Research note

Capacity assignment in computer communication networks with unreliable links Debashis Saha*, Amitava Mukherjee t and Sourav K Dutta*

This paper considers the problem of selecting the optimum capacities of the links in a computer communication network which employs unreliable links. Given the nodes, links, link probabil- ities, grade of service and cost functions of the network, the objective of this problem is to find the optimum link capacities that minimize the network design cost, subject to the constraint equation involving the grade of service. This is essentially a combinatorial opti- mization problem. A general methema- tical model for this problem is formulated and a set of feasible solu- tions is obtained using Lagrangean relaxation and subgradient optimization techniques. A simulation study has been performed to verify the model, and favourable results obtained for a variety of nontrivial networks.

Keywords: capacity assignment, unreliable links, optimization

This paper describes the network design problem with reliability con- straints where, given a finite set of nodes, links and link reliabilities, the objective is a cost-efficient selection of link capacities suffi- cient to satisfy the desired grade of service defined in terms of node-to-

*Department of Computer Science and En- gineering, Jadavpur University, Calcutta 700 032, India ~Department of Electronics and Telecommu- nication Engg, Jadavpur University, Calcutta 700 032, India Paper received." 2 May 1993; revised paper received." 6 July 1993

node traffic in normal and failed conditions.

A computer communication net- work (CCN) is a set of geographi- cally distributed autonomous computers connected by communi- cation links to exchange inter-com- puter messages. The topological design of a CCN is a long-standing optimization problem which has yet to be solved in its entirety, despite many partial solutions being made available in the literature 1'2. The objective of topological design of a CCN is, in general, to achieve a specified performance at a minimal cost. The design assigns the links as well as their capacities for connect- ing the network nodes and has several performance and economic implications. This is a very critical phase of network synthesis, as the expected network behaviour with respect to routing, flow control, reliability, survivability, etc. de- pends largely upon the network topology derived.

A variation of the topological design problem with reliability con- straints is one where links have non- zero failure probabilities, which pre- vents them from operating uni- formly at their maximum capacities. Problems such as these have not been addressed by researchers in the recent past 47. A variety of mathematical formulations of the problem with exhaustive and heur- istic algorithms to solve it have been developed with only partial success.

However, none of them, except Gavish et al. 4 has taken the relia- bility measure explicitly into ac- count, and Gavish 4 too only considered the reliability constraints as part of their formulation for fibre optic circuit networks. The formula- tion developed in this paper involves reliability constraints explicitly through the grade of service which is a direct function of the link failure probabilities.

The model used is based on the following arguments. During its operational phase, every CCN passes through a number of states which are characterized by sub- capacity (below full capacity) traf- fic flow through the links. The links are forced to operate under sub- capacity conditions for several rea- sons, including node and/or link failures (i.e. change in topology), network congestion, buffer over- flow, packet loss or duplication, an inefficient protocol, a bad routing strategy, etc. We account for all these factors leading to degrada- tion in network performance by specifying the link reliability which gives us the extent to which a link can be utilized to its maximum capacity over a period.

P R O B L E M D E F I N I T I O N

In general, the formulation of a topological design problem corres- ponds to different choices in perfor-

0140-3664/94/12/0871-05 ~ 1994 Butterworth-Heinemann Ltd computer communications volume 17 number 12 december 1994 871

Capac i t y ass ignmen t in ne tworks wi th un re l i ab le l inks: D Saha et al.

mance measures, design variables and constraints. In this paper, the following general formulation of the problem is selected:

Given

total number of network nodes (M) and links (N) total capacity of the network (C bit/s), coefficient vector (7~) of the de- sign cost, i = i ...... N reliability @3 of each link total effective capacity of the net- work under link failure (C bit/s)

Variables

each link capacity c~ (i = 1 ...... N)

Performance constraints

The grade o f service (g) of the ne twork is def ined as g = 1 - (C/C). The actual ne twork ca- paci ty which can be suppor t ed by the final design must be within (1 - g ) % of C.

Objective

To obtain a minimum cost net- work which satisfies the given conditions and constraints.

M A T H E M A T I C A L F O R M U L A T I O N

Based on the above discussion, the problem can be mathematically re- presented as follows:

N

Minimize: G = E (Ci]'i)~' i - I

where ~ is the computational cost parameter (~ > l):

U

Subject to: C = ~ C ~ i - I

N

C = ~ (Cp3 i=I

g = I - ( c / o

From the above expressions we obtain the Lagrangean relaxation of the problem:

L(2) = G + 2t (ZCip~ - C)

+ 2 2 ( E C i -- C)

+23(1 -- ( C / C ) - g)

Next we describe the sub-gradient optimization technique in the fol- lowing way.

