On the Expected Connection Lifetime and Stochastic Resilience of Wireless Multi-hop Networks

Transcription

1 On the Expecte Cnecti Lifetime an Stochastic Resilience of Wireless Multi-hop Networks Fei Xing Wenye Wang Department of Electrical an Computer Engineering North Carolina State University, Raleigh, NC 27695, USA Abstract To unerstan how noe mobility an Byzantine noe failures affect cnectivity of wireless multi-hop networks, this paper investigates resilience of geometric ranom graphs to lifetime-base noe failure an erives the expecte cnecti time before an en-user is isolate from the graph. Different from previous analytical stuies, which mainly focuse the so calle critical transmissi range, our stuy shes light the resilience analysis from the perspective of en-user s cnecti experiences. In the paper, we first introuce a simple but general noe behavior moel by a semi-markov process. Then we apply the theory of renewal process to the egree of a generic noe an analyze the stochastic property of noe cnecti time. At last, we provie the probability that the noe isolati event occurs within any en-user s lifetime an a close-form approximati of the network resilience. Our analysis an numeric simulati results inicate that networks with heavy-taile lifetime such as Weibull) istributis provies no improvement than those with light-taile e.g., Expential) istributis in terms of lger expecte cnecti lifetime for any en-user. Further, noe mobility has more significant impact than lifetime oes. I. INTRODUCTION Resilience of wireless multi-hop networks has attracte significant attenti in research literature [1], [2], [3]. A classical problem in this line of stuy is to unerstan the cnectivity of networks uner ranom failures [4], [5] an noe misbehaviors [6], [7]. To this en, many existing works assume inepenent noe behaviors an ranom link or noe failures an begin with an investigati into noe isolatis. For example, in [4], the cnecti resilience of a hoc networks, with regar to ranom noe failures, was analyze as a fault tolerance measure. In [6], the problem of noe isolati ue to misbehaving neighbors was first stuie, then the cnectivity of entire network was approximate by using the probability of iniviual noe isolati. These works all provie stochastic relatiship between network resilience an a variety of impacts, such as noe istributi, system size, cnectivity requirement, an noe failure probability; however, few works csier networking service availability from the perspective of en-users. This work is motivate by csiering the intrinsic behavior of en-users in wireless multi-hop networks. In these networks, especially mobile a hoc networks, en-users or noes) may enter or epart networks at any time an any locati; meanwhile, failures evelop ue to various reass, such as noe mobility, power epleti, or just voluntary user ecisi to leave. Thus, for each en-user, he/she may be more intereste in whether he/she will encounter isolati events, i.e., the case of no active neighbors, uring his/her lifetime in a network. In other wors, after a noe enters a network, as lg as the noe can enjoy a stable cnecti uring its lifetime, the networking service provie by the network is resilient for this noe. Therefore, we are intereste to investigate: What is the expecte cnecti time before a noe is isolate; what is the probability that a noe outlives all of its neighbors; what is the resilience of a network in terms of the satisfacti of en-users to their cnecti experience? To cope with the questis raise above, we start our investigati with a simple but generic noe behavior moel in which a noe transits its behavior between active an faile associate with arbitrary istribute transiti times. Then we use renewal process to moel the number of active neighbors of a noe at any time an erive the expecte cnecti time before a noe is isolate. Next, we use the probability of a noe outliving all its neighbors as the cornerste to provie an approximati of the resilience of the entire network. Here resilience generally refers the probability of a network having no noe isolati; while it also refers to the ability of an arriving user to stay cnecte to the rest of graph in this paper. Note that a similar methoology was use in [8] to aress the resilience for peer-to-peer networks; however, ctrary results are obtaine for wireless multi-hop networks in this work. Specifically, numeric simulati results show that heavy-taile lifetime istributi actually oes not provie improvement compare with expential lifetime istributi, an the noe mobility has more significant impact the expecte cnecti time. The rest of the paper is organize as follows: Secti II introuces our noe behavior moel an the assumptis use; Secti III erives the expecte cnecti time of a generic noe; Secti IV provies the close-form approximati of network resilience; followe by future works an cclusis in Secti V. II. MODELS AND ASSUMPTIONS In this secti, we introuce our noe behavior moel an explain the assumptis use later in the paper. First, we moel a wireless multi-hop network by a Geometric Ranom Graph [9], in which N noes are inepenently an uniformly istribute in a metric space, an a link exists between two noes if an ly if their istance is less than a preefine X/7/$ IEEE This full text paper was peer reviewe at the irecti of IEEE Communicatis Society subject matter experts for publicati in the IEEE GLOBECOM 27 proceeings.

