GI_G_1排队系统的队长和等待时间的瞬时分布_英文_

GI_G_1排队系统的队长和等待时间的瞬时分布_英文_ 第 23 卷第 3 期 数学理论与应用 V o l. 23 N o. 3 2003 年 9 月 Sep. 2003 M A TH EM A T ICAL TH EOR Y AND A PPL ICA T ION S Tran s ien t D istr ibut ion of the L ength and W a it ing t im e of Ξ 1 ƒƒQueue ing System G IG W ang Y im in L iM in ()Schoo l of M athem atics, Cen t ral Sou th U n iversity, Changsha, 410075 [ 1, 2 ]. , Abstract T he p resen t paper is a successio n of U sing M SPw e give a nw e app roach to calcu la te the 1 ƒƒt ran sien t distribu t io n of the length and w aiting t im e of G IGqueueing system Keywords queue length w aiting t im e t ran sien t distribu t io n ƒƒ1 排队系统的队长和等待时间的瞬时分布G IG 王益民 李 民 ()中南大学数学学院, 长沙, 410075 摘 要 本文是[ 1, 2 ]的继续, 在本文中利用马氏骨架过程给出了 ƒƒ1 排队系统的队长的瞬时分布的另 G IG 一新的计算方法和等待时间的计算方法。 关键词 队长 等待时间 瞬时分布 1 ƒƒ1 Tran s ien t d istr ibut ion of the length of G IGqueues , ( ? 0 ) , , , In th is sect io ndeno te by Σ0 Σ1 Σ2 the successive departu re t im es fo r the , ( ) ( ) [ 1 ], . .cu stom ers w hen they have f in ished their servicesand let Ηtbe Η1 tin ie ( ) > ( ) (1. 1)ΗtΗ1 t ( ) , (( ) , ( ) ) In o rder to study the t ran sien t distribu t io n of L tw e con sider L t Ηt in the , ( ) fo llow ingw h ich is a M arkov skeleton p rocess w ith Σn as it s skeleton t im e sequence O bv io u sly w e have ?+ ?(??) (1. 2)Σn n L et (, ) = (( ) ) = , < |(0) = , (0) = ) (1. 3)h ij ΗtP L tj tΣ1 L iΗΗ (n) ( , ) = ( ( ) = , < ( ) ( 0 = ) ) 1. 4 ( ) |0 = , P i j Ηt P L t j tΣ n L iΗ Η (, ) = (( ) = , |(0) = , (0) = ) (1. 5)P ij ΗtP L tj L iΗΗ T hu s (1) ( ) ( ) ( ) i j , = ij , 1. 6P Ηt h Ηt Ξ 候振挺教授推荐 收稿日期: 2002 年 12 月 5 日 卷数学理论与应用 第 23 44 () n( ) ( , ) (1. 7) P , ?tij Ηt P ij Η 1. 1 {(, ) , , = 0, 1, 2, , 0Φ ?, < + ?} Theorem p ij Ηtij tis the m in im al nonnegative so lu t io n ( ) :the un ique bounded so lu t io nto the fo llow ing nonnegative so lu t io n to s t (1- (- ) ) (- , - ) () (- )A su P 0j su tsdA Ηu dB su ((, ) = , ) +P 0j Ηth 0j Ηt 0 ??0? + 1- ? A - , - - - sΤ sΤts dA Τu - P k j dA Η u dB su t s s (() ( ) ) ( k ) ( ) ( ) ) (k = 1??0 ?0 u ? (, ) = (, ) + P ij Ηth ij Ηt? 3 ΗA - - , - dA t s P i+ k - 1 su ts u dB s su 1- A () k - 1(( ) ) ( ) ( ) ( ) k = 0??0 0 ( = 1, 2, , = 1, 2, , 0Φ < + ?) (1. 8)ij tΗ Proof It is easy to know (1- A ( t) , j = 0;Η ()1 t ()1. 9 (, ) = (, ) =P 0j Ηth 0j Ηt ( )( )j j - 1 ( ) ) t- s- A ( t- s) ) (1- B ( t- s) ) dA (s) , j Ε 1.Η (A ?0 0, j < i; ()1 t (, ) = (, ) =P ij Ηth ij Ηt ( )( )j - 1- 1 j - i ( ) ) t- s- A ( t- s) ) (1- B ( t- s) ) dA (s) , j Ε iΕ 1.Η (A ?0 ()1. 10 t s (, ) = (, ) +(1- - (- - ) (- () ) , ) () +P 0j Ηth 0j ΗtA su P 0j su tsdA Ηu dB su () ??