'
MAGYAR TUDOMÁNYOS AKADÉMIA
SZÁMÍTÁSTECHNIKÁI é s a u t o m a t i z á l á s i k u t a t ó in t é z e t e
OPERÁCIÓS RENDSZEREK ELMÉLETE IV. VISEGRÁDI TÉLI ISKOLA
ISBN 963 311 074 2
4
Tanulmányok 87/1978.
Szerkesztőbizottság :
GERTLER JÁNOS (felelős szerkesztő) DEMETROVICS JÁNOS (titkár)
ARATÓ MÁTYÁS, BACH IVÁN, GEHÉR ISTVÁN,
GERGELY JÓZSEF, KERESZTÉLY SÁNDOR, KNUTH ELŐD, KRÁMLI ANDRÁS, PRÉKOPA ANDRÁS
Felelős kiadó:
Dr VÁMOS TIBOR
MTA Számítástechnikai és Automatizálási Kutató Intézet MTA Számítástudományi Bizottsága
Konferencia szervező bizottsága:
ARATÓ MÁTYÁS (elnök) KNUTH ELŐD (titkár)
VARGA LÁSZLÓ
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A konferenciát a ’’Számítástechnika tudományos kérdései” c.
többoldalú akadémia együttműködés keretében rendeztük.
Конференция была проведена в рамках многостороннего сотрудничества академий
социалистических стран по проблеме
"Научные вопросы вычислительной техники"
Conference was held in the frame o f the multilateral cooperation of the academies of sciences of the socialist countries on Computer Sciences.
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T A R T A L O M J E G Y Z É K
Jacek Blazewicz, Jan Weglarz:
Scheduling to m eet deadlines with resource constraints — complexity results . . . . 5 Krámli András, Pergel József:
The statistical behavior of a characteristic of the insertion algorithm
’’Batched hashing and linear probing” ... ... 21 Benczúr András, Krámli András:
A stochastic m odel in presence o f intermittant ... 27 A. Wolisz, A. Krawet, J. Mierzwa, A. Nowakowski:
Operating system and data base for a small production control system ... 35 L. Simonfalvi:
Deadlock problems in a multicomputer interconnection system ... 53 Jürgen Dassow:
Some remarks on the algebra o f a u to m a ta ... 71 Roman Bednarz:
Statistical investigations o f cyber-73 multiaccess system... 79 L. Lakatos:
On a queueing problem in the theory o f operating systems... 93 Otto Spaniol:
Multifrequency aloha-type systems... 99 Dávid Gábor:
On the basic concepts o f a module language ... 117 Laura Bürger:
Alarm analysis: as a from of secondary data prcessing... 139 E. Vegh:
The structure and the efficiency o f the process-24k process control system... 155 Ruda Mihály:
Statistical information system with health service application... 167 Hans-Dietrich Gronau:
On the generation o f binary vectors by some closed sets o f boolean functions
(Linear functions and alternatives)... 173 Gáspár A ., Kocsis J., Lamm P., Visontay Gy.:
Az MTA tervezett számitógéphálózatával kapcsolatos tervezési szempontokról... 179
SCHEDULING TO MEET DEADLINES WITH RESOURCE CONSTRAINTS - COMPLEXITY
RESULTS
J a c e k B la z e w ic z , J a n W eglarz
I n s t y t u t A utom atyki, P o l i t e c h n i k a P o zn an sk a
ABSTRACT:
The p ro b lem to be c o n s id e re d i s one o f th e n o n p re e m p tiv e s c h e d u lin g o f t a s k s on p r o c e s s o r s , when e v e ry t a s k r e q u i r e s f o r i t s e x e c u tio n a d d i t i o n a l r e s o u r c e and h a s a s s o c i a t e d w ith i t an e x e c u tio n tim e , r e a d y tim e and a d e a d lin e b e f o r e w hich m u st be c o m p le te d . F o r th e s im p le s t c a s e o f s c h e d u lin g u n i t l e n g t h , in d e p e n d e n t t a s k s on two p r o c e s s o r s w ith one u n i t o f one r e s o u r c e ty p e a v a i l a b l e in th e sy ste m , an e f f i c i e n t a lg o r ith m f o r f i n d i n g a sc h e d u le w ith no t a s k l a t e , w henever such a s c h e d u le e x i s t s , i s p r e s e n t e d . I f we a llo w , i n th e l a s t p ro b le m , f o r an a r b i t r a r y number o f r e s o u r c e ty p e s a v a i l a b l e i n th e sy ste m , i t i s unknown w h e th e r o r n o t th e r e e x i s t s an e f f i c i e n t a lg o r ith m f o r s o lv i n g t h i s new p ro b lem . O th e r problem s a r e p ro v ed to be N P -co m p lete, h e n c e , c o m p u ta tio n a lly i n t r a c t a b l e . M oreover, a m ethod o f s o lv i n g th e th e g e n e r a l s c h e d u lin g p ro b lem w ith u n i t l e n g t h , in d e p e n d e n t t a s k s , b y th e r e d u c t i o n to th e problem o f n e tw o rk flo w w ith m u l t i p l i e r s , i s d e v e lo p e d .
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1 . INTROLU П TION
The problem t o b e c o n s id e r e d i s one o f s c h e d u lin g t a s k s on p r o c e s s o r s t o m e e t d e a d l i n e s . Such a c a s e o f te n a r i s e s i n p r o c e s s c o n t r o l and m o n ito rin g sy stem s i n w hich th e com puter m ust g u a r a n te e t h a t e v e ry t a s k w i l l be co m p leted b e f o r e some f i x e d tim e h a s e l a p s e d , a f t e r w hich th e o b ta in e d r e s u l t s w i l l be u s e l e s s . Such a problem h a s r e c e iv e d much a t t e n t i o n i n r e c e n t y e a rs [ 8 ,1 3 ,1 4 ,1 6 ] (see И a s a su rv e y J . How
e v e r th e u se o f a d d i t i o n a l r e s o u r c e s was n o t c o n s id e re d i n th e s e p a p e r s . A m ore g e n e r a l and more r e a l i s t i c model o f com
p u tin g sy s te m s , when some o f th e t a s k s may r e q u i r e th e u s e o f v a r io u s l i m i t e d r e s o u r c e s d u r in g t h e i r e x e c u tio n , was f i r s t
examined i n 0 , 1 0 , 1 2 ] .
I n [1 0 ] , s c h e d u lin g n o n p re e m p ta b le t a s k s to m inim ize sc h e d u l e le n g th was c o n s id e r e d . I t was shown t h a t a lm o st a l l p ro b lem s a r e N P -com plete and h en c e th e y a r e a s c o m p u ta tio n a lly i n t r a c t a b l e a s t h e t r a v e l l i n g sa lesm an p ro b le m , in И . bounds on s c h e d u lin g w ith l i m i t e d r e s o u r c e s were o b ta in e d . In
[1 2 ] th e s p e c i a l c a se was s tu d i e d i n w hich t h e r e i s o n ly one ty p e o f a d d i t i o n a l r e s o u r c e w hich ca n be sh a re d by a l i m i t e d number o f t a s k s . L a s t l y , i n И , th e problem o f s c h e d u lin g to m in im iz e mean flo w tim e was c o n s id e r e d . I t was p ro v ed t h a t th e same pro b lem s a s i n th e c a s e o f m in im iz in g
s c h e d u le le n g th a r e N p -c o m p le te. A p o ly n o m ia l i n tim e a lg o r ith m was a l s o g iv e n f o r t h e s im p le s t c a s e .
