LANG XIE POUL E HEEGAARD YUMING JIANG Adviser

LANG XIE, POUL E. HEEGAARD, YUMING JIANG Adviser: Frank, Yeong-Sung Lin Present by Wayne Hsiao

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 2

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 3

§ Telecommunication networks have become one of the critical infrastructures § It is critically important that the network is survivable § The ability of the network to deliver the required services in the face of various disastrous events § Disaster propagation is one of the most common characteristics of disastrous events and has serious impact on communication networks 4

§ Disaster propagation § A dynamic area-based event, in which the affected area can evolve spatially and temporally § For example, the 2005 hurricane Katrina in Louisiana, caused approximately 8% of all customarily routed networks in Louisiana outraged § The March 2011 earthquake and tsunami in east Japan, which cascaded from the center to Tohoku and Tokyo areas, damaged 1. 9 million fixed-lines and 29 thousand wireless base stations 5

§ Network design and operation need to consider survivability § This requires an understanding of the dynamical network recovery behaviors under failure patterns § To analyze the impact of disasters on the network as well as for estimating the benefits of alternative network survivable proposals, many mathematical models have been considered § However, up to now no much is known about the network survivability in the propagation of disastrous events 6

§ The present paper develops a network survivability modeling method, which takes into consideration the propagating dynamics of disastrous events § The analysis is exemplified for three repair strategies. § The results not only are helpful in estimating quantitatively the survivability, but also provide insights on choosing among different repair strategies 7

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 8

§ We focus on survivability as the ability of a networked system to continuously deliver services in compliance with the given requirements in the presence of failures and other undesired events § Network survivability is quantified as the transient performance from the instant when an undesirable event occurs until steady state with an acceptable performance level is attained § defined by the ANSI T 1 A 1. 2 committee 9

§ The measure of interest M has the value m 0 before a failure occurs. § ma is the value of M just after the failure occurs § mu is the maximum difference between the value of M and ma after the failure § mr is the restored value of M after some time tr § t. R is the relaxation time for the system to restore the value of M 10

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 11

§ Develop such a model particularly for networked systems where disastrous events may propagate across geographical areas § A network can be viewed as a directed graph consisting of nodes and directed edges § Nodes represent the network infrastructures § The directed edges denote the directions of transitions § The network is vulnerable to all sorts of disaster, which may start on some network nodes and propagate to other nodes during a random time 12

§ Suppose the number of nodes in the networked system is n § We consider a disastrous event, which occurs on these nodes in successive steps § The propagation is assumed to have ’memoryless’ property § The probability of disastrous events spreading from one given node to another depends only on the current system state but not on the history of the system § The affected node can be repaired (or replaced by a new one) in a random period § All the times of the disaster propagation and repair are exponentially distributed 13

§ The state of each node of the system at time t lies within the set {0, 1} § At the initial time t = 0, a disastrous event affects the 1 -st node and the system is in the state (0, 1, . . . , 1) § The disaster propagates from the node i − 1 to node i according to Poisson processes with rate λi § A disastrous event can occur on only one node at a time § Each node has a specific repair process which is all at once and the repair time period of node i is exponentially distributed with mean value μi 14

§ The state of the system at any time t can be completely described by the collection of the state of each node § Where Xi(t) = 0 (1 ≦ i ≦ n) if the event has occurred on the i-th node at time t, Xi(t) = 1 in the case when the event has not occurred on the i-th node at time t. 15

§ With the above assumptions, the transient process X(t) can be mathematically modeled as a continuous-time Markov chain (CTMC) with state space Ω = {(X 1, · · · , Xn) : X 1, · · · , Xn ∈ {0, 1}} § The state space Ω consists of total N = 2 n states § The process X(t) starts in the state (0, 1, . . . , 1) and finishes in the absorbing state (1, 1, . . . , 1) 16

§ Suppose that the system states are ordered so that in states 1, 2, . . . , Nf(Nf < N) the system has failure propagation and in states Nf +1, Nf +2, . . . , N the system is only in restoration phase § Then, the transition rate matrix Q = [qij] of the process {X(t), t ≧ 0} can be written in partitioned form as § where qij denotes the rate of transition from state i to state j 17

§ Let π(t) = {π i(t), i ∈ Ω } denote a row vector of transient state probabilities of X(t) at time t § With Q, the dynamic behavior of the CTMC can be described by the Kolmogorov differential-difference equation § Then the transient state probability vector can be obtained 18

NETWORK SURVIVABILITY UNDER DISASTER PROPAGATION (CONT. ) § Let Υi be the reward rate associated with state i § In our model, the performance is considered as reward § The network survivability performance is measured by the expected instantaneous reward rate E[M(t)] as 19

§ An infrastructure wireless network example 20

§ The state space of the chain is defined as S = {S 0 , . . . , SΦ} (Φ = 23 − 1) § State is described by a triple as (X 1, X 2, X 3) § Xi ∈ {0, 1} refers to the affected state of cell i, i = 1, 2, 3 § The set of possible states is 21

