An Adaptive Routing Technique for Channel Switching Networks

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AN A D A P T I V E R O U T I N G T E C H N I Q U E F O R CIIANNEL S W I T C H I N G NETWORKS

D . E . B e l l A . B u t r i m e n k n

R e s e a r c h M e m o r a n d a a r e i n f o r m a l p u b l i c a t i o n s r e l a t e d t o o n g o i n g o r p r o j e c t e d a r e a s of re- s e a r c h a t I I A S A . T h e v i e w s e x p r e s s e d a r e t h o s e of t h e a s t h o r , and do n o t n e c e s s a r i l y r e f l e c t t h o s e of I I A S A .

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An A d a p t i v e R o u t i n g T e c h n i q u e F o r C h a n n e l S w i t c h i n g N e t w o r k s

D . E . B e l l A. B u t r i m e n k o

1 . I n t r o d u c t i o n

T h e p e r f o r m a n c e o f a c o m m u n i c a t i o n s y s t e m d e p e n d s g r e a t l y o n t h e t e c h n i q u e u s e d f o r f i n d i n g g o o d r o u t e s f o r t r a n s m i t t i n g t h e i n f o r m a t i o n . I n o r d e r t o i m p r o v e t h e r o b u s t n e s s a n d

r e l i a b i l i t y o f s u c h s y s t e m s , we p r o p o s e h e r c a d e c e n t r a l i z e d o r d i s t r i b u t e d t e c h n i q u e t o c o n t r o l t h e i n f o r m a t i o n f l o w s . P r o c e d u r e s o f t h i s k i n d h a v e a l r e a d y b e e n p r o p o s e d [ l , 2 , 3 1 ; t h e f i r s t t w o b e i n q o r i e n t e d t o c h a n n e l s w i t c h i n q ( o r c i r c u i t s w i t c h i n q ) s y s t e m s h a v e p e r f o r m e d v e r y w e l l c o m p a r e d w i t h n o n - a d a p t i v e r o u t i n g t e c h n i q u e s .

I t seems t h a t e v e n b e t t e r r e s u l t s c o u l d b e o b t a i n e d i f t h e c h o s e n c r i t e r i a w e r e i n c o r p o r a t e d d i r e c t l y i n t o t h e a l g o r i t h m . F o r e x a m p l e , n e i t h e r jl] - n o r :2], b o t h of w h i c h m i n i m i z e t h e p r o b a b i l i t y o f l o s s e s , d i r e c t l y u s e t h e a c l ~ i e v e d p r o b a b i l i t i e s a s c o n t r o l p a r a m e t e r s . T h e f i r s t i s b a s e d o n t h e s h o r t e s t

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

T h e r o u t i n q t e c h n i q u e t o b e d e s c r i b e d h e r e i s b a s e d o n t h e s a m e p r i n c i p l e o f d i s t r i b u t e d c o n t r o l a s i n [ I ] . W e

c o n s i d e r a n e t w o r k i n w h i c h e v e r y n o d e ( e x c h a n g e ) r e c e i v e s c o n t r o l i n f o r m a t i o n o n l y f r o m i t s n e i c j h b o u r i n y n o d e s . I n e v e r y n o d e

t h e r e i s s t o r e d a s p e c i a l r o u t i n g m a t r i x a n d e v e r y n o d e e s t i m a t e s c o n t i n u o u s l y t h e p r o b a b i l i t y o f t h e t r u n k s t o e a c h o f i t s

n e i g h b o u r s b e i n q b l o c k e d , t h a t i s , b e i n g u n a v a i l a b l e f o r t r a n s - m i s s i o n , w h i c h w i l l d e p e n d o n t h e c o n g e s t i o n i n t h e s y s t e m . T h e r o u t i n g m a t r i x a t n o d e i h a s e n t r i e s W . . k w h i c h

1 I

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r e p r e s e n t t h e e s t i m a t e d p r o b a b i l i t y o f a m e s s a g e r e a c h i n g d e s t i n a t i o n k f r o m t h e n e i g h b o u r i n g n o d e j. C l e a r l y , when r o u t i n g a m e s s a g e t o d e s t i n a t i o n k t h e n o d e j , t o w h i c h i t s h o u l d b e s e n t n e x t , i s t h a t w h i c h m a x i m i z e s W . . among t h o s e

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n o d e s j f o r w h i c h t r u n k ( i , j ) i s u n b l o c k e d . I f a l l t r u n k s a r e b l o c k e d , t h e m e s s a g e i s considered t o b e l o s t .

