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There a r c two p o s s i b l e approaches t o t h e problem o f a n e t r ~ o r k of s e r v i c e f a c i l i t i e s , d l c ~ e an inclivicl.t~al c u s t o n e r p a s s e s t h r o u g h t h c n e t - work, queueing f o r s e r v i c e a t some o r a l l o f t h e s e f a c i l i t i e s . E i t h e r we produce mcthods by which t h e network may b e decomposed i n t o s m a l l manageable p a r t s , o r we must a t t e m p t t o s o l v e s i m u l t a n e o u s l y f o r a l l t h e p r o p e r t i e s of t h e network, a t a p a r t i c u l a r t i m e . A t p r e s e n t , t h e c l a s s o f networks f o r which comprehensive s o l u t i o n s a r e known i s s o r e s t r i c t e d
1
t h a t it i s u s u a l l y e s s e ~ l t i a l t o c o n s i d e r t h e f i r s t approach, a t l e a s t i n the' f i r s t i n s t a n c e .F i r s t we c o n s i d e r t h e problem o f decomposition p o i n t s , t h e p o i n t s
a t ~ ~ h i c h a s t r e a m of customers i s broken up i n t o s e v e r a l s u b s t r e a m s , 11eacl-
.-
irlg f o r d i f f e r e n t d e s t i n a t i o n s . C o n d i t i o n s a r e e s t a b l i s h e d f o r a c l a s s of clecoinposition switches, which e n s u r e t h a t if t h e i n p u t s t r e a m t o t h e s w i t c h i s a bfarkov renewal p r o c e s s , t h e n s o a r e each of t h e substreams.
We t h e n i n v e s t i g a t e tile bchaviou,. o f one o f t h e s e substreams a t a s u b s e q u e n t s e r v i c e f a c i l i t y . hhen t h e s e r v i c e t i m e s a r e exponenti.ally d i s t r i b u t e d , an a n a l y s i s of t h i s , t h e SB1/M/1 queue, based on t h e prop- e r t i e s of t h e busy c y c l e , consiclcrahly s i m p l i f i e s and e x t e n d s t h e r e s u l t s known f o r i t .
Some a t t e m p t i s made t o r e l a x t h e c o n d i t i o n on t h e s e r v i c e t i m e d i s t r i b u t i o n . S i n c e t h e r e i s always t h e p o s s i b i l i t y t h a t a I'oisson p r o - c e s s approximation t o t h e a r r i v a i stream may b e a c c e p t a b l e , t h e d u a l of
t h i s queue, where t h e sequence o f s e r v i c e t i m e s forms a hlarkov renewal
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p r o c e s s , i s c o n s i d e r e d . T h i s seems a y a r t i c u l a r ! y a p p r o p r i a t e g e n e r a l - i s a t i o n , s i n c e it allo~zrs f o r d i f f e r e n t c 3 a s s e s of s e r v i c e r e q u i r e m e n t s , perliaps r e f l e c t i n g t h e o r i g i n o f customers from v a r j o u s p a r t s of t h e n e t -
. .
nork. I n a d d i t i o n , some r e s u l t s a ~ c aprcsentcd f o r a queue i n wllich b o t h t h e arrjval t i m c s ancl s e r v i c e r c q u i r c m a ~ ~ t s depcnd on t h e sequence of customer t y p e s .
Since t h e o u t p c t from a queue i n a network w i l l form p a r t o r a l l of t h e i n p u t t o a subsequent que:le, rue c o n s i d e r , f i n a l l y , t h e d e p a r t u r e Processes from queues o f t h e t l p e s t h a t have been mentioned. Although, i n g e n e r a l , i t i s not p o s s i b l e t o d e s c r i b e t h e d e p a r t u r e stream from queues o f t h e s e ty-pes i n terms of a p a r t i c u l a r s t o c h a s t i c p r o c e s s , wc can f i n d t h e d i s t r i b u t i o n s of some of t h e parameters o f a very general d e p a r t - u r e p r o c e s s , i n p a r t i c u l a r , t h e number o f d e p a r t u r e s from t h e l a s t queue mentioned above.
