A Survey of Queueing t h e o r y

8/3/2019 A Survey of Queueing t h e o r y

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Annals of Operations Research 24(1990)29-43 29

A S U R V E Y O F Q U E U E I N G T H E O R YGuang-Hui HSUInstitute of Applied Mathematics, Chinese Academy o f Sciences, Beijing, P.R. China

AbstractThis paper is intended o give a survey of the main results of queueing theory in China.It consists of five parts: (1) the transient behaviour of typical queueing systems;(2) classical problems; (3) approximation theory; (4) model structure; (5) applications.

1. IntroductionThe research work on queueing theory in China began in the la te f i f t ies . The

theoret ical research was mainly concentrated on the t rans ient behavio ur of typicalqueueing systems, and some important results were obtained. In applications, manyproblems concerning te lephone sys tems, the text i le industry , communicat ions ,transportation, etc. were investigated, and the scientific basis for decision making wasprovided for practical departments. In recent years, the theoretical interest hasgradual ly changed to approximation theory , such as saturated service approximation ,weak convergence theory , approximate algori thms, matr ix analys is , e tc . , andapplication fields have been gradually extended to mining processes, seaports,te lecommunicat ion , mil i tary affai rs , computer des ign , e tc .

Th is paper is only a surve y of the main results of queueing theory in our country .For details of some parts, the reader is referred to Hsu's book [28].

2. The transient behaviou r of typical queueing systemsBai ley [ 1] and others investigated the transient beha viour of the queu e length atinstant t, q(t), and the busy period for the M/M/n sys tem. Karl in and McGregor [40],

Yue [67], and Saaty [52] investigated the transient behaviour of the queue length q(t).Yue's result is as follows.

THEOREM 1For the M/M/n system, suppose that q(0) = 0. Then for k > n, the probabili ty

that at instant t there are k customers present, including those being served, isc+i**1 f eU'a______~P k ( t ) = 2 h i . u~o(u ) dU' (c > 0),

J.C. Baltzer AG, Scientific Publishing Company


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30 Guang-Hui Hsu, Survey of queueing theory

wh ere a i s the roo t o fnlax2 - (;~ + u + n p ) x + ~, = O,

w i t h m i n i m u m a b s o l u t e v a l u e ,

1 I x a X ~ , n _ l _ . . , a s ~ ,- '( u+ p ) . . . ( u + i # ) ,~ p ( u ) - 1 - a ~ s=o

a n d ~ - i a n d # - ~ a r e the mean in te ra r r iva l t ime and the mean se rv ice t ime ,re spec t ive ly .F o r t h e G I / M / 1 sys tem , Co no l ly [12 ,13] inves t iga ted the queue l eng th q( t ) and

the bu sy pe r iod , and Tak~ics [55] inves t iga ted the queu e l en g th q( t ) , the q u e u e l e n g thi m m e d i a t e l y b e f o re t h e m t h a r ri v al , qm ' and the busy pe r iod . For the G I / M / n s y s t e m ,W u [ 64 ] e x a m i n e d th e q u e u e l e n g t h q ( t ) und er the in i t i al cond i t ion q (0 ) = 0 o r 1 . Forexam ple , fo r q (0 ) = 0 , he ob ta ined :

T H E O R E M 2

F o r t h e G I / M / n sys tem, suppose tha t q (0 ) = 0 . Then the Lap lace t rans fo rms o fPk(t), k = O, 1 . . . . . a r e

a - 11 ~ , ( _ 1 ) , B , - , ( z )Po'(z) = z + , : 1 z + r#l " f n ~9 ' ( - ) t , r ) [

+C~"Y~,:o ( n - r ) - n ~ z + r p( n - r) (1 - ~ ] .z + n g ( 1 - - - '

e ~ , ( z ) =n - 1 - ( r ) " ' - ' ( ' )r = r a Z - F F #

+ C C n Z m r r er = , ~ n - r j - - ~ z + r #

(n - r)(1 - ~) -]r

l < m < n - 1 ;

m > n ,, ~ (z ) = c ( ~z + n # ( 1 - ~ ) '


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Guang-Hui Hsu , Survey o f queueing theory 31

w h e r e ~ = ~ ( z ) is a u n i q u e s o l u t i o n i n I ~ 1 < 1 t o t h e e q u a t i o na * ( z + n # ( 1 - ~ ) ) = ~ , P ~ (z ) > 0 ,

a n d [ . ( n )a ' ( z ) AT) x- 1 _ a . ( z ) ~ - "- ~ n ( 1 - a ' ( z + r it ) ) - r( n ( 1 - ~ ) - r ) ( 1 - a ' ( z + r it ))

-1

+ 1 .

