ÿ Ÿ a þ î µ D ý û 31 9 Vol31 No9 2011 9 Systems ngineering Teory & Practice Sept 2011 : 1000-6788(201109-173-10 : O226 :!"#$ RCH *+-/01 %&'( GI/D M SP/1/N 2434 12 64748 1 94:<; 3 =<><? 2 (1 @BCDBFGIHGBJKLMGGN OP 610066; 2 @BQRGNIQGN SUT 643000; 3 VWFGYXZ[MGBJ\]^ _` 100084 bdcfefgffifjlknmfolpnqfrfsftfufvfwfxlynzfl nf~ olpn l f nƒf f f f ˆf fq RCH Šl nœl fžn l f GI/D M SP/1/N l f n f n l nšf fœf f f fˆf fž f f n l f n f f f fªl n«l f n n f f±l²n³f fµ n f~ n f f l n fļ f f ¹fºf»f¼ff¾f fà šf fœl náfâfãfäfåfl nçfãfolpnè fél Ê fëfì ÍfÎfÏÐifjlknmfolp f~ ; Žn l f ; ÑfÒf¼fÓ f f ; ; RCH Šl nœl Discrete-time GI/D M SP/1/N queuing system wit negative customer arrival and RCH killing policy bstract YU Miao-miao 12 TNG Ying-ui 1 FU Yong-ong 3 LIU Qiang-guo 2 (1 Scool of Matematics and Software Science Sicuan Normal University Cengdu 610066 Cina; 2 Scool of Science Sicuan University of Science and ngineering Zigong 643000 Cina; 3 Department of Computer Science & Tecnology Tsingua University Beijing 100084 Cina pplying te supplementary variable tecnique and embedded cain metod based on te iteration of conditional probability matrix we studied a discrete-time GI/D MSP/1/N queuing system wit negative customer arrival and RCH killing policy Tree kinds of queue lengt distributions namely te queue lengt distribution at positive customer pre-arrival arbitrary and outside observer s observation epocs are obtained Furtermore we also considered te waiting time distribution of te accessible positive customer Finally we presented several numerical examples under some special cases to demonstrate te correctness of te teoretical analysis of tis algoritm Keywords supplementary variable tecnique; embedded cain; discrete-time ian service process; negative customer; RCH killing policy 1 ÔÖÕ føfùfúfûfü f fýfþfßfàfáfâfãfäfåfæfçfèféfêfëfìfífîfïfðfàfñfòfófôfõ ÙfÚfÛfÜ öf føfùfúfûfüfæfýfþ î f é f f fû fî fâf fòfó fý fò fîfàfñ føfùfúfûfü f fî f! " # $ % Hunter [1] [2] û&' ( * î + - / 0 1 2 3 Glenbe [3] 4 ÛfÜ Ù ;9< =9>?9@ 9B C9D ÛfÜ 6nî 7 89 : f >? @ " # F Gfî H I J K Lf Mf 6nî N O P Q R SfÝfà T Uf Mf 6nî VXW Y Z [ \fö 4 f f6 ] ^ _ ` afîfß b c d Harrison Pitel [4] [] e f g û i j [6] * < k l b m n o p q r f s t u v w >? @? @ k ~ D >? @fûfü " # x y t ufû z M/G/1 f fý ƒ fé fî 2001 1 Yang û Cae [7] >? b Poisson fî ˆ Šfû Œ f fý r Ž : 2010-01-16 : š œ ž Ÿ (70871084; ª «Ÿ (200806360001; ± œ ž Ÿ ² ª ³ (10Z136 µ : ¹ º» ¼ ¾ -mail: mmyu7@163com; À Á»  ¾ à -mail: tangy@ uestceducn; Ä Å Ç ¼ È «¾ ; É Ê» Ë È