2 transmissi raius. Nevertheless, we o not cfine the system size N fixe uring the whole network lifetime; instea, we allow it to vary with time accoring to any arrival/eparture process as lg as N stays sufficiently large. Note that in this paper, we assume that networks are well-cnecte an noes moves inepenently with similar statistic properties. To escribe an iniviual en-user s behavior, we aopt the noe behavior moel propose in [6] but simplify it to a twostate semi-markov process in orer to keep our erivatis tractable. For each noe in the network, it behaves either in a cooperative C) stateorinafaile F) state. A noe is csiere to be cooperative if it works properly an obeys all routing an forwaring rules; otherwise, it is sai to be faile. Most possible reass for a noe to become faile may inclue but not limite to: moving out of a network, epleting all its power, malfunctiing unexpectely, or turning own ue to voluntary en-user s ecisi. Meanwhile, we also allow noe to recover from the faile state again into the cooperative state, by moving back in the network fiel, recharging/replacing batteries or just a restarting. Notice that the transiti time T ij i, j {C, F}) between states may be istribute arbitrarily, a semi-markov process is quite appropriate to escribe the noe behavior aforementie. The state iagram is epicte in Fig. 1, where p ij i, j {C, F}) are the transiti probabilities, an F ij t) are the cumulative istributi functis CDF) of the corresping transiti times. C Fig. 1. p cf, F cf t) p fc, F fc t) A semi-markov noe behavior moel. Note that in our moel, it can be easily prove that p cf = p fc =1, T cf = T c the sojourn time in state C), an T fc = T f the sojourn time in state F ). In the following ctext, we call T c an T f as a noe s Lifetime an Recovery-time, respectively. Although we allow en-user s lifetime istribute arbitrarily from heavy-taile such as Weibull) to Expential, we require certain statiarity of lifetime T c. This means that users joining the network at ifferent times shoul have their lifetimes rawn from the same istributi F Tc x). It is worth pointing out that the lifetime of a noe u is an absolute ccept, that is, no matter where noe u is or goes to, as lg as it is in the network an provies cooperative networking services for other noes, it is sai to be alive. However, from the perspective of any generic noe u, even if a neighbor, say v, keeps alive in the cooperative state, noe v will be treate as a failure when it moves out of u s transmissi range. Thus, noe u s cnecti time is not ly relate with each neighbor s lifetime T c but also relate with the link-time between u an v. In this paper, we efine the linktime between a pair of noe u an v as the maximum time that u an v are in each other s transmissi ranges, which is similar to the efiniti presente in [1]. For a generic noe u, we enote its link-time with its neighbor i by LT {u} i an may omit {u} if it is explicit in the ctext. F III. EXPECTED CONNECTION TIME ANALYSIS In this secti, we aress the first questi prompte in Secti I, that is: what is the expecte cnecti time, enote by TC, of a noe before it is isolate by its neighbors. A. Definiti of Cnecti Time To erive the expectati value of TC, enote by E[TC], for a generic noe u, we csier the following simple scenario without losing generality. We assume that a noe u arrives the network at time t = t an it has cooperative neighbors initially. As time goes, some neighbors of u may still be cooperative or may be recovere back to cooperative state from a previous failure) while others may be faile or have move out of u s transmissi range); in aiti, it is possible for u to have some new neighbors ue to arrivals in u s transmissi range). Nevertheless, it is reasable to cfine the