IEEE TRANSACTIONS ON RELIABILITY, VOL. 43, NO. 1, 1994 MARCH 59

Comment On “Cut Set Analysis of Networks Using Basic Minimal Paths and Network Decomposition”

V. C. Prasad Indian Institute of Technology, Delhi

V. Sankar, Student Member IEEE College of Engineering, Anantapur

K. S . Prakasa Rao, Senior Member IEEE Indian Institute of Technology, Delhi

Key Words - Basic minimal path, Network decomposition, Node cut set

Reader A i d s - General purpose: Bring out errors Special math needed for explanation: Graph theory Special math needed to use the results: Same Results useful to: Reliability theoreticians

Abstract - The network decomposition suggested by Jasmon (1985) fails to give all basic minimal paths of the network, depending on the number of nodes split. Consequently, some minimal node cut sets are not generated. Even those generated using these basic minimal paths need not be node cut sets.

COMMENTS

Jasmon [ 11 suggested a procedure for generating all basic minimal paths of a network using network decomposition. He decomposes the given network S into two connected subnet- works SI & S2 such that S1 contains the source (s), Sz con- tains the sink (t) and every edge of S lies either in S1 or in S2 (but not both). The basic minimal paths of each subnetwork are then generated. The basic minimal paths of S1 are combined judiciously with those of S2 to obtain all basic minimal paths of the original network. We show that this pro- cedure does not always generate all basic minimal paths. Therefore, if these paths are used to evaluate minimal node cut sets, then -

some minimal node cut sets are not generated at all, 9 some so called minimal node cut sets are actually not the

minimal node cut sets at all.

The following example illustrates these observations.

EXAMPLE

The sample network* is shown in Figure 1. Let this net- work S, be split at nodes 3, 4 and 5 which gives rise to

‘The figures are reproduced here from [l]; the figure number are the same as in [l] .

subnetworks SI & S2 as shown in figures 2a and 2b respective- ly. The Jasmon procedure [l] generates the following basic minimal paths of S1 & Sz:

S1 SZ

1. S-10-1-8-2-3 1. 3-t 2. S-7-5-1-8-2-3 2. 4-6-9-t 3. S-10-1-8-2-4 3. 5-6-9-t 4. S-7-5-1-8-2-4 5. S-10-1-5 6. S-7-5

Figure 1. A General Network Configuration

Fig. 2a Fig. 2b

Figure 2

Combining the paths of of SI & Sz having the same end- vertices, as in [ 11, the basic minimal paths of S are:

PI: S- 10- 1 - 8 -2 - 3 --t

P2: s-7-5-1-8-2-3-t

P4: s -7 -5 - 1 - 8 -2 -4 -6-9 - t

0018-9529/93/$3.00 01993 IEEE

60 IEEE TRANSACTIONS ON RELIABILITY, VOL. 43, NO. 1, 1994 MARCH

Ps: S-10-1-5-6-9-t Clarification Pa: S-7-9-6-5-t

Ghauth Jasmon, Member IEEE This procedure misses the following basic minimal paths: University of Malaya, Kuala Lumpur

P7: S- 10-1 -5-6-4-2-3- t

Pg: S-7-5-6-4-2-3-t. I have studied the comments by Prasad, Sankar, Rao. There is some confusion arising out of [ 11. In order to clear-up the situation, a revised Algorithm B is presented; it is the same as in [l], except that steps 6-9 have been added.

As a result of this, when the procedure of Jasmon & Kai [2] is used on P1 - P6, the following -

a. minimal node cut sets are missed: (1 4 9) and (4 8 9}, b. sets which are not minimal node cut sets, are reported as Algorithm B (Modijied) cut sets: (1 9) and (8 9}.

DISCUSSION

In the example, 3 nodes were split.

Lemma: If only 1 or 2 nodes are split, the Jasmon procedure [l] does not miss any basic minimal path.

Proof

Notation

P any basic minimal path vi,vj two vertices that are split to decompose the network

S into S1 & S, Pi portion of P lying in Si, i = 1,2 P‘ the portion of P which does not lie on P1 or P2 V f vertex adjacent to vi on P‘ V ” vertex adjacent to v f lying on P

The Jasmon procedure gives P1 & P2. Therefore P is obtained by [l], provided P = P1 U P2. If both P1 & Pz have the same end vertex, then it is evident that P’ is null. Therefore, let P1 end in vi, and Pz end in vi. v’ must lie in S2. Otherwise PI does not end in vi. v” cannot lie in S1, because P can re-enter only through vi. Therefore V” lies in S,. By the same argument the next vertex adja- cent to V ” lying on P must also lie in S,. Proceeding like this, the basic minimal path of P between v’ and vj can be obtained. Since v‘ lies in S,, it can be appended to P2 showing that P2 does not end in vi, which is a contradic- tion. Therefore, P‘ must be null. Q. E. D.

In this context, consider the following possibility. Let [s ,..., vj ,... vi] be a basic path in SI. Similarly let [t ,..., v)... ,vi] be a basic minimal path in S,. These two can not be combined to get a basic minimal path as it contains a loop. Therefore, when basic minimal paths of S1 are com- bined with those of S2, the resulting subgraphs might not be even minimal paths.