g K O ø î z - ~! o î G Y + ; c Ç S s Ì D < k 7 k p f * Z ` k Ø Ø S 174 ÌÎÍÎÏÑÐÎÒÎÓÎÔÎÕÎÖ 31 @ Ø â ã î ÛfÜ [8] GI/M/1 f Ù Ú e f g * Û b Ü Ý Þ ß H f fý r à á >? @fûfü GI/M/1 R ä å f ÛfÜ æ ç èfýf f fîfïf é 9ê ë d è ì ífîfý Ú 20 1fí Ø Ù Ú9>9?9@ Û Ü 99î î q9ï9ð 9ñ9 î9ï9 î 9R ø9 tencia Moreno [9 10] û9e9f føfùfú >? @fûfü føfùfú ò óôõ>9? @ [11] Geom/Geom/1 f fîfòfó Û Ü é î ø føfùfú >? @fûfü 9 9ò óô fò ð % ö q ù úfîfï ûýüþë ÿfý f fî (RC F ˆ Ù b m n o p fû Ü Ý 2fî Œ fí f s î? @fü ò ï 4 føfùfúfûfü? @? @ ú 6 9å S à Tfî t9u MfÝ ï 9M ÙfÚ à T fî t u é χ n n 1!fë sfò " 4 q # p $ t u % & * ' ( ñ * + - 436 7 8 9 >? @ B C @ 4D @ F I J / 0 1 2 : ; 4<6= % G H N 7 8 S T U V WX _ ` S T a b c436d e f g % L M G O P Q R Y6Z [ \ ] ^ Poisson 9 Bernoulli 9 i j k l m n 9 o p q r s g t u v 7 8 v 1 w H x y4z6 ~ ~ 90 ƒ 36 [12] Bocarov V ˆ Š S T ] ^ 7 8 9 MSP U 9 n Ž > Œ Ž > @ š Š 7 8 9 >? @ S œ ž Š Ÿ 1 ~ ª «Samanta [13] Gupta ± ² - _ ` Bocarov ~ ³ ] ^ MSP µ - ˆ Š _ ` S T 7 8 9 D-MSP U u v 7 8 9 ˆ» Š m ¼ ¹ º 1 w H x y4z6 ¾ k Ç È É j Ê _ ` S T 7 8 9 1 À Á D-MSP  à À Ä Å Ë e Ì Bernoulli 9 _ ` Í Î Bernoulli 9 _ ` PH m n 9 Ð Ñ _ ` S T O Ï Ò Ó Ô Õ 7 8 9 a b Ö k Ø Ù Ú 2 ÛÝÜßÞáàÝâÝãåäåæèçåéåêèë 21 D-MSP ì í î _ ` 7 8 9 (D-MSP Neuts [14] lfa [1] ï ð _ ` 9 ñ (D-MP a b Ö 7 8 ó ô _ ` - ö ø ù _ ` / L ò õ6 Ò Ó 7 8 9 ú û ü H ý ( þ _ ` S ÿ k 7 8 9 k = 0 1 ³ / D-MSP š Ú k p ú û T S ÿ í î S = 1 2 m Z k 9 ú û i 1 i k S ÿ Ø k m k + 1 ( þ : 1 k + 1 S ÿ ú û 9 7 8 7 8 9 i j Á / Ó ( þ U V L ij Ê 3 1 i m 1 j m; 2 k + 1 S ÿ ú û 9 7 8 7 8 9 i j Á / [ Ó ( þ U V M ij Ê 3 1 i m 1 j m! ' " # $ v % Ø 7 8 9 _ ` ~ m & 7 8 9 ˆ + -UV ' ( * Ú Ç m & / 0 1 L M Ê 3 L = (L ij M = (M ij 1 i m 1 j m ³ (L M " _ ` 7 8 9 (D-MSP 2 0 1 L M # $ (L + M e m = e m Å 3 e m * m 4 67 p : ; 1 = L + M m & 8 9 0 1 Z / 7 8 9? = ú û < G H L + M o ú û T S ú û - U V? @ W X 0 1 > π = ( π 1 π 2 π m π # $ π (L + M = π πe m = 1 ú û a b c43b e 1 G H " 7 8 Ž > Ž > 0 ñ G H a b Ö 43G 7 8 I J S 7 8 Ž > ü 2 C D [13] F / H H K L P M N Ï P 8 S T µ À ( þ N C û 7 8 9 W S s g T G U Å ; P Q L M x y4z6 R H X Y 7 8 Z O [ 36 IP V W z6 K P 8 z6 \ L M y S ] 1 y À ( þ B C 22 ^ _ ` a b c d e f : ;  C / / ñ RCH - g i GI/D-MSP/1/N a b Ö š u v : Ö 1 j k (S ðlm n o p q / ñ r ( þ S s t u (t t + (t = 0 1 2 7 8 _ y Á / / ñ v w x / ñ / x ( þ S s z u (t t Ì > ( 1; 2 Ç T ST > ~ W X [ / ñ 8 9 G T Ê U V W X a k =