§ Two repair strategies § Scheme 1: each cell has its own repair facility § Scheme 2: all cells share a single repair facility 22

§ Each cell i has its own repair facility with repair rate μi § Fig. 3 shows the 8 -state transition diagram of the CTMC model of the network example § The transition matrix is of size 8 × 8 and the initial probability vector is π = (1, 0, 0, 0, 0) 23

§ Given a disaster occurs and destroys BS 1, then all the users in cell 1 disconnect to the network § The initial state is (0, 1, 1) § The transition to state (0, 0, 1) occurs with rate λ 2 and takes into account the impact of disaster propagation from cell 1 to cell 2 § The CTMC may also jump to original normal state (1, 1, 1) with repair rate μ 1 24

§ On state (0, 0, 1), the CTMC may jump to three possible states § it may jump back to state (0, 1, 1) if the BS 2 is repaired (this occurs with rate μ 2) § it may jump to state (1, 0, 1) if the BS 1 is repaired (this occurs with rate μ 1) § the CTMC may jump to state (0, 0, 0) if the disaster propagates to cell 3 (this occurs with rate λ 3) 25

§ Let π(t) = [π(0, 0, 0)(t) · · · π(X 1, X 2, X 3)(t) · · · π(1, 1, 1)(t)] denote the row vector of transient state probabilities at time t § The infinitesimal generator matrix for this CTMC is defined as Λ which is depicted in Fig. 4 26

§ With Λ, the dynamic behavior of the CTMC can be described by the Kolmogorov differential- difference equation in the matrix form § π(t) can be solved using uniformization method § Let qii be the diagnoal element of Λ and I be the unit matrix, then the transient state probability vector is obtained as follows: 27

SCHEME 1 § Where β ≥ maxi|qii| is the uniform rate parameter and P = I+Λ/β. § Truncate the summation to a large number (e. g. , K), the controllable error ε can be computed from 28

§ In the situation with this repair strategy, all cells share the same repair facility § The repair sequence is the same as the propagation path § cell 1 → cell 2 → cell 3 § The set of all possible states in this situation is: 29

§ Accordingly, the transition diagram of the CTMC has the reduced 6 - state as illustrated in Fig. 5 30

§ The system is in each state k at time t, which is denoted by πk(t), k = 0, . . . , 5 § They can be obtained in a closed-form by the convolution integration approach § Inserting Eq. (8) into Eq. (2) we can derive 31

§ Continuing by induction, then we have 32

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§ We remark that simplification has been made in transition diagrams in Fig. 3 and Fig. 5 § A cell which is recovered from a hurricane is unlikely to be destroyed by the same hurricane 34

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 35

§ The expected instantaneous reward rate E[M(t)] gives the impact of users of the system at time t § Given the number of users Ni of each cell i, as defined, the reward rate for each state is easily found 36

§ The coverage radius of one BS is 1 km § For the three cells, we assume N 1 = 3000, N 2 = 5000, N 3 = 2000 § For the setting of propagation rates, We refer to the data from Hurricane Katrina situation report § The peak wind speed was reported as high as 115 mph (184 km/h) § The units of repair time of BS is hours § It is acceptable that the disaster propagation rates are more than two order of magnitude than repair rates 37

§ In Fig. 6, where the chosen repair strategy is Scheme 1 §d § Consider the scenario § The fault propagation rate is high (λ 2 = 5, λ 3 = 5), and the repair rates (μ 1 = 0. 04, μ 2 = 0. 08, μ 3 = 0. 12) are low § In this scenario, the fraction of active users is low (roughly 0. 07, 2 hours after the failure) § If the repair rates are relatively higher (μ 1 = 0. 36, μ 2 = 0. 72, μ 3 = 1. 08), the fraction of active users sharply increase § The effect of the fault propagation rate is not as evident for longer observation time (after 10 hours) 38

§ The plus-marked and dashed (blue) curves cross each other at time t ≈ 2 at Fig. 6 § If we account for up to roughly two hours after the disaster, the fault propagation rates affect the service performance more than the repair rates § In contrast, if we account for longer periods of time, the repairs rates yield more benefits than to have lower fault propagation rate 39

§ In the following, we compare three repair schemes § Scheme 1 § Scheme 2 § Scheme 3: same as Scheme 1 but with double repair rates 2μ 1, 2μ 2, 2μ 3 40

NUMERICAL RESULT (CONT. ) §d 41

§ Introduction § Network Survivability under Disaster Propagation § Numerical Result § Conclusion 42

§ We have modeled the survivability of an infrastructure- based wireless network by a CTMC that incorporates the correlated failures caused by disaster propagation § The focus has been on computing the transient reward measures of the model § Numerical results have been presented to study the impact of the underlying parameters and different repair strategies on network survivability 43

Thanks for Your Listening ! 44