62540 denning

Upload: punxkoko

Post on 04-Jun-2018

246 views

Category:

Documents


0 download

TRANSCRIPT

  • 8/14/2019 62540 Denning

    1/37

    T h e O p e r a t i o n a l A n a l y s is o f Q u e u e i n g N e t w o r k M o d e l s *P E TE R J . D E N N I N GComputer Sciences Department, Purdue Unwers~ty, West Lafayette, Indiana 47907J E F F R E Y P . B U Z E NBGS Systems, Inc., Box 128, Lincoln, Massachusetts 01773

    Q u e u e i n g n e t w o r k m o d e l s h a v e p r o v e d t o b e c o s t e f f e ct w e t o o l s f o r a n a l y z i n g m o d e r nc o m p u t e r s y s t e m s . T h i s t u t o r i a l p a p e r p r e s e n t s t h e b a s i c r e s u l t s u s i n g t h e o p e r a t i o n a la p p r o a c h , a f r a m e w o r k w h i c h a l lo w s t h e a n a l y s t t o t e s t w h e t h e r e a c h a s s u m p t i o n i s m e ti n a g i v en s y s t e m . T h e e a r l y se c t i o n s d e s c r ib e t h e n a t u r e o f q u e u e i n g n e t w o r k m o d e l s a n dt h e i r a p p h c a t i o n s fo r c a l c u la t i n g a n d p r e d i c t i n g p e r f o r m a n c e q u a n t i t m s T h e b a s i cp e r f o r m a n c e q u a n t i t i e s - - s u c h a s u t i l i z a t i o n s , m e a n q u e u e l e n g t h s , a n d m e a n r e s p o n s et u n e s - - a r e d e f i n e d , a n d o p e r a t m n a l re l a t i o n s h i p s a m o n g t h e m a r e d e r w e d F o l lo w i n g t h i s,t h e c o n c e p t o f j o b f l ow b a l a n c e is i n t r o d u c e d a n d u s e d t o s t u d y a s y m p t o t i c t h r o u g h p u t sa n d r e s p o n s e t u n e s . T h e c o n c e p t s o f s t a t e t r a n s i t i o n b a l a n c e , o n e - s te p b e h a v i o r , a n dh o m o g e n e i t y a r e t h e n u s e d t o r e l a t e t h e p r o p o r t i o n s o f t i m e t h a t e a c h s y s t e m s t a t e i so c c u pi ed t o t h e p a r a m e t e r s o f j o b d e m a n d a n d t o d e w c e c h a r a c t e n s t m s E f f ic m n t m e t h o d sf o r c o m p u t i n g b a s i c p e r f o r m a n c e q u a n t i t i e s a r e a l so d e s c r i b ed . F i n a l l y t h e c o n c e p t o fd e c o m p o s i t i o n is u s e d t o s tm p h f y a n a l y s e s b y r e p la c i n g s u b s y s t e m s w i t h e q u i v a l e n td e v i c e s. A l l c o n c e p t s a r e i l l u s t r a t e d l i b e r a l l y w i t h e x a m p l e sKeywords and Phrases b a l a n c e d s y s t e m , b o t t l e n e c k s , d e c o m p o s a b i l it y , o p e r a t i o n a la n a l y si s , p e r f o r m a n c e e v a l u a t i o n , p e r f o r m a n c e m o d e l in g , q u e u e l n g m o d e l s, q u e u e l n gn e t w o r k s , r e s p o n s e t u n e s , s a t u r a t i o n .CR Categorws: 8.1, 4.3

    I N T R O D U C T I O NQ u e u e i n g n e t w o r k s a r e u s e d w i d e l y t o a n -a l y ze t h e p e r f o r m a n c e o f m u l t i p r o g r a m m e dc o m p u t e r s y s t e m s . T h e t h e o r y d a t e s b a c kto the 1950s . In 1957 , Jackson publ i shed ana n a l y s i s o f a m u l t i p l e d e v i c e s y s t e mw h e r e i n e a c h d e v i c e c o n t a i n e d o n e o r m o r ep a r a l le l s e r v e r s a n d jo b s c o u l d e n te r o r e x i tthe sy s te m any wh ere [JACK57]. In 1963J a c k s o n e x t e n d e d h i s a n a l y s i s t o o p e n a n dc l o s e d s y s t e m s w i t h l o c a l l o a d - d e p e n d e n t* T h i s w o r k w a s s u p p o r t e d i n p a r t b y N S F G r a n tG J - 4 1 2 8 9 a t P u r d u e U n i v e r s i t y

    serv ice ra tes a t a l l dev ices [JACK63] . In1 96 7, G o r d o n a n d N e w e l l s i m p l i f i e d th e n o -ta t i o n a l s t r u c tu r e o f th e s e r e s u l t s fo r th es p e c i a l c a s e o f c l o s e d s y s t e m s [G O R D 67 ].B a s k e t t , e t a l . e x te n d e d th e r e s u l t s to i n -c l u d e d i f f e r e n t q u e u e i n g d i s c i p l i n e s , m u l t i -p l e c l a s s e s o f jo b s , a n d n o n e x p o n e n t i a l s er -v ice d i s tr ibut ions [BASK75] .T h e f i r st su c c e s s fu l a p p l i c a t io n o f a n e t -w o r k m o d e l t o a c o m p u t e r s y st e m c a m e i n1 9 6 5 w h e n S c h e r r u s e d th e c l a s s i c a l m a -c h i n e r e p a ir m a n m o d e l t o an a l y ze t h e M I Tt i m e s h a r i n g sy s t e m , C T S S [S C H E6 7]. H o w -e v e r , t h e J a c k s o n - G o r d o n - N e w e l l t h e o r y

    P e r m m s I o n t o c o p y w i t h o u t f e e a ll o r p a rt o f t hi s m a t er i a l is g r a n t e d p r o v i d e d t h a t t h e c o p i e s a r e n o t m a d eor distributed for direct co mm er ci al advantage the AC M copyright notice an d the title f the publication an dit s d a t e a pp e a r a n d n o t i c e i s g i v en t h at c o p y m g m b y p e r m i s s i o n o f t h e A ss o c la t l on fo r C o m p u t i n g M a c h i n e r y .T o c o p y o t h e r w ls e o r to r e p ub l i sh r e q m r e s a f ee a n d / o r s p ec i fi c p e r mi s s io n . 1 97 8 A C M 0 0 1 0 - 4 8 9 2 / 7 8 /0 9 0 0 - 0 2 2 5

    Com puting Surveys, V ol. 10, No. 3, Se pte m be r 1978

  • 8/14/2019 62540 Denning

    2/37

    226 P . J . D e n n i n g a n d J . P . B u z e nC O N T E N T S

    INTRODUCTION1 THE BASIS FOR OPERATIONAL ANALYSISOperatmnal Variables, Laws and TheoremsApphcattonAreasPrior Work m Operatmnal Analysts2 VALIDATION AND PREDICTION3 OPERATIONAL MEASURES OF NETWORKSTypes of NetworksBamc Operatmnal Quantities4 JOB FLOW ANALYSISVtslt RatmsSystem Response TuneExamplesBottleneck Analys~ExamplesSummary5 LOAD DEPENDENT BEHAVIOR6 SOLVING FOR STATE OCCUPANCIESState Trap~ttmn BalanceSolving the Balance EquationsAn ExampleAccuracy of the Analysm7 COMPUTATION OF PERFORMANCE QUANTITIESClosed System with Homogeneous Service Tm~esTermmal Driven System with Homogeneous Serwce TunesGeneral Systems8 DECOMPOSITONOffime ExperimentsApphcatmnsCONCLUSIONSACKNOWLEDGMENTSREFERENCES

    l a y d o r m a n t u n t i l 1 9 7 1 w h e n B u z e n i n t r o -d u c e d t h e c e n t r a l s e r v e r m o d e l a n d f a s tc o m p u t a t i o n a l a l g o r i th m s f o r t h e s e m o d e l s[ B u z E 7 1 a , B u z E 7 1 b , B U Z E 7 3 ] . W o r k i n gi n d e p e n d e n t l y , M o o r e s h o w e d t h a tq u e u e i n g n e t w o r k m o d e l s co u l d p r e d i c t t h er e s p o n s e t i m e s o n t h e M i c h i g a n T e r m i n a lS y s t e m ( M T S ) t o w i t h in 10% [ M o o R 7 1 ] .E x t e n s i v e v a l i d a t i o n s s i n c e 1 9 7 1 h a v e v e r i -f i e d t h a t t h e s e m o d e l s r e p r o d u c e o b s e r v e dp e r f o r m a n c e q u a n t i t i e s w i t h r e m a r k a b l ea c c u r a c y [ B u z E 7 5 , G I A M 7 6 , H U G H7 3,L I I ' s 7 7 , R O S E 7 8 ] . G o o d s u r v e y s a r e[ G E L E 76 a , KL E I 75 , KL E I 76 , a nd MON T 75] .M a n y a n a l y s t s h a v e e x p e r i e n c e d p u z z l e -m e n t a t t h e a c c u ra c y o f q u e u e in g n e t w o r kr e s u l t s . T h e t r a d i t i o n a l a p p r o a c h t o d e r i v -i ng t h e m d e p e n d s o n a s e ri e s o f a s s u m p -t i o n s u s e d i n t h e t h e o r y o f s t o c h a s t i c p r o c -e s se s :

    T h e s y s t e m i s m o d e l e d b y a s t a t i onar ys tochast ic process; J o b s a r e s tochas t i ca l ly independent ; J o b s t e p s f r o m d e v i c e t o d e v i c e f o l lo wa M ar k ov cha in ; T h e s y s t e m i s i n s tochast ic equi l ib-r ium; T h e s e r v ic e t i m e re q u i r e m e n t s a t e a c hd e v i c e c o n f o r m t o a n exponent ia l d i s -tr ibution; a n d T h e s y s t e m is ergodic-- i .e . , l o n g - t e r mt i m e a v e r a g e s c o n v e r g e t o t h e v a l u e sc o m p u t e d f o r s t o c h a s t i c e q u i l i b r i u m .

    T h e t h e o r y o f q u e u e in g n e t w o r k s b a s e do n t h e s e a s s u m p t i o n s i s u s u a l l y c al le d" M a r k o v i a n q u e u e i n g n e t w o r k t h e o r y "[ K L E I 7 5] . T h e i t a li c i z e d w o r d s i n t h i s l i s t o fa s s u m p t i o n s i l l u st r at e c o n c e p t s t h a t t h e a n -a l y s t m u s t u n d e r s t a n d t o b e a b l e t o d e p l o yt h e m o d e l s. S o m e o f t h e s e c o n c e p t s a r ed i f f i c u l t . S o m e , s u c h a s " e q u i l i b r i u m " o r" s t a t io n a r i t y , " c a n n o t b e p r o v e d t o h o l d b yo b s e r v i n g t h e s y s t e m i n a f in i t e t i m e p e r i o d .I n f a c t , m o s t c a n b e d i s p r o v e d e m p i r i c a l l y- - f o r e x a m p l e , p a r a m e t e r s c h a n g e o v e rt i m e , j o b s a r e d e p e n d e n t , d e v i c e t o d e v i c et r a n s i t i o n s d o n o t f o ll o w M a r k o v c h a i n s ,s y s t e m s a r e o b s e r v a b l e o n l y f o r s h o r t p e -r i o d s , s e r v i c e d i s t r i b u t i o n s a r e s e l d o m e x -p o n e n ti a l. I t is n o w o n d e r t h a t m a n y p e o p l ea r e s u r p r i s e d t h a t t h e s e m o d e l s a p p l y s ow e l l t o s y s t e m s w h i c h v i o l a t e s o m a n y a s -s u m p t i o n s o f t h e a n a ly s isI n a p p l y i n g o r v a l i d a t i n g t h e r e s u l t s o fM a r k o v i a n q u e u e i n g n e t w o r k t h e o r y , a n a -l y s t s s u b s t i t u t e o p e r a t i o n a l (i.e ., d i r e c t l ym e a s u r e d } v a l u e s fo r s to c h a s t ic p a r a m e t e r si n t h e e q u a t i o n s . T h e r e p e a t e d s u c c e s s e s o fv a l i d a t i o n s l e d u s t o i n v e s t i g a t e w h e t h e rt h e t r a d i ti o n a l e q u a t i o n s o f M a r k o v i a nq u e u e i n g n e t w o r k t h e o r y m i g h t a l s o b e re -l a t io n s a m o n g o p e r a t i o n a l v a r i a b le s , a n d , ifs o , w h e t h e r t h e y c a n b e d e r i v e d u s i n g d i f -f e r e n t a s s u m p t i o n s t h a t c a n b e d i r e ct l y v er -i f i e d a n d t h a t a r e l i k e l y t o h o l d i n a c t u a ls y s t e m s . T h i s h a s p r o v e d t o b e t r u e[ B u z E 7 6 a , b , c ; a n d D E N N 7 7 ] .T h i s t u t o r i a l p a p e r o u t l i n e s t h e o p e r a -t i o n a l a p p r o a c h t o q u e u e i n g n e t w o r km o d e l i n g . A l l t h e b a s i c e q u a t i o n s a n d r e -s u l ts a r e d e r i v e d f r o m o n e o r m o r e o f t h r e eo p e r a t i o n a l p r i n c i p l e s :