To s u m m a r i z e , t h e n , w e w i s h t o p r e s e n t a n a l g o r i t h m f o r s u c h n e t w o r k s w h i c h c o n v e r g e s t o t h e o p t i m a l r o u t i n g p o l i c y , u n d e r t h e a s s u m p t i o n s :

i ) o n l y l o c a l i n f o r m a t i o n i s a v a i l a b l e ; a n d

i i ) t h e b l o c k i n g p r o b a b i l i t i e s r e m a i n c o n s t a n t . (1.1) I n p r a c t i c e t h e s e a s s u m p t i o n s w i l l . b e f u l f i l l e d i f t h e t r a f f i c i s l i g h t , i n w h i c h c a s e t h e b l o c k i n g p r o b a b i l i t i e s w i l l b e a p p r o x i m a t e l y c o n s t a n t . I n o t h e r c a s e s t h i s p o l i c y may n o t b e o p t i m a l s i n c e t h e b l o c k i n g p r o b a b i l i t i e s w i l l d e p e n d p a r t i a l l y o n t h e r o u t i n q p o l i c y . The b e h a v i o u r o f t h i s a l g o r i t h m i n t h i s c a s e w i l l b e i n v e s t i g a t e d l a t e r b u t i t s e e m s l i k e l y t h a t i t w i l l y i e l d a qood i f n o t o p t i m a l p o l i c y .

2 . The A l g o r i t h m a n d I t s C o n v e r g e n c e 2 . 1 Example

L e t u s b e g i n w i t h a n e x a m p l e . I n F i g u r e 1 t h e r e i s a n e x a m p l e o f a n e t w o r k i n w h i c h t h e a i m is t o t r a n s m i t a c a l l f r o m n o d e 1 t o t h e d e s t i n a t i o n .

N O D E 2

NODE 1

NODE 3

FIGURE 1

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L e t P1, P 2 , P 3 b e t h e e s t i m a t e s a t e a c h n o d e o f t h e p r o b a b i l i t y o f s u c c e s s f u l l y t r a n s m i t t i n g a m e s s a g e t o t h e d e s t i n a t i o n . U s i n g c o n d i t i o n a l p r o b a b i l i t i e s we c a n s e e t h a t

Now i t i s c l e a r t h a t a t n o d e 1 i t i s b e t t e r t o t r a n s m i t t h e m e s s a g e t o n o d e 2 t h a n n o d e 3 s i n c e P2

.

P 3 , h e n c e

B u t s u p p o s e P 2 , P 3 w e r e n o t known e x a c t l y , b u t r a t h e r w e r e o n l y e s t i m a t e s . I f , i n o u r e x a m p l e , P 2 , P 3 w e r e c u r r e n t l y

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e s t i m a t e d t o b e 2 , r e s p e c t i v e l y , t h e p o l i c y a t n o d e 1 w o u l d b e t o t r a n s m i t m e s s a g e s t o n o d e 3 i n p r e f e r e n c e t o n o d e 2 , g i v i n g a new e s t i m a t e o f

a t n o d e 1..

W h a t w i l l b e p r o v e d h e r e i s t h a t f o r a l l n e t w o r k s t h e e s t i m a t e s o b t a i n e d i n t h i s way w i l l c o n v e r g e i f r e p e a t e d i t e r a t i v e l y , a n d t o t h e c o r r e c t s o l u t i o n , f r o m a n y s e t o f

i n i t l a 1 e s t i m a t e s . Incidentally, F i g . 1 i s a n a c y c l i c n e t w o r k a n d i t i s e a s y t o show t h a t s u c h a n a l g o r i t h m c o n v e r g e s f i n i t e l y i n n o m o r e t h a n n - 1 s t e p s ( n n o d e s ) ( u s e T h e o r e m 1 6 . 2 i n

t i a r a r y [4] )

.

T h i s a l g o r i t h m may b e r e p e a t e d o n c e f o r e a c h p o s s i b l e d e s t i n a t i o n s o we n e e d o n l y c o n s i d e r t h e c a s e w i t h o n e d e s t i n a t i o n .