B r ( z ) - c ~ n + l A r ( z ) ~ (nk)k = r + l a k ( z )m xn ( 1 - a * ( z + k # ) ) - k( n ( 1 - ~ ) - k ) ( l - a * ( z + k i t ) )

O < r < n - 1 ;A o ( z ) - 1 ;

r a * ( z + k l t )A t ( z ) - I - [ ' r > 1 ;k = l 1 - a ~

a n d a * ( z ) i s t h e L a p l a c e t r a n s f o r m o f th e d e n s i t y f u n c t i o n a ( t ) o f t h e i n t e r a r r i v a l t i m e .I n p a r t i c u l a r , i f a ( t ) i s e x p o n e n t i a l l y d i s tr i b u te d , t h e n t h e o r e m 2 is r e d u c e d t o

t h e o r e m 1 .N e x t , H s u [ 2 5 ] i n v e s t i g a t e d t h e q u e u e l e n g t h a t i n s t a n t t , q ( t ) , th e q u e u e l e n g t h

i m m e d i a t e l y b e f o r e t h e m t h a r r i v a l, q m ' t h e w a i t i n g t i m e a n d t h e n o n - i d l e p e r i o d , a n do b t a i n e d a ll th e e x p l i c i t e x p r e s s i o n s . L e t

P i k ( t) - P { q ( t ) = k [ q ( 0 ) = i } , i , k = 0 , 1 . . . . ; t > 0a n d

P,k (s ) =- e-StPik ( t ) d t , ~ s ) > 0 .

H s u o b t a i n e d t h e f o l l o w i n g t h e o r e m , w h i c h d e t e r m i n e d t h e P / k( t) 's .

T H E O R E M 3F o r t h e G I / M / n s y s t e m , w h e n I z l < 1 a n d ~ s ) > K ( a s u f f i c ie n t l y la r g e p o s i t i v e

n u m b e r ) ,


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3 2 G u a n g - H u i H s u , S u r v ey o f q u e u e i n g t h e o r y

s P : k ( s ) z ii = 0

1z - A * [ s + n / l ( 1 - z ) ]

{ n }- A * [ s + n / l ( 1 - z ) lP o ~ k ( s ) + E C j ( s , z) Pj* k ( s ) + C ( s , z ) ,j = l

w h e r e P . , * ( s ) , j = 0 , 1 . . . . n , a r e t h e u n i q u e s o l u t i o n s t o t h e f o l l o w i n g s y s t e m o f l i n e a ra l g e b r a i c e q u a t i o n s :

-Pd k (s ) + 0oo ( s )P:k ( s ) = - ~ ok Wok ( s ) ;i + I~ [ o ~ , ~ _ ~ + ~ ( s ) - a , . j 1 ~ 7 , ( s ) = - a S v /~ k ( s ) ,j= l l < i < n - 1 ;

n- f ( s ) P ~ k ( s ) + E C j ( s , f ( s ) ) P j * k ( s ) = - C ( s , f ( s ) ) ,j = l

o r , e x p r e s s e d e x p l i c i t l y :

P o ; : s) =[ & - 1 , k V / r - 1 , k ( s ) + J X l 9 I - 1 - i v + l v ]a ( s , f ( s ) ) 9 I - l a ~_ l , kV / j_ l ,~ ( s ) E__ (--1) ' f ; , t - 1-C(s,f(s))

l = l 0 1 - 1 , 0 ( $ ) "=- f C s ) + C1(',f('))Oo,C , ~ + ~=2.t(s,f(S))ot_,oCs) 'v---~2~ ~'~ "' -

1 - 1 i - 1 - 1P~tCs) 1 { _ 6~_ Lk ~ _ t ,~(s) _ ~" 6 ;. Lt ~,_ 1,k( s) ~ . . . . . ~ ,- "- - l ) J r , j - 10 i - l , o ( s ) i = 1 ~ ~ oj -2 v+l v * $ 1 < j - < n ,

w h e r e

i < ~ < ~ < . .. < n ,, ~ ;j[0 . , _ 1 , . , - i (s ) - ~ . , - 1 , i1[0 . , - 1 ,,~ - . , ( s) - an ,- 1 , . , ] . . . [Or, j 1- . . ( s) - ~ i , . . ]

