Adviser: Frank, Yeong-Sung Lin Present by Wayne Hsiao

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

 Introduction  Network Survivability  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  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 T1A1.2 committee 9

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

 Introduction  Network Survivability  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 λ i  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 μ i 14

 The state of the system at any time t can be completely described by the collection of the state of each node  Where X i (t) = 0 (1 ≦ i ≦ n) if the event has occurred on the i-th node at time t, X i (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, · · ·,X n ) : X 1, · · ·,X n ∈ {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,...,N f (N f < N) the system has failure propagation and in states N f +1,N f +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 q ij 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 Υ i 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,..., S Φ } ( Φ = 2 3 − 1)  State is described by a triple as (X 1, X 2, X 3 )  X i ∈ {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 μ i  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 = (1,0,0,0,0,0,0,0) 23

 Given a disaster occurs and destroys BS1, 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 BS2 is repaired (this occurs with rate μ 2 )  it may jump to state (1, 0, 1) if the BS1 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 (X1,X2,X3) (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 q ii 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 ≥ max i |q ii | 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  cell1 → cell2 → cell3  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 π k (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  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 N i 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  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) dd 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 2 μ 1, 2 μ 2, 2 μ 3 40

NUMERICAL RESULT (CONT.) dd 41

 Introduction  Network Survivability  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

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