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R e c a l l a l s o t h a t we a s s u m e t h a t t h e p r o b a b i l i t y p . . o f

1 I a r c ( i , j ) b e i n g u n b l o c k e d i s c o n s t a n t , i n d e p e n d e n t o f o t h e r a r c s a n d i n d e p e n d e n t w i t h r e s p e c t t o t i m e .

2 . 2 The A l g o r i t h m

L e t P = ( P l , .

. .

, P n ) b e t h e s e t o f n o d e p r o b a b i l i t i e s ( " o p t i m a l n o d e p r o b a b i l i t i e s " ) c o r r e s p o n d i n g t o a n o p t i m a l r o u t i n g p o l i c y ( i n t h e s e n s e o f a s s u m p t i o n ( 1 . 1 ) ) w h e r e Pi i s t h e p r o b a b i l i t y o f a m e s s a g e a t n o d e i b e i n g s u c c e s s f u l l y t r a n s m i t t e d t o t h e d e s t i n a t i o n a n d Pdestination i s a l w a y s e q u a l t o 1 . S u p p o s e t h a t a l l b u t Pk f o r some n o d e k a r e known a n d t h a t i t o n l y r e m a i n s t o f i n d P k . I t i s e v i d e n t t h a t t h e b e s t p o l i c y f o r a m e s s a q e a t k i s t o t r a n s m i t i t - t o t.he n o d e w h i c h , a m o n g s t t h o s e t o w h i c h t h e a r c s a r e u n b l o c k e d , g i v e s t h e b e s t o p p o r t u n i t y f o r r e a c h i n g t h e d e s t i n a t i o n . H e n c e , t h e p r o b a b i l i t y Pk may b e calculate^^

by t h e f o l l o w i n g r e c u r s i o n i n w h i c h we a s s u m e , w i t h o u t l o s s o f q e n e r a l i t y , t h a t P l

1

P 2

2

- ;, Pn :

F o r ~ o r n p ~ ~ c t n e s s , d e f i n e a v e c t o r F ( p , P ) o n a m a t r i x p o f a r c p r o b a b i l i t i e s a n d v e c t o r P o f n o d e p r o b a b i l i t i e s , h a v i n g c o m p o n e n t s

w h e r e

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T h e a l g o r i t h m c a n now b e s t a t e d q u i t e s i m p l y . B e g i n w i t h a n i n i t i a l g u e s s P O , 0 - < PO

-

^< 1 o f t h e n o d e p r o b a b i l i t i e s a n d p r o c e e d w i t h a r e c u r s i o n d e f i n e d b y

I t w i l l b e shown t h a t t h i s r e c u r s i o n c o n v e r g e s t o t h e o p t i m a l n o d e p r o b a b i l i t i e s f o r a l l i n i t i a l v a l u e s .

2 . 3 C o n v e r g e n c e

N o t e t h a t r e l a t i o n 2 . 2 . 1 may b e w r i t t e n

f o r some c o n s t a n t s a . w h i c h d e p e n d o n t h e p a r t i c u l a r p o l i c y k 1

s e l e c t e d . By o p t i m a l i t y

f o r a n y p o l i c y w h i c h g i v e s r i s e t o t h e a k j '

T h e o r e m 1. I f f o r t w o a r b i t r a r y v e c t o r s P , Q , P ( Q t h e n

P r o o f . F k ( p , P ) = C a P .

k j I f o r some a

j k j

< C a k j V j

- s i n c e P

2

Q

2

F k ( p I Q ) b y d e f i n i t i o n o f F

.

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C o r o l l a r y 1. I f PO

-

Q0 ( RO a r e t h r e e i n i t i a l v e c t o r s f o r t h e r e c u r s i o n t h e n P t _-< Q t

2

R~ f o r a l l t

2

^{0 .}

C o r o l l a r y 2 . I f pt _-< p t - I t h e n p t f l _-< pt a n d c o n v e r s e l y i f

t h e n

P r o o f . I f pt _-< p t - I t h e n b y t h e t h e o r e m

t h a t i s F ~ - < +p t ~

.

