TUFTS UNIVERSITY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING ES 152 ENGINEERING SYSTEMS Spring Lesson 4 Reliability Networks

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1 TUFTS UNIVERSITY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING ES ENGINEERING SYSTEMS Spring 0 Lesson 4 Reliability Networks Some Background Information: The reliability Rj of subtem j is the probability that it will continue to work over a given period of time. The failure rate of subtem j is therefore Fj Rj, and it is of course a probability as well. The failure of each subtem is considered to be independent of all the other subtems. A tem has a reliability structure which is related to but not identical to its physical structure.. If two or more subtems must all work in order for the tem to work then in the tem s reliability structure the subtems are in series. (The wireless mouse for the Mac has two AA batteries that are physically in parallel to provide adequate current. Since both must work for the mouse to work they are in series in the mouse s reliability structure.) If a tem s reliability structure consists of two subtems in series, with reliabilities R and R respectively, the tem s reliability is RR.. If two or more subtems must all fail in order for the tem to fail then in the tem s reliability structure the subtems are in parallel. If a tem s reliability structure consists of two subtems in parallel, with failure rates F and F respectively, the tem s failure rate is FF.. Note that in general F R (and R F). Let us consider the following problem Desirable Reliable Engineering Corporation (DREC) has produced and installed a control tem with the following reliability structure of subtems. A control signal input at A must be successfully processed by three () subtems before being output at B. In order to achieve a higher tem reliability a redundant design was used. s,, and are in series, in terms of the reliability structure, as are subtems 4,, and. In terms of reliability structure, subtems,, and are in parallel with subtems 4,, and. It is this parallel structure that produces the redundancy, since the overall tem works if either subtems,, and all work or if subtems 4,, and all work (or, of course, if all subtems work). The reliability structure is shown in Figure 4-. es Lesson 04

2 A B 4 Block Diagram Representation Of System Reliability Structure Figure 4- We will use block diagrams with the reliabilities shown explicitly, as in Figure 4-, and calculate the tem reliability. R 0.9 R 0.94 R 0.98 A B R 0.98 R 0.9 Reliability Block Diagram Figure 4- es Lesson 04

3 The path AB has a reliability of R RRR while the path A4B as a reliability R4 R4RR. Since the two paths are in parallel F FF4 ( R)( R4) Therefore R F FF4 ( R) ( R4) ( RRR)( R4RR) Recall that reliability is a measure of probability of failure within a given interval of time. That suggests the possibility of determining the initial mean time until failure of a tem. If R remains constant for each interval (i.e., the tem does not age, break in, m etc.) then the probability of the tem is still operating after m intervals is R. The a priori probability it fails in interval m is therefore F (m) R m ( R This is an example of a geometric distribution. Like its continuous cousin, the exponential distribution (to be seen later in this course) it has no memory. The failure rate in an interval is independent of the age of the tem. Therefore, the tem s mean time before failure is ). μ m F F F m F ( R F mf mr (+ R (+ R (m) ) m + R + R +...) +...) Of course a block diagram is just a digraph in disguise. We could represent the reliability structure of the tem as shown in Figure 4-, and note that if a subtem failure is es Lesson 04

4 represented by removal of the corresponding branch then the tem is working as long as B is reachable from A. R R R A R4 4 R R B Weighted Digraph (Network) Representation of Reliability Structure Figure 4- We can use this reachability concept as the basis of a simulation of tem reliability. The underlying digraph can be represented by the adjacency matrix A The reachability matrix for an n x n adjacency matrix is R B[(I + A + A A n )] where B is the boolean function. (B[x] if x 0, B[x] 0 if x 0.) This can be more efficiently calculated as n R B[( I + A) ]. Failure of a specific subtem corresponds to converting the corresponding to a 0 in es Lesson 04 4

5 the A matrix. Therefore if we determine if each subtem is working based on its reliability, set the A matrix accordingly, and compute the R matrix we have an appropriate basis for a simulation. In each case we will want to know if node is reachable from node. This corresponds to the r. Let us look at a problem that can not be decomposed into series and parallel subtems. This is shown in Figure 4-4 R R 0.9 R 0.9 R 0.94 Reliability structure not decomposable into series and parallel subtems Figure 4-4 One approach is to solve this problem twice, once with working and once with failed. This is shown in Figure 4-. es Lesson 04

6 Working R R 0.9 R 0.94 R 0.90 Failed 0.9 R 0.9 R 0.94 Figure 4- Case - working (probability 0.9): R, [-(-R)(-R)][-(-R4)(-R)] Case failed (probability 0.07): Therefore, R, -(-RR4)(-RR) 0.98 R 0.9*R, *R, An alternative approach is through simulation. To do this we will first represent the structure of Figure 4-4 as a digraph. Figure 4- shows the how to identify the appropriate nodes locations. es Lesson 04

7 R R R R 0.94 Identifying the Node Locations Figure 4- The resulting weighted digraph is shown in Figure Weighted Digraph Representation of System in Figure 4-4 Figure 4-7 This weighted digraph can be represented by the reliability matrix. The underlying digraph of course gives rise to the adjacency matrix A. Here is a run of the simulation. es Lesson 04 7

8 The VBA program is: Sub reliability() Dim rel(0, 0) As Single, a(0, 0) As Integer, c(0, 0) As Integer Dim reach(0, 0) As Integer, ccc(0, 0) As Integer, ppower As Integer Dim i As Integer, j As Integer, k As Integer, n As Integer, rrun As Long Dim maxrun As Long, work As Integer, totalwork As Integer Dim temrel As Single 'Read in number of runs and number of nodes in network Range("d").Select maxrun ActiveCell.Value Range("d4").Select n ActiveCell.Value Range("c").Select 'Read in reliability matrix R For i To n For j To n rel(i, j) ActiveCell.Value ActiveCell.Offset(0, ).Select ActiveCell.Offset(, -n).select For rrun To maxrun 'create A matrix for this run For i To n For j To n If Rnd() > rel(i, j) Then 'branch ij failed a(i, j) 0 Else 'branch ij is working a(i, j) End If 'create C (I + A) For i To n For j To n es Lesson 04 8

9 c(i, j) a(i, j) If i j Then c(i, j) c(i, j) + reach(i, j) c(i, j) ' create Reachability matrix R B[C^n-] For i To n For j To n reach(i, j) c(i, j) For ppower To n - 'create reach C^n- For i To n 'Note that we will not actually need to take boolean. For j To n 'We can use reach matrix rather than R matrix. For k To n 'ccc is temporary storage matrix ccc(i, j) ccc(i, j) + c(i, k) * reach(k, j) Next k For i To n For j To n reach(i, j) ccc(i, j) ccc(i, j) 0 Next ppower If reach(, n) > 0. Then 'element,n is not zero and node n reachable from node work Else 'node n not reachable from node work 0 End If totalwork totalwork + work Next rrun Range("d0").Select temrel totalwork / maxrun ActiveCell.Value temrel Range("a").Select End Sub Let s consider an even more complex reliability structure, shown in Figure 4-8. Reducing this structure to series and parallel blocks looks to be pretty difficult. Instead, let us determine the tem reliability by simulation. The node locations are already shown. es Lesson 04 9

10 R 0.70 R 0. R R 0. R 0.9 R 7 0. R 8 0. R Figure 4-8 The resulting network model is shown in Figure Weighted Digraph Representation of System in Figure 4-8 The input for the simulation is: Figure 4-9 es Lesson 04 0

11 . The resulting tem reliability is R es Lesson 04