We consider that c~(2) can be an optimal solution to the Lagrangean problem for a fixed vector 4. This vector 2 is a Lagrangean multiplier. We define the subgradient vectors with their coordinates as:

~1(2)= ~-'~ci- C iGN

' /22(4) = Z ciPi -- -C iEN

~3(2) = (l - c / c - g)

It has been shown m that the La- grangean multipliers are updated using the following expression:

~ k + l z z = 2 ~ + t k p ~ ( / = 1, 2, 3)

It has also been shown m that 2 k z converges to 2", provided that lim tk ---+ 0 and ~ ~. E~':0t /

Since an infinite series over ti is generated which is not applicable for a solution, an alternative expres- sion is given by Held et al. TM 12.

tk : 6,,(G - t ( ,~k)) / [I ///" II 2

The 6,. is a scalar quantity. Held et al.tl, 12 showed that if conditions (i) O< 6,~<2 and (ii) G~<L(2*) are satisfied~3 then we can write 2~ ~ 2*. Further, when G >/L(2*), then tk ~ 0.

The subgradient optimization procedure 9 invokes the following steps:

Step 1 Initialization Set G to an arbitrary large value Select an initial set of multipliers 4 0

Set the iteration counter k to 0 and the improvement counter to 0 Initialize 4" to 2 o Set the current best value of L(2) to 0 Set the stepsize 6 to 3 °.

Step 2 Soh, ing the Lagrangean relaxation

(i)

(ii)

The improvement and itera- tion counter are incremented by I. The Lagrangean relaxation problem using 2 k as the La- grangean multipliers is solved and then L(2k), c~ are found.

Step 3 Updating the parameters (i) If L(2*) is greater than the

current best value of L(2) then L(2) is replaced by L(2 k); set 4" t o 2 k and the improvement counter is reset to 1.

(ii) If c~ is feasible to the mini- mizing problem, then its as- sociated objective function for this minimizing problem is computed. If this value is less than the current value of G then G is set to this value. If the improvement counter has reached a prespecified upper limit then set a ---, 6/2 and the improvement coun- ter is set to 0 and again it is started from step 2(i). If iteration counter > a pre- specified limit, or if 3 < a prespecified limit, or if tk < a prespecified limit, or i f ( G - L(2*))/L(2*) < a pre- specified error tolerance, then the process terminates.

(iii)

(iv)

Step 4 Updating the multipliers We compute the new subgradients as follows:

i EN

~/12(2k) = Z ciPi -- -C iEN

i~3(2 ~) = (1 - C / c - g )

Compute the new stepsize as:

tk = a~ x (G - C(2 ~ ) ) / 1 1 / II 2

Compute the new multipliers:

k+J = m i n ( - I , 2 ~ + t t p ~ ) , /

l = 1,2,3

S e t k ~ k + !

Step 5 Go to step (2).

872 c o m p u t e r c o m m u n i c a t i o n s v o l u m e 17 n u m b e r 12 d e c e m b e r 1994

It has been found that the solution obtained by this procedure con- verges within a few hundred itera- tions to a value which is very close to L(,F). The rate of convergence and quality of bound generated depends to a large extent upon the limits set on the improvement and iteration counters.

Solving the Lagrangean relaxation

The Lagrangean relaxation can be rewritten:

L(a)= Z (iTi) Jr •1 Ci -- C iEN

The solution to this problem is as follows:

cl = [ ( - 21pl - 22)/~7~] t//~-I/

and

' ( ( ci= ( i /~7~))~1Pi--22-- ")'3

where i >~ 2.

t / (=- t)

The computational efficiency of the subgradient optimization proce- dure depends upon the solution of the Lagrangean relaxation problem. This is generated in Step (2ii) of the subgradient procedure.

We have solved the proposed formulation for various sets of data to illustrate the applicability of the technique to a number of non-trivial problems.

C O M P U T A T I O N A L E X P E R I M E N T S

This section deals with a set of design problems to verify the pro-

Capacity assignment in networks with unreliable links: D Saha et al.