S i m i l a r l y f o r t h e o t h e r c a s e . C o r o l l a r y 3 . I f f o r a n y t

-

^{0 ,}^{p t}

^'

p t f l o r pt

2

P t + l

-

t h e n t h e s e q u e n c e c o n v e r g e s .

P r o o f . T h e s e q u e n c e i s a l w a y s b o u n d e d ( b y z e r o a n d o n e ) a n d , i f t h e c o n d i t i o n s o f t h e c o l l o r a r y a p p l y , i s m o n o t o n i c b y C o r o l l a r y 2 . Hence i t c o n v e r g e s .

C o r o l l a r y 4 . T h e r e c u r s i o n c o n v e r g e s f o r i n i t i a l v a l u e s

PO = 1 a n d P O = 0 ( w i t h Pdestination 0 = 1 ) .

P r o o f . C o r o l l a r y 3 a p p l i e s t o t h e c a s e t = 0 .

T h e o r e m 2 . T h e a l g o r i t h m w i t h i n i t i a l v a l u e s P O = 1 c o n v e r q e s t o t h e o p t i m a l n o d e p r o b a b i l i t i e s .

P r o o f . By C o r o l l a r y 4 t h e s e q u e n c e c o n v e r g e s . By C o r o l l a r y 1 t h i s s o l u t i o n i s a n u p p e r bound f o r a i l s o l u t i o n s a n d h e n c e i s a n u p p e r b o u n d f o r t h e o p t i m a l n o d e p r o b a b i l i t i e s . B u t b y u s i n g t h e p o l i c y i m p l i e d by t h i s s o l u t i o n t h e u p p e r b o u n d

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may b e attained. Hence the solution is the vector of optimal node probabilities.

Theorem 3. The algorithm with initial values PO = 0 converges to the optimal node probabilities.

Proof. Note that in this case pkt may be interpreted as the maximum probability of reaching the destination in at most t steps. Hence as t + the recursion yields the maximum

probability that the destination may be reached in an infinite number of steps, which is exactly the meaning of the optimal node probabilities.

C o r o l l a r y . The recursion converges to the optimal node probabilities for all initial values PO.

Proof. Theorems 3 and 4 show that with initial values 0 and 1 the algorithm converges t o the optimal node probabilities. Since 0 - PO - 1 for all initial values, Corollary 1 implies the

result.

3. Computational Experience

T o investigate the performance of this algorithm over a range of blocking probabilities, a simulation proqram was written (Appendix 1). The algorithm was applied to the network as shown in Fig. 2.

Three kinds of initial situation were considered:

a) pi0 = I vi b ) p i O = O Vi.

C) ?;O = R where R = random number 0 / R 1

L

except for destination nodes, whose destination p r o b ~ b i l i t ~ e s are constant and equal to 1.

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T h e p r o b a b i l i t y o f b l o c k i n g w a s s e t c o n s t a n t a n d c h o s e n a t d i f f e r e n t i n t e r v a l s .

I n F i g . 3 t h e d e p e n d e n c e o f t h e n u m b e r o f i t e r a t i o n s o n p r e c i s i o n i s r e p r e s e n t e d t a k i n g t h e i n i t i a l s i t u a t i o n ( a ) w h e r e p r e c i s i o n i s d e f i n e d a s

F i g s . 4 a n d 5 show s i m i l a r c u r v e s f o r i n i t i a l s i t u a t i o n s ( b ) a n d ( c ) r e s p e c t i v e l y . From t h e s e c u r v e s w e c a n see t h a t t h e s y s t e m o f e q u a t i o n s c o n v e r g e s v e r y r a p i d l y a n d t h a t t h e n u m b e r o f i t e r a t i o n s g r o w s a s a l o g a r i t h m o f p r e c i s i o n .

T h e r e s u l t s d e s c r i b e d a b o v e a p p l y t o t h e c a s e when a l l

t h e e q u a t i o n s a r e c h a n g e d i n e v e r y s t e p a n d a l l Pk t

,

k = 1 , 2 ,

...,

a r e a v a i l a b l e f o r t h e c a l c u l a t i o n s i n s t e p t + l . I f we t h i n k o f t h e p r a c t i c a l i m p l e m e n t a t i o n o f t h e a l g o r i t h m , i t c o r r e s ~ l o n d s t o t h e s y n c h r o n o u s mode o f c o n t r o l . T h i s m e a n s t h a t i n e a c h n o d e s o m e c a l c u l a t i o n s h a v e t o b e made a t e v e r y i t e r a t i o n t o u p d a t e Pk t

.