()n+ 1 n 0 0 ? t s s (n) (k ) (1- (- ) ) (- , - ) (- (() (- ) (1111)? A sΤP sΤtsdA Τu dA Ηu dB su k j k = 1??0 ?0 u ? (, ) = (, ) + P 0j Ηth 0j Ηt? 1- 3 - - , - t s A su su ts dA ΗA P 1+ k - 1 u dB s ()n+ 1 () k - 1(( ) ) ( ) ( ) ( ) k = 1??0 0 ()112 1 , (1. 7) , . T hu sfrom imm ediately w e get ou r theo rem p rovedL et t< 0 0, () =()A x 113 1- Κt1- e, tΕ 0 1. 1. , thu s w e can ob ta in a special case of T heo rem w h ich can be u sed to calcu la te the ( ) 1 .ƒƒt ran sien t distribu t io n of L tof M G queue 2 ƒƒ1 Tran s ien t d istr ibut ion of the wa it in g t im e of G IGqueue ) ( ) ( In th is sect io n let W tbe the w aiting t im e including the service t im eof the cu stom er . (?0) , . , . arriving at t im e tD eno te Σ0 Σ1Σ2 by the successive arrival t im es of the cu stom ers , {( ) , ( ) , < + ?} () C learlyw e know W tΗttis a M arkov skeleton p rocess w ith Σn as it s skeleton .t im e sequence + ?() L et A B R and (, , ) = (( ) ?, < |(0) = ) , (211)h x tA P W tA tΣ1 W x 第 3 期 1 排队系统的队长和等待时间的瞬时分布ƒƒG I 45 G (, , ) = (() ?Ε |12), (0) = ) , (2q x tA P W Σ1 A tΣ1 W x 2. 1.L emma + (, , ) = (1- ( ) ) ( (- ) ) , (2. 3)h x tA A tIA x t t , (, ) =q x tA () () (2. 4)dB x dA s ??0(A - x - ) t+ Proof , , ) = (( ) ?|) =(, < (0) = h x tA P W tA tΣ1 W x + + ( (- ) ?) = ( (- ) ?) =, < ) ( < P x tA tΣ1 P x tA P tΣ1 + (- ) ) (1- ( ) ) (2. 5)IA x tA t (, , ) = (() ?Ε |, (0) = ) =q x tA P W Σ1 A tΣ1 W x t (() ?, ?|(0) = ) =P W sA Σ1 d sW x ?0t t + + () =dA s )+ ?) =?)(- ) ((- (- P x s Τ1 A dA sP Τ1 A x s ?0?0t () () ()dB x dA s2. 6+ () ??0 A - x - s , . T hu sthe lemm a is p roved L et (, , ) = (( ) ?|(0) = ) , (2. 7)P x tA P W tA W x 2. 1 2. 1[ 3 ], F rom L emm a and T heo rem ’ in imm ediately w e get + 2. 1 {(, , ) , 0 Φ , < + ?, ?() } Theorem p x tA x t A B R is the m in im al nonnegative () :so lu t io n also the un ique bounded so lu t io nto the fo llow ing nonnegative linear equat io n ? t + + ( (- ) )) () + (1- ( ) ) dA sA tIA x t (, , ) =(, - , ) (- (- )P x tA P y tsA dB y x t ?0?0+ (0Φ ?, ?) ) (2. 8)< + (x A B R References [ 1 ] , , ƒƒ1 ƒƒ1 , Hou Zhen t ingL i M inT ran sien t distribu t io n of the length of G IGand M Gququeing system s , 2003, 23 (1) , 119- 121.M athem atical T heo ry and A pp licat io n s [2 ] , . ƒƒ1 ƒƒ1 ,L i M inW ang Y im inT ran sien t distribu t io n of the length of G IGand M Gququeing system s , 2003, 23 (2) , 103- 105M athem atical T heo ry and A pp licat io n s [ 3 ] , Hou Zhen t ingM arkov skeleton p rocesses and app licat io n s to queueing system s A cta M athem aticae , , 2002, 18 (4) : 537- 552.A pp licatiae Sin icaEnglish Series