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

Network Survivability under Disaster Propagation:Modeling and AnalysisLang Xie, Poul E. Heegaard, Yuming Jiang Adviser: Frank, Yeong-Sung Lin Present by Wayne Hsiao

Agenda • Introduction • Network Survivability • Network Survivability under Disaster Propagation • Numerical Result • Conclusion

introduction • 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

Introduction (cont.) • Disaster propagation • Adynamic 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

Introduction (cont.) • Network design and operation need to consider survivability • This requires an understandingof the dynamical network recovery behaviors under failurepatterns • To analyze the impact of disasters onthe network as well as for estimating the benefits of alternativenetwork survivable proposals, many mathematical models havebeen considered • However, up to now no much is known about the networksurvivability in the propagation of disastrous events

Introduction (cont.) • The present paper develops a network survivability modelingmethod, which takes into consideration the propagatingdynamics of disastrous events • The analysis is exemplified for three repair strategies. • Theresults not only are helpful in estimating quantitatively thesurvivability, but also provide insights on choosingamong different repair strategies

Network survivability • 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

Network survivability (cont.) • The measure of interest M has the value m0before 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 • tR is the relaxation time for the system to restore the value of M

Network Survivability under Disaster Propagation • Develop such a model particularly for networkedsystems where disastrous events may propagate across geographicalareas • A network can be viewed as a directed graph consistingof nodes and directed edges • Nodes represent the networkinfrastructures • The directed edges denote the directions oftransitions • The network is vulnerable to all sorts of disaster,which may start on some network nodes and propagate to othernodes during a random time

Network Survivability under Disaster Propagation(cont.) • 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

Network Survivability under Disaster Propagation(cont.) • 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 iis exponentially distributed with mean value μi

Network Survivability under Disaster Propagation(cont.) • 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-thnode at time t, Xi(t) = 1 in the case when the event has not occurred on the i-th node at time t.

Network Survivability under Disaster Propagation(cont.) • With the above assumptions, the transient process X(t) can be mathematically modeled as a continuous-time Markov chain (CTMC) with state space Ω= {(X1, · · · ,Xn) : X1, · · · ,Xn∈ {0, 1}} • The state space Ωconsists of total N = 2nstates • The process X(t) starts in the state (0, 1, ..., 1) and finishes in the absorbing state (1, 1, ..., 1)

Network Survivability under Disaster Propagation(cont.) • Suppose that the system states are ordered so that in states1, 2, ...,Nf(Nf < N) the system has failure propagation andin states Nf +1,Nf +2, ...,N the system is only in restorationphase • 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

Network Survivability under Disaster Propagation(cont.) • Let π(t) = {π i(t), i∈Ω} denote a row vector of transientstate probabilities of X(t) at time t • With Q, the dynamicbehavior of the CTMC can be described by the Kolmogorovdifferential-difference equation • Then the transient state probability vector can be obtained

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

Network Survivability under Disaster Propagation(cont.) • An infrastructure wireless network example

Network Survivability under Disaster Propagation(cont.) • The state space of the chain is defined as S = {S0 , ..., SΦ} (Φ = 23 − 1) • State is described by a triple as (X1, X2, X3) • Xi ∈ {0, 1} refers to the affected state of cell i, i = 1, 2, 3 • The set of possible states is

Network Survivability under Disaster Propagation(cont.) • Two repair strategies • Scheme 1: each cell has its own repair facility • Scheme 2: all cells share a single repair facility

Scheme1 • 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 = (1,0,0,0,0,0,0,0)

Scheme1 • 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 λ2 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 μ1

Scheme1 • 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 μ2) • it may jump to state (1, 0, 1) if the BS1 is repaired (this occurs with rate μ1) • the CTMC may jump to state (0,0,0) if the disaster propagates to cell 3 (this occurs with rate λ3)

Scheme1 • 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

Scheme1 • 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:

Scheme1 • 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

Scheme2 • 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:

Scheme2 • Accordingly, the transition diagram of the CTMC has the reduced 6-state as illustrated in Fig. 5

Scheme2 • 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

Scheme2 • Continuing by induction, then we have

Scheme2

Scheme2 • We remark that simplification has been made in transition diagrams in Fig. 3 and Fig. 5 • Acell which is recovered from a hurricane is unlikely to be destroyed by the same hurricane

Numerical result • 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

Numerical result • The coverage radius of one BS is 1 km • For the three cells, we assume N1 = 3000,N2 = 5000, N3 = 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

Numerical result • In Fig. 6, where the chosen repair strategy is Scheme 1 • Consider the scenario • The fault propagation rate is high (λ2 = 5, λ3 = 5), and the repair rates (μ1 = 0.04, μ2 = 0.08, μ3 = 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 (μ1 = 0.36, μ2 = 0.72, μ3 = 1.08), the fraction of active userssharplyincrease • The effect of the fault propagation rate is not as evident for longer observation time (after 10 hours) • d

Numerical result(cont.) • 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

Numerical result(cont.) • In the following, we compare three repair schemes • Scheme 1 • Scheme 2 • Scheme 3:same as Scheme 1 but with double repair rates 2μ1, 2μ2, 2μ3

Numerical result(cont.) • d

Conclusion • 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

ThanksforYourListening!

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