maximum number of neighbors both cooperative an faile) to a number. Thus,atanytimet>, neighbor i 1 i ) can be csiere to be if it is cooperative or to be off if it is faile. In other wors, the neighbor failure/recovery proceure can be represente by an /off process X i t): { 1, neighbor i is cooperative at t X i t) =, neighbor i is faile at t. We illustrate the evoluti of neighbor /off processes X 1 t),,x t) in Fig. 2. X 1 t) X 2 t) X 1 t) X t) T c T f off RR TC TI t t 1 t 2 Fig. 2. The illustrati of neighbor /off processes. By using the notati in 1), the number of cooperative neighbors of noe u, calle as u s cooperative egree ifferent from the traitial noe egree) at time t is a ranom process, enote by Y t), an it is equal to: Y t) = X i t). 1) i=1 Similar to the efiniti in [6], a noe is isolate at such time t 1 t 1 >t ) when all of its neighbors are in the faile state see Fig. 2). Thus, we can formally efine the cnecti time TC as follows Definiti 1: For a generic noe u arriving at time t, u s cnecti time TC {u} is efine as the first hitting time to of u s egree Y t), i.e., TC TC {u} = inft >t : Y t) = Y t )= ), 2) X/7/$ IEEE This full text paper was peer reviewe at the irecti of IEEE Communicatis Society subject matter experts for publicati in the IEEE GLOBECOM 27 proceeings.

3 where is the initial egree of noe u at time t. In the later ctext, without losing generality, we set t = an = to make our erivati tractable an omit {u} in TC {u} for clarity. B. Derivati of Expecte Cnecti Time We erive the expecte cnecti time an use a similar metho that was provie in [8]. Csier that Y t) efine in 1)) can be viewe as an alternating renewal process [11], in which each stage corresping to Y t) > an each off stage to Y t) =. The evoluti of the alternating renewal process Y t) is illustrate in Fig. 3, in which TC {j} cnecti time) is the urati of the j-th stage an TI {j} calle isolati time) is that of the j-th off stage. Y t) Fig. 3. off TC {1} TI {1} TC {2} The illustrati of the alternating renewal process Y t). To etermine the expectati value of the first stage urati, i.e., E[TC {1} ], we first argue that TC {1} has the same istributi as all remaining cnecti times TC {j} j 2). It is obvious that TC {1} is ifferent from the other TC {j} in that the stage starts from Y t) = 1 ) while other stages start from Y t) =1. Nevertheless, when a noe arrives the network, it still nees a perio of time to iscover its cooperative neighbors through some neighbor iscovery mechanisms, such as irectial antenna [12] an HELLO message [13]. Although the time for neighbor iscovery may be quite short, compare with the cnecti time, it is still reasable to view the stage of TC {1} starting from Y t) =1. Thus, we have TC {1} an other TC {j} j>1) the same istributi asymptotically. By applying the steay state istributi to Y t) Corollary in [11], p.p. 297), we have the probability of Y t) > at any time t as follows: E[TC] lim P Y t) > ) t E[TC]+E[TI]. 3) Notice that the probability above can also be expresse as the probability that at least e cooperative neighbor is still alive, i.e., lim P Y t) > ) = 1 t p f, 4) where p f is the probability of noe u fining a neighbor in the faile state. Note that this probability is ifferent from the limiting probability of any noe being faile refer to [6]). To explain this iscrepancy, we csier that when noe u arrives the network at time t, e of its cooperative neighbors, say noe i 1 i ), has being cooperative in the network for some time. We suppose that u outlives i, then the cnecti time between u an i before the first iscnecti) is epenent the minimum between the t resiual lifetime of i an link-time between u an i i.e., LT {u} i ). Let RL i enote the resiual lifetime of neighbor i, T enote the urati of the process X i t) being in the state, then T = min{rl i,lt i }. Similarly, we can also view X i t) as an alternating renewal process, then we have p f expresse by p f = lim P Xt) =1) t E[T ]+, 5) where E[T ] an are the expecte time cnecting to any neighbor an expecte recovery-time, respectively. Then by substituting 5) into 4), an equating 4) an 3), E[TC] is solve as: E[TC] E[TI] 1+ E[T ) ) ] 1. 