T h i s m u s t t h e n b e t r a n s m i t t e d t o a l l n e i q h b o u r i n g n o d e s a n d i s s t o r e d b y t h e m . T f n o t a l l t h e new r e s u l t s a r e k n o w n , i t i s n o t p o s s i b l e t o c a l c u l a t e t h e n e x t Pk t + l . A c t u a l l y , t h e d e s c r i b e d c a l c u l a t i o n s s h o u l d b e c a r r i e d o u t b y t h e s a m e e q u i p m e n t a s i s r e s p o n s i b l e f o r t h e s w i t c h i n g a n d t h e r e i s n o r e a s o n t o a s s u m e t h e s y n c h r o n i z a t i o n o f i t .

I n F i g s . 6 t o 8 r e s u l t s a r e shown f o r a c a s e when n o t a l l c a l c u l a t i o n s a r e made s i m u l t a n e o u s l y . An i t e r a t i o n o f t h e a l g o r i t h m c o n s i s t e d o f c h o o s i n g o n e n o d e a t r a n d o m ( e q u a l p r o b a b i l i t i e s ) a n d c a l c u l a t i n g i t s new n o d e p r o b a b i l i t y , w h i c h w a s t h e n a v a i l a b l e f o r t h e n e x t i t e r a t i o n . A s e q u e n c e o f n ( n u m b e r o f n o d e s o f t h e n e t w o r k ) o f t h e s e i t e r a t i o n s i s c o u n t e d a s a s t e p i n t h e f i g u r e s f o r c o m p a r i s o n w i t h F i g s . 3 t o 5 .

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L o o k i n g a t t h e c u r v e s we c a n s e e t h a t t h i s d e s y n c h r o n i - z a t i o n d o e s n o t a l t e r t h e s p e e d o f c o n v e r g e n c e v e r y much.

We a l s o c o n s i d e r e d a n o t h e r model w h e r e e a c h n o d e i m i s s e d t h e c a l c u l a t i o n w i t h p r o b a b i l i t y m T h i s model c o r r e s p o n d s

i '

t o t h e s y n c h r o n o u s mode, b u t some n o d e s a r e a l l o w e d t o m i s s c a l c u l a t i o n when t h e e q u i p m e n t i s n o t a v a i l a b l e f o r t h e j o b ; m i s t h e p r o b a b i l i t y o f n o n - a v a i l a b i l i t y o f t h e e q u i p m e n t .

i

I n c o n c l u s i o n , we d e s c r i b e some a d d i t i o n a l i n v e s t i g a t i o n s n e c e s s a r y t o c l a r i f y t h e a p p l i c a b i l i t y o f t h i s a l g o r i t h m .

1. The main aim o f t h e a l g o r i t h m i s t o make a communication s y s t e m more r o b u s t a n d a d a p t i v e t o t h e c h a n g i n g o f t h e l o a d i n g a n d n e t w o r k s t r u c t u r e . T h i s l e a d s i m m e d i a t e l y t o a v a r i a b l e p r o b a b i l i t y of t r u n k b l o c k i n g , pi j , w h i c h we h a v e c o n s i d e r e d a s c o n s t a n t .

Two p r o b l e m s now a r i s e :

a ) Measurement ( a n d p e r h a p s p r e d i c t i o n ) o f p . .

.

1 3 b ) I n w h a t way d o e s t h e f l u c t u a t i o n o f p . .

1 I a f f e c t t h e c o n v e r g e n c e o f t h e a l g o r i t h m . 2 . The n e x t s t e p f o r i n v e s t i g a t i o n s h o u l d b e t h e

s i m u l a t i o n o f communication c h a n n e l s w i t c h i n g n e t w o r k s c o n t r o l l e d by t h e h e l p o f t h i s a l g o r i t h m and by c o m p a r i s o n w i t h t h e c o n t r o l o f o t h e r s

( e . g . [I, 2 , 3 1 ) .

(13)

FIGURE 3 eps = 2-*

THE NUMBER OF ITERATION STEPS DEPENDS OIV eps

(14)

FIGURE 4

(15)

0 . 6 > p,, ? 0 3 - - - 0 . 4 > p , , 2 0 . 1

- -

_{0 . 2} _>_{p , ,}_{2 0}

X

-,x* I.