Î W v H ~ W P Š 7 Ž 4 Ç 7 9ƒ ˆ : Š Œ Ž RCH GI/D MSP/1/N š œ 17 Pr T = k k 1 r W X 3 > Ç / / 1/λ; ñ T S T œ H Pr T = k = η η k 1 (k 1 η = 1 η; W X [ 8 9 G 7 ž T T 1 Ÿ η : Œ ; : Œ Ž ; : Œ Ž ; : œ ª «1 ± (S ² ³ µ 7 8 º» W ¼ 4 ¹ þ t + t S ÿ 7 8 Á / m & 2 0 1 (L M _ ` 7 8 9 7 8 I j ¾ FCFS o 7 8 / ; / / ñ x j ¾ - g i À Á b t 7 8 Ö S RCH / > / / ñ Ö p  þ Ö Ã 7 Q R ; È Ä 7 8 D6Ö Ã 6 / Å N Ž / ; p 9 > ~ 7 / / ñ Ö S ÿ t(t = 1 2 ú û i Ç 7 ¼ i u v õ6z 8 9 G N(t: * t S ÿ Ö 43 / 1 È Ä 7 8 ( / ; U(t: * t S ÿ È É S T / ñ ; : J(t: * t S ÿ 7 8 I 7 8 Ž > k Ç Í Ê N(t U(t J(t Ë Ì _ ` S T a b Ö S ÿ t ú û U V Ú Z P nj (u t = Pr N(t = n U(t = u J(t = j 0 n N 1 j m x C Î Ï P n (u t = P n1 (u t P n2 (u t P nm (u t 0 n N u 0 C 9 3Ð f g Z P n (u= lim P n (u t= t P n = lim t P N(t = n= k + Ñ Ò = @ û - Ö 43 / : lim P n1(u t lim P nm (u t 0 n N u 0; t t lim P N(t = n J(t = 1 lim P N(t = n J(t = m t t b Ó W X 0 n N; Pn (z= P n (uz u z 1 0 n N; a(z = a k z k z 1; I m u=0 W ¼ 0 m * m & Ô 0 1 Õ k=1 0 1 e m * m 4 1 67 3 ÖØ ØÙØÚØÛØÜÞÝàßàáÞâàãàäÞåàæ 9 Ö 43 Å 3 ³ / 1 7 ç Ö K è ú û : ; Ö S T «t t+1 K è
Ô S ' Î ' 176 œà àéþêàëàìàíàîàï ú û - 9 Î U V W i ñ ò ó ô Ö K è ú û ü [ H k õ 67 ö W P 0 (u 1 t + 1 = P 0 (u t + P 1 (u tη + P 1 (u t ηm + P 1 (0 tηa u M + P 0 (0 ta u η +P 0 (0 ta u ηm + P 2 (u tηm (1 P n (u 1 t + 1 = P n 1 (0 t ηa u L + P n (u t ηl + P n (0 tηa u L + P n (0 t ηa u M +P n+1 (u tηl + P n+1 (u t ηm + P n+1 (0 tηa u M + P n+2 (u tηm 31 ð 1 n N 3 (2 P N 2 (u 1 t + 1 = P N 3 (0 t ηa u L + P N 2 (u t ηl + P N 2 (0 tηa u L + P N 2 (0 t ηa u M +P N 1 (u tηl + P N 1 (u t ηm + P N 1 (0 tηa u M + P N (u tηm +P N (0 tηa u M (3 P N 1 (u 1 t + 1 = P N 2 (0 t ηa u L + P N 1 (u t ηl + P N 1 (0 tηa u L + P N 1 (0 t ηa u M @ û - (6 (10 ø ù +P N (u tηl + P N (u t ηm + P N (0 tηa u L + P N (0 t ηa u M (4 P N (u 1 t + 1 = P N 1 (0 t ηa u L + P N (0 t ηa u L + P N (u t ηl ( H t S ~ v67 ö W P 0 (u 1 = P 0 (u + P 1 (uη + P 1 (u ηm + P 1 (0ηa u M + P 0 (0a u η + P 0 (0a u ηm +P 2 (uηm (6 P n (u 1 = P n 1 (0 ηa u L + P n (u ηl + P n (0ηa u L + P n (0 ηa u M + P n+1 (uηl +P n+1 (u ηm + P n+1 (0ηa u