6) Next, we erive E[T ].LetF RL t) an F LT t) be the istributis of i s resiual lifetime RL i an link-time LT i, respectively, then E[T ] is: E[T ]= 1 F RL t))1 F LT t))t. 7) We can further represent the CDF of the resiual lifetime F RL t) as follows, by applying basic renewal process theory Theorem-6.12 [11] p.p. 285): F RL t) =P RL i <t)= 1 E[T c ] 1 FTc x) ) x, 8) where E[T c ] is just the expecte lifetime. By substituting 8) into 7), we obtain E[T ] as a functi of E[T c ], F Tc t), an F LT t): E[T ]= 1 F LT t)) 1 1 F T c x))x) t. 9) E[T c ] Next, we erive E[TI]. Let the first instant of the stage of Y t) = starts at time t 1, as shown in Fig. 2, we know that there are 1 alreay-faile neighbors being in recovery an e neighbor just faile at time t 1. Thus, TI is the minimum time for the last neighbor to recover or for any e of the going recoveries to finish. Let RR i 1 i 1) enote the Resiual Recoverytime of any alreay-faile neighbor i, then TI can be expresse as TI = min{rr 1,RR 2,,RR 1,T f }. Further, if let RR = min{rr 1,RR 2,,RR 1 }, then TI = min{rr,t f } refer to Fig. 2). Let F RR t) an F Tf t) be the istributis of RR an T f, respectively, then E[TI] is: E[TI]= We can further express F RR t) by 1 F Tf t))1 F RR t))t. 1) F RR t) = P RR <t)=1 1 P RR i <t)) 1 = F T f x))x) 1, 11) X/7/$ IEEE This full text paper was peer reviewe at the irecti of IEEE Communicatis Society subject matter experts for publicati in the IEEE GLOBECOM 27 proceeings.

4 where is the expecte recovery-time. By substituting 11) into 1), we have E[Tf ] E[TI]= 1 z ) 1z =, 12) where z = 1 F T f x))x. Finally, we obtain the following result. Theorem 1: In a wireless multi-hop network moele by the geometric ranom graph efine in Secti II), the expecte cnecti time of any generic noe is: E[TC] 1+ E[T ] ) 1 ), 13) where E[T ] is given by 9). Next, we will use numerical simulatis to reveal the impacts of noe mobility an noe lifetime istributi the expecte cnecti time. C. Numerical Simulati Results From 13), we know that the expecte cnecti time is relate with the link-time istributi F LT t), noe lifetime istributi F Tc t), expecte lifetime E[T c ], expecte recovery-time, an average noe egree. We will emstrate the impacts of these parameters E[TC]. In our numerical simulatis, we use expential functi for F LT t), i.e., F LT t) =1 e λt, where 1/λ is the expecte link-time. For F Tc t), we use expential functi first, i.e., F Tc t) =1 e µt, where E[T c ]=1/µ. In aiti, we also use a two-parameter Weibull functi to moel F Tc t), i.e., F Tc t) =1 e t/β)α with E[T c ]=β Γ1 + 1/α) where Γ ) is the Gamma functi. Base the settings above, we have E[TC] simplifie immeiately to the following special case for expential link-time an lifetime: ) ) 1+ 1, 14) E[TC] 1 λ + µ) where λ an µ are as mentie above. In Fig. 4a), we show the impact of the expecte link-time, ranging from 1s to 1s, the expecte cnecti time E[TC]. It is clear that E[TC] increases ramatically when the expecte link-time increases. Further, by increasing the expecte lifetime E[T c ] from 3s to 12s, it is shown that E[TC] can be increase correspingly; however, when the expecte link-time is small < 5s), the increase is trivial. This inicates that noe mobility impacts E[TC] more than noe lifetime an becomes a ominant factor especially when link-times are short. Fig. 4b) shows the same results but in the