Table l Problem descriptions

Prob&m Nodes A r c s Totalcapacity Totaleffective capaci~

1 4 4 18,000.0 13320.0 2 4 4 18,000.0 7200.0 3 12 18 96,000.0 55656.0 4 12 18 96,000.0 42288.0

gram, as well as to study the effects of various parameters on the net- work design. Two problems are generated randomly based on the number of nodes and arcs and the link capabilities. The test networks are described in Table 1 and shown in Figures I and 2. The link reliabil- ities are written against the link capacities in Tables ~ 5 . For exam- ple, the problem 1 (see Table 1) is formulated as:

Given N = 4 , C=18 ,000bi t / s , [7i] : 1 .0 , for simplicity's sake, and [Pi] = [0.9, 0.7, 0.5, 0.1] Performance constraints g ~< 0.3, i.e. C ~> 13,320 bit/s

The computational result is:

[Ci]= [7364.79, 5919.89, 4474.99, 1585.19]

Closer scrutiny of the link capacities reveals that the link capacity is

4 3

0 3 0

4 2

1 0" 0 1 2

Figure 1

directly proportional to the link reliability, i.e. (Ci/Pi) = c o n s t a n t .

This observation can be further substantiated by the graph shown in Figure 3. This result has been vindicated by the even larger design problems viz. problem 3 4 in Ta- blel. The cost function is of the form ~(Ci ' / i ) ~. Table 1 is supple- mented by Figures 1 and 2, where problems 1 and 2 correspond to Figure 1, and problems 3 and 4 to Figure 2.

E F F E C T S O F L I N K S U R V I V A B I L I T Y

To study the effect of the prob- ability of link failures on the capacity requirements, various net- work topologies with differing link reliabilities have been tested using the program outlined in Saha et al. 8. The results of the test runs show that a link with a higher chance of failure should be as- signed less capacity to avoid degra- dation in the overall grade of service. This implies that less traf- fic should be encouraged through a failure-prone link, which gives an important clue to the network designers. The cost, however, in- creases as the total network relia- bility (EpJ decreases. (Details can be found elsewhere8'9.)

8 7 5 4

.O ,3 O. 9 0 811-0

oo/r,, ,u ,.

/ \ O" ,7 "0 12 11

2

3

Figure 2

computer communications volume 17 number 12 december 1994 873

Table 2 Table 4

Problem 1 Node = 4 arc = 4 C = 18000.0 C b a r = 133200.0 c[1] = 7364.799-998 p[1] = 0.900000 c[2] = 5919.899998 p[2] = 0.700000 c[3] = 4474.999998 p[3] = 0.500000 c[4] = 1585.199998 p[4] = 0.100000 Minimum cost = 1.11824e + 08

Table 3

CONCLUSION

Problem 2 Node = 4 arc = 4 C = 18000.0 C bar = 7200.0 c[l ] = 4814.800-000 p[ l ] = 0.900000 c[2] = 4169.900000 p[2] = 0.700000 c[3] = 3525.000000 p[3] = 0.500000 c[4] = 2235.200001 p[4] = O. 100000 Min imum cost = 5.79921e + 07

Problem 3: C = 96000.0 C_bar = 55656.0 c[l] = 6001.499959 c[2] = 4974.174959 c[3] = 5764.424959 c[4] = 5448.324959 c[5] = 5764.424959 c[6] = 4420.999959 c[7] = 4737.099959 c[8] = 7028.824959 c[9] = 8530.299959 c[10] = 4025.874959 c[l 1] = 4974.174959 c[l 2] = 3788.799959 c[13] = 3630.749959 c[14] = 5369.299959 c[l 5] = 6554.674959 c[16] = 8530.299959 c[17] = 4737.099959 c[18] = 2208.299959 Minimum cost = 5.61421e + 08

Node = 12 a rc= 18 p[l] = 0.600000 p[2] = 0.470000 p[3] = 0.570000 p[4] = 0.530000 p[5] = 0.570000 p[6l = 0.400000 p[7] = 0.440000 p[8] = 0.730000 p[9] = 0.920000 p[lO] = 0.350000 p[11] = 0.470000 p[12] = 0.320000 p[131 = 0.300000 p[l 4] = 0.520000 p[l 5] = 0.670000 p[16] = 0.920000 pll 7] = 0.440000 p[18] = O. 120000