M + P n+2 (uηm 1 n N 3 (7 P N 2 (u 1 = P N 3 (0 ηa u L + P N 2 (u ηl + P N 2 (0ηa u L + P N 2 (0 ηa u M +P N 1 (uηl + P N 1 (u ηm + P N 1 (0ηa u M + P N (uηm +P N (0ηa u M (8 P N 1 (u 1 = P N 2 (0 ηa u L + P N 1 (u ηl + P N 1 (0ηa u L + P N 1 (0 ηa u M +P N (uηl + P N (u ηm + P N (0ηa u L + P N (0 ηa u M (9 P N (u 1 = P N 1 (0 ηa u L + P N (0 ηa u L + P N (u ηl (10 u Z G ú x zp 0 (z = P 0 (z+p 1 (zη+p 1 (z ηm +P 1 (0ηa(zM +P 0 (0a(zη+P 0 (0a(z ηm +P 2 (zηm P 0(0 P 1 (0η P 1 (0 ηm P 2 (0ηM (11 zp n(z = P n 1 (0 ηa(zl+p n(z ηl+p n (0ηa(zL+P n (0 ηa(zm +P n+1(zηl +P n+1(z ηm +P n+1 (0ηa(zM +P n+2(zηm P n (0 ηl P n+1 (0ηL P n+1 (0 ηm P n+2 (0ηM 1 n N 3 (12 zp N 2(z = P N 3 (0 ηa(zl+p N 2(z ηl+p N 2 (0ηa(zL+P N 2 (0 ηa(zm +P N 1 (zηl+p N 1 (z ηm +P N 1(0ηa(zM +P N (zηm +P N(0ηa(zM P N 2 (0 ηl P N 1 (0ηL P N 1 (0 ηm P N (0ηM (13 zp N 1 (z = P N 2(0 ηa(zl+p N 1 (z ηl+p N 1(0ηa(zL+P N 1 (0 ηa(zm +P N (zηl ³ (11 (1 ø ù +P N(z ηm +P N (0ηa(zL+P N (0 ηa(zm P N 1 (0 ηl P N (0ηL P N (0 ηm (14 zp N (z = P N 1(0 ηa(zl+p N (0 ηa(zl+p N (z ηl P N(0 ηl (1 u ù û e m x > ¼ [ H N Pn (ze m = ü [a(z 1] z 1 N P n (0e m (16
X 7! % Ê g ü H W 9ƒ ˆ : Š Œ Ž RCH GI/D MSP/1/N š œ 177 (16 u S ý þ > Pn = Pn1 P nm 4õ Bayes z 1 ÿ g k % Ø @ û - (0 n N * L-Hospital x N P n (0e m = λ (17 / ñ Ö 3 / UVW X Pn = lim P N(t = n J(t = 1 U(t = 0 lim P N(t = n J(t = m U(t = 0 t t P N(t = n J(t = 1 U(t = 0 P N(t = n J(t = m U(t = 0 = lim lim t P U(t = 0 t P U(t = 0 P n (0 1 = = N λ P n(0 0 n N (18 P n (0e m 3 (11 (1! i ü k õ ÿ Z x P 0 i z = 1 (18 P n P n T? Ö P n Ù P n (0 T? Ö ÿ ñ 67 P n (1= u=0 P n(u=p n P N = λp N 1 ηl (I m ηl 1 (19 P N 1 = [ λp N 2 ηl + ( λp N 1 +P N (ηl+ ηm λp N 1 ηl] (I m ηl 1 (20 P N 2 = [ λp N 3 ηl+( λp N 2 +P N 1 λp N 1 (ηl+ ηm+λp N 1ηM λpn 2 ηl] (I m ηl 1 (21 P n = [ λp n 1 ηl+( λp n +P n+1 λp n+1 (ηl + ηm + λp n+1 ηm + P n+2 ηm õ' λp n ηl λp n+2 ηm] (I m ηl 1 n = N 3 N 4 1 (22 N P n = π À Å 3 π H p @û- Ö 3 õ (19 (22 < / ñ / i B 9 ƒ Y @ û - Ù S ÿ Ö 43 U V W X67 ' / P n Ò Ó š Y ' 4 àöà àã τ r (r 1 * " r Ç S ÿ # $ ñ N r = N(τr * " r Ç # $ ñ k ( J r = J(τr * " r Ç Ö Ž + Ÿ # $ ñ * õ ¹ b Ó k % - ð Ê ú û / 0 Ω = (i j i = 0 1 N j = 1 2 m m & 2 0 1 (L M D-MSP? @ W U VWX6 7 ò Ö &% ³ P n ( P n # $ ' Ê (N r J r r 1 P = P0 P 1 P N Pn = Pn1 P nm = lim P N r = n J r = 1 lim P N r = n J r = m r r p 1 2 3 4 P 6 7 ' 8 9 : ; < = > P P = P P e m(n+1 = 1? 