log-log scale to show the linearity. In Fig. 5a) an 5b), we show the positive impact of E[T c ] E[TC] an the negative impact of E[TC]. Itis intuitively shown that the slower neighbors recover from the faile state, the shorter a noe is cnecte with the network. An interesting result here is that the impact of a smaller egree is almost similar as that of a lger recovery-time. Now, we use Weibull lifetime functi an show the impact of the expecte link-time the expecte cnecti time E[Tc]=3s E[Tc]=6s E[Tc]=12s Expecte link time s) a) Linear rawing E[Tc]=3s E[Tc]=6s E[Tc]=12s Expecte link time s) b) Log-log rawing Fig. 4. Impact of expecte link-time, =1, = 3s. E[Tf]=3s E[Tf]=6s Expecte lifetime s) a) = =5 = Expecte lifetime s) b) = 3s Fig. 5. Impact of expecte lifetime, λ =.1. in Fig. 6a) an 6b). In the Weibull functi, α = 2 an β = E[T c ]/Γ1 + 1/α). The result inicates that there is no improvement gaine if the lifetime is heavy-taile istribute. An the expential lifetime can actually provie better upper boun for E[TC] E[Tc]=3s E[Tc]=6s E[Tc]=12s Expecte link time s) a) Weibull lifetime Linear) E[Tc]=3s E[Tc]=6s E[Tc]=12s Expecte link time s) b) Weibull lifetime Log-log) Fig. 6. Impact of expecte link-time, =1, = 3s. IV. STOCHASTIC ANALYSIS ON RESILIENCE In the previous secti, we stuie the expecte cnecti time for any generic noe, which provies iniviual user s the expectati how lg they can have a stable cnecti before the first iscnecti. In this secti, we further investigate the probability that a noe is isolate an the network resilience in terms of the global cnectivity. A. Probability of Noe Isolati In this paper, the noe isolati event is the case that a generic noe u outlives all its neighbors, which is equivalent to the fact of u s lifetime T c {u} being greater than u s cnecti time TC. Thus, the probability of noe isolati, enote by γ, is formally efine as follows X/7/$ IEEE This full text paper was peer reviewe at the irecti of IEEE Communicatis Society subject matter experts for publicati in the IEEE GLOBECOM 27 proceeings.

5 Definiti 2: For a generic noe u, the probability of u being isolate is γ = P TC < T c {u} ). If we know the CDF of TC,sayF TC t), we can use the following equati to calculate γ: P TC < T {u} c )= F TC t)f Tc t)t, 15) where f Tc t) is the PDF probability ensity functi) of T c noe lifetime). However, the exact istributi of TC is generally ifficult to represent in a close-form since it epens the evoluti of the cooperative egree, i.e., ranom process Y t) efine in 1). Fortunately, we have alreay touche the isolati probability γ in Secti III-B. Notice that γ efine above is actually equivalent to the probability that the cooperative noe egree becomes, then we have γ = lim P Y t) =) ), 16) t E[T ]+ where is the average noe egree not cooperative egree). In aiti, if we substitute 16) into 13), we can obtain ETC) 1 γ ) 1), 17) which yiels: γ ) 1. 18) + E[TC] Equati 18) provies new insight into the relatiship between the noe isolati probability an expecte cnecti time. It can be easily preicte that as E[TC] increasing, the isolati probability γ ecreases in logarithm. B. Analysis Network Resilience In this subsecti, we utilize the earlier erive metric γ to characterize the evoluti of wireless multi-hop networks uner ranom failures. In particular, we aress the questi: what is the probability that a wireless multi-hop network survives N en-users from potential noe isolatis uring their entire lifetimes? We call this probability as network resilience an enote it by Ψ in the later ctext. To aress the questi aforementie, we take a close look at some useful results in ranom graph theory, especially the stuies geometric ranom graph. It has been prove that with high probability, the network becomes k-cnecte