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U NIVERSITY OF T EXAS A T E L P ASO A Simple Algorithm for reliability evaluation of a stochastic-flow network with node failure By Yi-Kuei Lin 1 Oswaldo Aguirre

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I NTRODUCTION Networks are series of points or NODES interconnected by communication paths or LINKS G = (N,L) Where: N= number of nodes L= number links 1≤ N ≤ ∞ 0≤ L ≤ ∞ 2 G = (7,7)

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I NTRODUCTION (CONT’D…) Network applications: Distribution networks Transportation networks Telecommunication networks Network problems Shortest path Network flow Network reliability 3

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P ROBLEM DESCRIPTION 4 Network reliability: The probability that a message can be sent from one part of the network to another

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P ROBLEM DESCRIPTION (CONT’D…) 5 Binary state Multistate Links have two states 0/1 Insufficient in obtaining reliability models that resemble the behavior of the system Components can have a range of degraded states X = (x 1,x 2,x 3,…….x n ) More accurate results to real behavior

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M ETHODOLOGY (CONT’D…) 6 Minimal cut vector (MC): It is a set of components for which the repair of any failed components results in a functioning system a1a1 a1a1 a5a5 a6a6 a3a3 a4a4 a7a7 a9a9 a 10 a8a8 Minimal Cuts: 1.a 1,a 5 2.a 1,a 7 3.a 5,a 8 4.a 2,a 3,a 5 5.a 1,a 4,a 6 6.a 2,a 6 7.a 2,a 7 8.a 6,a 8 9.a 7,a 8

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M ETHODOLOGY (CONT’D…) 7 Minimal path vector (MP): A minimal path vector is a path vector for which the failure of any functioning components results in system failure a1a1 a1a1 a5a5 a6a6 a3a3 a4a4 a7a7 a9a9 a 10 a8a8 Minimal Paths 1.a 7,a 1,a 8,a 2,a 10 2.a 7,a 1,a 8,a 3,a 9,a 6,a 10 3.a 7,a 5,a 9,a 6,a 10 4.a 7,a 5,a 9,a 4,a 8,a 2,a 10

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A LGORITHM 8 a1a1 a1a1 a5a5 a6a6 a3a3 a4a4 a7a7 a9a9 a 10 a8a8 Find the system reliability. When the network can transmit at least 5 messages or demand (d)>4 Using minimal path sets Minimal Paths 1.a 7,a 1,a 8,a 2,a 10 f 1 2.a 7,a 1,a 8,a 3,a 9,a 6,a 10 f 2 3.a 7,a 5,a 9,a 6,a 10 f 3 4.a 7,a 5,a 9,a 4,a 8,a 2,a 10 f 4

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A LGORITHM (CONT’D…) 9

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10 Step 1: find solutions that satisfy the following conditions Each flow ( f j ) <= max capacity of the Minimal path (MP j ) f1 <= Max cap MP1 ( a7,a1,a8,a2,a10)=(6,2,5,3,5) <= 2 f2 <= Max cap MP2 ( a7,a1,a8,a3,a9,a6,a10)=(6,2,5,3,4,3,5) <= 2 f3 <= Max cap MP2 ( a7,a5,a9,a6,a10)=(6,3,4,3,5) <= 3 f4<= Max cap MP2 ( a7,a5,a9,a4,a8,a2,a10)=(6,3,4,3,5,3,5) <= 3 a1a2a3a4a5a6a7a8a9a10 2323336545

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A LGORITHM (CONT’D…) 11 Step 1: find solutions that satisfy the following conditions ( fj | ai MPj) <= Max Cap. Of Component i ( fj | a1 MPj) = f1 + f2 <= 2 ( fj | a2 MPj) = f1 + f4 <= 3 ( fj | a3 MPj) = f2 <= 2.... ( fj | a10 MPj) = f1 + f2 + f3 +f4 <= 5 f1+f2+f3+f4=5

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A LGORITHM (CONT’D…) 12 Step 1: find solutions that satisfy the following conditions (2,0,3,0),(2,0,2,1),(1,1,2,1),(1,1,1,2),(0,2,1,2) and (0,2,0,3) Step 2: Transform F into X (a 1,a 2,a 3,a 4,a 5,a 6,a 7,a 8,a 9,a 10 ) a1= f1 + f2 a4= f4a7=a10=f1+f2+f3+f4 a2= f1 + f4 a5= f3+ f4a8=f1+f2+f3 a3= f2 a6= f2+f3a9=f2+f3+f4 Thus: X1 = (2,2,0,0,3,3,5,2,3,5) X2 = (2,3,0,1,3,2,5,3,3,5) X3 = (2,2,1,1,3,3,5,3,4,5) X4 = (2,3,1,2,3,1,5,4,4,4) X5 = (2,2,2,2,3,3,5,4,5,5) X6 = (2,3,2,3,3,2,5,5,5,5)

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A LGORITHM (CONT’D…) 13 Step 3: Remove non minimal ones (X) to obtain lower boundary points X 1 =( 2, 2,0, 0,3, 3,5,2,3,5 ) X 2 =( 2, 3,0, 1,3, 2,5,3,3,5 ) X 1 =( 2,2,0,0,3,3,5,2,3,5 ) <= X 3 =( 2,2,1,1,3,3,5,3,4,5 ) X 1 =( 2,2,0,0,3, 3,5,2,3,5 ) X 6 =( 2,3,2,3,3, 2,5,5,5,5 )

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A LGORITHM (CONT’D…) 14 Step 4: Obtain Reliability of the system After selecting only 2 vectors: X1 = (2,2,0,0,3,3,5,2,3,5) X2 = (2,3,0,1,3,2,5,3,3,5) The reliability of the system can be evaluated using the inclusion exclusion formula P(X1 U X2 ) = P(X1) + P(X2) – P(X1X2) The reliability that the system can send at least 5 units of flow is 0.824241

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Q UESTIONS 15

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