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I n t h i s p a p e r , we s u rv e y r e s u l t s o b ta in e d i n th e f i e l d o f s c h e d u lin g t a s k s to m eet d e a d lin e s u n d e r r e s o u r c e c o n t r a i n t s
D l -
We c o n s id e r th e f o llo w in g model o f a com puting sy ste m . T here a r e n t a s k s T-^, T g , . . . , T , w hich a r e to be p ro c e s s e d on m i d e n t i c a l p r o c e s s o r s P-^, P 2 , . . . , Pm. T here i s a ls o g iv e n a s e t o f a d d i t i o n a l r e s o u r c e s ^ = (к -^ ,Р 2 , . . . » R ^ • P o r each r e s o u r c e t h e r e e x i s t s a bound m^ ( in te g e r ^ w hich g iv e s th e t o t a l amount o f t h a t r e s o u r c e a v a i l a b l e a t any g iv e n tim e . P o r e v e ry t a s k T ^, i = l , 2 , . . . . , n , t h e r e a r e g iv e n : e x e c u tio n tim e / V r e a d y tim e r ^ , d e a d lin e d i and v e c t o r R ^ T ^ ( i n t e g e r ) c o n ta in in g r e s o u r c e r e q u i r e m ents (the 1 - t h com ponent o f t h i s v e c t o r R^ » d e - n o te s th e number o f u n i t s o f R1 r e q u ir e d by t a s k T ^) . A p a r t i a l o r d e r ^ , s p e c if y in g p re c e d e n c e c o n s t r a i n t s i s d e
f in e d on th e t a s k s e t . T. < T. means t h a t T. c a n n o t b e g ib
■*“ J J
u n t i l Ti i s f i n i s h e d .
L et u s r e c a l l some d e f i n i t i o n s w hich w i l l be u s e f u l i n th e f o llo w in g . P o r e v e ry t a s k T^ i n th e s c h e d u le , we d e n o te by
i t s c o m p le tio n tim e . The s c h e d u le w i l l be c a l l e d o p tim a l i f a l l t a s k d e a d lin e s a r e r e s p e c te d i n i t , t h a t i s f o r e v e ry T^ th e c o n d i tio n ^ d^ i s f u l f i l l e d . A s c h e d u lin g a lg o r ith m i s a p ro c e d u re t h a t p ro d u c e s a sc h e d u le f o r e v e ry g iv e n s e t o f t a s k s . By a p re e m p tiv e s c h e d u lin g a lg o r ith m , we mean to a llo w th e i n t e r r u p t i o n o f th e e x e c u tio n o f t a s k s i n a s c h e d u le . When i n t e r r u p t i o n s a r e n o t a llo w e d , we c a l l an a lg o r ith m nonpreem p-
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t i v e . In t h i s p a p e r we w i l l he co n c e rn e d w ith n o n p reeip p tiv e s c h e d u lin g a lg o r ith m s o n ly . By o p tim a l s c h e d u lin g a lg o r ith m we mean h e re th e a lg o r ith m w hich f i n d s an o p tim a l s c h e d u le , w henever one e x i s t s .
A d i s t i n c t i o n w i l l he made betw een some s c h e d u lin g s u b -p ro b lem s w hich d i f f e r from ea ch o th e r by th e v a l u e s o f p roblem p a ra m e te rs / t h a t i s by p ro b lem i n p u t s / . T h is i s due to th e f a c t t h a t some a lg o r ith m s a p p ly o n ly to c e r t a i n r e s t r i c t e d c l a s s e s o f th e s e i n p u t s . The c o m p le x ity o f a s c h e d u lin g su b p ro b lem a ls o d ep e n d s s t r o n g l y on i t s i n p u t . Bor c o n v e n ie n c e ,
t o d e n o te such a su b p ro b lem , we a d o p t, w ith s l i g h t m o d if ic a t i o n , th e n o t a t i o n o f [1 5 ] : njm 1 4 k , w here :
n - number o f t a s k s ; m- number o f p r o c e s s o r s ;
- problem p a r a m e te rs su c h a s : s=k к r e s o u r c e ty p e s . Ш1 = r ( r u n i t s o f r e s o u r c e R-^) , ^ ( a r b i t r a r y p re c e d e n c e c o n s t r a i n t s betw een th e t a s k s ) , f o r e s t
(p re c e d e n c e c o n s t r a i n t s betw een th e t a s k s such t h a t th e a s s o c i a t e d p re c e d e n c e g ra p h form s a f o r e s t ) , r ^ / 0 ( p o s s ib ly n o n - e q u a l r e a d y tim e s f o r t a s k s ) , ц = 1
^ m i t p r o c e s s in g t i m e s ) , p r (p re e m p tiv e a lg o rith m ) , n o n p r (n o n p re e m p tiv e a lg o r ith m ) ;
- o p t i m a l i t y c r i t e r i o n , f o r example s c h e d u le le n g th
We w i l l be co n cern ed w ith p ro b le m s w hich a r e fo rm u la te d i n su c h a way an answ er t o them may o n ly be " y e s" o r "n o ". As
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i s known £ .1 ,7 ,1 1 ] t h i s i s n o t r e s t r i c t i v e , s in c e e v e ry s c h e d u lin g p ro b lem can he f o rm u la te d i n such a m an n er. F o l
lo w in g t h i s , we may d e n o te p roblem o f s c h e d u lin g t o m eet d e a d lin e s by «C-Oi d r
The m ain s u b j e c t o f o u r i n t e r e s t w i l l be th e c o m p u ta tio n a l c o m p le x ity o f th e s c h e d u lin g p ro b le m s . F o r some p ro b lem s i t w i l l be p o s s i b l e to g iv e p o ly n o m ia l i n tim e a lg o r ith m s w hich
can be hence u se d d i r e c t l y i n o p e r a t in g sy ste m s. However, m ost o f th e c o n s id e r e d p ro b lem s can be p ro v e d to be H P -co m p lete, th u s f o r t h e i r o p tim a l s o l u t i o n we can u se e n u m e ra tiv e m ethods su ch a s b ra n c h and bound o r dynam ic program m ing, s in c e no
s u b s t a n t i a l l y b e t t e r method i s l i k e l y to e x i s t . T hus, when th e N P -co m p leten ess o f th e p roblem i s p ro v e d , we can u se ap p ro x im a te a p p ro a c h e s i n s c h e d u lin g p ro c e d u r e s o f o p e r a t in g sy s te m s .
The p a p e r i s c o n s tr u c te d i n th e f o llo w in g way. I n S e c tio n 2 an o p tim a l and p o ly n o m ia l i n tim e a lg o rith m i s g iv e n f o r th e problem n |2 | s = l , m ,= l, Т ^ = 1 , n o n p r | Ci ^ d i . S e c tio n 3 d e a ls w ith N P-com plete p ro b le m s. I t i s shown t h a t p ro b lem s n I 2 J s = l , f o r e s t , t ^=1, n o n p r |c i ^ d i , n | 2 \ s = l , т-^=1,
< , Т ±=1, n o n p r |C i $ d i and n I 3\ s * l , ^ * 1 , n o n p r|C i ^ d i a r e N P -co m p lete. The c o m p u ta tio n a l c o m p le x ity o f th e re m a in in g problem n I 2 I s= k, r ^ O , *1^=1, n o n p r j c ^ d ^ ^ i s s t i l l open.
I n S e c tio n 4 a method f o r s o lv i n g th e g e n e r a l p ro b lem o f s c h e d u lin g u n i t l e n g t h , in d e p e n d e n t t a s k s i s d e v e lo p e d . T h is i s done by r e d u c t i o n to th e p ro b lem o f n etw o rk flo w s w ith
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m u l t i p l i e r s . F i n a l l y , th e problem o f m in im iz in g maximum l a te n e s s i s b r i e f l y exam ined.
2. ALGORITHM FOR SCHEDULING TASKS ON TWO PROCESSORS
In t h i s S e c t i o n we p r e s e n t a s im p le , p o ly n o m ia l i n tim e , a l g o rith m f o r s c h e d u lin g u n i t l e n g t h , in d e p e n d e n t t a s k s on two p r o c e s s o r s when e v e ry t a s k may r e q u i r e n o t more th a n a u n i t o f one ty p e o f a d d i t i o n a l r e s o u r c e . T h at i s , we c o n s id e r th e problem n | 2 | s = l , m ^ l , t i = l попрг(С ± C d ^ . An a lg o r ith m f o r s c h e d u lin g t a s k s i n t h i s c a s e may be d e s c r ib e d a s f o llo w s . A lg o rith m 1 .
1° Form th e l i s t o f th e t a s k i n n o n d e c re a s in g o r d e r o f d ^ . Renumber t a s k s a c c o rd in g to t h i s o r d e r . S e t t : = 0 .
2° A ssig n t a s k T-^ to a p r o c e s s o r a t moment t and remove i t from th e l i s t .
3° A ssig n t a s k T-^ to th e seco n d p r o c e s s o r i n th e f o llo w in g manner :
a / I f R^ ( t-j) = 1 , choose th e f i r s t t a s k on th e l i s t f o r w hich R-^ (T-j) = 0. I f t h e r e i s no su ch t a s k le a v e th e
second p r o c e s s o r i d l e .