    A l l q u a n t i t i e s s h o u l d b e d e f i n e d s o a sComput ing Surveys, Vol 10, No, 3, Septem ber 1978

  • 8/14/2019 62540 Denning

    3/37

    The Operational Analysis of Queueing Network Models 227t o b e precisely measurable, a nd a l l a s -s u m p t i o n s s t a t e d s o a s t o b e directlytestable. T h e v a l i d i t y o f r e s u l t s s h o u l dd e p e n d o n l y o n a ss u m p t i o n s w h i c h c a nb e t e s t e d b y o b s e r v i n g a r e a l s y s t e mf o r a f i n i te p e r i o d o f ti m e . T h e s y s t e m m u s t b e flow balanced--i.e ., t h e n u m b e r o f a r r i v a l s a t a g i v e nd e v i c e m u s t b e ( a l m o s t ) t h e s a m e a st h e n u m b e r o f d e p a r t u r e s f r o m t h a td e v i c e d u r i n g t h e o b s e r v a t i o n p e r i o d . T h e d e v i c es m u s t b e homogeneous--i.e., t h e r o u t i n g o f j o b s m u s t b e i n d e -p e n d e n t o f l o c al q u e u e l e n g t h s , a n d t h em e a n t i m e b e t w e e n s e r v i c e c o m p l e -t io n s a t a g iv e n d e v ic e m u s t n o t d e p e n do n t h e q u e u e l e n g t h s o f o t h e r d e v i ce s .T h e s e o p e r a t i o n a l p r in c i p le s , w h i c h w i ll b ed i s c u s s e d a t l e n g t h i n l a t e r s e c t i o n s , l e a d t ot h e s a m e m a t h e m a t i c a l e q u a t i o n s a s t h et r a d i t i o n a l M a r k o v i a n a s s u m p t i o n s . H o w -e v e r , t h e o p e r a t i o n a l a s s u m p t i o n s c a n b et e s t e d , a n d t h e r e a r e g o o d r e a s o n s t o b e -l ie v e t h a t t h e y o f t e n h o l d . T h i s i s w h yo p e r a t i o n a l q u e u e i n g n e t w o r k a n a l y s i s e x -p l a i n s t h e s u c c e s s o f v a l i d a t i o n e x p e r i-

    m e n t s . I t i s n o w p o s s i b l e t o u s e t h eq u e u e i n g n e t w o r k t e c h n o l o g y w i t h m u c hm o r e c o n f i d e n c e a n d u n d e r s t a n d i n g .1. THE BASIS FOR OPERATIONALANALYSIST h r o u g h o u t t h i s p a p e r w e w ill b e c o n c e r n e dw i t h d e r i v i n g e q u a t i o n s t h a t c h a r a c t e r i z et h e p e r f o r m a n c e o f a c t u a l c o m p u t e r s y s -t e m s d u r i n g g i v e n ti m e p e r io d s . T o d o t h is ,w e n e e d a m a t h e m a t i c a l f r a m e w o r k inw h i c h w e c a n d e f i n e f o r m a l v a r i a b l e s , f o r -m u l a t e h y p o t h e s e s , a n d p r o v e th e o r e m s .T h e t h e o r y o f s t o c h a st i c p r o c e s s e s h a st r a d i t i o n a l l y b e e n u s e d a s s u c h a f r a m e -w o r k . M o s t a n a l y s e s o f p e r f o r m a n c e b e g i nw i t h t h eStochastic Hypothesis: T h e b e h a v i o ro f t h e r e a l s y s t e m d u r i n g a g i v e n p e r io do f t i m e i s c h a r a c t e r i z e d b y t h e p r o b a -b i l i t y d i s t r i b u t i o n s o f a s t o c h a s t i cp r o c e s s .S u p p l e m e n t a r y h y p o t h e s e s a r e u s u a l ly a ls om a d e . T h e s e h y p o t h e s e s , w h i c h c o n c e r nt h e n a t u r e o f t h e s t o c h a s t i c p r o c e ss , ty p i -c a l l y i n t r o d u c e c o n c e p t s s u c h a s s t e a d y

    s t a t e , e rg o d i c i ty , i n d e p e n d e n c e , a n d t h e d is -t r i b u t i o n s o f s p e c if ic r a n d o m v a r i ab l e s . A l lt h e s e h y p o t h e s e s c o n s t i t u t e a stochasticmodel.O b s e r v a b l e a s p e c t s o f t h e r e a l s y s t e m - -e .g ., s t a t e s , p a r a m e t e r s , a n d p r o b a b i l i t y d i s -t r i b u t i o n s - - c a n b e i d e n t i f i e d w i t h q u a n t i -t i e s i n t h e s t o c h a s t i c m o d e l , a n d e q u a t i o n sr e l a t i n g t h e s e q u a n t i t i e s c a n b e d e r i v e d .A l t h o u g h f o r m a l l y a p p l i c a b l e o n l y t o t h es t o c h a s t i c p r o c e s s , t h e s e e q u a t i o n s c a n a l s ob e a p p l i e d t o t h e o b s e r v a b l e b e h a v i o r o ft h e s y s t e m i t s el f u n d e r s u i t a b l e l im i t i n gc o n d i t i o n s [ B u z E 7 8 a ] .S t o c h a s t i c m o d e l s b e s t o w b o u n t i f u lb e n e f i t s . I n d e p e n d e n t a n d d e p e n d e n t v a r i -a b l e s c a n b e d e f i n e d p r e c i s e l y , h y p o t h e s e sc a n b e s t a t e d s u c c in c t ly , a n d a c o n s i d e r a b l eb o d y o f t h e o r y c a n b e c a ll ed o n d u r in ga n a ly s i s. H o w e v e r , s t o c h a s t i c m o d e l i n g h a sc e r t a i n d i s a d v a n t a g e s , t h e m o s t i m p o r t a n tb e i n g t h e i m p o s s i b i li t y o f v a l i d a ti n g t h eS t o c h a s t i c H y p o t h e s i s a n d t h e s u p p le m e n -t a r y h y p o t h e s e s t h a t d e p e n d o n it .T h e S t o c h a s t i c H y p o t h e s i s is an a s s er -t i o n a b o u t t h e c a u s e s u n d e rl y i n g t h e b e h a v -i o r o f a r e a l s y s t e m . B e c a u s e o n e c a n n o tp r o v e a s s e r t e d c a u s e s b y s t u d y i n g o b s e r v e de f fe c t s, t h e t r u t h o r f a l s e h o o d o f t h e S t o -c h a s t ic H y p o t h e s i s a n d i ts d e p e n d e n t s u p -p l e m e n t a r y h y p o t h e s e s - - f o r a g iv en s y s t e ma n d t i m e p e r i o d - - c a n n e v e r b e e s t a b l i s h e db e y o n d d o u b t t h r o u g h a n y m e a s u re m e n t . 1T h i s i s t r u e e v e n i f m e a s u r e m e n t e r ro r i sa s s u m e d t o b e z e r o a n d e v e r y c o n c e iv a b l em e a s u r e m e n t i s a s s u m e d t o b e t a k e n .

    T h u s , a n a n a l y s t c a n n e v e r b e c e r t a i nt h a t a n e q u a t i o n d e r i v e d f r o m a s t o c h a s t i cm o d e l c a n b e c o r r e c t l y a p p l i e d t o t h e o b -s e r v a b l e b e h a v i o r o f a r e a l s y s t e m .Ope rat iona l Var iab les , L aw s, and Th eo rem sH y p o t h e s e s w h o s e v e r a c i ty c a n b e e s t a b -l is h ed b e y o n d d o u b t b y m e a s u r e m e n t w illb e c a l l e d operationally testable. O p e r a -t i o n a l a n a l y s i s p r o v i d e s a r i g o r o u s m a t h e -m a t i c a l d i s c i p l i n e f o r s t u d y i n g c o m p u t e rs y s t e m p e r f o r m a n c e b a s e d s o le l y o n o p e r -a t i o n a l l y t e s t a b l e h y p o t h e s e s .] F o r e x a m p l e o n e c a n n e v e r e st ab li sh t h r o u g h m e a -s u r e m e n t t h a t a s e t o f o b s e r v e d s e rv i ce t i m e s t s o r isn o t a s a m p l e f r o m a s e q u e n c e o f i n d e p e n d e n t e x p o -n e n t ia l l y d i s t ri b u t ed r a n d o m v a r i ab l e s .

    Com put ing Surveys , Vol . 10 , No. 3 , Septe mb er 1978

  • 8/14/2019 62540 Denning

    4/37

    228 P . J . D e n n i n g a n d J . P . B u z e nI n o p e r a t i o n a l a n a l y s i s t h e r e a r e t w ob a s ic c o m p o n e n t s t o e v e r y p r o b l e m : a s y s -

    t e m , w h i c h c a n b e r e a l o r h y p o t h e t i c a l , a n da t i m e p e r i o d , w h i c h m a y b e p a s t , p r e s e n t ,o r f u tu r e . T h e o b j e c t i v e o f a n a n a l y s i s ise q u a t i o n s r e l a t i n g q u a n t i t i e s m e a s u r a b l e i nt h e s y s t e m d u r i n g t h e g i v e n t i m e p e r i o d .T h e f i n i t e t i m e p e r i o d i n w h i c h a s y s t e mi s o b s e r v e d i s c a l le d t h e o b s e r v a t i o n p e r i o d .A n o p e r a t i o n a l v a r i a b l e i s a f o r m a l s y m b o lt h a t s t a n d s f o r t h e v a l u e o f s o m e q u a n t i t yw h i c h is m e a s u r a b l e d u r in g t h e o b s e r v a t i o np e r i o d . I t h a s a s i n g l e , s p e c i f i c v a l u e f o re a c h o b s e r v a t i o n p e r i o d .O p e r a t i o n a l v a r i a b l e s a r e e i t h e r b a s i cq u a n t i t i e s , w h i c h a r e d i r e c t l y m e a s u r e dd u r i n g t h e o b s e r v a t i o n p e r i o d , o r d e r i v e dq u a n t i t i e s , w h i c h a r e c o m p u t e d f r o m t h eb a s i c q u a n t i t i e s . F i g u r e 1 s h o w s a s i ng l e -s e r v e r q u e u e i n g s y s t e m w i t h f o u r b a s i cq u a n t i t i e s :T - - t h e l e n g th o f t h e o b s e r v a t i o n p e r io d ;A - - t h e n u m b e r o f a r ri v a ls o c c u rr in g d u r -i ng t h e o b s e r v a t i o n p e r i o d ;B - - t h e t o ta l a m o u n t o f t im e d u r in gw h i c h t h e s y s t e m i s b u s y d u r i n g t h e

    o b s e r v a t i o n p e r i o d ( B _< T ) ; a n dC - - t h e n u m b e r o f c o m p l e ti o n s o c c u rr in gd u r i n g t h e o b s e r v a t i o n p e r i o d .F o u r i m p o r t a n t d e r i v e d q u a n t i t ie s a r e

    f f i A / T , t h e a r r i v a l r a t e( j o b s / s e c o n d ) ;X = C / T , t h e o u t p u t r a t e( j o b s / s e c o n d ) ;U ffi B / T , t h e u t i l i z a t i o n ( f r a c t i o no f t i m e s y s t e m i s b u s y ) ; an dS ffi B / C , t h e m e a n s e r v i c e t i m ep e r c o m p l e t e d jo b .

