FutureUniversity Hakodate, he was an associate profes-sor in Tohoku University, Japan, was an assistantprofessor in the Graduate School of InformationScience, Japan Advanced Institute of Science andTechnology (JAIST) . His current research interestsinclude wireless sensor networks, optical networks,

and network coding, etc. Dr. Jiang has authored and coauthored more than160 publications in journals, books and international conference proceedings,which include IEEE/ACM Transactions on networking, IEEE Transactions onCommunications, and IEEE Journal of Selected Area on Communications.Dr. Jiang was also the winner of the Best Paper Award of WCNC08 and theICC2005-Optical Networking Symposium. He is a senior member of IEEE.(Email: jiang@fun.ac.jp)

Hiroki Nishiyama Hiroki Nishiyama received hisM.S. and Ph.D. in Information Science from TohokuUniversity, Japan, in 2007 and 2008, respectively.He was a Research Fellow of the Japan Society forthe Promotion of Science (JSPS) until finishing hisPh.D, when he then went on to become an AssistantProfessor at the Graduate School of InformationSciences at Tohoku University. He has received BestPaper Awards from the IEEE Global Communi-cations Conference 2010 (GLOBECOM 2010) aswell as the 2009 IEEE International Conference on

Network Infrastructure and Digital Content (IC-NIDC 2009). He was alsoa recipient of the 2009 FUNAI Foundation’s Research Incentive Awardfor Information Technology. His active areas of research include, trafficengineering, congestion control, satellite communications, ad hoc and sensornetworks, and network security. He is a member of the Instituteof Electronics,Information and Communication Engineers (IEICE) and an IEEE member.

Nei Kato Nei Kato received his M.S. and Ph.D.Degrees in Information Science from Tohoku Uni-versity, Japan, in 1988 and 1991, respectively. Hejoined the Computer Center of Tohoku University in1991 and has been a full professor at the GraduateSchool of Information Sciences since 2003. He hasbeen engaged in research on computer networking,wireless mobile communications, image processingand neural networks, and has published more than200 papers in journals and peer-reviewed conferenceproceedings.

He currently serves as the chair of the IEEE Satellite and Space Commu-nications Technical Community (TC), the secretary for the IEEE Ad Hoc &Sensor Networks TC, the vice chair of the IEICE Satellite CommunicationsTC, a technical editor for IEEE Wireless Communications (since 2006), aneditor for IEEE Transactions on Wireless Communications (since 2008), andas an associate editor for IEEE Transactions on Vehicular Technology (since2009). He also served as a co-guest-editor for IEEE WirelessCommunicationsMagazine SI on “Wireless Communications for E-healthcare”, asymposiumco-chair of GLOBECOM’07, ICC’10, ICC’11, ChinaCom’08, ChinaCom’09,and the WCNC2010-2011 TPC Vice Chair.

His awards include the Minoru Ishida Foundation Research EncouragementPrize (2003), the Distinguished Contributions to Satellite CommunicationsAward from the IEEE Satellite and Space Communications Technical Com-mittee (2005), the FUNAI information Science Award (2007), the TELCOMSystem Technology Award from the Foundation for ElectricalCommunica-tions Diffusion (2008), the IEICE Network System Research Award (2009),and many best paper awards from prestigious international conferences suchas IEEE GLEBECOM, IWCMC, and so on.

Besides his academic activities, he also serves as a member on theTelecommunications Council expert committee, the special commissionerof the Telecommunications Business Dispute Settlement Commission forthe Ministry of Internal Affairs and Communications in Japan,and as thechairperson of ITU-R SG4 in Japan. Nei Kato is a member of the Institute ofElectronics, Information and Communication Engineers (IEICE) and a seniormember of IEEE.