1 @ B C D 0 n N F G H @ U V < Y I P J K L M N O P Q R S T W X P Z [ \ ] ^ _ ` a b c d U V P Z > g J K e f Z i j&k l m n ] ^ o p a 2 q N O P Z @ U V r s t u v w = > x Q R S T n y z : D V ( S n (k (n 0 k 1 m ~ I& i j ƒ S n (k Z J K n @ ˆ Z 6 7 Š : r n Œ r + 1 n Ž Z k n a m &š œ ž Ÿ n n Ž r n Ž œ Z i F a r + 1 n Ž œ Z j; ij
V µ w q r Ì ö Ì Ð Ø T 178 ª«31 D V ± S n (n 0 m ~ I& i j ƒ (S n Z ² ³ ij J K n @ I 6 7 Š : a r Œ r + 1 n Ž &š œ ž Ÿ n n Ž r n Ž œ Z i F a r + 1 n Ž œ Z j 3 œ Ž Z ž ¹ º :» Ž ¼ Z T L Ž ¾ ¼ ZÁÀÃÂ Ä p Å Ç È z Š Z < É H d = > _ S n = a k S (k n n 0 (23 k=1 @ U V S n J Ê N O P Q R S T Z _ ` Ë G H S n Z 6 7 J D Ì u Ï Í Î @ U Å Ñ Ò Ó Ô 8 Õ S n (k л Ž Z Ö Ø η Z Bernoulli Ö» Ž Z 8 9 œ a Ù Ú Z Ž W X a n &š œ ž Z Ž Ø Ë 8 9 = > Û ζ º Ü Ý ( Þ ß 2: 1 U V (P ζ = 0 J(t + 1 = j J(t = i (1 i m 1 j m J K a n &š à Ž ž Å i á â j; 2 U V (P ζ = 1 J(t + 1 = j J(t = i (1 i m 1 j m J K a n &š 1 n Ž ž Å i á â j; 3 U V (P ζ = 2 J(t + 1 = j J(t = i (1 i m 1 j m J K a n &š 2 n Ž ž Å i á â j ã Õ Ç Û n @ U V ä Ð å (P ζ = 0 J(t + 1 = j J(t = i = ηl (P ζ = 1 J(t + 1 = j J(t = i = ηm + ηl (P ζ = 2 J(t + 1 = j J(t = i = ηm : æèçêéêëêìêíêîêïêéêëêðêñ : æèòêóêôêõêìêíêîêïêéêëêðêñ 2 êøêùêúêûêüêýêþêÿ m a k n Z n ¼ Å b» Ž Z i œ Ž Z Ø Ë 8 9 0 1 2 Û º Ü @ < D ã G H = > U V Ì S (k 0 = ηls (k 1 0 k 1 (24 S (k 1 = ηls (k 1 1 + ( ηm + ηl S (k 1 0 k 1 (2 S (k n = ηls(k 1 n + ( ηm + ηl S (k 1 n 1 + ηms (k 1 n 2 n 2 k 1 (26 8 9 (24 (26 I& S (0 0 = I m S n (0 = 0 m (n 1 n > 2k S n (k = 0 m (23 Ì Ç Ì G H S n (n 0 Z 6 7 J U V U V S n (n 0 m ~ S n (n 0 @ U V J Ê N O P Q R S T Z n _ ` ( Ë ˆ Z i j ƒ Sn Z J K n @ I 6 7 Š : a r Œ r + 1 n Ž ij &š œ ž Ÿ (n + 1 n Ž r n Ž œ Z i F a r + 1 n Ž œ Z j c d U V S n (n 0 Z 6 7 J m Š! " Ì (24 (26 ã H S (k (z = S (k (z = = > S (k n zn S(z = ¹ n U V S n z n [ S (1 (z] k = [ ηl + ( ηm + ηl z + ηmz 2 ] k
R Å Å < w w Ð q Ð Ì ; 9 # $ % & ' : ( êçêéêë * + RCH - / 0êî GI/D MSP/1/N 1 2 3 4 6ê ê 179 7 8 9 9 a (27 Ì : z = 1 ; (28 Ì Í Î Ç 8 9 c d S(z = S n z n = a k S n (k z n = S(z = k=1 a k k=1 a k [ ηl + ( ηm + ηl z + ηmz 2 ] k k=1 S (k n z n (27 S(1 = S n = a k [ ηl + ( ηm + ηl + ηm] k (28 k=1 S n = S(1 n S k k=0 = j œ Q R >? @ B C ¼ N O P Z Q R S T P = S 0 S 0 0 0 0 0 S 1 S 1 S 0 0 0 0 S 2 S 2 S 1 0 0 0 S N 2 S N 2 S N 3 S 