*

-

-.x-**

, x- /x- /x-

, x / - x -

F I G U R E 5

(16)

' .x

x'?"

-

¹ ^>^{P , ~}^{L O .}⁸

- ----.

,/ ^I

^0.9^>^{P , ~}

^20.6

t

^/ ^I

^{- -}

^0.7^>^{p . .}^I^I ^?^0.4

F I G U R E 6

(17)

-

1 > p . . 2 0 . 8

I J

---

_0.9 ^{- ,}

> p , , ? 0.6 ,2

X Y

F I G U R E 7

(18)

-

₁_>_{P , ~}₂_{0 . 8}

^/.

X

/

- - - 0.9 > p l j 2 0 . 6 /

/' X

--

0.7 > p . . L O . 4 /

/x/

^,

I J

,,.

^X^'

/ ,

F I G U R E 8

(19)

APPENDIX 1

Description of the Implemented Alqorithms

This proqram was used to test the converqence of the alqorithm.

It beqan with EPS beinq 1 , and if DIFA becomes less than EPS, then EPS becomes EPS/2 until a maximum of 100 iterations has been done or DIFA becomes equal to zero. To avoid numerical errors, the proqram was run in double precision and addition was done in an increasinq order.

For real time simulation this does not seem to be necessarv and thus the Droqram would be faster.

List of the most important variables:

Inteqers ITER:

ITERM :

KANAL :

KANALA :

KANALE :

KNOT :

KNOTZ :

NBKNOT:

NKANAL:

NKNOT :

NKNO'TZ :

Number of current iteration Maximum number of iterations Current channel

Number of the first channel of current node Number of the last channel of current node Nuntber of current node

Number of current adiacent node Wicihest node number

Number of channels Number of nodes

tiiqhest number of destination nodes.

Inteqer arravs

KN (60) : KN(1) number of node at the end of Channel I

M(17) : ~ ( 1 ) . M(I)

+

1 . M(I+l)

-

1 . numbers of the channels leavina node I.

Reals PB:

PBH:

Lower bound of PBLOK if created Unner bound of PBLOK if created.

(20)

Real arrav

PBLOK ( 60) : PBLOK(1) ~ r o b a b i l i t v of channel I beinq blocked

Double precision

DOSKN : Addition variable DIFA:

DIF:

Hiqhest difference between old and calculated value of PDOS (node.

destination) for all nodes and all destinations

Difference between old and new value of PDOS for current node and current destination.

Double precision array

PDOS(16.16.2): PDOS (node. destination. L) : ~ r o b a - bilitv of reachinq destination from node

PMEM B (16): workina arrav.

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I N P U T A N D I N l T l A L l S l N G EVERY THING

O U T P U T O F M , K N , P B L O K

yF+?

DIFA: 0

L = M D I T E R + l 2 ) - 1 KNOT= 1,NKN

,

K A N A L E = D I F A < E P S 7

, ^,

M [ K N O T + 1 1-1 O U T P U T

0 1 ~ 1 - P B L O K ( K n N A L ) PMEMB [ N B K N O T I = PDOSl N B K N O T ,

7---

K A N = K A N A L A

D I F A 1 0

o-@

S H E E T 1

(22)

D O S K N = 0

N B K N O T = K N ( K A N A L ) DOSKN =DOSKN +

PME MB(NBKN1

$ E mi . N

&

I

K N O T Z , L ) )

1

D I F >DIFA

PDOS(KN0T. KNOTZ, L N I = D O S K N

S H E E T 2

(23)

H 5 ~ t I ) CDC h 6 0 0 FTfd V 3 . 0 - P 3 5 5 O P T s l 0 9 / 1 2 / 7 4 P N Q f i H 4 ⁴ U S H E D ( I h l P U T . O U T P I I T . T A P F ~ = I N P U T v T A P F 2 = O U T P U T )

l ~ i . J ( l - L t P i i t C I S l U l u H t L t ' r P 1 * r ' 2

I ) i J l J t i l . t P N t C l S I O N P F A E M t ~ ~ O D ~ K h r D I F v D I F A ~ P O O S ~ F P S r D l t > l F ' l h " ' S I V \ I f i l u ( b 0 ) r C ( 1 I ) r P : ) O S ( L h r l h r ? ) v P M E M H ( 1 h ) r P I J L O K ( 6 O ) i 4 t n u ( 1 . 4 1 1 ) Pb3ePdt-l