O i - t , o s ) O , , 1 , o ( s ). . . n . _ 1 , o ( s )

l < _ i < j < _ n - 1 , O < v < j - i ;

f i ? = o ~ ,i + - i ( s ) - 8i~o i - t ,o ( s )

l < _ i < j < n - 1 ,

nC j ( s , z ~ - ~ " z~+ tO ' " l ( s ) - A ' [ s + n l ~ l " " 1 < - j


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G u a n g - H u i H s u , S u r v e y o f q u e u e i n g t h e o r y 33

z k + l 1 - A * [ s + n / t ( 1 - z ) ] k > n ;C ( s , z ) - s + n # ( 1 - z ) '

z i + 1 1 1 t i k (S ) + z n + Z l l t ~ k (S , Z ) , k < n ,i=k

O ~ j ( s ) - O ~ j ( s , 0 ) ,

O 0 ( s , z ) - n/ . t e [ s+nu (1-z / luOi j ( s , u )du ,0

l l t i j ( s , u ) - i e - S X [ 1 - A ( x + u ) ] ( j ) ( 1 - e - ~ ) i - J e - J l ~ d x ,0

V ~ j ( s ) - v / ~ j ( s , O ) ,

V * j ( s , z ) - S n# e - Is + ' u (1 - z ) l u V i j ( s , u ) d u ,0

1, i = j ;6 i j - - O , i ~ j ,

tS ik -* ~ { 1 , k < i ;t~ iJ = O , k > i ,j = kf ( s ) i s a un ique so lu t ion in Iz l < 1 to the equ a t ion

z = A * ( s + n B ( 1 - z )),A ~ = f e - S X d A ( x ) ,

O-

a n d A ( x ) is the distribution function of the interarfival t imes.In pa r ti cu lar , i f q (0) = 0 , then theorem 2 can fo l low f rom theorem 3 .


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3 4 G u a n g - H u i H s u , S u r v e y o f q u e u e i n g t h e o r y

L e t

P i~m ) - P { q m = k l q o = i } , m = O , 1

r / ( m ) l RH s u o b t a i n e d t h e f o l l o w i n g t h e o r e m w h i c h d e t e r m i n e d t h e r i k ~ "T H E O R E M 4

F o r t h e G I / M / n s y s t e m , w h e n I z l < 1 a n d 0 < I u I < t ( a s u f f i c i e n t l y s m a l l p o s i t i v en u m b e r ) ,

e l k . - = z k + l + u ( z ) V j k ( u ) ,m =O i = 0 g - - u A * [ n p ( 1 - z ) ] i = o

w h e r e V ~ k (U ) , j = O , 1 . . . . . n - 1 , a r e t h e u n i q u e s o l u t i o n s t o t h e f o l l o w i n g s y s t e m o fl i n e a r a l g e b r a i c e q u a t i o n s :

h + ls [ u a h j - - tS hj ] l ~ k ( u ) = - - rj= O

n - 1U s d j ( b ( u ) ) V j k ( u ) = - [ b ( u ) ] k + l ,j = o

O < h < n - 2 ;

o r , e x p r e s s e d e x p l i c i t l y :- [ b ( u ) 1 * + l - l = l a l-- a7 77 7- ,t. v ' - l , ~ ' - j ~ l > ' - , , k 2 0 JVok(U = n -~ d t ( b ( u ) ) [ l - z I - l - j ]u d o ( b ( u ) ) - ~ . ~ ( u a t - l , o - 5 , v , l ~V + lt v& ~ , o ) + _ _ Z C - ~ j - ~ , o - ~ -l,O j ~ f o ~ - J s s , , - ~I = l " = =

v ~ , ( , , ) - 1 , ] , ~ i _ ~ . , + ~ Z ~ ~ - ~ - , , , + ~- ~ t - ~ , , ~ ( - l ) :'j,j_~u a j - l , j L I = l v= O

w h e r eI j - I j - | - I 1 }~ , _ ~ , o ) Y . ( - 1 Y * ~ f i , ~ _ ~ V o ~ ( ~ ) ,( . a j - l , o - 6 j - l o ) + ~ ( u a l - l o -

I= l v = Ol < _ j < _ n - 1

si < n l < n 2 < . . . < n v < j( u a n l - 1 5 - ~ n , - 1 , i ) ( u a n 2 - 1 , n , - ~ n 2 - 1 ,n l ) . . . ( u a j , n ~ - ~ Sj,n ~)

u V + 1 a i - 1 , j a n 1 - 1 , h i a n 2 - 1 ,n 2 . a n y - 1 , n v

: o : _ u a i i - a j iJ l j u a i _ 1 , i1< i< j< n - 2 ,l < i < j < n - 2 ,

O < v < j - i ;


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n - 1d j ( z ) - ~ a i j z j + l

i = m a x ( j - 1 , 0 )A * ( n l t ( 1 z ) ) z j + n + l , . ,- - - z an_ l , j ( z ) , 0 < j < n - 1 ,

