Optimum topological layout of communication networks

M~'roelec~'on. Relh~b., VoL 29, No. 3, pp. 387-391, 1989.

0026-2714/8953.00 + .00 © 1989 Pergamon Press plc

Printed in Great Britain.

OPTIMUM TOPOLOGICAL LAYOUT OF COMMUNICATION NETWORKS

F . BEIGHELT a n d A. STARK

Department of Mathematics, University of Information Electronics Mittweida, 9250 Mittwelda, German D. R.

ABSTRACT The t o p o l o g i c a l b a s i c s t r u c t u r e undirected ~aph withunreliable

o f a c o m m u n i c a t i o n n e t w o r k i s g i v e n by a n e d g e s . The r e l i a b i l i t y of the network is

t o be i n c r e a s e d by s u c c e s s i v e l y i n c r e a s i n g t h e r e l i a b i l i t y o f e d g e s . The paper presents an effective algorithm for obtaining maximum network reliability under a cost constraint. The algorithm is applied to the redundance allocation problem. Numerical examples are presented.

KEYWORDS Network; reliability; topological structure; cost constraint; llnk allocation INTRODUCTION Let G = (V,E)

be a n u n d i r e c t e d

connected graph with the node set V =

(1,2t...,m) and the edge set E = (elte2,...,en). B~ definition, each edge is a 2-point subset of V. G is assumed to be free of loops. The graph G determines the topological basic structure of the communication network under consideration. Assumptions q. The edges are unreliable. The state of edge ei can be ch-racterized by a Boolean variable X i (Xi = I when ei operates, X i = 0 when ei is failed). 2. The random variables Xq, X2~ -.., X n are statistically independent. 3- The nodes are absolutely reliable. By G and these assumptions a stochastic network N is defined. In this paper~ N is interpreted as a communication network. There are several useful reliability criteria for Np for instance the two-termi~Ll reliability and the overall reliability. The two-terminal reliability with respect to two n o d e s s a n d t - d e n o t e d by P ( s p t ) - i s t h e p r o b a b i l i t y t h a t a comm,m ~ c a tion between s and t is possible, that means that a path between s and t exists which consists only of operating edges. The overall reliability denoted by P(VIV) - is the probability that a communication between all -

node pairs (s,t) is possible. Let Pi = P(Xi = I) be the reliability (more exactly$ availability) of •i. Then the reliabilit7 R of N is a function 387

388

F. BEICHELTand A. STARK

of p = (pl,p2,...0Pn): R = R ( p ) . An e f f e c t i v e m e t h o d f o r c o m p u t i n s R ( p ) presented in (Beichelt and SproB, 1987)s see also (Beichelt, 1988) for further relevant references.

is

Problem The network reliability R(p) is to increase b7 g/adually increasing edge reliabilities. Let pi(mi) be the reliability of e i after m i steps, Pi = Pi(O). Then the reliability increase of e i due to step mi+1 is di(ml) = Pi(mi+1) - Pi(mi)P m i = 0 , 1 1 0 . o Let further m = (ml0m21...tmn) be the "step number vector" and p(m) = (Pl(ml)0P2(m2)0...,pn(mn)) be the "element r e l i a b i l i t y vector',. hen a step n er veotor ': i s to find so that

R(p(~))

:

~(p(~)), (I)

n

~

mic i ~ 0,

where c i is the cost for increasing the reliability of e i in a step, and C is the total cost which can be spent for increasing the network reliability (c i is assumed not to be dependent on mi). ~athematical methods for exact solving problems of type (1)are surveyd by Tillmenn et al. (1980). However, it easy to see that optimization problems of this type belong to the class of NP-complete problems (Bulfin and Liu, 1985). This implies that it is highl~ unlikely that there exist exact solution algorithms with computation times that increase less than exponentially with problem size. Hence the development of algorithms for the approximatlve solution of (I) is Justified when these algorithms require polynomially computation times. In what follows, such a "heuristic" algorithm is presented. As demonstrated by Stark (1987), this algorithm is from the computatlonally point of view generally more effective than known ones (Aggarwal et al., 1982 a,bl Dale and Winterbottom, 1986). The approach used can easily be generalized to the case when there is more than one constraint. SOLUTION PROCEDURE The approximate solution of problem (q) is carried out in stages. Each stage includes the l-step increase of the reliability of one edge. The edges will generally change from stage to stage. If a step number vector m is realized and in the subsequent stage the reliability of • i is increased, then let Di(P(m)) be the corresponding increase of network reliability. Let further

%(p(m)) = _ -

max

i=~2,...t n

Di(p(m))/c i - _

For given m, qk(p(m)) is the maximum possible gain of network reliability per unit cost in the subsequent stage. This gain is obtained, if in this stage the reliability of ek is increased. The consequent application of this approach leads to the followingalgorit hm. For being a n o b v l o u s compromise between reliability and cost this algorithm is expected to yield

Topological layout of networks

goodapproximative

solutions

of

(1).

Its

application

389

requires

the

knowledge

of R(~) for an~ D, G, Pi' di(mi)' and oil i = 1,2,...,n! m i = 1,2,... AiKorithm 1.

Initialize m i = O,

i = q,2,...,n!