Q l l k < J d M L a r ( ? P ? . O

~ I K I A ' V L \ L = b L PI K IU (; r = i rl r.JKIJ~lT= l h J T t r ~ f i = L!)II

P F A n ( l . 7 ? ? ) ( K N ( l ) r l = l r h O ) r ( M ( I ) * I = l r 1 6 ) * ( P H L O K ( I ) d = l v 6 O )

. 3 ? / l - O h b 1 0 T (~UIi/2012/1bI2/40F2.?/20Fil.2)

C

C r { k C P l J \ r O , r L ? Ir't. lulrMt3tW OF T 1 E ADdACENT NODES OF NODE I I S AS YFT C S T ( l i ? E o I I ~ y ( 1 ) :.I I N O ~ (GETS N O R M A L I Z E L ) .

(.

0 0 3 C O l = i l r N K N O T

~ ' f l ( I ) = v ~ I - l ) + k l ( I ) 3 / ' 0 CON I 1 IU:JE

l l ~ j 3~ 1 I = 1 .1~lhNCl1

M ( N K N O T - L * 2 ) = M ( : J Y N O I - I + 1 1 + 1 -32 1 CIIN r J 1 \ 4 1 I E

'."

^(I⁾⁼^I

'4 ( 5 1 6 I I J I ) ~ + i = I J K 4 b I U L * 1

u ! m ' I T F ( ? r 5 2 3 ) ( ~ ( l ) r l = l r l 7 ) v ( K h ( I ) v 1 = 1 v 6 0 )

37.3 F O R ~ n r ( l T ( ~ ~ x r I 3 ~ / J O ( 2 X ~ ~ r l ) / 3 0 t ~ H * ~ ~ / / / / / 3 O ~ l H * ~ ~ 4 0 t.k's= I. { ) I ) O

C r: I F T t i F v h ~ . : ) t S u t PuLc)K COML FI?OYE THE. I N P U T T H I S 0 0 - L O O P I S O M I T T E D * t :

I J ( ~ 7 0 K,UNAL= 1 vNl(ANI1L 7 2 l i = i + u ) \ k ( Or 1 ) )

J F ( * - k ' 4 H ) l . l . r ! 2 1 I F ( l , t i - 0 ) r J 3 . c ! 2 * ? c ! r ' 3 P Y L O h ( K A N P L ) =I?

? [ I C ' , ~ k J l I &I# I!-

c k I T E ( ? r b 2 2 ) ( P H L U K ( 1 ) r l = l r h U ) h?? F O F ? ~ ~ I I T ( h ( ! d F h . J / ) / )

C

(; l r l F S T % w l T ~ \ ~ G d ~ i L ! r t S F O R POOS P H t CHEATEL,

L

( 1 ; ) I + ⁴ ^KI,IO ^I= A * v r t \ o r

( 1 0 1 Q 4 K N U T / = ~ * N r ( N O T L

I F ( K J ( I T . ~ U . K I U C ~ T ? ) b 0 1 0 1'35 PO\)\ (1(:.10T V I ( R I O ~ Z ~ Z ) =QANF ( O r 0 )

r i ) O \ ( K ' L ! ) T , K : ~ O ~ I 2. I ) =PL:OS ( K l d O T t K l u O T Z r ? ) ($0 1 1 1 1 9 4

\-r., ~ ' ~ J I ~ ) s ( r c , d o T ~ n h l o l L , 1 ) =l . u

l J 0 0 S ( r < ~ ~ U T r h N O T L r c ' ) = l . U 1 ~ r o , l i 4 1 ' 1 t .

1 9 3 (;l.Jld I 1 I U . l t .