1 S 0 0 S N 1 S N 1 S N 2 S 2 S 1 S 0 @ U V 6 = > D j U V P F O? G @ B C D S N 1 S N 1 S N 2 S 2 S 1 S 0 8 ; H Ð H P n w F G H I 8 J K L Ð H P n MONOPOQOROSUTWVYX[ZW\W]U^W_WÙaWbWcUd e f Þ e œ f g e J Z R KiL ÐiH y Pn out (n 0 = ß 9 1 K n Z f g e J j a t + Œ (t + 1 r j f g e J Œ I 8 J t ; Ð k Z Q R l m n ã o p = > _ Ì P 0 = P out 1 M + P out 0 (29 P n = P out n+1m + P out n L 1 n N 1 (30 P N = P out N L (31 (29 (31 Ì 8 H P out N = P NL 1 (32 P out n = ( P n P out n+1m L 1 n = N 1 N 2 1 (33 P0 out = P 0 P1 out M (34 (32 (34 Ì 8 Õ I 8 Ð k q Z @ : 8 9 G H R á r F s t f g e J œ Ž Zu @ 9 w Õ Ð H v a» Ž C K Ñ Ò Ž a œ Z xzÿ Ð H i b G Z ; ~ < xzÿ Ð H Z? ã Z Ü a Ç Ÿ Ð H ¼ Í Î 8 9 Ù Ì ƒ Little G H Ž a œ xzÿ Z 6 O OTOVOZO\U]W^W UŴRW UcWd Å b C K œ Z Š Œ D Z Ž Ži œ Ž Ø Ëi b i b N 1 n O œ e f j 9 4 f& Š Z @ U V S n (k (n 0 k 1 @ < n š 8 O œ Z Ž Z Ð H
9 > µ µ º š Ð 8 O B c 1760 ª«31 T q J K n 8 O œ Ž a K & Z y ˆ Z Ð H Å w k = P T q = k k 0 b» Ž œ RCH T ž Ÿ T q Z = > Ð H J w 0 = w k = ( P 0 + P 1 η e m 1 P loss P 1 S(k 1 0 ( ηm+ ηηl+p2 S(k 1 + N 1 n=3 m P loss J K œ Z @ P n S(k 1 n 1 ( ηm + ηηl+p n S(k 1 n 2 P loss = P N e m (3 1 ( ηm + ηηl+p2 ( S(k 1 0 ηm + ηηm +η 2 L ( ηm + ηηm +η 2 L e +Pn m S(k 1 n 3 η2 M (36 1 P loss 7 O O O D q z s t Z i ª o K s t Z «á e f Matlab Ø s t d» Ž RCH T ž Z ¹ n C K œ Z Ø = J 1 J 2 Ð å c d Ÿ R Ü Geom/D MSP/1/N œ P H/D MSP/1/N œ a Û º? J Z K L Ð H ± I œ Ž Z " Ð H J& Ø ² t ¼ ³zy Ø Z Ó Ô = 1 (Geom/D MSP/1/N Ž Z Ö Ø λ = 032 Z Bernoulli Ö» Ž Z Ø η = 01 Z x Ð H œ U V á 2 ~ D-MSP I J Ê œ Z Ë N = 1 j Ç L = ( Ø O 032 034 046 012 M = ( B ¼ Z ² t = ¹ J 1 018 016 014 028 1 Geom/D MSP/1/1» ¼ êþ ¾ êü À Á Â Ã Ä Å Ç È É» ¼ êøêù ü Ê Ë̾ Í ÄÌÅ n P nj P nj Pnj out k w k j = 1 j = 2 j = 1 j = 2 j = 1 j = 2 0 02306 02142 02306 02142 01827 0108 0 04694 1 01402 01060 01402 01060 01620 01340 1 01077 2 00774 0096 00774 0096 00932 00712 2 00860 3 00431 00332 00431 00332 0017 00398 3 0068 4 00240 0018 00240 0018 00288 00221 4 0046 00134 00103 00134 00103 00160 00123 0043 6 00074 0007 00074 0007 00089 00069 6 00346 7 00041 00032 00041 00032 0000 00038 7 00276 8 00023 00018 00023 00018 00028 00021 8 00220 9 00013 00010 00013 00010 0001 00012 9 0017 10 00007 0000 00007 0000 00009 00007 10 00140 11 00004 00003 00004 00003 0000 00004 11 00111 12 00002 00002 00002 00002 00003 00002 12 00089 13 00001 00001 00001 00001 00001 00001 13 00071 14 00001 00001 00001 00001 00001 00001 14 0006 1 00000 00000 00000 00000 00000 00000 1 0004 2 (P H/D MSP/1/N» Ž á Ž á ± I Ø U V J Ê Z Ô Ó Ô Ž Z á? Ï Ð J K (T α Z 3 ~ Ä Ñ PH á I& T = œ Z Ë N = 16 j Ø O 02 03 02 0 03 00 03 00 04 B ¼ Z ² t = ¹ J 2 α = (03 01 06 : ŒiÎ 1