Pi(mi ) = Pi' i = 1,2,...,n! = (mq,m2,...,mn),

= (pI(=I), p2(m2), ..., pn(~)), = (d~(=1+O, d2(~+I),..., ~ ( ~ + I ) ) .

d(p)

2. Let I = (i! c i ~ C, 1 i = n). If I = ~, go to 9. 3. Gompute qi(p) for all i e I. 4.

Find

a number k with

the

property

qk(p ) = m a x qi(p ). 5.

iEI

-

Let m k = mk+q.

e. ~et pk(mk) = pk(mk_1) + dk(mk), C = C-c k7. Replace in p the kth component Pk(mk-q) by Pk(mk)" Replace in d(p) the kth component dk(mk) by dk(mk+q). 8 . Go to 2 . 9. An approximative solution of (1) is found. It is given by the actual step number vector m. Stop. 10. OPTIMUMPARALLRLREDUNDANCEALLOCATION A possible application of the algorithm is the redundance allocation problem: Each edge e i = (u,v) ma~ consist of one or more direct links between the nodes u and v. The reliability increase of • i in a step is realized by installing a further llnk between nodes u and v. A direct commumication between u and v is possible if and only if at least one llnk of e i operates. Therefore, each edge behaves as a parallel system from the reliabilit7 point of view. Any llnk of e i has the reliability Pi and its installation gives rise to cost ci, i = 1,2,...,n. In what follows, it is assumed that at the beginning each edge consists of exactly one 14nk and that all links operate independently from each other. Hence, mi+q Pi(mi) = 1 - (1 - pl ) , mi di(mi) = pi(1 - pi ) • Problem

(1)

to

edge

each

bility

MR

29 : 3 - G

is

ma~ now b e f o r m u l a t e d so that

obtained~

under

a given

as follows: cost

How man~ l i n k s

constraint

must

maximum n e t w o r k

be added relia-

390

F. BEICHELTand A. STARK

Example The reliability of the network given in Fig. 1 is to increase by allocating additional parallel links as described above. Table 1 contains the corresponding llnk reliabilities Pi and the cost c i. e3

4

e4

I

5

3 Fig. 1. Example network with 5 nodes and 7 edges. Table 1. Numerical parameter of the example network i

1

2

3

4

5

6

7

Pi cA

0.80

0.50

0.85

0.75

0.75

0.80

0.85

4

4

5

3

2

4

5

Applying the algorithm yields nearly optimum parallel llnk allocations with respect to the two-terminal reliability P(1,5) and the overall reliability P(V~V). Table 2 presents the results. In this table, m i is the number of additional lines installed in edge e i. The vectors m have been computed for several C-values. At first sight, the comparatively large ~ump of the reliability when 0 is increased from 3 to,4 ma~ be surprising. However, this is due to the high importance of e I with respect to the rellabillty criteria considered. Table 2. Application of the algorithm to the example network ~o-termine.l reliability 0

mI

m2

m3

m4

m5

m6

~

Overall reliability P(1,5)

mI

m2

m3

m4

m5

m6

m7

P(V~V)

o

q

q

q

q

q

q

q

0.8506

q

q

I

I

q

I

q

0.838?

3 g 6 10

q 2 2 2

I 1 1 1

q q 1 1

2 q 1 1

I q 2 2

q q q 2

q q q 1

0.8644 0.9248 0.9311 0.9548

q 2 2 2

q 1 1 1

I 1 q q

2 q 1 q

q 1 2 2

I 1 1 2

q 1 1 1

0.8571 0.9154 0.9282 0.9529

15 20 28

2 2 2

2 3 4

1 1 1

2 q q

q 3 2

2 2 2

1 1 2

0.9764 0.9842 0.9941

3 3 3

1 1 1

1 1 1

2 3 4

1 2 2

2 2 2

1 q 2

0.9777 0.9871 0.9942

53

2

9

1

1

2

2

3

0.9997

6

1

1

5

3

g

2

0.9998

Topologicallayout of networks

391

REYE~ Aggaz~al, E . K . , Y . 0 . Chopra and J . S . Bajwa ( 1 9 8 2 a ) . T o p o l o g i c a l l a ~ o u ~ s of l~n~s for optimizing the s-t reliability in a computer communication s~stem. Hicroelectr. Reliab., 22, 341-345. Aggarwal, K.K., Y.C. Chopra and J.S. BaJwa (1982b). Topological is~outs of links for optimizing ~he overall reliabilit7 in a computer comm-n4cation s~stem. Microelectr. Reliab., 22, 347-351. Beichelt, F. (1988). Zuverl~ssigkeit strukturierter STsteme (Reliabillt7 of complex s~stems). Verlag Technlk, Berlin. Beichelt, F. and L. SproS (1987). An improved Abraham-method for generating disjoint sums. IEEE Trans. Reliab., 36, 70-74. Bulfin, R.L. and Oh. Y. Liu (1985). Optimal allocation of redundant components for large s~stems. IEEE Trans. Rellab., 34, 241-247. Dale~ Oh. and A. Winterbottom (1986). Optimal allocation of effor~ to improve s~stem reliability. IEEE Trans. Reliab. t 35, 188-191. Gopal, L., K.K. Aggarwal and J.S. Gupta (1979). A new approach to optimal redundanc~ allocation for complex s~stems. Microelectr. Reliab., 19, 387-390 • Kontoleon, J.M. (1979). Optimum link allocation of fixed topolo~ ne@works. I~W~ Trans. Reliab., 28, 145-147. Stark, A. (1987). Zuverl~ssigkeitstheoretisohe Importanz yon Ubertragungskan~len in Kommunikatlonsnetzen (Reliabilit~ theoretic importance of transmission cbennels in communication networks). Diploma thesis. Univel~sit~ of Information Electromlcs Mittweida, Mittweida. Tillmann, F.A., Oh. Hwang and W. Kuo (1980). Optimization of S~stem Reliability. Marcel Decker, Inc., New York, Basel.