C

C i'.lO* r d t I r t k A T J 3 i ' i S T A k r S

(24)

- 2 2 -

C U C 6 6 0 0 F T N V 3 . 0 - p 3 5 S O P T = l 0 9 / 1 2 / 7 4 G R A 0 0 0 2 0

,,

C T r i t V & l U F S O F P M E Y H AHF P f J T T t l ) 1 N I N C R E A S I N G ORDER i

1 4 1 I V = I r

( I ~ J I /tr k A i * ~ A l . = t i a t \ ~ l rKANALF

~ . l t 4 K ~ \ l l ~ l = K N (KAIqAl-)

I \ ! ? h ~ t ~ = K i \ l ( l\R1411L- I )

I F ( P l * F * n q ( k W K r J ) .LE.PMEMY ( N H K Y O T ) ) ti0 T O 1 4 0 h k 1- r ' = k - ' M ~ - l 4 ~ 1 ( ltll3K'Y )

P r ' i t v ' ' 1 ( t J r j K i V ) = k ' M t 1 4 H ( NHKNOT )

P . I t b ' 1 3 ( P . : b h p ' ( J T ) = n k 1-P I V = I v + l

I , - t r l C g l U ? 1 I d l J F

I F ( I V . ( ; T . O I G O

' r o

^{1 4 1}

1 3 5 '\L\hAL=KANALA.tiANALE

h n s r . . o ~ = K I I ANAL)

I . , ~ ) S K , J = P M c M h ( I V ~ ~ ~ P J U ~ ) + 0 O C > K ~ 1. 3 5 C;!PJ

r

¹^{: d l}^IF

~ ) l r = ) ! A ' < % ~ ~ J L I ~ ~ I \ J - ~ \ ) O S ( K l d f l T 9 ~ l ~ ~ ) l l r L ) 1 I F ( I ) I F . ( ~ T . I J I F A ) L , I F A = D I F

- 0 4 I J o r 1 5 ( r 1 \ i l ' . l . K \ U T ( . L I \ I ) =UO5I<N 13<1 ( l ) : x l I l # \ l l l E

I 4 1 r c 1 1 , i I * I . I ~ 7 : ) C d i \ ; T l t d i l t

, i 1 1 + l F ( ! ~ ~ i M ~ ( l J L t l B ) - F . u S 1 1 $ 1 7 1 3 7 r l I ) 1:37 I F ( ~ ~ l F f i . t i : . u . l i l ) C l ) ~ i L ~ l t ~ 1 3

(25)

- 2 3-

CUC 6 6 0 0 FTN V 3 . 0 - P 3 5 5 O P T = ] 0 9 / 1 2 / 7 4

r + > = t P S / ? . O ! J I !

1 ' ( I ) ~ ~ V S ( L J I F A ) - + P ~ ) 1 3 7 . 1 3 ! * 1 0 1 0 (:I)IV

r

I I V I I ~ ~

IN^[ l t . ( s t 7 ) 1 T k n t k P b ~ D I F ~

7 F \ J H ~ * ~ T ( ! ~ t( 1 3h a ) * ' + t i ~ ~ A C h t I 5 t q d H I T E R . - S C I i R I T T F I U OHNE E R F O L G REENDET

~ : + + F P > = . l J l l . . 3 . r > t i l ) ~ k l ~ = . l ) 1 1 . 3 ) 1 3 I L ~ T T t ( ? * I * ) P n q P H H . i ) I F A

1 4 F ( I H r b l ~ J I ( / c ' L ' ~ Pt>l-nK L l L G I / , J 1SCHEhl r F b . 3 r 4 H I J N O r F 6 . 3 r h H * * * * Q * / / e l h r l 3 ( l H U ) r f I r U , 3 d r l 3 ( 1 H s ) )

10 7 >TOP

c. ^lY^{i )}

(26)

References

111 Butrimenko. A. Adaptive Routinq Techniques and Simulation of Communication Networks.

Fourth International Traffic Conaress, Munich, 1970.

!r2

1

Lasarev, W.G. and Parshennkov, N. A Game Method for Dvnamic Control of Communication Networks

(in Russian). Postroeniie U ~ r o v l i a u s h i c h Ustroisv e Svstem. "Nauka", 1974.

31 Butrimenko, A. Routinq Techniques for Messaae Switchina Networks with Messaqe Outdatinq.

S v m ~ o s i u m on Comvuter Communication Networks and Teletraffic. Polvtechnic Institute of Brooklvn. 1972.

I 4

1

^Harary,F. Graph Theory. Addison-Wesley,

1970.