Ъ/ I f R^ ^ Т д )= 0 , choose th e f i r s t t a s k on th e l i s t w ith r e s o u r c e r e q u ir e m e n t R^ (Т -,)= 1 , i f e i t h e r 1=2, o r f o r
e v e ry к , 1 < k ^ 1 , th e f o llo w in g c o n d itio n b e in g f u l f i l l e d :
t+ l+ f C k - l) /2*| £ dk ( 1 )
I f th e above c o n d i tio n i s n o t f u l f i l l e d f o r some k , choose t a s k T2 .
4 fk] d e n o te s t h e s m a l l e s t i n t e g e r n o t l e s s th a n X
- и -
4° Remove th e a s s ig n e d t a s k s from th e l i s t and s e t t : = t + l . I f th e r e a r e any t a s k s on th e l i s t , renum ber them and go to s te p 2 ° .
L e t u s n o te t h a t i n th e above a lg o r ith m fo rm u la (1^ i s u s e d t o check w h eth e r o r n o t i t w i l l be p o s s i b l e t o sc h e d u le t a s k s o p tim a lly i f t a s k T^ w ith R-^ (T-^) =1 i s a s s ig n e d .
We w i l l new p ro v e th e o p t i m a l i t y o f A lg o rith m 1.
Theorem 1
A lg o rith m 1 i s o p tim a l f o r th e p ro b lem n | 2 | s = l, m ^ l , t i = l , non p r(C i £ d ±.
P ro o f
L e t u s assume t h a t t h e r e e x i s t s an o p tim a l sc h e d u le 1C* and l e t sc h e d u le o b ta in e d a f t e r u s in g A lg o rith m 1 d i f f e r from i t . We w i l l p ro v e t h a t ÎC i s a l s o o p tim a l.
А /
P i r s t , l e t u s assum e t h a t i n A a d i f f e r e n t assig n m e n t i s made th a n t h a t f o llo w in g S te p 3 o f th e a l g o r it h m . The f o llo w in g c a s e s may o c c u r.
i . S chedule IT i s n o t c o n s i s t e n t w ith S te p 3a o f A lg o rith m 1 . Such a c a s e may o n ly o c c u r when T^ (w ith R-^ (Т-^) =1J i s p ro c e s s e d a lo n e d e s p i t e th e f a c t t h a t t' s w ith
% ( л ) =o
a r e a v a i l a b l e . I t i s o b v io u s t h a t by a s s i g n i n g one o f th e s e ta s k s to th e second p r o c e s s o r we can im prove th e s c h e d u le . Thus, J t A /l i s a l s o o p tim a l.
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i i . S chedule & i s n o t c o n s i s t e n t w ith S te p 3b o f A lg o rith m 1 . Thus, i t i s th e c a se t h a t i n Jl a t tim e t t h e r e a re
p r o c e s s e d : T^, f o r w h ic h R^ ( Тд) =0 ( c f . S te p s 2 and 3b o f A lg o rith m l ) , and T2 f o r which d e s p i t e th e f a c t t h a t c o n d i t i o n ( i ) i s f u l f i l l e d . L e t u s n o te t h a t we may, w ith o u t d e l a y i n g any t a s k , p r o c e s s T^ i n th e p o s i t i o n f o r m e rly a s s ig n e d t o T2 , w here T1 i s th e f i r s t t a s k on th e l i s t w ith R^ ( t^ sI . T his f o l l o w s from t h e f a c t t h a t th e r e i s no such Tk , l < k < l , f o r w h ich R-^(Т ^)= 1, and h en c e we may b e g in p r o c e s s in g T2 a t moment t+1 , m oving th e b e g in n in g o f T^ to moment t+2 , e t c . F i n a l l y , i n th e p o s i t i o n fo rm e r
l y a s sig n e d t o we p r o c e s s ( o r Т^_2 i f t =1^ • A l l o f th e s e c h a n g e s do n o t c a u s e th e s c h e d u le to l o s e i t s o p t i m a l i t y , b e c a u s e f o r e v e ry r a s k Tk , l < k < l , c o n d i tio n
C i ) i s f u l f i l l e d .
We c o n s id e r now o t h e r c a s e s i n w hich JL and JC may be d i f f e r e n t . N o tice t h a t th e s e c a s e s a r e c h a r a c t e r i z e d bjr th e f a c t t h a t i n th e o p t i m a l sc h ed u le T^ i s p r o c e s s e d f i r s t and Tk a f t e r i t , d e s p i t e th e f a c t t h a t d ^ C d , . The f o llo w in g s i t u a t i o n s may t h u s o cc u r.
i i i . R-^ (Тд^ = R C . Tk V ^ i s o d i o u s t h a t i n t h i s c a s e we can a l ways exchange t a s k s Тд^ and T^ and su c h a change w i l l m a in ta in t h e o n -tim e s t a t u s o f b o th t a s k s .
i v . R^ (t-J =1. R^ (т ^ ) = 0 . I f t a s k Тд i s p ro c e s s e d in p a r a l l e l w i t h in J t , th e n i f :
- d ^ ^ d ^ an d c o n d itio n ( i ) i s f u l f i l l e d , th e n th e o b t a i n - ed sc h e d u le Л i s c o n s i s t e n t w ith A lg o rith m 1 .
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V .
/ L e t u s n o te t h a t i f c o n d i tio n ( i ) was n o t f u l f i l l e d th e n sc h e d u le IT c o u ld n o t he o p t i m a l / ;
- d ^ ^ d ^ , th e n a c c o rd in g to i i i . , r e p l a c i n g T^ and Tk we do n o t c a u se th e s c h e d u le to l o s e i t s o p t i m a l i t y and t a s k s T^ and T^ w i l l he p ro c e s s e d i n p a r a l l e l . I f T^ i s p r o c e s s e d a lo n e i n 3 T , th e n T^. may he p r o c e s s e d i n p a r a l l e l on th e second p r o c e s s o r
R^ = °, =1* t a s k T i s p r o c e s s e d i n
p a r a l l e l w ith T-^ i n Л , th e n i f
- d . ^ d, , i t i s p o s s i b l e to change th e o r d e r o f p r o c e s s - in g such t h a t T y and th e t a s k w hich was p ro c e s s e d i n p a r a l l e l w ith i t i n 1 C , a r e p ro c e s s e d f i r s t and i n t h e i r p r e v io u s p o s i t i o n t a s k s T-^ and T^ and th e new s c h e d u le i s a ls o o p tim a l;
- dj < dk and C =0, th e n we may r e p l a c e T^ and T^ m a in ta in in g t h e i r o n -tim e s t a t u s ;
- d ^ d^ and R^ С ^ ) = 1 , th e n X i s c o n s i s t e n t w ith A lg o rith m 1 / c f . S tep 3a o f th e a l g o r i t h m / .
T h u s, we have p ro v ed th e o p t i m a l i t y o f A lg o rith m 1 , s in c e i f t h e r e e x i s t s any o p tim a l sc h e d u le X , u s in g s t e p s i , i i , i i i , i v , and V , we may g e t , i n a f i n i t e number o f s t e p s , an o p tim a l s c h e d u le Jv w hich i s c o n s i s t e n t w ith A lg o rith m 1 .
• j
I t i s e a sy t i v e r i f y t h a t A lg o rith m 1 ta k e s 0 (n ) tim e .
3 . HP-COMPLETE SCHEDULING PROBLEMS
I n t h i s S e c tio n we g iv e theorem s c o n c e rn in g th e c o m p le x ity o f t h e re m a in in g s c h e d u lin g p ro b le m s. P r o o f s may be found i n
0 ).
- 14 -
Theorem 2
Problem n | 2 | s = l , f o r e s t , *£^=1, n o n p r j c ^ d ^ i s N P -com plete.
Theorem 5
Problem n | 2 | s = l , т-^=1, ^ , T i = l, n o n p r j c ^ d i i s NP-com
p l e t e . Theorem 4
Problem n |3 | s = l , *tV=l, n o n p r j c h ^ d ^ i s N P-com plete
¥e g iv e h e r e a l s o an i n t e r e s t i n g r e s u l t c o n c e rn in g th e p o ljr- n o m ial r e d u c i b i l i t y among th e problem o f m in im iz in g maximum l a t e n e s s and th e p ro b lem o f s c h e d u lin g to m eet d e a d lin e s £ 5]] .