    T h e b a s i c q u a n t i t i e s ( A , B , C ) a r e t y p i c a lo f " r a w d a t a " c o l l e c te d d u r in g a n o b s e r v a -t i o n , a n d t h e d e r i v e d q u a n t i t i e s (~ , X , U , S )a r e t y p i c a l o f " p e r f o r m a n c e m e a s u r e s . " A l lt h e s e q u a n t i t i e s a r e v a r i a b l e s w h i c h m a yqueue

    FIGURE 1

    s e r v e r

    B,TSingle server queuelng system.

    c h a n g e f r o m o n e o b s e r v a t i o n p e r i o d t o a n -o t h e r .I t i s e a s y t o s e e t h a t t h e d e r i v e d q u a n t i -t i e s s a t i s f y t h e e q u a t i o nU = X S .

    T h u s , i f t h e s y s t e m i s c o m p l e t i n g 3 j o b s /s e c o n d , a n d i f e a c h j o b r e q u i r e s 0 .1 s e c o n do f s e r v ic e , t h e n t h e u t i li z a ti o n o f t h e s y s t e mi s 0 . 3 o r 3 0 % . A n e q u a t i o n s u c h a s t h i s ,w h i c h e x p r e s s e s a n i d e n t i t y a m o n g o p e r a -t i o n a l q u a n t i t i e s , i s c a l l e d a n o p e r a t i o n a ll a w o r o p e r a t i o n a l i d e n t i t y . T h i s is b e c a u s et h e r e l a t io n m u s t h o l d i n e v e r y o b s e r v a ti o np e r i o d , r e g a r d l e ss o f t h e v a l u e s o b s e r v e d .T h e i d e n t i t y U = X S i s c a l l e d t h e u t i l i z a -t i o n l a w . W e w i l l e n c o u n t e r v a r i o u s o t h e ro p e r a t i o n a l l a w s l a t er .N o w , s u p p o s e t h a t w e a ss u m e t h a t t h en u m b e r o f a r ri v a ls i s e q u a l t o t h e n u m b e ro f c o m p l e t io n s d u r i n g t h e o b s e r v a t i o n p e -r io d . T h a t is , w e a s s u m e

    A ffi C.T h i s a s s u m p t i o n is ca ll ed j o b f l o w b a l a n c eb e c a u s e i t i m p l i e s ~ ffi X . J o b f l o w b a l a n c eh o l d s o n l y i n s o m e o b s e r v a t i o n p e r i o d s .H o w e v e r , i t i s o f t e n a v e r y g o o d a p p r o x i -m a t i o n , e s p e c i a l l y i f t h e o b s e r v a t i o n p e r i o di s l on g , b e c a u s e t h e r a t i o o f u n f i n is h e d t oc o m p l e t e d j o b s , ( A - C ) / C , i s t y p i c a l l ys m a l l. J o b f l o w b a l a n c e i s a n e x a m p l e o f a no p e r a t i o n a l l y t e s t a b l e a s s u m p t i o n : i t n e e dn o t h o l d i n e v e r y o b s e r v a t io n p e ri o d , b u ta n a n a l y s t c a n a l w a y s t e s t w h e t h e r o r n o ti t d o e s - - o r h o w m u c h e r r o r is m a d e b ya s s u m i n g i t d o e s .

    U n d e r t h e a s s u m p t i o n o f j o b f lo w b a l-a n c e , i t is e a s y t o s e e t h a tU ffi AS.T h i s i s a n e x a m p l e o f a n o p e r a t i o n a l t h e o -r e m : a p r o p o s i t i o n d e r i v e d f r o m o p e r a t i o n a lq u a n t i t i e s w i t h t h e h e l p o f o p e r a t i o n a l l yt e s t a b l e a s s u m p t i o n s .I n a s t o c h a s t i c a n a l y s i s o f F i g u r e 1 ,w o u l d b e i n t e r p r e t e d a s t h e r e c i p r o c a l o ft h e m e a n t i m e b e t w e e n a r r i v a l s , S a s t h em e a n a m o u n t o f s e rv i c e r e q u e s t e d b y j o b s ,a n d U a s t h e s t e a d y - s t a t e p r o b a b i li t y t h a tt h e s y s t e m h a s a t l e a s t o n e j o b i n i t . T h es t a t e m e n t U = A S i s a li m i t t h e o r e m f o rs t o c h a s t i c s t e a d y s t a t e [ K L E I 7 5 ] . I n g e n -e r a l , a s t e a d y - s t a t e s t o c h a s t i c t h e o r e m i s as t a t e m e n t a b o u t a c o l l e c t i o n ( e n s e m b l e ) o f

    Computing Surveys, Vol 10, No 3, September 1978

  • 8/14/2019 62540 Denning

    5/37

    T h e O p e r a t io n a l A n a l y s i s o f Q u e u e in g N e t w o r k M o d e l spossible infinite behavior sequences, but itis not guaranteed to apply to a particularfinite behavior sequence. An operationaltheorem is a stat emen t about the collectionof behavior sequences, finite or infinite,that satisfy the given operational assump-tions: it is guaranteed to apply to everybehavior sequence in the collection. (Fordetailed comparisons between stochasticand operational modeling, see [BouH78,BuzE78a].)A p p l i c a t i o n A r e a sThere are three major applications for op-erational results such as the utilization law:

    P e r f o r m anc e C a l c u l a t ion . Operationalresults can be used to compute quan-tities which were not measured, butcould have been. For example, a mea-surement of U is not needed in a flow-balanced system if k and S have beenmeasured. Con s i s t ency Check ing . A failure of thedata to verify a theorem or identityreveals an error in the data, a fault inthe measurement procedure, or a viola-tion of a critical hypothesis. For ex-ample, U ~ kS would imply an error ifobserved in a flow-balanced system. P e r f o r m anc e P r e d i c t i on . Operationalresults can be used to estimate per-formance quantities in a future timeperiod (or indeed a past one) for whichno directly measured data are avail-able. For example, the analyst can es-timate k and S for the future timeperiod, and then predict that U willhave the value kS in that time period.(Although the analyst may find waysof estimating U directly, it is ofteneasier to calculate it indirectly fromestimates of k and S.)

    The first two applications are straightfor-ward, but the third is actually a two-stepprocess. The first step is e s t i m a t i n g thevalues of k and S for the fu ture time period;the second step is ca lcu la t i ng U. Our pri-mary concern in this paper is deriving theequations which can be used for perform-ance calculation, consistency checking, andthe second step in performance prediction.Parameter estimation, the first step in

    229performance prediction, is a problem of in-ducti on-in ferrin g the characteristics of anunseen par t of the universe on the basis ofobservations of anoth er finite part. Gardnerhas an interesting discussion of why no onehas found a consistent system of inductivemathemati cs [GARD76]. Various techniquesfor dealing with the parameter estimationproblem will be discussed throughout thispaper.P r i o r W o r k i n O p e r a t i o n a l A n a l y s i sMan y textbooks illustrate the ideas of prob-ability with operational concepts such asrelative frequencies and proportions oftime. In addition, the derivations of man ywell-known results in the classical theory ofstochastic processes are based, in part, onoperational arguments. However, the ex-plicit recognition th at operational analysisis a separate branch of applied mathemat-ic s- qu it e apart from the theory of stochas-tic processes--is a more recent develop-ment.The concept of operational analysis as a

    separate mathematical discipline was firstproposed by Buzen [BuzE76b], who char-acterized the real-world problems thatcould be treated with operational tech-niques, and derived operational laws andtheorems giving exact answers for a largeclass of practical performance problems. Atabout the same time, operational argu-ments leading to upper and lower boundson the satura tion behavior of computer sys-tems were presented by Denning and Kahn[DENN75a]. These arguments were the op-erational counterpart of similar results de-veloped by Muntz and Wong [MUNT74].The only operational assumption used atthis point was job flow balance.These early operational results dealt ex-clusively with mean values of quantitiessuch as throughput, response time, andqueue length. The theory was soon ex-tended so tha t complete operational distri-butions-as well as mean values--could bederived for operational analogs of thebirth-death process and the M/ M/ 1queueing process [BUzE76a, BUZE78a].These extensions introduced two newanalysis techniques: the application of

    Com putin g Surveys, Vol. 10, No. 3, Sep tem ber 1978

  • 8/14/2019 62540 Denning

    6/37

    230 P . J . D e n n i n g a n d J . P . B u z e n" f lo w b a l a n c e " i n t h e l o gi ca l s t a t e s p a c e o ft h e s y s t e m (a s c o n t r a s te d w i t h t h e p h y s i c a ls y s t e m it se lf ) a n d t h e h o m o g e n e i t y a s s u m p -t i o n s , w h i c h a r e t h e o p e r a t i o n a l c o u n t e r -p a r t s o f M a r k o v i a n a s s u m p t i o n s i n s t oc h a s -ti c t h e o r y . T h e s e t e c h n i q u e s f o r m t h e b a s i sf o r t h e o p e r a t i o n a l r e a t m e n t o f m a n y p r o b -l e m s w h i c h a r e c o n v e n t i o n a l l y a n a l y z e dw i t h e r g o d i c M a r k o v i a n m o d e l s .T h e r e s u l t s i n [ B u z E 7 6 a a n d B u z z 7 8 a ]a p p l i e d o n l y to s i n g l e - re s o u r c e q u e u e i n gs y s t e m s . T h e s a m e a n a l y s i s t e c h n i q u e sw e r e a p p l i e d t o m u l ti p l e -r e s o u r c e q u e u e i n gn e t w o r k s b y D e n n i n g a n d B u z e n[D E N N7 7a ], w h o s h o w e d t h a t t h e " p r o d u c tf o r m s o l u ti o n ," e n c o u n t e r e d i n M a r k o v i a nq u e u e i n g n e t w o r k s , h o l d s i n g e n e r a lq u e u e i n g n e t w o r k s w i t h f l o w b a l a n c e a n dh o m o g e n e i t y ; t h is r e s u l t is m o r e g e n e r a lt h a n c a n b e d e r iv e d i n t h e M a r k o v i a nf r a m e w o r k . T h i s w o r k a l s o i n t r o d u c e d an e w o p e r a t i o n a l c o n c e p t , " o n l i n e ffi o f f li n eb e h a v i o r , " w h i c h c h a r a c t e r i z e s t h e w a y a n -a l y s t s u s e d e c o m p o s i t i o n t o e s t i m a t e p a -r a m e t e r s o f d e v i c e s a n d s u b s y s t e m s . T h eo p e r a ti o n al t r e a t m e n t o f q u e ue i n g n e t w o r km o d e l s i s d i s c u s s e d i n d e t a i l i n t h e r e s t o ft h i s p a p e r . A d d i t i o n a l p o i n t s a b o u t t h e t h e -o r y a n d a p p l i c a ti o n s o f o p e r a t i o n a l a n a l y s ish a v e b e e n g i v e n i n [ B O U H 7 8 , B U Z E 7 7,B u z E 7 8 a ] .2 . V A L I D A T I O N A N D P R E D I C T I O NW e h a v e n o t e d t h r e e u s e s o f m o d e l s i ns t u d y i n g c o m p u t e r p e r f o r m a n c e : c a l c u l a -t io n , c o n s i s te n c y - c h e c k i n g , a n d p r e d i c t i o no f p e r f o r m a n c e m e a s u r e s . V al i da t i on r e f e r st o e x t e n s i v e t e s t in g o f a m o d e l t o d e t e r m i n ei t s a c c u r a c y i n c a l c u l a t i n g p e r f o r m a n c em e a s u r e s . Predic t zon r e f e r s t o u s i n g a v a l -i d a t e d m o d e l t o c a l c u l a t e p e r f o r m a n c em e a s u r e s f o r ' a t i m e p e r i o d ( u s u a l ly i n th ef u tu r e ) w h e n t h e v a l u es o f p a r a m e t e r s r e -q u i r e d b y t h e m o d e l a r e u n c e r t a i n .F i g u r e 2 i l l u s t r a t e s t h e s t e p s f o l l o w e d i na t y p i c a l v a l i d a t i o n . F i r s t , t h e a n a l y s t r u n sa n a c t u a l w o r k l o a d o n a n a c t u a l s y s t e m .F o r t h e o b s e r v a t i o n p e r i o d , h e m e a s u r e sp e r f o r m a n c e q u a n ti ti e s, s u c h a s t h r o u g h p u ta n d r e s p o n s e t i m e , a n d a ls o t h e p a r a m e t e r so f t h e d e v i c e s a n d t h e w o r k l o a d . T h e n t h ea n a l y s t a p p l i e s a m o d e l t o t h e s e p a r a m -e t e r s, a n d c o m p a r e s t h e r e s u l t s a g a i n s t t h e