Table 5

I n t h i s r e s e a r c h n o t e , a d e s i g n

t e c h n i q u e f o r C C N s w i t h n o n - z e r o

l i n k f a i l u r e p r o b a b i l i t i e s is c o n s i d -

e r e d . T h e n e t w o r k d e s i g n p r o b l e m

u n d e r r e l i a b i l i t y c o n s t r a i n t s h a s

b e e n f o r m u l a t e d m a t h e m a t i c a l l y a s

a n o n l i n e a r p r o g r a m m i n g p r o b l e m

w h i c h c a n b e s o l v e d b y t h e L a g r a n -

g e a n - b a s e d s u b - g r a d i e n t o p t i m i z a -

t i o n t e c h n i q u e . T h e u n r e l i a b l e l i n k s

m o d e l n o t o n l y t h e f a i l u r e o f t h e

l i n k s t h e m s e l v e s , b u t a l s o r e p r e s e n t

t h e f a i l u r e o f t h o s e n e t w o r k c o m p o -

n e n t s w h i c h m a y d i r e c t l y o r i n d i r -

e c t l y l e a d to a n i n c r e a s e in t r a f f i c

i n t e n s i t i e s , s l o w e r t r a n s m i s s i o n

s p e e d s , l o w e r t h r o u g h p u t a n d a fa l l

in e f f e c t i v e n e t w o r k c a p a c i t y . U n r e -

l i ab l e l i n k s c a u s e p a c k e t s t o b e

r e c e i v e d in e r r o r , o r l a t e o r n o t a t

al l d u e to u n a c c e p t a b l e d e l a y , c o n -

g e s t i o n , b u f f e r o v e r f l o w , h i g h ca l l -

b l o c k i n g p r o b a b i l i t i e s , a n d so o n .

T h u s , in a d d i t i o n to t h e l i n k c a p a -

c i t i e s , t h e e f f e c t o f l i n k p e r f o r m a n c e

d e g r a d a t i o n m u s t b e t a k e n i n t o

a c c o u n t in a n e t w o r k d e s i g n b y

m e a n s o f l i n k f a i l u r e p r o b a b i l i t i e s .

T h e d e s i g n w a s g e n e r a l i z e d b y

a s s i g n i n g d i f f e r e n t p r o b a b i l i t i e s t o

d i f f e r e n t l i nks , a n d a n a n a l y t i c

e x p r e s s i o n w a s p r o p o s e d f o r t h e

e x p e c t e d g r a d e o f s e r v i c e f o r a

g i v e n n e t w o r k t h a t r e f l e c t s t h e

e f f e c t o f b o t h c a p a c i t y a n d f a i l u r e

p r o b a b i l i t y o f t h e c o m m u n i c a t i o n

l inks . T h e s u b g r a d i e n t o p t i m i z a t i o n

893 .4998

Problem 4: C = 96000.0 C bar = 422888.0 c[1] = 5061.500-001 c[2] = 4346.175001 c[3] = 4896.425001 c[4] = 4676.325001 c[5] = 4896.425001 c[6] = 3961.000001 c[7] = 4181.100001 c[8] = 5776.825001 c[9] = 6822.300001 c[10] = 3685.875002 c[ 11] = 4346.175001 c[12] = 3520.800002 c[ 13] = 3410.750002 c[ 14] = 4621.300001 ell 5] = 5446.675001 c[l 6] = 6822.300001 c[17] = 4181.100001 ell 8] = 2420.300002 Minimum cost = 4.04823e + 08

N o d e = 12 a rc= 18 p[l] = 0.600000 p[2] = 0.470000 p[3] = 0.570000 p[4] = 0.530000 p[51 = 0.570000 p[6] = 0.400000 p[7] = 0.440000 p[8] = 0.730000 p[9] = 0.920000 p[lOl = 0.350000 p[111 = 0.470000 p[ l 21 = 0.320000 p[131 = 0.300000 p[141 = 0.520000 p[l 5] = 0.670000 p[l 6] = 0.920000 p[17] = 0.440000 p[181 = 0.120000

780 .0998

. N

'~ 866.6998

U L 553.2997 <

439 .8997

I 0 .10 0 .26 0 . 3 4 0 .42 0 .50 0 .59 0 . 6 7 0 .75 0 .83 0 .91 0 . 9 9

P r o b a b i l i t i e s

Figure 3

C a p a c i t y a s s i g n m e n t in n e t w o r k s w i th u n r e l i a b l e l inks: D S a h a et al.

874 c o m p u t e r c o m m u n i c a t i o n s v o l u m e 17 n u m b e r 12 d e c e m b e r 1994

Capacity assignment in networks with unreliable links: D Saha et al.

technique is then used to obtain a minimum cost assignment which satisfies the design specifications. Computational results indicate good performance of the algorithm that converges quickly to a subopti- mal solution.

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