~ [ ( i + l ) e - / a x ) i + 1a q ( y ) - - t ]- \ J e - J / ~ ( 1 - - i d A ( x + y ) ,a i j - a i j ( 0 ) ,

a~ ( z ) - ~ n l l e - n~ y ( 1 - Z )a i j ( y ) d y ,0

1 , h = k ;S h k - - O , h ~ k ,

a n d b ( u ) i s a un i q ue so l u t i on i n I z l < 1 to t he equ a t i onz = u A * ( n # ( 1 - z ) ) .A f t e r 1965 , some scho l a r s f r om o t he r coun t r i e s a l so i nves t i ga t ed t he G I / M / n

sys t em, e . g . B ha t [ 2 ] i nves t i ga t ed q ( t ) a n d r e d u c e d t h e p r o b l e m t o s o l v i n g a s y s t e mo f l i n e a r a l g e b r a ic e q u a t i o n s s i m i la r t o t h at i n t h e o r e m 3 a b o v e , b u t h e d i d n o t s o l v ei t exp l i c i t l y . D e Smi t [ 14 ] i nves t i ga t ed q m ' q ( t ) , t h e w a i t i n g t i m e a n d t h e n o n - i d l ep e r i o d , a n d d e r i v e d a l l t h e e x p l i c i t e x p r e s s i o n s b y d i f f e r e n t m e t h o d s .

I n o r d e r t o c h a r a c t e r i z e t h e d e g r e e o f b u s y t i m e f o r m a n y - s e r v e r s y s t e m sas e f f i c i en t l y a s pos s i b l e , H su [26] i n t roduced t he c onc ep t o f a k -b usy pe r i o d(k > - ( n - 1 )) , w h i ch i s de f i ned a s the t i me pe r i od beg i nn i ng a t a cus t om er ' s a r r iva lw hi ch t r ans f e r s t he qu eue l en g t h t o n + k f rom n - 1 + k and l a s t i ng un t i l t he qu euel eng t h pas ses t o n - 1 f rom n fo r t he f i r s t t i me . For t he G I / M / n s y s t e m , h e d e r i v e dt h e e x p l i c i t e x p r e s s i o n s o f t h e p r o b a b i l i t y l a w o f t h e k - b u s y p e r i o d . F o r t h e n o n - i d l ep e r io d , t h e c o n c e p t o f t h e k - n o n - i d le p e r i o d m a y b e i n t r o d u c e d s i m i l a rl y , a n d t h ee x p l i c i t e x p r e s s i o n s o f t h e i r p r o b a b i l i t y l a w f o r t h e G I / M / n s y s t e m h a v e a l s o b e e nde r i ved ( s ee H su [28] ) .

P r abh u [51] cons i d e r ed f i r s t pas sage t i mes , i .e . t he f i r s t pas sag e s t eps o f thee m b e d d e d M a r k o v c h a i n . T h e s e a r e d i s c re t e r a n d o m v a r i ab l e s . C o h e n [ 1 1] i n v e s t ig a t e dt he f i r s t pas sage t i mes fo r t he G I / M / 1 a nd M / G / 1 s y s t e m s , r e s p e c t i v e l y ; b u t f o rG I / M / 1 , o n l y h i s f i r st p a s s a g e t i m e s t o t h e u p p e r b o u n d a r e j u s t r e a l t i m e p e r i o d s ; f o rM / G / 1 , o n l y h i s f i rs t p a s s a g e t im e s t o t h e l o w e r b o u n d a r e j u s t r e a l t i m e p e r i o d s .H o w e v e r , H s u a n d Y a n [ 3 5 , 3 6 ] i n v e s t i g a te d t h e r e a l t i m e p e r i o d s - t h e f i rs t e n t r a n c e


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3 6 Guang-Hui Hsu, Survey o f queueing theory

t i m e s f o r t h e M/G~1 and GI]M/n sys t ems , r e spec t i ve l y , and de r i ved a l l t he exp l i c i te x p r e s s i o n s . H e r e , t h e f i r s t e n t r a n c e t i m e t o t h e u p p e r ( l o w e r ) b o u n d i s d e f i n e d t o b e g i na t t he i n i t i a l t i me and t o l a s t un t i l t he queue l eng t h i s equa l t o some uppe r ( l ow er )bound fo r t he f i r s t t i me . For t he f i r s t en t r ance t i me t o t he uppe r bound , t hese au t hor se x t e n d e d C o h e n ' s r e s u l t s f o r GI/M/1 t o t he r e su l t f o r the G I / M / n sys t em; fo r t he f i r ste n t r a n c e t i m e t o th e l o w e r b o u n d f o r M / G / 1 , t h e y o b t a i n e d a m u c h s i m p l e r r e s u l t t h a nC o h e n ' s b y a d i f f e r e n t m e t h o d ; a n d f o r t h e f ir s t e n t r a n c e t im e t o th e u p p e r b o u n d f o rM / G / 1 a n d f o r th e f i r st e n t ra n c e t i m e t o t h e l o w e r b o u n d f o r GI/M/n , t he i r r e su l ts w eree n t i r e l y n e w .