ë n š Ð q : ö º " 9 ö 9 # $ % & ' : ( êçêéêë * + RCH - / 0êî GI/D MSP/1/N 1 2 3 4 6ê ê 1761 2 P H/D MSP/1/16» ¼ êþ ¾ êü À Á Â Ã Ä Å Ç È É» ¼ êøêù ü Ê Ë̾ Í ÄÌÅ n P nj P nj Pnj out k w k j = 1 j = 2 j = 1 j = 2 j = 1 j = 2 0 0270 0221 02788 0249 02292 0189 0 020 1 01427 0109 01407 01044 01686 01373 1 01070 2 00671 0008 00662 0001 00827 00621 2 0081 3 00319 00241 00314 00237 00391 00296 3 00620 4 0011 00114 00149 00113 0018 00140 4 00472 00072 0004 00071 0003 00088 00066 0039 6 00034 00026 00033 0002 00042 00032 6 00273 7 00016 00012 00016 00012 00020 0001 7 00208 8 00008 00006 00008 00006 00009 00007 8 0018 9 00004 00003 00004 00003 00004 00003 9 00120 10 00002 00001 00002 00001 00002 00002 10 00092 11 00001 00001 00001 00001 00001 00001 11 00070 12 00000 00000 00000 00000 00000 00000 12 0003 13 00000 00000 00000 00000 00000 00000 13 00040 14 00000 00000 00000 00000 00000 00000 14 00031 1 00000 00000 00000 00000 00000 00000 1 00023 16 00000 00000 00000 00000 00000 00000 16 00018 gie J 1 Z Ø? iži Ž Z ÐiHi iò yióiª a sitii= ³ÔyÕ i Ø Z Ö Geom/D MSP/1/1 œ Ž s Ø K L Ð H I 8 Ðik q Z KiL ÐiH ¾i m Ù Ú Œ Û Bernoulli Ü O C K œ Z BST ª Ý Þ 9 ß Ø t Î 1 n à g D z Ÿ e á s t f Ð Ò â Z Å b J ã ä å a J 1 D J 2 Í Î c d Ÿ Ï O œ Ž Z @ f Ð Ð H æ a ç è C s t é Ziê ; > Í Î Ž 60 k=0 w k 10000 m = w z Ÿ e á Ð H Z Ì s t (3 (36 Z ª Ï O œ Ž a K & Z ì ß : D Ì Ï á ¹ º s t I Ù Ì (3 (36 ; < Z ; I L á Û Z Little Ì ; [W q ] = [W q ] = kw k k=1 N (n 1P out n=1 n λ(1 P loss G H Ï Ø t Î 2 Ð å ~ Ç D ¹ º s t Í Î Ž ¹ º [W q ] = 18736 m ± î z e á ï ð Z D s t ñ Z e m D s t Z = ¾ í 12 x 10 4 PH/D MSP/1/16 4 PH/D MSP/1/16 1 3 08 3 P loss 06 04 [W q ] 2 2 1 02 1 0 00 01 01 02 η 3 òêøêù ó ô õ ö» ¼ ø ù õêü ú û 0 00 01 01 02 η 4 òêøêù ó ô õ ö È É» ¼ê êøêù Ê Ë ¾ ÍêüÌú û