    m e a s u r e d p e r f o r m a n c e q u a n t it ie s . I f, o v e rm a n y d i f f e r e n t o b s e r v a t i o n p e r i o d s , t h ec o m p u t e d v a l u e s c o m p a r e w e l l w i t h a c t u a l( m e a s u r e d ) v a l u e s , t h e a n a l y s t w il l c o m e t ob e l ie v e t h a t t h e m o d e l is g o od . T h e r e a f t e r ,h e w i ll e m p l o y i t c o n f i d e n t l y f o r p r e d i c t i n gf u t u r e b e h a v i o r a n d f o r e v a l u a t i n g p r o -p o s e d c h a n g e s in t h e s y s te m .T h e s c h e m e o f F i g u r e 2 is u s e d t o v a l i d a tem a n y t y p e s o f m o d e ls , i n c lu d i n g h ig h l y d e -t a i l e d d e t e r m i n i s t i c m o d e l s , s i m u l a t i o nm o d e l s , a n d q u e u e i n g n e t w o r k m o d e l s . I ng e n er a l, th e m o r e p a r a m e t e r s u s ed b y t h em o d e l , t h e g r e a t e r i s i ts a c c u r a c y in s u c hva l i da t i ons .P e r f o r m a n c e p r e d i c t i o n t y p i c a l ly fo l lo w st h e s c h e m e o f F ig u r e 3. T h e a n a l y s t b e gi n sw i t h a s e t o f w o r k l o a d a n d d e v i c e p a r a m -e t e r s f o r a p a r t i c u l a r o b s e r v a t i o n p e r i o d ,k n o w n a s t h e base l ine per iod . H e t h e nc a r r i e s o u t a m o d i f i c a t i o n a n a l y s i s t o e s t i -m a t e t h e v a l u e s t h e s e p a r a m e t e r s a r e e x -p e c t e d t o h a v e i n t h e projec t ion per iod ,w h i c h is a n o t h e r t i m e p e r io d f o r w h i c h h e

    ~ ( )eosuem4m Perocmotceo~ao~ODEL VALID~F IGU R E 2 . T yp i c a l va l i d a t i o n s c h em e .

    ?

    MODIFCATONANALYSIS

    F IG U R E 3 . T y p i c a l p e r f o r m a n c e p r e d i c t i o n s c h e m e .

    Com puting Surveys, Vol 10, No 3, Septem ber 1978

  • 8/14/2019 62540 Denning

    7/37

    The Operational Analysis of Queueing Network Modelsdesires to know performance quantities. (Inthe projection period, the same system maybe processing a changed workload, or achanged system may be processing thesame workload, or both.) The analyst ap-plies the validated model to calculate per-formance quantities for the projection pe-riod. If the modification is ever imple-mented, the predictions can be validated bycomparing the actual workload and systemparameters against the project values (#1)and the actual performance quantitiesagainst the projected quantities (#2).A variety of invariance assumptions areemployed in the modification analysis.These assumptions are typically that deviceand workload parameters do not changeunless they are explicitly modif ied--the an-alyst may assume, for example, that themean disk service time will be invariant ifthe same disk is present in both the baselineand projection periods, or that the meannumber of requests for each disk will beinvariant if the same workload is present inboth periods. Th ough usually satisfactory,such assumptions can lead to trouble if agiven change has side effects--for example,increasing the number of time-sharing ter-minals may unexpectedly reduce the batchmultiprogramming level even though thebatch workload is the same.The wise analyst will make all his invar-iance assumptions explicit. Otherwise, hewill have difficulty in explaining a failure inValidation #1, which will cause a failure inValidation #2-- eve n though previous testsof the model were satisfactory (Figure 2).In some prediction problems there is noexplicit baseline period. In these cases, theanalyst must estimate parameters for theprojection period by other means. For ex-ample, he can estimate the mean servicetime for a disk from published specifica-tions of seek time, rotation time, and d atatransfer rate; and he can estimate the meannumber of disk requests per job from ananalysis of the source code of representativeprograms. Usually, however, the modifica-tion analysis is more accurate when it be-gins with a measured baseline period.A model's quality depends on the numberof parameters it requires. The more infor-mation t he model requires about the work-

    231load and the system, the greater the accu-racy a ttainable in its calculations. However,when there are many parameters, theremay be a lot of uncertainty about whetherall are correctly estimated for a projectionperiod; the confidence in the predictionsmay thereby be reduced. Queueing networkmodels isolate the few critical parameters.They permit accurate calculation and cred-ible prediction.Additional issues of performance calcu-lation and parameter estimation will be dis-cussed as the y arise throughout the paper.(See also [BuzE77, BUZE78a].)

    3 . O P E R A T I O N A L M E A S U R E S O FN E T W O R K SFigure 1 illus trated a single resourcequeueing model consisting of a queue anda service facility. This model can be usedto represent a single input/output (I/O)device or central processing unit (CPU)within a computer system. A model of theentire computer system can be developedby connecting single-resource models in thesame configuration as the devices of anactual computer system. A set of intercon-nected single-resource queueing modelscomprises a multiple-resource queueingnetwork.T y p e s o f N e t w o r k sFigure 4 shows two of K devices in a mul-tiple-resource network. A job enters thesystem at IN. It circulates around in thenetwork, waiting in queues and having ser-vice requests processed at various devices.When done, it exits at OUT. The networkis operationally connected in that each de-vice is visited at least once by some jobduring the observation period.The model assumes tha t no job overlapsits use of different devices. In practice, fewapplications programs ever achieve morethan a few per cent overlap between CPUand I/O devices: the error introduced bythis assumption is usually not significant.2The model also assumes that a device is2 Measure ments taken at the Purdue Umverslty Com-puter Center reveal that the average overlap of CPUand I/O within a job is between 4 and 6 per cent.

    Computing Surveys, Vol 10, No 3, September 1978

  • 8/14/2019 62540 Denning

    8/37

    232 P . J . D e n n i n g a n d J . P . B u z e nKN DevicesOobs

    . ~ - ~ D ev ice X ~ ~ / / ~

    / q lJX j

    ' q o I q o i . . . . . . q loIN O U T

    ( c l o s e d )F IG U R E 4 . T w o d e v i c e s i n a q u e u e i n g n e t w o r k .

    . . .

    busy if a request is pending ther emno partof the system can block progress in anotherpart. Th is assumption is not met by all realsystems; for example, the CPU might beunable to continue if an I/O buffer is full.A job is in queue at device i if it iswaiting for or receiving service there. Welet n, denote the number of jobs in queueat device i, and N = n l + + n K denote thetotal nu mber of jobs in the system. Thes y s t e m o u t p u t r a t e , X o , is the number ofjobs per second leaving the system. If thesystem is o p e n , X o is known and N variesas jobs enter or leave the system. If thesystem is c l o s e d , the numb er of jobs N isfixed; this is modeled by connecting theoutput back to the input, as suggested bythe dashed arrow in Figure 4.An analysis of an open system assumesthat X0 is known and seeks to characterize

    the distribut ion of N. An analysis of a closedsystem begins with N given and seeks todetermine the resulting X0 along the OU T/IN path. Other quantities such as queuelengths and response times at the devicesmay be sought in both cases.

    Example: Figure 5 shows a common typeof network, the central server. Device 1is the CPU, devices 2, , K are I/O sta-tions. A job begins with a CPU serviceinterval (burst) and continues with zero ormore I/O service intervals which alternatewith further CPU bursts. The quantitiesqu are called the routing frequencies andthe S, the mean service times. Definitionsfor these quantities will be given shortly.In the closed central server network ofFigure 5, a new job enters the system assoon as an active job terminates. This be-

    Computing Surveys, Vol 10, No 3, September 1978

  • 8/14/2019 62540 Denning

    9/37

    The Operational Analysis of Queueing Network Models 233

    IN

    q l O + q 1 2 + + q l K f f i I

    i q l o

    Xo

    q l KSK

    q l 2 ~ ~ - - ~ - ~S2

    OUT

    F IG U R E 5. C e n t r a l s e r v e r n e t w o r k .

    havior typically occurs in a batch process-ing system operating under a backlog. Thethrou ghpu t of the system under these con-ditions is denoted by X0.Time sharing systems which are drivenby interactive terminals can also be repre-sented as closed networks. Figure 6 depictsthe structure. The model is separated intotwo {open) subnetworks: the central sub-system, which consists of I/O devices andthe CPUs, and the terminal subsystem.Each terminal is manned by a user whoalternates between thinking and waiting.In the thinking state, the user is contem-plating what job next to submit, and thecentral subsystem is performing no workfor him. On submitting a next job, the userenters the waiting state, where he remainsuntil the central subsystem completes thejob for him. The mean time a user spendsin a thinking interval is called the thinktime; we denote it by Z. The mean time a

    user spends in a waiting interval is calledthe response time (of the central subsys-tem); we denote it by R. Since users thinkindependently, the think time Z is indepen-dent of M. Because jobs delay each otherwhile contending for resources in the cen-tral subsystem, R is a function of M.It is also possible to define mixed systemswhich are open for some workloads andclosed for others. Figure 7 illustrates a typ-ical case. The interactive workload com-prises the jobs associated with the M inter-active terminals. The batch workload com-prises jobs submitted by other means, forexample, remote job entry stations. Thenumber of interactive jobs in the network(including the termina l subnetwork) is fixedat M, but the number of batch jobs may bevariable. The batch thro ughput (Xo) isgiven, but the interactive throughput (Xo')depends on X0 and on the other parametersof the network.

    C o m p u t m g S u r v e y s , V o l 1 0, N o . 3 , S e p t e m b e r 1 9 7 8

  • 8/14/2019 62540 Denning

    10/37

    234 P. J . Denn i ng an d J . P . Buz en;

    T E R M I N A LS U B S Y S T E M

    - ~ N ~

    M T e r m , n o l sZ T h i n k T , m e

    C E N T R A L S U B S Y S T E M

    [ i O U TK D e v i c e s N J o b s O~N 0);C,~--number of times a job requests ser-vice at device j immedia tely aftercompleting a service request at de-

    vice i.These are similar to data specified in Figure1, but here we are not requiring device i tobe a single server. If we treat the outs ideworld as device O , we can define alsoAoj-- number of jobs whose first servicerequest is for device j;C,o--number of jobs whose last servicerequest is for device i.

    We will assume th at Coo = 0, because other-wise there would be jobs that used no re-sources before departing. However, it ispossible that C, > 0 for any device i sincea job could request another burst of servicefrom a device which had just completed aC o m p u t i n g S u r v e y s , V o l 1 0 , N o 3 , S e p t e m b e r 1 9 7 8

  • 8/14/2019 62540 Denning

    11/37

    T h e O p e r a t io n a l A n a l y s i s o f Q u e ue i ng N e t w o r k M o d e l sr e q u e s t f o r t h a t j o b . T h e n u m b e r o f c o m -p l e t i o n s a t d e v i c e i i s

    KC,=ECv, t = l . . . . . K.j - - 0

    T h e n u m b e r o f a r r iv a ls t o , a n d d e p a r t u r e sf r o m , t h e s y s t e m a r e , r e s p e c t i v e l y ,K KA0 = ~ Ao~, C o = ~ C , o .j - - 1 t - I

    F r o m F i g u r e 4 i t is c l e a r t h a t A o = C o i n ac l o s e d s y s t e m .I n t e r m s o f t h e s e b a si c d a t a , f o u r d e r i v e do p e r a t i o n a l q u a n t i t i e s a r e d e f i n e d :U , ffi u t i l i z a t i o n o f d e v i c e i

    = B , / T .S , ffi m e a n s e rv i c e t i m e b e t w e e nc o m p l e t i o n s o f r e q u e s t s a td e v i c e i

    = B, /C~X, = output rate of requests fromd e v i c e i

    = C , / Tq u ---- r o u t i n g f r e q u e n c y , t h e f r a c t i o no f j o b s p r o c e e d i n g n e x t t od e v i c e j o n c o m p l e t i n g as e r v i c e r e q u e s t a t d e v i c e if C , / C , , i f / - - 1 . . . . K

    = --LAoj/Ao, i f i = 0 .

    n t ( t )A/ \

    B ,( t )

    654 -3 -2 .I -

    0 5 I 0 1 5 2 0FIGURE 8. Exam ple of a device's operation.