when the minimum noe egree in the communicati geometric ranom) graph becomes k [9] Theorem-6.1.2, p.p. 64). In other wors, a wireless multi-hop network is cnecte if an ly if every noe has its cooperative egree Y t)) greater than, citial a sufficiently large system size N. Thus, the resilience of a network can be approximate by Ψ lim 1 P Y t) = )) N. 19) t Since lim t P Y t) =)is the isolati probability γ, by substituting 18) into 19), we finally have Ψ 1 ) 1 ) N. 2) + E[TC] From 2), it is clear that the ratio between the expecte recovery-time an expecte cnecti time i.e., /E[TC]) etermines the value of resilience given fixe maximum) noe egree an system size N. Specifically, suppose E[TC], then for any number <ψ<1 we can obtain from 2) that the ratio shoul satisfy E[TC] < 1 ψ 1 N ) so that the resilience is no less than the given number, i.e., Ψ ψ, asymptotically. V. CONCLUSIONS AND FUTURE WORKS In this work, we analyze the stochastic property of noe cnecti time an provie a close-form approximati. In aiti, we erive the isolate probability an the network resilience as functis of the expecte cnecti time. Our analysis inicates that networks with heavy-taile lifetime such as Weibull) istributis provies no improvement than those with light-taile e.g., Expential) istributis in terms of lger expecte cnecti lifetime for any enuser. Instea, noe mobility has more significant impact than lifetime oes. In our future works, we will take potential noe misbehaviors into csierati to investigate their impact the cnecti time an network resilience. Since this work is cucte flat wireless multi-hop networks with uniform noe istributi, we will also csier whether en-users an network can achieve better resilience in the networks with clusters or scale-free property. REFERENCES [1] B. Awerbuch, D. Holmer, C. Nita-Rotaru, an H. Rubens, An Oneman Secure Routing Protocol Resilient to Byzantine Failures, in Proc. of ACM Workshop Wireless security, 22, pp [2] S. Lee, Y. Yu, S. Nelakuiti, Z. Zhang, an C. Chuah;, Proactive vs Reactive Approaches to Failure Resilient Routing, in Proc. of IEEE INFOCOM 4, Mar. 24, pp [3] H. Yang, F. Ye, Y. Yuan, S. Lu, an W. Arbaugh, Towar Resilient Security in Wireless Sensor Networks, in Proc. of ACM MobiHoc 5, May 25, pp [4] T. Dimitar, F. Sja, M. Jani, an G. Aksenti, Cnecti Resilience to Noes Failures in A Hoc Networks, in Proc. of 12th IEEE Meiterranean Electrotechnical Cference, Croatia, May 24. [5] C. Bettstetter, On the Minimum Noe Degree an Cnectivity of a Wireless Multihop Network, in Proc. of ACM MobiHoc 2, Jun. 22, pp [6] F. Xing an W. Wang, Moeling an Analysis of Cnectivity in Mobile A Hoc Networks with Misbehaving Noes, in Proc. of IEEE ICC 6., vol. 4, 26, pp [7], Analyzing Resilience to Noe Misbehaviors in Wireless Multihop Networks, in Proc. of IEEE WCNC 7., 27. [8] D. Lear, V. Rai, an D. Loguinov, On Lifetime-Base Noe Failure an Stochastic Resilience of Decentralize Peer-to-Peer Networks, in Proc. of ACM SigMetrics 5, 25. [9] P. Santi, Topology Ctrol in Wireless A Hoc an Sensor Networks. John Wiley an Ss Inc., 26. [1] N. Saagopany, F. Bai, B. Krishnamachari, an A. Helmy, PATHS: Analysis of PATH Durati Statistics an their Impact Reactive MANET Routing Protocols, in Proc. of ACM MobiHoc 3, Jun. 23. [11] J. Mehi, Stochastic Processes. John Wiley an Ss Inc., [12] S. Vasuevan, J. Kurose, an D. Towsley, On Neighbor Discovery in Wireless Networks with Directial Antennas, in Proc. of IEEE INFOCOM 5, 25. [13] C. E. Perkins, E. M. Beling-Royer, an S. R. Das, A hoc On-Deman Distance Vector AODV) Routing, IETF, RFC [Online]. Available: X/7/$ IEEE This full text paper was peer reviewe at the irecti of IEEE Communicatis Society subject matter experts for publicati in the IEEE GLOBECOM 27 proceeings.