Theorem 5
n lml ' Ч Ijman œ n lm|M Ci £ di*
P ro o f
G iven th e p ro b lem n |m| Л/{ Ьт а х w ith d e a d lin e s d* and th e v a lu e L o f ,aximum l a t e n e s s f o r w hich a " y e s -n o " q u e s tio n i s an
sw ered, we c a n c o n s t r u c t a n in s t a n c e to n|m |A .|c d ± i n th e f o llo w in g way: P u t a l l t h e p a r a m e te rs th e same a s i n
n |m |X |L max, and p u t d ± := d ' + L, i = l , 2 , . . . , n .
I t i s c l e a r t h a t th e p ro b le m n|m I M Lmaj, h a s a s o l u t i o n w ith v a lu e L i f and o n ly i f t h e problem n |m \X |c i ^ d i h a s an answ er " y e s " .
Q
F o llo w in g S e c t io n s 2 and 3 we may sum m arize th e c o m p le x ity o f s c h e d u lin g t o m eet d e a d l i n e s i n th e f o llo w in g T ab le 1.
- 15 -
T a b l e 1
C om plexity o f s c h e d u lin g problem s t o m eet d e a d lin e s
P r o b l e m c o m p l e x i t y
m 9
m l R e f e r e n c e s
0 ( n 2 ) 2
L * 1 0 S = 1 n ^ - 1 f
o p e n 2
L - 1 0 S = 1 -
N P - c o m p l e
t e 2
V f o r e s t s = 1 - Í 5 ]
N P - c o m p l e
t e 2
n = 1
- s = 1 m „ « 1
1 [ 5 ]
N P - c o m p l e
t e 3
ИII
0 S = 1 -
[ 5 ]
4 . REDUCTION TO THE NETWORK FLOW PROBLEM
As we have s t a t e d , th e problem n |2 |s = k , r ^ O , L i =->
n o n p r j c ^ ^ d ^ s t i l l re m a in s open. I n t h i s S e c t i o n , we p r e s e n t a method f o r s o lv i n g t h i s problem » b u t we do n o t c la im t h a t i t can be s o lv e d i n p o ly n o m ia l tim e . F u r t h e r , we w i l l show how th e p roblem n jm |s= k , r ^ O , t ^ = l , n o n p r jc ^ ^ d ^
can be s o lv e d . Namely, we re d u c e i t to th e p ro b lem o f n e tw o rk flo w s w ith m u l t i p l i e r s . The l a t t e r can be fo rm u la te d a s f o l l o w s .
L e t G be a d i r e c t e d graph w ith v e r t i c e s s ^ jS g ^ v ^ ,. . . »v^
and a r c s e-, , e « , . . . ,e . L e t w" "(v ) be th e s e t o f a rc s d i r e c t e d
-L i_ 4.
in t o v e r t e x v and w+( v ) , th e a r c s d i r e c t e d away from v . G w i l l be s a id to d e n o te a n etw o rk w ith m u l t i p l i e r s i f :
a / The s o u r c e , s ^ , o f th e n e tw o rk h a s no incom ing a r c s , i . e . w” ( s 1)= 0 .
- 16 -
Ъ / The s i n k , s 2 , has no o u tg o in g a r c s , i . e . w+ ( s 2) =0.
с / With e v e r y v e r te x v.^ /e x c lu d i n g t h e so u rc e and s i n k / th e re c o r r e s p o n d s an i n t e g e r h ^ O , c a l l e d i t s m u l t i p l i e r ,
d / To each a r c , e^, t h e r e c o rre s p o n d s a n i n t e r v a l
We a re r e q u i r e d to f i n d a flow v e c t o r ф а ^ ф ^ , ^ . . . , ) su c h t h a t :
1 / а 1 $ ф 1 <.Ъ1
27 h (v ) i t w “Cv^ " I & Ы ф1
^ i l w ” (s
2
}f o r a l l V , v/s-, , v / s , i s m axim ized
ф1 , i = l , 2 , . . . , q , a r e i n t e g e r s .
S o lv in g th e p ro b le m n |2 |* = k , r ^ / 0 , Ц =1, n o n p r ^ C h ^ L we c o n s t r u c t a n e tw o r k in th e f o llo w in g way. E ach a rc e^ h a s a s s o c i a t e d w ith i t an i n t e r v a l £ ° . d , t h a t means e i t h e r ф^=0 o r ф^=1. We d i s t i n g u i s h t h r e e g ro u p s o f v e r t i c e s . A ll h (y j a r e e q u a l t o 1 ex cep t some v e r t i c e s fro m th e seco n d g ro u p .
The f i r s t group r e p r e s e n t s tim e i n t e r v a l s i n th e s h e d u le . For exam ple v e r te x 1 r e p r e s e n t s tim e i n t e r v a l ^ ) , ÏJ , v e r t e x 2, tim e i n t e r v a l L1 - 2] , and so o n . I t i s o b v io u s t h a t th e o p tim a l s c h e d u le must b e no lo n g e r t h a n D d.imax= m ax^d^ . So th e number o f v e r t i c e s i n t h e f i r s t g ro u p i s equal t o d . „ .0 ^ u imax
The second g ro u p o f v e r t i c e s r e p r e s e n t s t h e p o s s i b i l i t y o f p r o c e s s in g t a s k s i n p a r a l l e l . T h a t i s , we h a v e a v e r t e x c o r re s p o n d in g to t h e p a r a l l e l p r o c e s s i n g o f T. and T . /w e d e n o te
- 17 -
i t by T ^ / i f and o n ly i f B ^ T j ) + * ^ 0 ^ ) 1 = 1 , 2 , . . . , s , and b o th d , ^ r . and d , ^ r , . S in c e i n t h i s way we s im u la te th e
J- J J •*-
p r o c e s s in g o f two t a s k s i n p a r a l l e l , we have a m u l t i p l i e r 2 a s s o c i a t e d w ith each such v e r t e x . Of c o u rs e , we a ls o have n v e r t i c e s t h a t c o rre s p o n d to th e p r o c e s s in g o f s i n g l e t a s k s . / T h e i r m u l t i p l i e r s a r e e q u a l to 1 / . The t o t a l number o f v e r - t i c e s o f th e second group i s 0 n2
We draw an a r c j o i n i n g v e r t e x к C.k-1 » 2 ». . . . » d j ^ ^ ) o f th e f i r s t group t o th e v e r t e x T .T . ( o r T . ) , o f t h e second g ro u p i f and o n ly i f t a s k s T . and T. ^ o r s i n g l e T. ) can be p r o c è s -
X J X
sed i n th e tim e i n t e r v a l i . e . b o th r^=m ax b ’r 4 <
£ k - l and d ^ m in ^ L , d Л ^ k .
The t h i r d g roup o f v e r t i c e s c o n t a i n s n v e r t i c e s which c o r re sp o n d to t a s k s . We draw an a r c jo i n i n g a v e r t e x from th e
second group w ith v e r t e x T^ from th e t h i r d g ro u p i f and o n ly i f t h i s f i r s t i s one o f th e f o llo w in g : T. o r T .T ., j = l , 2 , . . . . , n . I n t h i s way we a r e s u re t h a t a l l ta s k s w i l l be p r o c e s s e d .
I t i s c l e a r t h a t th e maximal flo w j _ £ ^ ( s ) ф^ i s e q u a l to n and ca n be a c h ie v e d i f and o n ly i f a l l t a s k s meet t h e i r d e a d l i n e s . The o p tim a l s c h e d u le i s c o n s tr u c te d on th e b a s i s o f th e o b ta in e d a r c f lo w s .
- 18 -
5 . PINAL REMARKS
I n t h e c la s s o f p ro b le m s o f s c h e d u lin g t o m eet d e a d lin e s u n d e r r e s o u r c e c o n s t r a i n t s th e o n ly problem f o r w h ich an o p tim a l and p o ly n o m ia l i n tim e s c h e d u lin g a lg o rith m i s known to e x i s t i s n ^ 2 ^ s = l, m^=l, 1 ^ = 1 , n o n p r j c k ^ d ^ The more c o m p lic a te d p ro b lem w ith an a r b i t r a r y number o f r e s o u r c e t y p e s s t i l l re m a in s o p e n . O ther p ro b le m s have b e e n proved to b e N P -com plete.