    N o t e t h a t , f o r a n y i , q ,o + qtl + . . . + qtK =1 . N o t e t h a t q,0 i s a n o u t p u t r o u t i n g f r e -q u e n c y ( f r a c t i o n o f c o m p l e t i o n s f r o m d e -v i c e i c o r r e s p o n d i n g t o t h e f i n a l s e r v i c er e q u e s t o f s o m e j o b ) a n d q0j i s a n i n p u tr o u t i n g f r e q u e n c y ( f r a c t io n o f a r r i v a ls t ot h e s y s t e m w h i c h p r o c e e d f ir s t t o d e v i c e j ) .N o t e a l s o t h a t t h e s y s t e m o u t p u t r a t e isd e f i n e d a s X o - - C o / T . I t i s e a s y t o d e d u c et h e o p e r a t i o n a l la w

    KXo = ~ X,q,o.t - 1

    N o t e t h a t X 0 , X 1 . . . . . X r c a n n o t b e i n t e r -p r e t e d a s " t h r o u g h p u t s " b e c a u s e n o a s -s u m p t i o n o f j o b f lo w b a l a n c e h a s b e e nm a d e . I t i s c l e a r t h a t t h e u t i l iz a t i o n la wU, = X,S ,

    h o l d s a t e v e r y d e v ic e .W e l e t n , d e n o t e t h e q u e u e l e n g t h a td e v i c e i ; i t i n c l u d e s j o b s w a i t i n g f o r a n d

    2 3 5

    r e c e i v i n g s e rv i c e . S o m e t i m e s w e w r i t e n , (t )t o m a k e e x p l ic i t t h e t i m e d e p e n d e n c e . ( A ne x a m p l e n , ( t ) a p p e a r s i n F i g u r e 8 . ) T o c a l -c u l a t e m e a n q u e u e l e n g t h a n d r e s p o n s et i m e a t a d e v i c e , a n a l y s t s u s u a l l y i n t r o d u c et h e b a s ic m e a s u r e W ,, w h i c h i s t h e a r e au n d e r t h e g r a p h o f n , (t ) d u r i n g t h e o b s e r -v a t i o n p e r io d . S i n c e f ~,, t h e m e a n q u e u el e n g t h a t d e v i c e i, i s t h e a v e r a g e h e i g h t o ft h i s g r a p h ,ft, = W JT .T h e m e a n r e s p o n s e t i m e a t d e v i c e i , d e -n o t e d b y R , , i s a l so r e l a t e d t o W , i n a s i m p l ew a y . N o t e t h a t W~ c a n b e i n t e r p r e t e d a st h e t o t a l n u m b e r o f " j o b - s e c o n d s " a c cu -m u l a t e d a t d e v i c e i d u r i n g t h e o b s e r v a t i o np e r i o d ( i f j j o b s a r e a t a d e v i c e f o r s s e co n d s ,j s j o b - s e c o n d s a c c u m u l a t e ) . R , is d e f in e d a st h e a v e r a g e a m o u n t o f t i m e a c c u m u l a t e d a td e v i c e i p e r c o m p l e t e d r e q u e s t . T h u sR, = W, /C, .

    A n i m m e d i a t e c o n s e q u e n c e o f t h e s e d e fi -n i t i o n s i s t h e o p e r a t i o n a l l a w6, = X~R,,

    w h i c h i s c a l l e d L i t t l e ' s L a w .E x a m p l e : F i g u r e 8 s h o w s d e v i c e t a n d ap o s s ib l e o b s e r v a t i o n o f i t s q u e u e l e n g t h f o ra p e r i o d o f 2 0 s e c o n d s . T h e b a s ic m e a s u r e sa r e

    A, = 7 job s, B, = 16 seco nd s, C, ffi 10 job s.N ote th a t n ,(0) f fi 3 and th a tn,(20) ffi n,(0) + A , - C~ = 0.

    T h e b a s i c o p e r a t i o n a l p e r f o r m a n c e m e a s u r e sa r e

    C o m p u t i n g S u r v e y s , V o l 1 0, N o . 3, S e p t e m b e r 1 9 78

  • 8/14/2019 62540 Denning

    12/37

    2 3 6 P . J . D e n n i n g a n d J . P . B u z e nU, -- 16/2 0 St = 16/ 10 X , ffi 10/20

    = 0.80 --- 1.6 -- 0.5s e c o n d s j o b s / s e c o n dT h e t o t a l a r e a u n d e r n,(t) i n t h e o b s e r v a t i o nper iod i s

    W~ -- 40 job-s eco nd s.T h u s t h e m e a n q u e u e l e n g t h isht = W , / T ffi 2 jo bs ,

    a n d t h e m e a n r e s p o n s e t i m e p e r s e r v i c ecom ple t ion i s :R, f f i WJC, = 4 seconds .

    4 . J O B F LO W A N A L Y S I SG i v e n t h e m e a n s e r v i c e t i m e s (S ,) a n d t h er o u t i n g f r e q u e n c i e s (q,j), h o w m u c h c a n w ed e t e r m i n e a b o u t o v e ra l l d e v i ce c o m p l e t i o nr a t e s ( X J o r r e s p o n s e t i m e s ( R J ? T h e s eq u e s t i o n s a r e u s u a l l y a p p r o a c h e d t h r o u g ht h e o p e r a t i o n a l h y p o t h e s i s k n o w n a s t h e

    P r m c i p l e o f J o b F l o w B a l a n c e : F o re a c h d e v i c e i , X t i s t h e s a m e a s t h et o t a l i n p u t r a t e t o d e v i c e i .T h i s p r i n c i p l e w i l l g i v e a g o o d a p p r o x i m a -t io n f o r o b s e r v a t i o n p e r i o d s l o n g e n o u g ht h a t t h e d i f fe r e n c e b e t w e e n a r r i v a ls a n dc o m p l e t i o n s , A t - C , is s m a l l c o m p a r e d t oC~. I t w i ll b e e x a c t i f t h e i n i t i a l q u e u e l e n g t hn ~(0 ) i s t h e s a m e a s t h e f i n a l n , ( T ) . C h o o s i n ga n o b s e r v a t i o n p e r i o d s o t h a t t h e i n i ti a l a n df in a l s t a t e s o f e v e r y q u e u e a r e t h e s a m e isn o t a s s t ra n g e a s i t m i g h t s e e m . T h i s n o t i o nu n d e r l i e s t h e h i g h l y s u c c e s s f u l " r e g e n e r a -t i o n p o i n t " m e t h o d o f c o n d u c t i n g s i m u l a -t i o n s [ I G L E 7 8 ] .W h e n j o b f lo w is b a l a n c e d , w e r e f e r t ot h e X , a s d e v i c e t h r o u g h p u t s . E x p r e s s i n gt h e b a l a n c e p r in c i p l e a s a n e q u a t i o n ,

    KC j f A j = Z C ,j , t = O . . . . gt--O

    ( N o t e t h a t j o b f lo w b a l a n c e a l l o w s u s tos u b s t i t u t e C oj f o r A o~ .) W i t h t h e d e f i n i t i o nqtj = C J C , , w e m a y w r i t eKCj = E C,q,j.

    tmOE m p l o y i n g t h e d e f i n i t i o n X ~ f f i C , / T , w eo b t a i n

    J O B F L OW B A L A N C E E Q U A T I O N SKX+ffi Y,X,q,~, / f f i O . . . . gtmO

    I f t h e n e t w o r k i s o p e n , t h e v a l u e o f X 0 i se x t e r n a l l y s p ec i f i e d a n d t h e s e e q u a t i o n sw i ll h a v e a u n i q u e s o l u t i o n f o r t h e u n -k n o w n s X ,. H o w e v e r , i f t h e n e t w o r k isc l o se d , X o i s i n i t ia l l y u n k n o w n , a n d t h ee q u a t i o n s h a v e n o u n i q u e s o l u t i o n ; t h i s c a nb e v e r i f i e d b y s h o w i n g t h a t t h e s u m o f t h eX j - e q u a t i o n s f o r j ffi 1 . . . , K r e d u c e s t o t h eX o - e q u a t i o n . I n a c l o s e d n e t w o r k , t h e r e a r eK i n d e p e n d e n t e q u a t i o n s b u t K + 1 u n -k n o w n s . N o n e t h e l e s s , t h e j o b f lo w b a la n c ee q u a t i o n s c o n t a i n i n f o r m a t i o n o f c o n s id e r -a b l e v a l u e .V i s i t R a t i o sT h e " v i s i t r a t io , " w h i c h e x p r e s s e s t h e m e a nn u m b e r o f r e q u e s t s p e r j o b f o r a d e v i ce , c a na l w a y s b e c a l c u l a t e d u n i q u e l y f r o m t h e j o bf lo w b a l a n c e e q u a t i o n s . W i t h t h e m e a n s er -v i ce ti m e s , t h e y c a n b e u s e d t o d e t e r m i n et h e t h r o u g h p u t s a n d r e s p o n s e t im e s o f s ys -t e m s u n d e r v e r y l i g h t o r v e r y h e a v y l o ad s .D e f i n e

    V , = X , / X o ;V~ i s t h e j o b f lo w t h r o u g h d e v i c e t r e l a t i v et o t h e s y s t e m ' s o u t p u t f l o w . O u r d e f i n i t i o n si m p l y t h a t V , ffi C, /Co , w h i c h i s t h e m e a nn u m b e r o f c o m p l e t i o n s a t d e v i c e i f o r e a c hc o m p l e t i o n f r o m t h e s y s t e m . S i n c e V , c a nb e i n t e r p r e t e d a s t h e m e a n n u m b e r o f v is it sp e r j o b t o d e v i c e i , w e c a ll it t h e v i s i t r a t i o .T h e r e l a t i o n X , f f i V, Xo i s a n o p e r a t i o n a ll aw , c a l l e d t h e F o r c e d F l o w L a w . I t s t a t e st h a t t h e f l o w i n a n y o n e p a r t o f t h e s y s t e md e t e r m i n e s t h e f l o w s e v e r y w h e r e in t h e s y s -t e m .E x a m p l e : C o n s i d e r t h e p e r f o r m a n c e q u e s -t i o n : " J o b s g e n e r a t e a n a v e r a g e o f 5 d i s kr e q u e s t s a n d d i s k t h r o u g h p u t i s m e a s u r e da s 1 0 r e q u e s t s / s e c o n d ; w h a t i s t h e s y s t e mt h r o u g h p u t ? " T h i s q u e s ti o n se e m s m o m e n -t a r t l y fr iv o l o u s , s i n c e n o t h i n g i s s ta t e d a b o u tt h e r e l a t io n b e t w e e n t h e d i sk a n d a n y o t h e rp a r t o f t h e s y s t e m . Y e t t h e f o r c e d f lo w la wgives the answer p r ec i se ly . Le t subsc r ip t tr e f e r to the d isk:

    Xo ffi X ,/ V ,ffi 10 r eque s t s / se co nd5 r e q u e s t s / j o bffi 2 job s /se co nd .O n r e p l a c i n g e a c h X , w i t h V, Xo i n t h e j o bf lo w b a l a n c e e q u a t i o n s , w e o b t a i n t h e

    Computmg Surveys, Vol. 1O, No. 3, September 1978

  • 8/14/2019 62540 Denning

    13/37

    T h e O p e r a t i o n a l A n a l y s i s o f Q u e u e i n g N e t w o r k M o d e l sV I S I T R A T I O E Q U A T I O N S

    1 7 0 = 1 KV ~ = q o j + ~ V,q,j, 1 = 1 . . . . . K

    T h e s e a r e K + 1 i n d e p e n d e n t e q u a t i o n sw i t h K + 1 u n k n o w n s : a u n i q u e s o l u t i o n i sa l w a y s p o ss i b le a s s u m i n g t h e n e t w o r k i so p e r a t i o n a l l y c o n n e c t e d . T h e s e e q u a t i o n ss h o w t h e r e l a t i o n b e t w e e n t h e n e t w o r k ' s" c o n n e c t i v e s t ru c t u r e , " r e p r e s e n t e d b y t h eq, j , a n d t h e v i s i t r a t io s . A l t h o u g h V~ =X , / X o i s a n o p e r a t i o n a l l a w , t h e iT, s a t i s f yt h e v i s it r a t i o e q u a t i o n s o n l y i f j o b f lo w i sb a l a n c e d i n t h e n e t w o r k .

    E x a m p l e : T h e c e n tr a l s er v e r n et w o r k(F igure 5) has these job f low equa t ions :X o = X l q l oX I f X o + X 2 + . . . + X ~X , = X I q I , , i f f i 2 . . . . . K .