3 . Class i ca lproblemsT h e P o i s s o n p r o c e s s , i. e. t h e s i m p l e s t re a m , i s t h e m o s t f u n d a m e n t a l p r o c e s s i n

queue i ng t heory . I t w as de f i ned by K hi n t ch i ne [42 ] a s a s t a t i ona ry , o rde r l y , f i n i t es t r e a m w i t h o u t a f t e r - e f f e c t s . A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r a s t r e a m t o b ea s i m p l e s t r e a m w h i c h h a s o f t e n b e e n u s e d i n q u e u e i n g t h e o r y i s t h a t th e i n t e r ar r iv a lt i m e s a r e m u t u a l l y i n d e p e n d e n t a n d i d e n t i c a ll y e x p o n e n t i a l l y d i s tr ib u t e d . M a n ys c h o l a rs h a v e d i s c u s s e d i ts p r o o f , e .g . D o o b [ 17 ] g a v e a r i g o r o u s p r o o f f o r t h e n e c e s -s i ty , and Pa rz en [50] gave a r i go rous p r oo f f o r t he su f f i c i ency . Fan g e t a l. [18 ] cons i de r edt he s t r uc t u r e o f a m ore g ene ra l s t r eam - a s t a t i ona ry s t ream w i t ho u t a f t e r - e f f ec t s - andp r o v e d t h a t a n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r a s t r e a m t o b e a s t a t io n a r y s t r e amw i t hou t a f t e r - e f f ec t s i s t ha t t he i n t e r a r r i va l t i mes a r e i ndependen t and i den t i ca l l ye x p o n e n t i a l l y d i s t r i b u t e d , a n d t h a t t h e n u m b e r s o f c u s t o m e r s a r r i v i n g a t e a c h i n s t a n ta r e a l so i ndep end en t and i den t i ca l l y d i s t r i bu t ed and a r e i ndep end en t o f t he i n t e ra r r i va lt im e s . L a t e r , C o h e n [ 1 1] a l s o g a v e a r i g o r o u s p r o o f f o r th e s u f f i c i e n t c o n d i t i o n o f th es i m p l e s t r e a m , a n d B i l l in g s l e y [3 ] g a v e a p r o o f f o r th e n e c e s s i t y a n d s u f f ic i e n c y .R e c e n t l y , H s u a n d L i u [ 3 3] g a v e a m u c h s i m p l e r r ig o r o u s p r o o f f o r t h e n e c e s s a r y a n ds u f f i c i e n t c o n d i t i o n o f t h e s i m p l e s t r e a m , a n d d i s c u s s e d s o m e o t h e r n e c e s s a r y a n ds u f f i c i e n t c o n d i t i o n s f o r a r e n e w a l p r o c e s s t o b e a P o i s s o n p r o c e s s .

A n o t h e r c l a s s ic a l p r o b l e m i s a n a ss e r ti o n m a d e b y P a l m i n 1 9 4 3 ( s ee K h i n t c h i n e[42] ) , t ha t i s , f o r t he M/M/oo s y s t e m , s u p p o s i n g e a c h c u s t o m e r i s s e r v e d b y t h e f r e es e r v e r w i t h t h e s m a l l e st n u m b e r , t h e n t h e l o s s o f p r o b a b i l i t y P t o t h e r t h s e r v e r( r > 1 ) i s a l w ays g r ea t e r t han P _ 1, t he p roba b i l i t y to t he ( r - 1 ) s t s e rve r , i f t he i npu t st o t hese s e rve r s a r e o f i den t i ca l in t ens i t y . K h i n t ch i ne [42 ] p rov ed on l y t ha t P > P~ , bu tH o u [ 2 3 ] g a v e t h e p r o o f o f t h e t ru t h o f t h e a s s e rt io n .