REPERENCES
1 . AhOjA.V., H o p c r o f t , J . E . , U llm an,D .D . î The D esign and
A n a ly s is o f C om puter A lg o rith m s , A ddison-W esley, R e a d in g , M a ss., 1974
2 . B la z e w ic z ,J . ; S c h e d u lin g d ep e n d e n t t a s k s w ith d i f f e r e n t a r r i v a l tim e s t o m eet d e a d l i n e s , i n E .G e le n b e , H .B e iln e r / e d s . / M o d e llin g and P e rfo rm an c e E v a lu a tio n o f Computer System s, N o rth H o lla n d , Amsterdam, 1 9 7 6 , 1976, p p . 5 7 -6 5 . 3 . B îa z e w ic z ,J . : D e a d lin e s c h e d u lin g o f t a s k s - a s u rv e y ,
F o u n d atio n s o f C o n tro l E n g in e e rin g 1 , No4, 1976, p p . 2 o 3 -2 l6 .
4 . B îazew icz, J . : Mean flo w tim e s c h e d u lin g u n d e r r e s o u r c e con
s t r a i n t s , R e p o r t No P R -1 9 /7 7 , I n s t i t u t e o f C o n tro l E n g in e e r
i n g , T ech n ical. U n i v e r s i t y o f Poznan, 1977
5 . B ïa z e w ic z ,J . : S c h e d u lin g w ith d e a d lin e s and r e s o u r c e con
s t r a i n t s , R e p o r t N o P r-2 5 /7 7 , I n s t i t u t e o f C o n tro l E n g in e e r
i n g , T e c h n ic a l U n i v e r s i t y o f Poznan, 1977
6 . B lazew icz, J . : S im ple a lg o r ith m s f o r m u l t i p r o c e s s o r s c h e d u l
in g to m eet d e a d l i n e s , I n f o r m a tio n P r o c e s s i n g L e t t e r s 6, N o .5, 1977
- 19 -
7 . C offm an,E .G ., j r . / e d / : Computer and Jo b /S h o p S c h e d u lin g T heory. J . W iley and S ons, New Y ork, 1976.
8 . D h a ll ,S .K ., L i u ,C .L .: On r e a l - t i m e s c h e d u lin g problem / t o a p p e a r /
9 . G arey ,M .R ., G ra h am ,R .L .: Bounds f o r m u lt ip r o c e s s o r s c h e d u lin g w ith r e s o u r c e c o n s t r a i n t s , SIAM, J .o n Com puting 4, 1 9 7 5 .
1 0 . G a re y ,M .R .» Jo h n so n ,D .S .: C om p lex ity r e s u l t s f o r m u l t i p r o c e s s o r s c h e d u lin g u n d e r r e s o u r c e c o n t r a i n t s , SIAM J .o n Comput
in g 4, 1975, p p . 3 9 7 -4 1 1 .
1 1 . K arp ,R .M .: R e d u c i b i l i t y among c o m b in a to r ia l p ro b lem s, i n R.
M i l l e r and J .T h a t c h e r / e d s . / C om p lex ity o f Comp. C om putat 1972.
1 2 . K ra u s e ,K .L .,S h e n ,V .Y .,S c h w e tm a n ,H .D .: A n a ly s is o f s e v e r a l t a s k - s c h e d u lin g a lg o rith m f o r a model o f m u ltip ro g ram m in g com puter s y s te m s , J.ACM. 22, 1975, p p . 5 2 2 -5 5 o .
1 3 . L a b e to u lle , J . : Some theorem s on r e a l - t i m e s c h e d u lin g , i n E .G e le n b e , R.M ahl / e d s . / Computer A r c h i t e c t u r e and N e tw o rk s, 1974.
1 4 . L a b e t o u l l e , J . : R e a l tim e s c h e d u lin g i n a m u lt ip r o c e s s o r e n v iro n m e n t, / t o a p p e a r /
1 5 . L e n s tr a ,J .K .,R in n o o y Kan, A .H .G ., B ru c k e r, P . : C om plexity o f m achine s c h e d u lin g p ro b le m s, A n n .D is c re te M ath, / t o a p p e a r / 1 6 . L iu ,C .L .j L a y la n d ,J .W .: S c h e d u lin g a lg o r ith m s f o r m u l t i
program m ing i n a h a rd r e a l - t i m e e n v iro n m e n t, J.ACM 2 o ,N o .l . 1973. p p . 4 6 -6 1 .
■< ‘ *■
THE STATISTICAL BEHAVIOR OP A CHARACTERISTIC OF THE INSERTION ALGORITHM
"BATCHED HASHING AND LINEAR PROBING"
A. K rá m li J . P e r g e l
I n t h i s n o te we i n v e s t i g a t e th e s t o c h a s t i c b e h a v io r o f some s t a t i s t i c s a r i s i n g from th e i n s e r t i o n a l g o r i t h m " b a tc h e d h a s h in g and l i n e a r p r o b in g " / s e e [ l ] / . I n th e l i t e r a t u r e / s e e
e . g . [2] / t h e r e a r e r e s u l t s f o r th e p r o b a b i l i t y d i s t r i b u t i o n o f th e m ost im p o r ta n t c h a r a c t e r i s t i c s o f s u c h a l g o r i t h m s , b u t t h e i r tim e e v o l u t i o n - from th e p o in t o f view o f common d i s t r i b u t i o n s - i s l e s s ex am in ed . I n [3] t h e r e i s p ro v e d th e
M a rk o v ity o f th e p r o c e s s o f th e number o f c o n n e c te d f u l l i n t e r v a i s i n th e c a s e o f " h a s h in g an l i n e a r p r o b in g " , m oreover t h e
t r a n s i t i o n p r o b a b i l i t i e s a r e d e te r m in e d . Now we g e n e r a l i z e t h e s e r e s u l t s to th e c a s e o f b a tc h e d h a s h in g .
F i r s t we g iv e a s h o r t d e s c r i p t i o n o f t h e a l g o r it h m . L e t a f i l e c o n s i s t o f N b lo c k s / b u c k e t s / and e a c h b lo c k can c o n t a i n к r e c o r d s . I f a new r e c o r d i s t o be i n s e r t e d , t h e n a h a s h f u n c t i o n k « ) e O , - , N) a s s i g n s a b lo c k a d d r e s s t o лД . I f th e l l (Д ) - t h b lo c k c o n t a n in s l e s s th a n к e l e m ents i s a f i x e d n a t u r a l n u m b e r/, th e n t h e new r e c o r d i s p la c e d i n t o t h i s b lo c k . I f i t c o n t a i n s к r e c o r d s , th e n th e r e c o r d w i l l be p la c e d i n th e f i r s t b lo c k o f th e se q u en ce
ll ОД) , 1г (/Д ) + 4 j ... C o n ta in in g l e s s th e n к r e c o r d s .
/T h e s i g n "+" i s u n d e r s to o d mod N, w hich means t h a t th e f i l e i s c o n s id e r e d t o be c i r c u l a r / . The se q u e n c e
- 2 2 -
( 4 = 1 , 2 , . . . , ^ i s su p p o se d t o be a s e q u e n c e o f in d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d random v a r i a b l e s . L et us in tr o d u c e some n o t a t i o n s . The v e c t o r
0 (t) c: (0^ (t)
d e s c r i b e s t h e s t a t e o f t h e f i l e a f t e r th e i n s e r t i o n o f th e t - t h r e c o r d ;0 L[i) i s th e num ber o f r e c o r d s i n th e b lo c k i . The v e c t o r I ( i ) - ( ( Д . . . ^ U) ) i s th e s t a t i s t i c s / a f u n c t i o n o f 0 (0 / th e s t o c h a s t i c b e h a v i o r o f w hich i s t o be i n v e s t i g a t e d .
«
= th e num ber o f b lo c k s c o n t a i n i n g i r e c o r d s .(0 I i f i < t
= th e num ber o f c o n n e c te d i n t e r v a l s o f b lo c k s c o n t a i n i n g к r e c o r d s i f i = k N o tic e t h a t î ( t ) i s d e te r m in e d by t I (i) {» (^)
b u t f o r th e s a k e o f th e s i m p l i c i t y o f t h e t r e a t m e n t we assum e i t a s a com ponent o f ^ (t) •
Theorem 1 . The v e c t o r p r o c e s s f GO i s M arkov.