    S e t t i n g X, = V, Xo, t h e s e e q u a t i o n s r e d u c eto1 = Vlqlo

    V ] = I + V 2 + . . . + V KV , = Vlq~,, t = 2 . . . . . K .

    I t i s e a s y t o s e e t h a tV1 -- 1/qloV , = q l , /q l o , i = 2 . . . . K .

    In th i s case , on ly K of the poss ib le rou t ingf requ enc ies q~j are non zero; th ese q~, can bede te rm ined un iq ue ly i f the 17, a r e g iven.T h i s i s n o t s o i n a g e n e r a l n e t w o r k , w h e r eK v i s i t r a t ios a r e insuf f ic ien t to de te rminet h e ( K + 1 ) 2 u n k n o w n r o u t i n g f r e q u e n c i e s.A s w e s h a l l s e e , a l l t h e p e r f o r m a n c e q u a n -t i t ie s c a n b e c o m p u t e d u s i n g o n l y t h e v i s i tr a t i o s a n d t h e m e a n s e r v i c e t i m e s S , a sp a r a m e t e r s . T h e v i si t r a t io e q u a t i o n s a r eu s e d t o p r o v e t h a t t h i s is s o. In p r a c t i c e , t h ea n a l y s t m a y b e a b l e to e x t r a c t th e v i si tr a t io s d i r e c t l y f r o m w o r k l o a d d a t a , t h e r e b ya v o i d i n g c o m p u t i n g a s o l u t i o n o f t h e v i s itr a t i o e q u a t i o n s .

    S y s t e m R e s p o n s e T i m eO n e m e t h o d o f c o m p u t i n g th e m e a n r e -s p o n s e t i m e p e r j o b , R , f o r a n o p e n o r c l o s e ds y s t e m is to a p p l y L i t t l e 's l aw t o t h e s y s t e ma s a w h o l e ,

    R = ~ f / X o ,

    2 3 7w h e r e f il = f h + . . . + f i x. I f / g / o r X o a r e n o tk n o w n , a n a l t e r n a t e m e t h o d c a n b e u se d .S i n c e h , = X , R , f r o m L i t t l e 's la w a t d e v i c ei , a n d X , = V , X o f r o m t h e f o r c e d f l o w l aw ,w e h a v e f i , / X o = V , R , . T h i s r e d u c e s [ V / X ot o t h e G e n e r a l R e s p o n s e T t m e L a w :KR f Y. V~R,.

    t l lT h i s l a w h o l d s e v e n i f j o b f lo w i s n o t b a l -a n c e d .L i t t l e ' s l a w c a n b e u s e d t o c o m p u t e t h ec e n t r a l s u b s y s t e m ' s r e s p o n s e t i m e R i n t h et e r m i n a l d r i v e n s y s t e m o f F i g u r e 6. T h em e a n t i m e f o r a u s e r t o c o m p l e t e a t h in k -w a i t c y c l e i s Z + R . W h e n j o b f l o w i sb a l a n c e d , X 0 w i ll d e n o t e t h e r a t e a t w h i c hc y c l e s a r e c o m p l e t e d . B y L i t t l e ' s l a w ,( Z + R ) X o m u s t b e t h e m e a n n u m b e r o fu s e r s o b s e r v e d t o b e i n a t h i n k - w a i t c y c l e ;b u t a l l t h e u s e r s a r e i n s u c h c y c l es , h e n c e ,M = ( Z + R ) X o . T h e r e f o r e ,

    R f f i M / X o - Z .T h i s i s c a l l e d t h e I n t e r a c t i v e R e s p o n s eT i m e F o r m u l a .Examples

    T h i s s e c t i o n ' s t h r e e e x a m p l e s i l l u s t ra t e p e r -f o r m a n c e c a l c u l a ti o n a n d p e r f o r m a n c e p r e -d i c t io n u s in g t h e o p e r a t i o n a l l a w s s u m m a -r i z e d i n T a b l e I . T h e f i r s t e x a m p l e i l l u s -t r a t e s a s i m p l e p e r f o r m a n c e c a l c u l a t i o n ; af ew m e a s u r e d d a t a a r e u s e d t o fi n d t h em e a n r e s p o n s e t i m e . T h e s e c o n d e x a m p l ei l l u s t ra t e s a p e r f o r m a n c e c a l c u l a t i o n f o r as y s t e m w i t h a n i n t e r a c t i v e a n d a b a t c hw o r k l o a d ; i t a l s o i l l u s t r a t e s a p e r f o r m a n c ep r e d i c t i o n , e s t i m a t i n g t h e e f f e c t o f t r i p l e d

    T A B L E I . O P ERATI ONAL EQUATIO NS *U t t h z a t t o n L a w U , = X , S ,L t t t l e ' s L a w f ~ ffi X , R ,F o r c e d F l o w L a w X ~ ffi V , X o

    KO u t p u t F l o w L a w X o ffi ~ X ,q ,o

    KG e n e r a l R e s p o n s e T t m e L a w R ffi ~ V ,R ,I n t e r a c t t v e R e s p o n s e T i m e R - M / X o - ZF o r m u l a ( A s s u m e s f l o w b a l -a n c e )* O p e r a t i o n a l d e r i v a t i o n s f o r m o s t o f t h e s e e q u a t i o n sw e r e fL rst p r e s e n t e d I n [ B u z E 7 6 b ] .

    Com putmg Surveys , Voi. 10 , No. 3 , Septem ber 1978

  • 8/14/2019 62540 Denning

    14/37

    238 P. J . Denn ing and J . P . Buzenbatch throughput on interactive responsetime. The third example illustrates a morecomplex predict ion problem, estimating theeffect of consolidating two separate timesharing systems; it illustrates the use ofinvariance assumptions in the modificationanalysis.

    For the first example, we suppose thatthese data have been measured on a timesharing system:Each job generat es 20 disk requests;Th e disk utilization is 50%;The mean service time at the disk is 25milliseconds;There are 25 terminals; andThink time is 18 seconds.

    We can calculate the response time afterfirst calculating the throughput. Let sub-script i refer to the disk. The forced flowand utilization laws implyXo = X,/V, ffi uJv, s,.

    From the data,(.5)Xo ffi - - ffi 1 job/second.(20)(.025)

    From the interactive response time for-mula,R ffi 20/1 - 18 ffi 2 seconds.

    Our second example considers a mixedsystem of the type shown in Figure 7. The sedata are collected:There are 40 terminals;Think time is 15 seconds;Inter active response time is 5 seconds;Disk mean service time is 40 milliseconds;Each interactive job generates 10 diskrequests;Eac h ba tch job ge nerates 5 disk requests;andDisk utilization is 90%.

    We would like to calculate the thro ughpu tof the batch system and then estimate alower bound on interactive response timeassuming that batch throughput is tripled.The interactive response time formula givesthe interactive throughput:

    Xo' ffi M/(Z + R')ffi 40/(15 + 5)ffi 2 jobs/second.Let subscript i refer to the disk. The diskthroughput is X, + X,', where X, is the b atchcomponent and X / i s the interactive com-ponent. Th e utilization law implies

    X, + X; ffi U,/S,ffi (.9)/(.04)ffi 22.5 requests/second.The forced flow law implies that the inter-active compo nent is

    X, ffi V,'Xo' ffi (10)(2) ffi 20 requests/second,so that the batch component is

    X, -- 22.5 - 20 ffi 2.5 requests/second.Using th e fo rced flow law again, we find thebatch throughput:Xo ffi X J V , ffi 2.5/5 ffi 0.5 jobs/second.

    Now consider the effect of tripling thebatc h throughput. If X0 were changed to 1.5jobs/second without changing V, then X,would become V~0 ffi 7.5 requests/second.Assuming that the increased throughputdoes not change the disk service time, themaximum completion rate at the disk is1/S~ ffi 25 requests/ second ; this implies t ha tthe completion rate of the interactive work-load, X/ , cannot exceed 25 7.5 ffi 17.5requests/second. Therefor e

    Xo' = X,'/V;

  • 8/14/2019 62540 Denning

    15/37

    T h e O p e r a t m n a l A n a l y s i s o f Q u e u e i n g N e t w o r k M o d e l si n g s y s t e m s ; e a c h i s b a s e d o n a s w a p p i n gd i s k w h o s e m e a n s e r v i c e t i m e p e r r e q u e s tis 4 2 m s e c . T h e m e a n t h i n k t i m e i n b o t hs y s t e m s is Z - 1 5 s e c o n d s . T h e s e d a t a h a v eb e e n c o l l e c t e d :

    Sys tem A Sys tem B16 te rmin a l s 10 t e rm ina l s2 5 d i sk r e q u e s t s / j o b 1 6 d i s k r e q u e s t s / j o b80% disk u t i l i za t ion 40% d isk u t i l i za t ionI n o r d e r t o r e d u c e d i s k r e n t a l s , m a n a g e -m e n t i s p r o p o s i n g t o c o n s o l i d a t e t h e t w os y s t e m s i n t o o n e w i t h 2 6 t e r m i n a l s a n du s i n g o n l y o n e o f t h e d is k s. W e w o u l d l i k et o e s t i m a t e t h e e f f e ct o n t h e r e s p o n s e t i m e sf o r t h e t w o c l a s s e s o f u s e r s .

    W e l e t s u b s c r i p t i r e f e r t o t h e d i sk , a n du s e p r i m e d s y m b o l s to r e f e r t o S y s t e m B .T h e f o r m u l a X0 = U,/V~S, g i v e s t h r o u g h -p u t s f o r t h e t w o s y s t e m s :(.8)X o - - - -(25)(.042)

    = 0 .7 7 j o b s / s e c o n d ( S y s t e m A )(.4)Xo' (16)(.042)

    = 0 .6 0 j o b s / s e c o n d ( S y s t e m B )T h e r e s p o n s e t i m e s a r eR --- 16/ (.77 ) - 15

    = 5 .8 second s (Sys tem A)R 'ff i 10/( .6) - 15

    = 1 .1 second s (Sys tem B)O v e r a n o b s e r v a t i o n p e r i o d o f T s e c o n d st h e r e w o u l d b e X , T d i s k r e q u e s t s s e r v i c e di n S y s t e m A , a n d X { T i n S y s t e m B ; t h ef r a c t i o n o f al l d i s k r e q u e s t s w h i c h a r e A -r e q u e s t s w o u l d b e

    X~ T/(X,T + X{T ) ffi U,/ (U, + U{) = 2/3 .I n o r d e r t o e s t i m a t e t h e e f f e c t o f c o n s o l -i d a t i o n , w e n e e d t o k n o w t h e d i s k c o m p l e -t i o n r a t e s f o r e a c h w o r k l o a d w h e n b o t hw o r k l o a d s s h a r e t h e o n e d i s k . B e c a u s e t h ec h a r a c t e r i s t i c s o f t h e u s e r s a n d t h e d i s k a r et h e s a m e b e f o r e a n d a f t e r t h e c h a n g e , i t i sr e a s o n a b l e t o m a k e t h is in v a r i a n c e a s s u m p -t io n : I n t h e c o n s o l i d a t e d s y s t e m , 2 / 3 o f t h ed i s k r e q u e s t s w i l l c o m e f r o m t h e A - u s er s . I ti s a l s o r e a s o n a b l e t o a s s u m e t h a t t h e d i s ku t i l i z a t i o n w i l l b e n e a r l y 1 0 0 % i n t h e c o n -s o l i d a t e d s y s te m . T h i s i m p l ie s t h a t t h e t o t a ld i sk t h r o u g h p u t w i ll b e 1 / S , - - 1 / ( . 0 4 2 ) =

    2 3 92 4 r e q u e s t s / s e c o n d . O f th i s t o t a l , t h et h r o u g h p u t s o f t h e t w o t y p e s o f u s e r s a re

    X, = (2/3)(24)= 16 r eq ue s t s / se co nd (A-use rs )

    X { = ( 1 / 3 ) ( 2 4 )ffi 8 r equ es t s / s eco nd (B-use r s)

    T h i s i m p l i e s t h a t t h e s y s t e m t h r o u g h p u t sa r eXo = XJV~

    = 16/25ffi 0 .64 jo bs /se co nd (A-users)

    Xo' = X / / V /= 8 / 1 6- - 0 .5 job s / se co nd (B-use r s)a n d t h a t t h e r e s p o n s e t i m e s a r e

    R = 16/(.64) - 15= 10 seco nds (A-users)

    R '= 10 / ( .5 ) - 15= 5 seconds (B-use r s )