4. Approximation theoryT h e e s t im a t i o n o f th e c o n v e r g e n c e r a t e o f l im i t t h e o r e m s f o r s y s te m s i n h e a v y

t r a f f i c i s a n i m p o r t a n t s u b j e c t i n a p p r o x i m a t i o n t h e o r y . F o r t h e GI/GI1 s y s t e m w i t hs e r v i c e t r a f f ic p = 1 , N a g a e v [ 4 7] p r o v e d t h a t t h e B e r r y - E s s e e n r a t e o f c o n v e r g e n c ef o r t h e w a i t i n g t i m e c a n b e o b t a i n e d , i .e .,


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G u a n g - H u i H s u , S u r v e y o f q u e u e i n g t h e o r y 37

Wnsup P ty-~/-m < x - ~ e-y2/2dy = O(m-112),

if cr2 = E ( v - u ) 2 > O, E l y - ul 3 < ~, where v is the service time and u is the interarrivaltime. However, for the G I / G / n system with p > 1, by the Skorohod embedding method,Kennedy [41] obtained only a slower rate of convergence. Then Sawyer [53] provedthat the rate of convergence obtained by the Skorohod embedding method cannot befaster than 0 ( n - 1 / 4 ) , which is certainly much worse than the Berry-Esseen rate. Jin andWang [39] made a more precise estimation by the method of characteristic functions,and eventually proved that for the G I / G / 1 system with p > 1, the Berry-Esseen rateof convergence for the waiting time can still be obtained. Further, Jin [38,37] con-sidered the G I / G / n system with p > 1, and by means of some established inequal, tiesand modified systems, proved that with all the theorems of the Berry-Esseen type forthe queue length, the waiting time and the virtual waiting time can be obtained if someappropriate conditions concerning the first and third moments of interarrival timesand service times are added. For instance, the theorem of the Berry-Esseentype for the queue length at the mth arrival, qm' is as follows: if p - E v / ( n E u ) > 1 ,t7 2 =- E ( u - E u ) 2 > O , t7 2 - ( v - E v ) 2 > O , E l u - E u l 3 < ,~ a n d E I v - E v l 3 < ~, then

1 < r f m x } - 1 __up P { q m - 1 - -~ r n _ ~ -.~- y 2 / 2 d y = O(m-1/2),

where r 2 - n ( tr v 2E u + n c r ~ E v ) / ( E v ) 3.Another important subject in approximation theory is the study of functional

central limit theorems for systems in heavy traffic by means of the theory of conver-gence of probability measures. An extensive literature on this subject exists. Wang andDing [60] considered the system in heavy traffic with delayed feedback, where boththe input and service are general, and the customer having been served returns witha positive probability and through a delayed service device joins the waiting line again.They derived the functional central limit theorems for the idle period, the queue length,the virtual waiting time, and the sojourn time. Wang and Wu [61,62] investigated theweak convergence for systems with priority. The former deals with many-server systemswith preemptive-resume priority, and the latter deals with many-server systems withpreemptive-repeat priority and single-server systems with nonpreemptive priority. Zhangand Wang [69] considered many-channel many-server systems in heavy traffic wherethe customer joins the shortest waiting line, and derived weak convergence theoremsfor the queue length, the load, the waiting time, and the departure process.

Another type of systems in heavy traffic is the saturated service system, i.e. thesystem with p < 1 and p --- 1. Kingman [43] proved that for the saturated service systemG I / G / 1 , t h e stationary waiting time is asymptotically exponentially distributed, and


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38 Guang-Hui Hsu , Surve y o f queueing theory