P r o o f . F i r s t we g iv e th e p o s s i b l e ty p e s o f t r a n s i t i o n s
| ( t ) — |(i+<)
I . T here i s a u n iq u e i such t h a t § (t+^) =» f^ ( ^ ) - “f
' l - 4 ' l - 4
5. ( t+4) = Í : (OH J L
f o r ev ery ^ -éç. i - 4) t
and
j. (444) = f U)
I I .
! Л t u b f , ( 0 , f, ( И =
*L i ]L M-4
f e - 4 and,( * * < ) = 1 . ( 0
r *
f o r e v e ry j < li- 'i
III. ^ (4 + -0 = an d t h e r e is a u n i q u e Í 6
I
- 23 -
su c h t h a t + ^ W H f ^
and $. Î: M(U4J= i .“ C- ( 0 ( 4) f o r e v e ry j. Af î - { } i
O u
IV.
and V.
a n i ^ _ ( И "
f • ( tM ) = !^. ( О f o r ev e ry ^ ^ ^ - 4 u + o = ? ,( * ) - < “ d Ц ь , ) - <
:k 4 ' ' k
and |j (t) f o r e v e ry j < t - \ .
For th e p r o o f o f th e M a rk o v ity o f th e p r o c e s s
fft)
wehave t o v e r i f y th e r e l a t i o n
(2) p ( f ( m ) I Î W , f u - ' ) , . . . ) = P ( f (t+<) I Í (tí)
R e l a t i o n (2 ) f o r th e t r a n z i t i o n s o f ty p e I i s a sim p le co n se q u e n c e o f th e sym m etry o f t h e i n s e r t i o n a l g o r it h m . I n th e re m a in in g 4 c a s e s th e p r o o f o f (2) c a n be c a r r y e d o u t a n a lo g o u s ly / r e p l a c i n g fo rm u la (3) by th e c o r r e s p o n d in g o n e s / t h e r e f o r e we do i t o n ly f o r t r a n s i t i o n s o f ty p e V .
F or a c o n f i g u r a t i o n 0 we d e n o te by { è ^ s ) th e se q u e n c e o f c o n n e c te d i n t e r v a l s o f f u l l b lo c k s and by
^0 th e sum o f th e l e n g t h s o f th e c o n n e c te d i n t e r v a l s o f f u l l b l o c k s , w hich a r e f o llo w e d by a u n iq u e b lo c k c o n t a i n i n g к - l r e c o r d s and a f u r t h e r i n t e r v a l o f f u l l b l o c k s . By th e d e f i n i t i o n o f th e i n s e r t i o n a l g o r it h m :
(3) P ( S fcM 0 (D ) = Í I
N
- 2 4 -
t h e s e t o f a l l c o n f ig
(0 I
)•••) ^0(i) J
r ùj ? < ) • • • ) ^ (4- 'j
L e t u s d e n o te by L /l *> '
u r a t i o n 0(t) f o r w hich t h e se q u en ce { ^§(t) >" ' ) i s a p e r m u ta tio n o f th e n a t u r a l num bers ^ j .. lv ,
By th e f o rm u la (3) a n d t h e sym m etry o f th e i n s e r t i o n a l g o r i t h m a t e v e ry moment t u n d e r t h e p e r m u ta tio n s o f th e a d d r e s s e s p r e s e r v i n g th e s e t o f n a t u r a l num bers
t«*2
m v ol (V9ë ( t ) m"I lJ( 0 s { v m
m ) “ • ) ht)
we g e t t h e p r o p e r t i e s :
d o es n o t depend on th e
(4) P C t ' j .
o r d e r o f num bers
m and
(5 ) P ( i fcM - ^
|U ) .
The p r o p e r t y (4) re m a in s v a l i d u n d e r e v e ry c o n d i t i o n on th e p a s t ^ j )--• , w hich t o g e t h e r w ith p r o p e r t y (5) p ro v e s (2 ) f o r t r a n s i t i o n s o f ty p e
V .
C * W , (t)) d ep e n d s o n ly on th e s t a t e
A b s t r a c t
In this talk the authors prove that a statistics of the record insertion algorithm "batched hashing and linear pro
bing" forms a Markov process.
- 25 - >
R e f e r e n c e s
[Ï] B la k e , I . P. - Konheim , A. G ., B ig B u c k ets A re /A re N o t/
B e t t e r ! , JACM, 24 / 1 9 7 7 / 4 . 5 9 1 -6 0 7 .
[2З K n u th , E . E . , The A rt o f Com puter P rogram m ing, Vol. 3 , A d d iso n -W e sle y , R eading-M enlo P ark-L ondon-D on M i l l s 1973.
[3] K rá m li, A . j P e r g e l , J . , A p p ro x im a tio n o f a r e c o r d i n s e r t i o n a l g o r i t h m by random w alk p r o c e s s , Problem s o f C o n tro l and I n f o r m a tio n Theory 6 /4 2 0 7 -2 1 1 .
András Krámli József Pergel
Computer and Automation Institute Coordination and Scientific Hungarian Academy of Sciences Secretariat, Computing
Center of the Hungarian Planning Office
i
A STOCHASTIC MODEL IN PRESENCE OP INTERMITTANT FAILURE
B enczúr A ,, K rá m li A .,
In p a p e r £ e ] we p u b lis h e d o u r f i r s t r e s u l t s c o n c e rn in g a s t o c h a s t i c m odel f o r th e b e h a v io u r o f a co m p u ter system i n p r e s e n c e o f i n t e r m i t t e n t f a i l u r e s . We have d e d u c e d our m odel from l o g i c a l c o n s i d e r a t i o n s o f w orks M and Г г ] assum ing t h a t th e f a i l u r e s ca n be l o c a l i z e d t o t h e l o s s o f t h e c o n te n t o f c e n t r a l memory /q u ic k r e c o v e r y / . Led by p r a c t i c a l e x p e r ie n c e s f i r s t we w ere i n t e r e s t e d i n tim e consum ing u p d a t e p r o c e s s e s w hich a r e t y p i c a l f o r b a tc h s y s te m s .
The c o s t o f an u p d a te r u n i s a convex / n o n l i n e a r / f u n c t i o n o f th e e x t r a tim e c a u se d by f a i l u r e s , t h e r e f o r e we h a d to i n v e s t i g a t e n o t o n ly th e mean v a lu e o f t h i s e x t r a tim e , b u t th e a s y m p to tic b e h a v io u r o f i t s p r o b a b i l i t y d i s t r i b u t i o n .
Our m odel i s v e ry c l o s e t o m odels g iv e n by Chandy И an d G elenbe [ 4 ] - Por th e s i m p l i c i t y o f tr e a tm e n t we assume th e tim e to be d i s c r e t e , and we n e g l e c t th e tim e o f c r e a t i n g a ch eck p o i n t /w h ic h i s p r a c t i c a l l y e q u a l to a c o n s t a n t v a l u e / .
F u r t h e r we assum e t h a t th e r e c o v e ry tim e i s eq u al t o th e tim e
Yt
o f th e n o rm al w o rk in g o f th e sy ste m a f t e r th e l a s t ch e ck p o i n t / Yb meana th e age o f th e s e r v e r i n [ 4 l / , i f d u r in g th e re c o v e ry no f a i l u r e o c c u r s . In t h e o p p o s ite c a s e , when a f a i l u r e o c c u r s , th e n th e sy ste m r e t u r n s a g a in t o t h el a s t c h e c k p o in t , e . t . c . - so i n o u r model f a i l u r e s a r e a llo w e d d u r in g th e re c o v e ry p r o c e d u r e .
- 28 -
We d e n o te by M t h e seq u en ce o f tim e i n t e r v a l s b e tw e e n two c o n s e c u t i v e f a i l u r e s , w h ich fo rm s a se q u e n c e o f i . i . d . random v a r i a b l e s , i . e . th e f a i l u r e p ro c e s s i s a g e n e r a l ren ew a l p r o c e s s . The s e q u e n c e Ш o f tim e i n t e r v a l s b e t w een two c o n s e c u t i v e c h e c k p o in t s m e asu red by th e tim e o f th e n o rm a l w orking o f th e s y s te m i s a s e q u e n c e o f i . i . d . a n d bounded
F t y •
random v a r i a b l e s w ith th e common d i s t r i b u t i o n f u n c t i o n We assume t h a t t h e p r o c e s s e s M and a re t o t a l l y in d e p e n d e n t.