    N o t e t h a t t h e t w o t y p e s o f u s e r s e x p e r i e n c ed i f f e r e n t r e s p o n s e t i m e s . T h i s i s b e c a u s et h e B - u s e r s , w h o g e n e r a t e l es s w o r k f o r t h ed i sk , a r e d e l a y e d le s s a t t h e d i s k t h a n t h eA - u s e r s .O n c e a g a i n i t is w o r t h n o t i n g e x p l i c i t l yt h a t t h e p a r a m e t e r s V,, ]7,', S,, a n d Z a r ea s s u m e d t o b e i n v a r ia n t u n d e r t h e p r o p o s e dc h a n g e . T h e c a r e fu l a n a l y s t w i l l c h e c k t h ev a l i d i t y o f t h e s e a s s u m p t i o n s . T h e a s s u m p -t io n t h a t t h e r a t i o o f S y s t e m A t o S y s t e mB t h r o u g h p u t s is i n v a r i an t u n d e r t h ec h a n g e s h o u l d b e a p p r o a c h e d w i t h c a u t i o n ;i t is t y p i c a l o f t h e a s s u m p t i o n s a s k i ll e da n a l y s t w i l l m a k e w h e n g i v e n i n s u f f i c i e n td a t a a b o u t t h e p r o b l e m . W e w il l p r e s e n t a ne x a m p l e s h o r t l y i n w h i c h a f a s t e r C P U a f-f e c t s t w o w o r k l o a d s d i f f e re n t l y : t h e r a t i o o fi n t e ra c t iv e t o b a t c h t h r o u g h p u t c h a n ge s .B o t t le n e ck A n a l y s isT h i s s e c t i o n d e a l s w i t h t h e a s y m p t o t i c b e -h a v i o r o f t h r o u g h p u t a n d r e s p o n s e t i m e o fc l o se d s y s t e m s a s N , t h e n u m b e r o f j o b s i nt h e s y s t e m , i n c re a s e s. W e w i ll a s s u m e t h a tt h e v i s i t r a t i o s a n d m e a n s e rv i c e t i m e s a r ei n v a r i a n t u n d e r c h a n g e s i n N .N o t e t h a t t h e r a t i o o f c o m p l e t io n r a t e sf o r a n y t w o d e v i c e s is e q u a l t o t h e r a t i o o ft h e i r v i s i t ra t i o s :

    Com putmg S urveys, Voi. 10, No. 3, September 1978

  • 8/14/2019 62540 Denning

    16/37

    240 P. J . Dennin g and J . P. Buz enX, IXj = V V ~ .

    S i n c e / . 7 , ffi S a s i m i l a r p r o p e r t y h o l d s f o ru t i l i z a t i o n s :

    u , / v ~ = v , s , / y ~ s , .Our invariance assumptions imply thatthese ratios are the same for all N.Device i is saturated if its utilization isapproximately 100%. If U, ffi 1, the utiliza-tion law implies th at

    X, = l / S , ;this means th at the s atura ted device is com-pleting work at its capaci ty- -an average ofone request each S, seconds. In general,U, -< 1 and X,

  • 8/14/2019 62540 Denning

    17/37

    The Operational Analy sisThis is a product of a waiting time at theterminals (Z) and a saturation job flowthrough the terminals (1/VbSD; by Little'slaw, Mb denotes the mean number of think-ing terminals when the system is saturated.The response time asymptote crosses themini mum response time R0 at

    Mb* = (Ro + Z}/VbSb = N* + Mb.when there are more than Mb* terminals,queueing is certain to be observed in thecentral subsystem.Notice that the response time asymp-totes and intersections M0 and M0* dependonly on M, Z, V0, and S0. The only assump-tions needed to compute them are job flowbalance and invariance of the visit ratiosand mean service times under changes inload. Note also that when Z = 0 theseresults yield the response time asymptotesof a closed central system. These resultsmay be extended to include the case whereservice times are not strictly invariant, buteach S, approaches some limit as the queuelength at device i increases [MUNT74,DENN75a].To summarize: the workload parametersor the visit ratio equations allows the ana-lyst to dete rmine the visit ratios, V,. Devicecharacteristics allow determination of themean service time per visit, S,. The largestof the products V,S, determines the bottle-neck device, b. The sum of these productsdetermines the smallest possible responsetime, R0. The system through put is 1/VoSbin saturation. The saturation point N* ofthe central subsystem is Ro/VbSb; andN* + Z/VoSo terminals will begin to satu-rate t he terminal driven system.An analysis leading to sketches such asFigures 9 and 10 ma y give some gross guid-ance on effects of proposed changes. Forexample, reducing V,S, for a device whichis not a bottleneck (e.g., by reducing theservice time or the visit ratio) will not affectthe bottleneck; it will make no change inthe asymptote 1/VoSo and will generallyproduce only a minor change in minimalresponse time R0. Reducing the productV,S, for all the bottleneck devices will re-move the bottleneck(s); it will raise theasymptote 1/VbS0 and reduce R0. However,this effect will be noticed only as long asVbSb remains the largest of the V,S,: too

    of Queueing Network Models 241much improvement at device b will movethe bottleneck elsewhere. These points willbe illustrated by the example of the nextsection.The property that 1/VbSb limits systemthroughput was observed by Buzen forMarkovian central server networks[BvzE71a]. It was shown to hold under verygeneral conditions by Chang and Laven-berg [CHAN74]. Muntz and Wong used i t inbottleneck analysis of general stochasticqueueing networks [MUNT74, MUNT75];Denning and Kah n derived the operationalcounterpart [DENN75a]. Response timeasymptote s were observed by Scherr for hismodel of CTSS [SCHE67], and by Moorefor his model of MTS [Moon71]. Theconcept of satura tion point was introducedby Kleinrock [KLEI68], who also studiedall these results in detail in his book[KLEI76].ExamplesThis section illustrates the applications ofbottleneck analysis for the three cases ofFigures 11 through 13. For each, we con-sider a series of questions as migh t be posedby a computing center's managers, whoseek to und erstand the present system andto explore the consequences of proposedchanges.Figure ll(a) depicts a central server sys-tem driven by a set of interactive terminals.The visit ratio equations for this networkare

    V0= 1 = .05VIVl = Vo + V~ + V3v2 = .55V,V3 = .40V~

    iFIGURE ll(a).

    q J2

    Q*a$ a 0 4 t . ~

    An example system.

    Computing Surveys, Vol. 10, No 3, September 1978

  • 8/14/2019 62540 Denning

    18/37

    2 4 2Sec,

    2.2 -;

    P. J. Denning a nd J. P. Buz en

    i j / /2 0 2 2Ms M i *FIGURE ll(b) . Respons e t ime curve.

    ~ M

    C P U DISKe c .

    2 4 . . . . . . .. . . . . . .. . . . . . . .. . / / DRUM-- %0

    ~ M21 22 23 50 63F I G U R E 11 (C ). R e s p o n s e t i m e a s y m p t o t e s .

    T h e s o l u t i o n i sV 1 = 2 0 , V ~ = l l , V 3 = 8 .

    T h e V,S, p r o d u c t s a r eV~S~= (20)(.05)

    ffi 1 . 0 0 s e c o n d s ( T o t a l C P U t i m e )V2S2= (11)(.08)

    - - .8 8 s e c o n d s ( T o t a l D i s k t i m e )V~S3 ffi (8 )(.04 )= .3 2 s e c o n d s ( T o t a l D r u m t im e )

    T h e s e p r o d u c t s s u m t o t h e m i n i m a l r e -s p o n s e t i m eR0 ffi 2.2 sec on ds.

    T h e l a r g e s t p r o d u c t i s ~ S 1 ; t h e r e f o r eb ffi 1 a n d t h e C P U i s t h e b o t t l e n e c k . ( T h es y s t e m i s " C P U b o u n d . " )F i g u r e 1 1(b ) sh o w s t h e a s y m p t o t e s o f t h er e s p o n s e t i m e c u r ve . T h e n u m b e r o f t h in k -i ng t e r m i n a l s i n s a t u r a t i o n i s

    M1 = Z/VISI = 20 t e r m i na l s .T h e s a t u r a t i o n p o i n t o f t h e c e n t r a l su b s y s -t e m i s

    N* ffi Ro/V]S1 = 2 .2 jobs .T h e n u m b e r o f t e r m i n a l s re q u i r e d t o b e g i ns a t u r a t i n g t h e e n t i r e s y s t e m i s

    M l* ffi 22.2.Q u e s t i o n : Throughput is measured asO. 715jobs~second and mean response timeas 5.2 seconds. Wh at is the mean n umbe rof users logged in during the observationperiod? T h e i n t e r a c t i v e r e s p o n s e t i m e f o r -m u l a c a n b e so l v e d f o r t h e ( m e a n ) n u m b e ro f a c t i v e t e r m i n a l s ,

    M = ( R + Z ) / X offi (5.2 + 20)/(.715)ffi 18 t e r m ina l s .Q u e s t i o n : Is 8-second response timefeasible wh en 30 users are logged in? I fnot, what minimum amount of CPUspeedup is required? S i n c e t h e r e s p o n s et i m e a s y m p t o t e r e q u i r e s t h a t , f o r M ffi 3 0,

    R >_ (30)(1.00) - 20 ffi 10 sec on ds ,t h e 8 - s ec o n d r e q u i r e m e n t c a n n o t b e m e t . I f$ 1 ' i s t h e s e r v ic e t i m e o f a f a s t e r C P U , w en e e d

    MV1SI' - Z ~ 8 seco nds ,o r

    $1 ' < .047 secon ds .T h i s g i v e s a s p e e d u p f a c t o r o f S1/SI' - 1.07;t h e n e w C P U m u s t b e a t l e a s t 7% f a s te r .S i n c e V]S]' ffi ( 2 0 ) ( .0 4 7 ) = .9 3, t h e s y s t e mw o u l d s ti ll b e C P U - b o u n d w i t h t h i s f a s t e rp r o c e s s o r ( s e e F i g u r e 1 1 (c )) ; t h e r e f o r e t h ec h a n g e i s f e a s ib l e .

    Q u e s t i o n : Is lO-second response timefeasible wh en 50 users are logged in? I fnot, what minimum amount of CPUspeedup is required? I f t h e C P U w e r e i nf i-n i t e l y f a s t (S 1 ffi 0 ) , t h e d i s k w o u l d b e t h eb o t t l e n e c k ( s ee F i g u r e 1 1 (c )) . I n t h i s c a s e

    R >_ MV2S~ - Z.F o r M ffi 50 ,

    R _> (50)( .88) - 20 = 24 sec on ds.T h u s , n o a m o u n t o f C P U s p e e d u p w i l lm a k e 1 0 - s e c o n d r e s p o n s e f e a s i b l e w h e nM = 5 0.

    O u r s e c o n d e x a m p l e c o n c e r n s t h e 2 5 - t e r -m i n a l t i m e s h a r i n g s y s t e m o f F i g u r e 12 . Am e a s u r e m e n t h a s r e v e a le d t h a t j o b s r e q u ir e

    Computmg Surveys, Vol 10, No. 3, September 1978

  • 8/14/2019 62540 Denning

    19/37

    The Operational Analysis of Queueing Network Models 2 4 3

    25 T e r m m a l l

    FIGURE 12.I l o T lI I

    R= 511A t i m e s h a r i n g s y s t e m .

    a m e a n t o t a l C P U t i m e o f 2 40 m s e c , t h a tC P U u t i l i z a t io n is 3 0% , a n d t h a t r e s p o n s et i m e i s 5 s e co n d s . T h e t h r o u g h p u t a n dt h i n k t i m e a r eX o f f i U / V SI

    ffi .30) / .24) ffi 1.25 jobs / secondZ ffi M/Xo - Rffi 25/1.25 - 5 -- 15 seco nds .