1. Treat the source node in S1 as a sink and nodes 3, 8, 10 as source nodes. This means that the path tracing algorithm of [2] is used except that tracing must continue to all the source nodes and not terminate when a source node is encountered.

2. Treat the nodes 3, 8, 10 as source nodes. 3. The minimal path trees for S1& S2 are obtained using

121. 4. All the paths of S1 ending with the nodes 3 , 8 , 10 are

joined with the paths of S2 ending with the same number respec- tively. For example the path 1-2-3 of S1 is joined with path 3-4 of 52 and so on.

5. From the set of paths in step 4, eliminate those paths that do not fulfill the requirements of a basic minimal path as outlined in [l, section 21.

6. a. Starting with one subnetwork, say S1, treat one par- titioning node as a source and the other partitioning nodes as sinks and find the basic minimal paths [3]. Repeat this process for the other partitioning nodes.

b. Repeat step 6a for subnetwork 52. 7. Paths obtained in step 6 are then combined. 8. Paths obtained for S1 and 52 in step 3 are now com-

bined with paths in step 7 to obtain paths with the original sink/source nodes.

9. Repeat step 5 for all paths obtained in step 8. a

REFERENCES

[l] G. B. Jasmon, “Cut set analysis of networks using basic minimal paths and nehvork decomposition”, IEEE Trans. Reliability, vol R-34, 1985

[2] G. B. Jasmon, 0. S. Kai, “A new technique in minimal path and cut set evaluation”, IEEE Trans. Reliability, vol R-34, 1985 Jun, pp 136-143.

Oct, pp 303-307.

(continued on page 64)

IEEE TRANSACTIONS ON RELIABILITY, VOL. 43, NO. 1, 1994 MARCH

Y. Hochberg, A.C. Thamane, Multiple Comparison Procedures, 1987; John Wiley * Sons. H.J. Chen, K. Vanichbuncha,“Simultaneous upper confidence bounds for distances from the best two-pammter exponential distribution”, a m - mun. Stutist.-7heory Meth, vol 18, 1989, pp 3019-3031. A.C. Cohen, B.J. Whitteny, Parametric Estimtion in Reliability and Life Spun Models, 1988: Marcel Dekker. N. Balakrishnan, A.C. Cohen, Order Statistics and Inference, 1990; Academic Press.

AUTHORS

Dr. Narinder Kumar; Dept. of Statistics; Panjab University; Chandigarh - 160 014 INDIA.

Narinder Kumar is a research student and a faculty member in the Dept. of Statistics, Panjab University, chandigarh. He received his MSc (1988), MPhil (1989), and PhD (1994) from Panjab University. His research interest is in ranking & selection problems.

Dr. Neeraj Misra; Dept. of Statistics; Panjab University; Chandigarh - 160 014 INDIA.

Neeraj Misra is a faculty member in the Dept. of Statistics, Panjab Univer- sity, Chandigarh. He received his MSc (1985) and PhD (1990) from Indian Institute of Technology, Kanpur. His research interests include ranking & selec- tion, and related estimation problems.

Dr. Amar Nath Gill; Dept. of Statistics; Panjab University; Chandigarh - 160 014 INDIA.

Ampr Nath Gill is a faculty member in the Dept. of Statistics, Panjab University, Chandigarh. He received his MSc (1984), MPhil(1985), and PhD (1990) from Panjab University, Chandigarh. His research interests include rank- ing & selection problems, and reliability.

Manuscript TR91-048 received 1991 April 4; revised 1991 October 29; revised 1991 April 1; revised 1992 December 9.

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Comment On “Cut Set Analysis of Networks Using Basic Minimal Path and Network Decomposition”

(Continued from page 60)

AUTHORS

Dr. V. C. Prasad, Professor; Dept. of Electrical Engineering; Indian Institute of Technology; Haw Khas, New Delhi-110016 INDIA.

V. C. Prasad took Bachelor’s in Electrical -ring in 1968, Master’s in 1970, and PhD in 1976. His intemrs are Computer aided design, graph theory, and reliability.

V. Sankar, Assistant Professor; Dept. of Electrical Engineering; College of Engineering; Anantapur, INDIA.

V. Sankar obtained his Bachelor’s in Electrical Engineering in 1978 and Master’s in Power Systems in 1980. He is working for his PhD in I.I.T. Delhi. His interests are power systems and reliability.

Dr. K. S . Prakasa Rao, Professor; Dept. of Electrical Engineering; Indian In- stitute of Technology; New Delhi-1 10016 INDIA.

K. S. Prakasa Rao obtained his Bachelor’s in Electrical Engineeering in 1964, Master’s in Power Systems in 1966, and PhD in 1974. His interests are various aspects of power systems, including reliability.

Dr. Ghauth Jasmon; Dept. of Electrical Engineering; University of Malaya; 59100 Kuala Lumpur, MALAYSIA.

Ghauth Jasmon (S’81, M’82): For biography, see [l].

Manuscript TR91-184 received 1991 October 18; author reply received 1992 February.

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