I f the f a i l u r e p r o c e s s i s Markov r e n e w a l / i . e . i s g e o m e tr ic a lly d i s t r i b u t e d w i t h th e p a r a m e te r
p / ,
th e n we c a n r e p la c e t h e c o s tUrt)
o f th e r e c o v e r y in tr o d u c e d i n [4 ] by t h e e x p e cte d tim eU Y t )
o f th e r e c o v e r y p ro c e d u re d i s t u r b ed by random f a i l u r e s ( h i s th e age o f th e s e rv e r" ).So, from t h e p o in t o f v ie w o f e x p e c te d e x tr a tim e c a u se d by f a i l u r e s o u r m o d e l can be re d u c e d t o G e le n b e ’ s o n e .
But i f we a r e i n t e r e s t e d i n th e p r o b a b i l i t y d i s t r i b u t i o n o f t h i s e x tra t i m e , o r th e f a i l u r e p r o c e s s i s n o t Markov r e n e w a l, then o u r m o d el i s m ore a d e q u a te .
I t i s easy t o c a l c u l a t e t h a t u n d er t h e above a s s u m p tio n s ï ,
C Yfc )
>C p ^
s e e [ 5] . So, by £4] th e s t a t i o n a r y p r o b a b i l i t y o f t h e norm al w o rk in g o f th e ^ s y s te m i s e q u a l to 0 , i f th e momentum g e n e r a t o r f u n c t i o n J>~ P. ^ o f t h e d i s t r i b u t i o n f u n c t i o n R j ) does n o t e x i s t f o r*= <
4
• Q his f a c t makes n a t u r a l o u r c o n d i t i o n on th e b o u n d ed n ess o f •L e t us i n t r o d u c e th e random p r o c e s s e s
{ 9 | J K J
é w = - 9 k
The random v a r i a b l e 0 ^ i s e q u a l to t h e d i s t a n c e betw een
th o s e c h e c k p o in ts from w here th e k - l - s t and th e 1с-Ь» r e c o v e r i e s were made, and 8 ^ i s th e e x t r a tim e c a u se d by th e
к -th f a i l u r e .
n i ( 4 , i s the
J=< *
num ber o f f a i l u r e s o c c u r in g up to th e n - t h c h e c k p o in t. )
I f , th e n { } and f ®lc }
a r e se q u e n c e s o f i . i . d . random v a r i a b l e s , and Anscombe’ s Set "1Л = min ^
n f a
c e n t r a l l i m i t th e o re m c a n be a p p l ie d t o th e norm ed sums
V" S k . E e k
k . 4 'n J * ~
I f
P С V « Н О
f o r e v e ryl £
max , th e n t h e se q u e n c e ca n be embedded i n t o a n i r r e d u c i b l e a p e r i o d i c Markov c h a in { E le î w ith f i n i t e s t a t e s p a c e , t h e r e f o r e t h e r ee x i s t s a u n iq u e s t a t i o n a r y d i s t r i b u t i o n f o r Ä 1 . I t c a n be shown t h a t i f th e p r o c e s s | i s s t a t i o n a r y , th a n t h e p ro c e s s
£ 0^1
i s a l s o s t a t i o n a r y . S etE 6 ^ = ftl, E Cj
w here th e e x p e c t a t i o n i s ta k e n on t h e b a s i s o f s t a t i o n a r y d i s t r i b u t i o n s . The s t a t i o n a r y se q u e n c e s a t i s f i e s t h e
- 30 -
s t r o n g m ix in g c o n d i t i o n , a n d R ozanov’ s c o n d i t i o n s e e h i o f t h e v a l i d i t y o f th e c e n t r a l l i m i t th e o re m . U nder f u r t h e r c o n d i t i o n s on p r o c e s s e s { % ] and ( U we can p ro v e th e c e n t r a l l i m i t th eo rem f o r th e sums
t o o :
í * k í h a s f i n i t e 8 - t h moment a n d
F f ÜL ~ lT0 ) 0 0 >th e n t h e r e e x i s t s a p o s i t i v e number ^ k
s u c h t h a t
lim
Oo
p(
21
lc«4____________* v T \T
u '
2
We can f o r m u l a t e th e q u e u e in g p ro b le m s a n a ly s e d by G elenbe i n o u r r e c o v e r y model t o o .
We assum e t h a t th e q u eu e can be a r b i t r a r i l y lo n g , th e p r o b a b i l i t y o f t h e a r r i v a l o f a r e q u e s t a t th e moment t i s e q u a l to p < 1 in d e p e n d e n tly o f th e s t a t e o f th e sy ste m and t h e s e r v ic e tim e - m easured by th e tim e o f th e n o rm a l w orking o f th e system - i s g e o m e t r i c a l l y d i s t r i b u t e d . L e t be t h e le n g th o f t h e queue a t t .
The p r o c e s s C M c a n be r e g a r d e d a s a d o u b ly s to c h a s t i c Markov c h a i n th e t r a n s i t i o n p r o b a b i l i t i e s o f w h ich depend on t h e r e a l i z a t i o n s o f p r o c e s s e s { * « , ] and /h e r e ^ d o e s n o t c o n t a i n th e tim e , when th e queue i s e m p ty /. The p r o c e s s t u c a n b e embedded i n t o a Markov c h a in t Ы w ith c o u n ta b le s t a t e s p a c e . U sin g th e e le m e n ta ry m ethods o f renew a l th e o r y t h e r e c a n be p ro v e d th e f o l lo w in g :
- 31 -
THEOREM 2 . The p r o c e s s £ ^ h a s s t a t i o n a r y d i s t r i b u t i o n i f and o n ly i f p КП
A b s t r a c t
T h is t a l k g iv e s an exam ple f o r th e a p p l i c a t i o n o f c e n t r a l l i m i t th e o re m s i n th e c a l c u l a t i o n o f th e p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n o f th e e x t r a - t i m e consum ed by r e c o v e r y p ro c e d u r e s e n s u rin g th e r e l i a b l e w o rk in g o f DBMS.
- 32 -
R e f e r e n c e s
[ i ] CODASYL C o m itte e Data B ase Task Group R e p o rt, A s s o c i a t i o n f o r Com puting M a c h in e ry , A p r i l , 1971.
U 'i POSSUM, B.M. , Data b a s e i n t e g r i t y a s p ro v id e d by a p a r t i c u l a r d a ta b a s e management s y s te m ,
P r o c e e d in g s o f I P I P W orking C o n feren c e on Data B ase M anagem ent, C a rg e se C o r s i c a , P r a n c e , 1-5 A p r i l , 1974.
[зЗ Chandy, K.M. , Browne, I . C . , D i s s l y , W.R. U h rin g - " A n a ly t
i c a l m odels f o r R o l l back an d R ecovery S t r a t e g i e s i n D a ta Base S y stem s" - IEEE T r a n s a c tio n s on
S o ftw a re E n g in e e rin g - V o l. 1 , n° 1, - p p . l o o - l l o - M arch 1975.
W G elen b e, E . On th e optimum c h e c k p o in t i n t e r v a l / M a n u s c r i t / M V oid, H ., S jjö r g e n ,B .H ., O p tim al Backup o f Data B a s e s : A
S t a t i s t i c a l I n v e s t i g a t i o n s , B IT , 13, 1973 pp . 2 3 3 -2 4 1
Гб] R é n y i, A ., On t h e c e n t r a l l i m i t th e o re m f o r th e sum o f a random number o f in d e p e n d e n t random v a r i a b l e s ,
A cta M a th . Acad, S e i . H u n g a r.v . 11, i 96 0 . p p .9 7 - l o 2 .