ß ß 1762 ª«31 ß 3 ß 4 Ð å ü ý Ÿ» Ž η Ï œ @ Ï O œ Ž Ziþ ÿ ß&Ï ß d P loss [W q ] ¹ n Ë Ï Ø η Z G Ø η Z» Ž n Å b» Ž Ï T K t a Ù Ú Z Ž W X œ Z ß Ï F @ ë b 0 Ž w O œ C K ¼ Å b» Ž Z Ï ñ a Ù Ú Z Ž Z 9 ß K Ž a K & Z æ ß Ï 8!#" ÄiÑ%$%& G Ë%' iö b %(% U V iîirif Z%*%+%-0/%' iš Ÿi» Ž á 1 D-MSP Z Š C K œ 2 å b 3 œ 4 á 1 j 4 D-MSP ß 6 Ç 7 Ú 4 «d á 1 4 8  ªi 9iª æ : á 1& 4 ; < Øi = ³ > = G 4? v @ 9 Ç f Ð 4 B œ f C D &š f F 4 G H I J%K í%l M %4 ;%< N O P%Q R%S%T%U%V%L%M%W%X%Y%Z %[%\%]%^%_%` 4%a%b%c%d%: L%M%C%e%f%g%%i%j%k%l 4%m%n%o%p%q%r s t u v c d w x y z v ~ ƒ [1] Hunter J J Matematical Tecniques of pplied Probability Vol II [M] New York: cademic Press 1983 [2] ˆ Š Œ Ž [M] : š œ 2008 Tian N S Xu X L Ma Z Y Discrete-time Queueing Teory[M] Beijing: Science Press 2008 [3] Gelenbe Queues wit negative arrivals[j] Journal of pplied Probability 1991 28: 24 20 [4] Harrison P G Pitel Te M/G/1 queue wit negative customers[j] dvances in pplied Probability 1996 28(2: 40 66 [] Zu Y J nalysis on a type of M/G/1 models wit negative arrivals[c]//proceeding of te 27t Stocastic Process Conference UK 2001 [6] ž Ÿ ª «[M] : š œ 1999 Si D H Te Density volvement Metod of Stocastic Models[M] Beijing: Science Press 1999 [7] Yang W S Cae K C note on te GI/M/1 queue wit poisson negative arrivals[j] Journal of pplied Probability 2001 38: 1081 108 [8] ± ² RC ³ µ GI/M/1 ¹ º» ¼ [J] ¾ š š À 2008 29: 360 364 Zu Y J Gu Q F GI/M/1 queue wit RC strategy of negative customers and working vacation[j] Journal of Jiangsu University 2008 29: 360 364 [9] tencia I Moreno P Te discrete-time Geo/Geo/1 queue wit negative customers and disasters[j] Computers & Operations Researc 2004 31: 137 148 [10] tencia I Moreno P single-server G-queue in discrete-time wit geometrical arrival and service process[j] Performance valuation 200 9: 8 97 [11] Œ Á Â Ã Ä Å Ç Geom/Geom/1 [J] ¾ š š À 2007 28: 266 268 Zu Y J Ma L Qu Z F et al class of Geom/Geom/1 discrete-time queueing system wit negative customers[j] Journal of Jiangsu University 2007 28: 266 268 [12] Bocarov P P Stationary distribution of a finite queue wit recurrent flow and ian service[j] utomat Remote Control 1996 7: 66 78 [13] Samanta S K Gupta U C Caudry M L nalysis of stationary discrete-time GI/D MSP/1 queue wit finite and infinite buffers[j] 4OR: Quarterly Journal of Operations Researc 2009(7: 337 361 [14] Neuts M F Matrix-geometric Solutions in Stocastic Models[M] Baltimore: Te Jons Hopkins University Press 1981 [1] lfa S Discrete time queues and matrix-analytic metods[j] Top 2002(10: 147 210