conjectured that it would be true for the G I / G / n system. Later, this conjecture wasproved by K011erst~m [44]. Luo [46] improved the result of K011erstr0m by obtainingthe same conclusion under weaker conditions.The approximate algorithm is also an interesting subject in approximationtheory. Liu [45] considered the M/G/1 system with feedback. By the method ofregenerative processes, he gave the approximation algorithms for distributions of thequeue length, the waiting time and the sojourn time, and did numerical computationsfor some typical systems. After comparing the results of his computations with simu-lations, the efficiency of these algorithms was affirmed.The matrix analysis method developed by Neuts [48] is a powerful tool forstudying queueing systems and other stochastic models, and can provide efficientalgorithms for solving these problems. We used this method to investigate the matchedqueueing system with a double input, the expectation of the first passage time to thestates of level 0 for Markov processes of the GI/M/1 type, and the maximum queuelength in a busy period for some queueing systems.Hsu et al. [31,32] considered the matched queueing system with a double input,which is a generalization of ordinary queueing systems. There are two independentinputs, the customers of which can be served only when they are matched accordingto the proportion 1 :r, where r is a random variable. For both the Poisson inputs andthe PH-distributed service times, the necessary and sufficient conditions for theergodicity of the four-dimensional state process were discussed, and the algorithms forthe four-dimensional state process, the waiting time, the busy period, and the non-idleperiod were given respectively.Hsu and He [30] discussed the expectation of the first passage time to the statesof level 0 for GI/M41 type Markov processes. By using the approximation of Markovprocesses with a finite number of states to one with a denumerably inf'mite number ofstates, the expression of the expectation was derived. Then, the result was applied tofind the mean busy periods for some related queueing systems.He [22] investigated the maximum queue length in a busy period for queueingsystems. By means o f the random walk, the distributions of the maximum in matrixform for GI/M/1 , M/G/1 e tc . systems were derived, respectively. These results areconvenient for numerical computation.Finally, we must mention the subject of comparison and continuity in approx-imation theory. Hsu and Liu [34] investigated the comparison of so-called generalizedstandard systems, where the interarrival times and the service times are mutuallyindependent sequences of independent random variables. On the basis of some com-parison theorems, they discussed the system continuity in a sense; for example, for thegeneralized standard M/G/1 system where the service times {v} do not need to beidentically distributed, if the distribution of v converges weakly to a distribution B,then under some conditions the queue length and the waiting time converge weaklyto the stationary queue length and the stationary waiting time in the standard M / G / 1system, respectively, where the interarrical times and the service times are mutuallyindependent sequences of independent and identically distributed random variables,and the distribution of the service times is B.


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Guang-Hui Hsu, Survey of queueing theory 39

5 . M o d e l s t r u c t u r e

Wu [63] investigated the transient behaviour of the queue length at any instantand the busy period for the GI/EJ1 system, and derived their explicit expressions.Cao and Yan [5] investigated the M/G/1 system, where the probabilities ofacceptance are dependent on the queue length.For the GI/M/n system with batch service, Hsu [24] investigated the waitingtimes for random order of service and for the last-come-first-served discipline, respec-tively.

Han [20,21] examined the stationary and transient behaviour for a system withqueue-length dependent arrivals, derived the distributions of the queue length and thebusy period, and discussed an optimization problem on the total loss in an infinitelylong period.Wu et al. [65] considered some discrete systems (M/D/1, M/M/c, D/M/1,'etc.)and obtained the distributions and the expectations of the quantities concerned.

Zhang [68] investigated the GI/M/o, with batch arrival, where the number ofcustomers in a batch is a random variable. He obtained both the transient and limitingdistributions for the queue length immediately before the arrival and for the queuelength at any instant, respectively.

Chen [6] examined the M/G/1 system with batch arrival and batch service,where the number of customers in a batch of arrival is a random variable and thenumber of customers in a batch of service is fixed. He discussed the classification ofstates for the embedded Markov chain, and obtained the expressions of the generatingfunction of the limiting distribution and of the mean queue length.

Chen [7] considered the optimal control of the input process for the M/G/1system, where there are many classes of customers, and the rewards for serving dif-ferent classes of customers are different. How to choose customers served to maximizethe rewards obtained in a unit time was the problem studied, which was reduced to aMarkovian decision programming. For the case of finite waiting room, an algorithmof a linear programming for solving the problem was given; for the case of infinitewaiting room, an approximation theorem was established.Cheng and Cao [9] investigated the M/G/1 system, where the service station maybe out of order but can be repaired. Many quantities which axe of interest to bothqueueing theory and reliability theory were discussed.Cao [4] investigated the machine maintenance problem where the service stationmay be out of order but can be repaired, and where both the lifetimes of machines andthe service station have exponential distributions and both their repair times havegeneral distributions.

Wang [59] considered the many-server system with semi-Markovian input andexponential service times, and obtained the results of the transient behaviour of thequeue length immediately before the arrival, the queue length at any instant and thewaiting time, which are generalizations of both ~inlar's results [10] on the single-serversystem and Hsu's results [25] on the GI/M/n system.


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4 0 Guang-Hui Hsu, Survey of queueing theory

Ti an [56] i nves t i ga t ed t he s t r uc t u r e o f a b i nomi a l i npu t , and ob t a i ned i t sn e c e s s a r y a n d s u f f ic i e n t c o n d i t io n . T i a n [5 8] c o n s i d e r e d t h e t r a n s ie n t b e h a v i o u r o f t h eq u e u e l e n g t h f o r t h e s i n g l e - s e rv e r s y s t e m w i t h b i n o m i a l i n p u t a n d e x p o n e n t i a l s e r v i c et i mes , and de r i ve d t he exp l i c i t expres s i on . I n pa r t i cu l a r , i f t he t i me i n t e rva l con s i de r edt ends t o i n f i n i t y , t hen t he t r ans i en t d i s t r i bu t i on o f t he queue l eng t h fo r t he M/M/1s y s t e m i s o b t a i n e d , w h i c h h a s b e e n d e r i v e d b y T i a n [ 5 7 ] . T h i s d i s t r i b u t i o n s e e m s t ob e d i f f e r e n t f r o m t h e o r d i n a r y o n e e x p r e s s e d b y B e s s e l f u n c t io n s w i t h p u r e l y i m a g i n a r ya rgument ( s ee H su [28] ) , bu t t hey a r e i n f ac t equ i va l en t .