    Question: The CPU utilization seemslow. What effect would a cheaper CPU ofhalf speed have on response time? Install-i ng a C P U o f h a l f s p e e d c a n n o t i n c r e a s es y s t e m t h r o u g h p u t , n o r c a n i t r e d u c et h r o u g h p u t b e l o w h a l f i ts o r ig i n a l v a l u e . ( I fa l l s e r v i c e t i m e s , i n c l u d i n g Z , w e r e d o u b l e d ,t h r o u g h p u t w o u l d b e e x a c t l y h a l f t h e o r ig -i n a l v a l u e . ) T h e r e f o r e ,

    0 .625 _< Xo ~ 1 .25 jo bs / se co nda f t e r t h e c h a n g e . ( W i t h U ~ = XoVIS~ t h i si m p l i e s 0 . 3 _< /31 -< 0 .6 a f t e r t h e c h a n g e . )A p p l y i n g t h e r e s p o n s e t i m e f o r m u l a ,

    5.0 _ R _< 25.0 sec on ds.T h e s l o w e r C P U w i l l h a v e n o e f f ec t o nr e s p o n s e t i m e i f s o m e o t h e r d e v i c e is s a t u -r a t e d ( n o c h a n g e i n X o); o t h e r w i s e , i t c o u l dc a u s e r e sp o n s e t i m e t o in c r e a s e b y a s m u c ha s a f a c t o r o f f i v e.T h i s e x a m p l e i l l u s tr a te s w h y s y s t e m b o t -t l e n e c k s c a n c o n f u s e t h e u n w a r y a n a l y s t . I fs o m e d e v ic e ( n o t t h e C P U ) i s s a t u r a t e d ,l o w e r i n g C P U s p e e d w i ll i n c r e a s e C P U u t i-l iz a t i o n w i t h o u t o b s e r v a b l e e f f e c t o n r e -s p o n s e t im e . C P U u t i li z a t i o n c a n b e a d e -c e p t iv e m e a s u r e o f a s y s t e m ' s p e r f o r m a n c e .

    O u r t h i r d e x a m p l e c o n c e r n s t h e s y s t e mo f F i g u r e 13, w h i c h h a s t w o w o r k l o a d s . I tw i ll i l l u s t r a te h o w a f a s t e r d e v i c e m a y a f f e c t

    p e r f o r m a n c e a d v e r s e l y . E a c h b a t c h j o b r e -q u i r e s o n e d i s k - s w a p f o l l o w e d b y a n u n i n -t e r r u p t e d C P U e x e c u t io n b u r s t a v e r a g i n g 1s e c o n d . E a c h i n t e r a c t i v e j o b r e q u i r e s a na v e r a g e o f 10 p a g e s w a p s f r o m t h e d is k ,e a c h f o l l o w e d b y a s h o r t C P U b u r s t a v e r -a g i n g 1 0 m s e c .P r i m e d s y m b o l s r e f e r to t h e i n t e r a c t i v ew o r k l o a d . I t i s e a s y t o s e e f r o m F i g u r e 13t h a t t h e b a t c h v i s i t r a t i o s a r e V , ffi 112 = 1 ,a n d t h e i n t e r a c t i v e v i s i t r a t i o s a r e V f ffi V2'ffi 10. T h e t o t a l o f t i m e s r e q u i r e d b y j o b s a tt h e d e v i c e s a r e :Disk CPUB a t c h V~S~ ffi .09 s ec . V2S2 ffi 1.0 sec.I n t e r a c t i v e V,'S,' ffi .90 se c. V2'$2' fi .1 se c.

    E v i d e n t l y t h e i n t e r a c t i v e w o r k l o a d i sd i s k - b o u n d a n d t h e b a t c h w o r k l o a d C P U -b o u n d . T h i s i s a g o o d m i x t u r e o f j o b s i n th es y s t e m .Question: A measurement reveals thatthe CP U is saturated, and that interactiveresponse t ime is 4 seconds. Wh at ~s batchthroughput? Disk utilization? W e c a n s o lv et h e i n t e r a c t i v e r e s p o n s e t i m e f o r m u l a fo rt h e i n t e r a c t i v e t h r o u g h p u t :Xo' ffi M/(R' + Z)

    ffi 25 / (4 + 30 ) = . 735 j ob s / s ec on d .S i n c e X o ' = X2'/V2', t h e i n t e r a c t i v e c o m -p o n e n t o f C P U t h r o u g h p u t i s X2' ffi 7.35r e q u e s t s / s e c o n d , a n d t h e u t i l i z a t i o n d u e t oi n t e r a c t i v e j o b s i s

    U2' ffi X{S2' = (7.35)(.01) ffi .074.S i n c e t o t a l u t i l i z a t i o n i s 1 . 0 0 , t h e c o m -p o n e n t d u e t o b a t c h j o b s m u s t b e U s = .9 26 .

    T h u s t h e b a t c h t h r o u g h p u t i sXo = X1 ffi X2 = U2/$2 = .9 26 j o b s / s e c o n d .T h e u t i l i za t io n o f t h e d i s k i s

    X,S~ + X{SI' = (.926)(.09) + (7.35)(.09) ffi .745.Question: An analysis of batch back-logs reveals that the computing centerneeds to support a batch throughput of atleast 4.S jobs~second. Is this feasible in thepresent system? I f t h e r e w e r e n o i n t e r a c t i v e

    j o b s , t h e h i g h e s t p o s s ib l e C P U b a t c ht h r o u g h p u t w o u l d b e X2 = 1/82 = 1 j o b /s e c on d . T h e r e q u i r e d b a t c h t h r o u g h p u tc a n n o t b e a c h i e v e d .Question: A CP U 5 times faste r is avail-able. Wh at happen s if batch throughput of

    Computing Surveys, Vol 10, No. 3, Sep tem ber 1978

  • 8/14/2019 62540 Denning

    20/37

    2 4 4 P. J . Denn ing and J . P . Buzen

    I N I ~ - . 1~ L , .9

    M = 2 5 , . ,I N Z = 3 0 s e c . - - I ~ - I T D I S K - c J - ~ T l c P u x 2 ' ~_ .I =2 ;OUT

    S = ' = . 0 9 s e c . S z ' = .0 1 s e c .S I = . 0 9 s e c . S z = 1 . 0 s e c .

    F I G U R E 1 3 .

    I n t e r o c h v e W o r k l o a d4 ~ B o t c h W o r k l o a d

    A system with tw o workloads.

    4 .5 jobs~se cond i s ach ieved wi th th i s CPU?W i t h t h e n e w C P U , t h e b a t c h C P U b u r s tb e c o m e s $ 2 ffi .2 s e c o n d , a n d t h e i n t e r a c t i v eC P U b u r s t $ 2' = .0 0 2 s e c o n d . W i t h a f o r c e db a t c h f l o w o f X 0 ffi X 1 = X 2 = 4 . 5 j o b s /s e c on d , t h e b a t c h c o m p o n e n t s o f d i sk a n dC P U u t i l iz a t i o n w o u l d b e

    U~ = X~S~ = (4.5 )(.09 ) ffi .41U2 ffi X2S2 ffi (4.5)(.20) = .90

    T h i s g i v es b o u n d s f o r t h e i n t e r a c t i v e c o m -p o n e n t s o f t h r o u g h p u t :Xl' ffi UI'/S~' ~- (1 - .41)/( .09)

    = 6 . 6 1 r e q u e s t s / s e c o n dX2' = U2'/$2'

  • 8/14/2019 62540 Denning

    21/37

    The Operational Analys is of Queueing Net work Models 245S u m m a r yBy augmenting the basic operational defi-nitions with the assumption t hat job flow isbalanced in the system, the analyst can usevisit ratios, via the forced flow law, to de-termine flows everywhere in the network.Response times of interactive systems canalso be estimated. Table I summarized theprincipal equations.When the available information is insuf-ficient to determine flows in the network ata given load, the analyst can still approxi-mate the behavior under light and heavyloads. For light loads the lack of queueingpermits determining response time andthroughpu t directly from the products V~S,.For heavy loads, a saturating device limitsthe flow at one point in the network,thereby limiting the flows everywhere;again, response time and th roug hpu t can becomputed easily. For intermediate loads,further assumptions about the system areneeded.5 . L O A D D E P E N D E N T B E H A V IO RThe examples of the preceding section werebased on assumptions of invariance for ser-vice times, visit ratios, and routing frequen-cies. These assumptions are too rigid formany real systems. For example, if the mov-ing-arm disk employs a scheduler that min-imizes arm movement, a measurement ofthe mean seek time during a lightly loadedbaseline period will differ significantly fromthe average seek time observed in a heavilyloaded projection period. Similarly, thevisit ratios for a swapping device will differin baseline and projection periods havingdifferent average levels of multiprogram-ming.These two examples illustrate load de-pendent behavior. To cope with it, the an-alyst replaces the simple invariance as-sumptions with conditional invariance as-sumptions that express the dependence ofimportant parameters on the load. Insteadof asserting tha t the disk's mean seek timeis invariant in all observation periods, theanalyst asserts that the mean seek time isthe same in any two intervals in which thedisk's queue length is the same. Th at is, theaverage seek time, whenever the disk's

    queue length is n (for any integer n), isassumed to be the same in both the baselineand the projection period, but the propor-tion of time tha t th e queue length is n maydiffer in the two periods. Similarly, theswapping device's visit ratio whenever themultiprogramming level is N is assumed tobe the same in both the baseline and theprojection period, but the proportion oftime tha t the multiprogramming level is Nmay differ in the two periods.Tables II and III summarize the opera-tional concepts needed to express condi-tional invariants and to work with loaddependent behavior. Table II shows thateach of the basic quantities (C,j, B,) is re-placed with a function of the load. ThusC,j(n) counts the number of times t at whichjobs request service at devicej immediatelyon comple ting a service request at device i,given that n, ffi n just before each suchtime t. The funct ion T,(n) specifies the totaltime dur ing which n, ffi n.Table III shows the various operationalmeasures which can be derived from thebasic quantities of Table II. There are twonew concepts here. The first is the servicefunction, S~(n) ffi 1/X,(n), which measuresthe mean time between completions whenn, = n; if device i can process several servicerequests at once, S,(n) can be less than themean amount of service required by a re-quest. The second concept is the queuelength distribution, p,(n), which measuresthe proportion of time during which n, ffi n.That the mean queue length f~, ffi W, / T isequivalent to the usual definition E,>onp,(n) can be seen from the definition ofW, in Table II.The method of partitioning the dat a ac-cording to time intervals in which n,(t) = nis called stratified sampling. The sets ofintervals in which n,(t) ffi n are called thestr ata of the sample. All data in the samestratum are aggregated to form the mea-sures of Tables II and III.Our analytic methods can deal with onlytwo kinds of load dependent behavior: adevice's service function may depend onthe length of that device's queue; the visitratios and routing frequencies may dependon the total number of jobs in the system.Thu s quanti ties like q,~(n) = C~(n)/C,(n) or

    Com put ing Surveys . Vol . 10, No. 3 , Septem ber 1978

  • 8/14/2019 62540 Denning

    22/37

    246 P . J . D e n n i n g a n d J . P . B u z e nTABLE II. BASXCMEASURES

    C o m p l e t i o n C o u n t sInterdevice

    Device conditionalDevice unconditional

    A r r w a l C o u n t sTo a device

    To the systemB u s y T t m e sConditional

    UnconditionalR o u t i n g F r e q u e n c w s

    Originating in systemOriginating outside system

    A c c u m u l a t e d W a i t in g T t m e

    iff i l . . . . K jff i0 . . . . K

    C,j(n)

    C, (n)C,Ao~

    AoT,(n)B ,

    qvqo~W ,

    ffi N u m b e r o f t i m e s t a t w h i c h a j o b r e q ue s t s s e r v ic e a t d e vi c e jnex t after com ple tin g a service request at devlce i given ns n just before t.

    K= ~ C . j ( n )j -offi ~ C,(n )

    n>0

    ffi Number of times t a t which an arrwing job uses devicej for itsfirst service request.E

    == ~Aojj l lffi Tota l time dunng which n, ffi n.= ~ T, (n ) = T - T, (0)n>0

    1 ~ C,j(n) [Undefined if C, ffi 0]--~ C,n>Offi AoJA o [Undefined ff Ao -- 0]= ~ n T , ( n )n:>O

    T A B L E Ill. O P E R A T I O N A L P E R F O R M A N C E M E A S U R E Siffil . . . . K yffi0 . . . K[Any quantity whose denominator would be zero is undefined ]

    R e q u e s t C o m p l e t t o n R a t e sConditional X, (n) ffi C , (n ) / T , (n )Unconditional, dewce* X, - C J TKUnconchtlonal, system Xo = ~. X,q,0ffiC o l T#--lM e a n S e r v i c e T i m e B e t w e e n C o m p l e t i o n sConditional S, (n) ffi T , (n ) /C, ( n)Unconditional S, ffi B , / C ,Q u e u e S i z e D t s t r t b u t w n p , ( n ) ffi T , ( n ) / TU t t h z a t i o n U , ffi B , / T ffi 1 - p ,