- 33 -
[7] Ибрагимов, И.А.; Лични^Ю. В^ Независимые и стационарно связанные величиныj НАУКА,
Москва, ^ 6 5 \
[е] Benczúr,A*, Krámli, A., A note on data base integrity Acta Cybernetica, Tom. 3 , Base. 3 , Szeged, 1 9 7 7 .
p p . 181-185
OPERATING SYSTEM AND DATA BASE FOR A SMALL PRODUCTION CONTROL SYSTEM A.Wolisz, A.Krawet, J.Mierzwa, A.Nowakowski
1 .INTRODUCTION
Authors of this paper took part in the project of developing a dispatcher-aiding computer system for the soaking pits divi^:v sion of iron and steelworks. By now, the system has been suc- cesfully operating for 6 months, and considerable amount of experience on it*s exploitation has been obtained. The problem of systèmes functional characteristic, and scope of application software for such a plant will be discussed in a separate pa
per, being mentioned here briefly to give some impression of the problem whitch was solved. Cn the other hand, some solutions introduced in the structure of operating system, application software and data base organisation are believed to be of more general type-pertaining possibly to a larger class of discrete type production control systems.
2.PLANT DESCRIPTION
The soaking pits division is a crucial part of the so called
"hot steel line" in a steelwork, situated between the melting shop and slabbing mill. The division should provide the slabfc . bing mill with a constant stream of properly heated ingots, in
dependently of possible perturbances in melting shop operation.
In order to do so, it is provided with a local stock of ingots /a local buffer/ and cooperates closely with a cold ingots m a ga zine.
The scheme of material flow in soaking pits division is presen
ted in fig.1.
In the considered case the soaking pits division consisted o.f 40 pits, each of them allowing for the average load of 100 tons and being recharged in average once during a shift.
- 36 -
Ingots were delivered in batches corresponding to consecutive casts, consisting of 300-350 tons of steel. Up to 3000 tons of steel could be hold on the local stock. Up to twenty mould ty
pes and 150 types of steel were to be considered.
f ig . 1
3.SYSTEM FUNCTIONS
The computer system was assumed to operate in a real-time off-line mode, being used by a dispatcher in a conversational V/ay. The main functions were following ones :
a/ keeping track of the material flow through the plant, in
cluding ;
- perserving information about the load of every soaking pit
- perserving information about the local stock of ingots - constant displaying of the pit*s actual state
b/ suggesting the way of heating and time to be spend in the pits, including :
- classification of ingots to various groups - charging temperature calculations
- determining the number of process phases and their dura
tion
- 37 -
с/ heating monitoring,consisting in displaying the time-to- completion for every phase and alarming the dispatcher if any process phase has not been completed in proper time d/ reporting - including shift reports and current reports All the information about the division's operation is intro
duced by the dispatcher through an alfanumeric keyboard,
in conversational units corresponding to elementary technolo
gical operations.
4.SYSTEM CONFIGURA TION
The scheme of the utilized computer system configuration is presented in fig.2.
fig. 2 The equipm ent c o n s i s t o f :
- C e n tr a l P r o c e s s i n g U n it w ith 16 kword c o r e memory /C PU / - 2 V id e o -D is p la y s /VD1 and VD2/ w ith a lp h a n u m e ric K ey b o ard s - C h a ra c te r P r i n t e r /С Р /
- P ap er Tape R e a d e r /PTR / - P ap er Tape Punch /Р Т Р /
P e r i p h e r a l s w ere u t i l i s e d in th e f o llo w in g way :
- The PTR was used f o r i n t r o d u c i n g th e code and d a ta e x c l u s i v ely
- The PTP was used f o r o b t a i n i n g th e h i s t o r y o f ev ery h e a t i n g
- 38 -
cycle, to be later used as a source information for accoun
tancy, technology analysis and quality inspection
- The CP was used for reports printing and obtaining hard co
py of some actions
- The VD1 was used for conversation - it is data introducing and obtaining information about the suggested decisions - The VD2 was used for displaying the state of soaking pits
division
5. GENERAL SOFTWARE ORGANISATION
We shall now discuss briefly the software and data base organisation, presented in fig.3 following representation in
troduced in /1/.
Permanent DB Variable DB 03 DB
f i g . 3
- 39 -
A detailed discussion of some main parts will follow in con
secutive sections. The data base is devided into three main parts :
a/ The permanent data base /PDB/
PDB contains information about the mathematical models in
cluded in the system, eg. the cooling model, the heating model. The PDB may be changed exclusively through reading from the paper tape under the operating systems initial start module supervision. It is of the read-only type for all programs included in the system,
b/ The variable data base /VDB/
VDB contains information about the actual state of the >
plant, thus enabling monitoring of the material flow and soaking pits operation,
с/ The Operating System data base /OSDB/
OSDB contains all information desired for program schedu
ling, buffers for inter-program communication, and device management tables.
Program modules included into the system have been devided into several levels /fig.3/ grouping those having equal ac
cess rights to other modules and various parts of data base.
Arrows represent the privilege of calling lower levels by m o dules occupying higher levels.
a/ The lowest level consisting of I/O traffic control routine . / and drivers for peripheral devices^ handles the I/O re-* ■-* „
quests. All the peripherals except of video-displays ope
rate in a fully asynchronous way, with speed controled by the CPU. The interface standard is in agreement with the one utilised in Hewlett-Packard computers /2/. Interfacing of the video-display /Videoton VT 340/o-type/ is different because in the SEND mode and ACTIVISATION LOOP their ope
ration speed is computer independent and information loss has to be avoided. Detailed discussion of this problem is given in the Appendix.
b/ The FORTRAN LIBRARY contains a set of routines for standard mathematical and logical operations. It includes also a special program, the FORMATTER, used as the single interfa-
- 4 0 -
ce between I/O requests of the applications programs and the device management level. FORMATTER allows for data in
put and output in the desired, predefined format.
с/ The Application Library / А Ы В /
А Ы В contains a set of routines prepared to operate on VDB.
They perform data storage and retrieval in specific data structures used in VDB, like packing and unpacking several data items into one memory location. They perform a number of sorting and searching operations according to the defi
ned key. A L I B contains also a subroutine for editing stan
dard massages and input data control routines for data correctness checking.
d/ The Application Modules, are designed as conversational units for man-machine communication. Eighteen such modules included in the system may be requested by the dispatcher through appropriate operations codes. Those modules are closely connected with various technological situations, about which either data are to be introduced, or some in
formation is to be obtained from the system. After choo
sing the proper module the dialog is controled by the com
puter, which asks consecutive questions. The dispatcher has to introduce numerical or alphanumerical data of defi
ned type when requested. As sometimes a logical-type an
swer /YES or N0/ is needed so called DECISION POINTS have been defined, were the answer YES, NO or CHANGE THE MODULE
>M0DULE CODE < is to be given, '//hen no application module is being executed, a pseudo module - waiting for a new mo
dule request is in progress. Both ALIB and Application Mo
dules are written in FORTRAN.
e/ Alarm Module is utilised on operátoros request, communica
ted thrugh the CPU switch register, in the case of peri
pheral devices failure, in order to arrange proper recon
figuration. This module nay be also called when the device improper operation has been established by the Operating System. 1 • It haust access to the device management level,
to break the improper I/O operation and clear the device s ta tus.
- 41 -
f/ The Time-Tracking Module, is in charge of monitoring the time-dependent events, emiting proper signals if a module connected with such an event should be executed,
g/ The Job Scheduler, is in fact a sophisticated interval
clock^s driver. The utilised CPU does not have a real-time clock, but only an interval clock generating interrupts with frequency 1/sek. The Scheduler is being executed with
such a frequency, analysing the state of the computer sys
tem, possible requests for modulées execution, signals from the time-tracking module which it activates every minute and other signals edited, while application module was exe
cuted. It organises also a real-rtime clock in the way of interval counting. After the analysis and testing of some I/O devices conditions, a proper program is started /inclu
ding the possibility of resuming the interrupted one/.
Remark : As modules f and g operate interrupting the execution of lower levels, which are not of reentrant type, neither the time-tracking module nor the scheduler may use any of the a, b, c levels modules.
The scheduler includes an initialisation section, seting sys
tèmes starting conditions according to the needs represented by the switch register. There is possible to start with any combination of following actions :
- reading the code from PTR device
- reading the content of PDB from the PTR device - clearing the content of VDB
Afterwords the initial state of VDB can be introduced in a conversational mode, and the clock is set, starting normal operation.
6.DATA BASS DESCRIPTION
In the previous section the data base has been briefly descri
bed. Nov/ some more details on it^s organisation shall be given.
The PDB consists of 14 files, defined as either vectors or two dimensional matrices. In the files data are organised in most cases in a relational structure, of either simple or hierar