C h e n [8 ] e x a m i n e d t w o q u e u e i n g m o d e l s w h e r e t h e n u m b e r o f s e rv e r s is q u e u e -l e n g t h d e p e n d e n t , a n d o b t a i n e d t h e s t a t i o n a ry d i s tr i b u ti o n o f th e q u e u e l e n g t h a n d o t h e ra s s o c i a t e d p e r f o r m a n c e m e a s u r e s .

6 . A p p l i c a t i o n sB y m e a n s o f st o c h a s ti c s im u l a t io n , H s u a n d D o n g [ 29 ] a n d t h e O R D i v i s i o n e t

a l . [ 49 ] i nves t i ga t ed t he op t i ma l ma t ch r e l a t i on be t w een shove l s , t r ucks and d i scha rgepos i t i ons fo r l oad i n g and t r anspor t a t i on p roces ses i n open-p i t m i n i ng . O n t h i s bas i s,t h e y i n v e s t i g a t e d t h e o p t i m a l a l l o c a t i o n o f t r u c k s w h e n t h e t r a c k s w e r e d i v i d e d i n t oa n u m b e r o f f le e t s, e a c h w o r k i n g w i th o n e o f th e s h o v e l s f i x e d l y ; t h e y a l so i n v e s t ig a t e dt h e o p t i m a l r e a l l o c a t i o n o f t r u c k s a n d t h e c o m p u t a t io n o f th e t o t a l p r o d u c t i o n i nc o n s i d e r a t io n o f t h e r e a l l o c a ti o n w h e n t h e s h o v e l s m i g h t b e o u t o f o r d e r ; a n d t h e yi n v e s ti g a te d t h e c o r r e s p o n d i n g p r o b l e m s f o r m a n y t y p e s o f t r u c k s. T h i s r e s e a r c h p r o v i d e da n e f f i c i e n t a n d s c i e n t if i c m e t h o d f o r th e t e c h n i c a l d e s i g n a n d t h e p r o d u c t i o n m a n a g e -m e n t o f a n o p e n - p i t m i n e .

Y a n g a n d C h e n [6 6] i n v e s ti g a te d t h e d e s i g n o f t h e l o a d i n g s y s t e m o n a c o a lw ha r f . C oa l i s t r anspor t ed t o t he coa l ya rd by t r a i ns and t r anspor t ed o u t by sh i ps . Tra i nsa n d s h i p s a r e re g a r d e d a s t w o k i n d s o f c u s to m e r s w h o d e m a n d t o b e s e rv e d f o r l o a d i n ga n d u n l o a d i n g , r e s p e c t iv e l y . H o w e v e r , th e t r a in c a n b e u n l o a d e d o n l y i f t h e y a r d w i t hf i n i te capac i t y i s no t f u l l, and t he sh i p can b e l oaded o n l y i f t he ya rd i s no t em pt y . B ys i mul a t i on , som e quan t i t i e s f o r t h i s sys t em w ere ca l cu l a t ed , w h i ch p rov i ded t he bas i sf o r w h a r f d e s i g n .

S u n e t a l. [ 5 4 ] g a v e t h e f l o w c h a r t a n d p r o g r a m m i n g f o r t h e s i m u l a t io n o f ag e n e r a l m o d e l o f q u e u e i n g s y s t e m s , a n d i n v e s t ig a t e d i ts a p p l i c a ti o n t o t h e d e s i g n o fa n a u t o m a t i c m a c h i n i n g l i n e .

H s u [ 27 ] i n v e s ti g a t e d th e p e r f o r m a n c e a n a ly s i s o f m e m o r i e s f o r a c o m p l e xc o m p u t e r s y s t e m , s u c h a s a n a r r ay c o m p u t e r , a v e c t o r c o m p u t e r , e t c . T h e r e a r e nm e m o r i e s , r e g a r d e d a s a r r a n g e d i n a c i rc l e. T h e C P U m a k e s r e f e r e n c e t o r ( 0 < r

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A Survey of Queueing t h e o r y