GORDON AND NEWELL QUEUEING NETWORKS AND COPULAS
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1 Yugoslav Journal of Operaons Research Vol 9 (009) Number 0- DOI:0.98/YUJOR0900C GORDON AND NEWELL QUEUEING NETWORKS AND COPULAS Danel CIUIU Facul of Cvl Indusral and Agrculural Buldngs Techncal Unvers of Cvl Engneerng Buchares Romana. Romanan Insue for Economc Forecasng Buchares Romana. dcuu@ahoo.com Receved: December 007 / Acceped: Ma 009 Absrac: In hs paper we have found an analcal formula for a copula ha connecs he numbers N of cusomers n he nodes of a Gordon and Newell queueng newor. We have consdered wo cases: he frs one s he case of he newor wh nodes and he second one s he case of he newor wh a leas 3 nodes. The analcal formula for he second case has been found for he mos general case (none of he consans from a ls s equal o a gven value) and he oher parcular cases have been obaned b lm. Kewords: Gordon and newell queueng newors copula.. INTRODUCTION A Jacson queueng newor (see [74]) s an open queueng newor wh nodes where he arrvals from ousde newor a he node s ep( λ ) he servce a he node s ep( ) and afer fnshes s servce a he node a cusomer goes o he node wh he probabl P or leaves he newor wh he probabl P 0. We now (see [74]) ha he arrvals from nsde or ousde newor a he node are ndependen wh he dsrbuon ep( Λ ) where Λ s he soluon of he ssem P Λ + λ =Λ = () = A Gordon and Newell queueng newor (see [5]) s a closed queueng newor wh nodes and cusomers. The servce me n he node has he dsrbuon N
2 0 D. Cuu / Gordon and Newell Queueng Newors and Copulas ep( ) and afer he servce n hs node he cusomer goes o he node wh he probabl. We have noced ha he mar P as above s he ranson mar of P an ergodc Marov chan (see [6]). If we denoe b ( p ) = he ergodc probabl we now ha n α N = P N = n... N = n = (... ) () where s proporonal o n = N = p and α N (... ) s compued such ha PN ( = n... N = n) =. ( ) Obvousl because N = N he above random varables N are no = ndependen. The depend hrough a copula hs erm havng he followng defnon (see [09]). n Defnon. A copula s a funcon C :0 [ ] [ 0] such ha: a) If here ess such ha = 0 hen C (... n ) = 0 b) If = for an hen C (... n) = and c) C s ncreasng n each argumen. We have he followng heorem (see [09]). Theorem. (Slar). Le X X... X n be random varables wh he cumulave dsrbuon funcons F F... F n and he common cumulave dsrbuon funcon H. In hs case here ess a copula C such ha H (... n) = C( F... Fn( n)). The copula C s well defned on he carhesan produc of he margnals F F... F n. Defnon. ([0]). If n= he copula C s Archmedean f Cuu ( ) < u for an u (0) and CCuv ( ( ) w) = CuCvw ( ( )) for an uvw [ 0]. If n he copula C s Archmedean f here ess a n Archmedean copula C and a Archmedean copula C such ha Cu (... un) = C( C( u... un ) u n ). In [] mehods o smulae Archmedean copulas have been presened and n [3] algorhms o smulae queueng ssems wh a channel wh arrvals and servces dependng copulas have been shown. For an n copula C we have (see []) W(... ) C(... ) mn(... ) where (3) n n n n W(... n) = n+ (3 ) = s he lower Fréche bound and mn s he upper Fréche bound.
3 D. Cuu / Gordon and Newell Queueng Newors and Copulas 03. THE COPULA THAT CONNECTS N We now fnd a copula C such ha P( N n... N n ) = C( F( n )... F ( n )) where. We have N and for = F s he cdf of he dscree random varable = = <... < CFn (... F( v)) = Fn ( ) PN ( n... N n) + + PN ( n... N n) and from here we oban = + + = = <... < + CF ( ( n)... F( n)) Fn PN ( n. (4)... N n ) P( N n... N n ) In he same manner we compue PN ( n... N n) and we oban he recurrence formula + = + = = <... < CF ( ( n)... F( n)) Fn CF ( ( n). (4 )... F ( n )) P( N n... N n ). Proposon. If = he copula ha connecs he wo nodes s he lower Fréche bound C = W. Proof: Consder F and F he margnals of N and N. Frs we noce ha f n+ n < N we have F( n) + F( n) < f n+ n = N we have F( n) + F( n) = and f n+ n N we have F( n ) + F ( n ). These relaons come from he fac ha f n + n N we have + n N F ( n ) F ( n ) = P( n < N < N n ) and f n we have F ( n ) + F ( n ) = P( n N N n ). If n n N + < we have obvousl CF ( ( n) F( n)) = PN ( n N n) = 0 because N + N = N wh he probabl. If n + n N we appl he formula (4) and from he same reason we have PN ( n N n) =0. We oban CF ( ( n) F( n)) = F( n) + F( n) n hs case. I resuls ha C = W and he proposon s proved. Consder now. We compue frs F( n ) and F ( u) = h( u). We buld he Gordon and Newell queueng newor wh wo nodes and he same margnal
4 04 D. Cuu / Gordon and Newell Queueng Newors and Copulas F for hs. Frs we leave he sae correspondng o he node wh he ranson probables P and P and we group he oher saes (see [6]). If ( p ) s he = ergodc probabl of he saes Marov chan we oban he ergodc Marov chan wh he ergodc probabl ( p ). Ne f we denoe b = we se he servces n hese wo nodes o ep( ) and ep( ) n n p PN ( = n) = α ( ) where = and N equvalen o =. We denoe b =. I resuls ha p =. We oban p =. Obvousl = s n+ n + N + = f F( n) = and n + f N + = ln( u+ ) f ln h( u) = ( N + ) u f = (5) (5 ) We wan o fnd an analcal form for he copula Cu (... u ) wh. Frs we can see ha I resuls ha α N (... ) n + F( n) =. (5 ) α (... ) N n PN ( n... N n) = P Nl ( n + ) (6) = where l s a node of he newor (no necessarl beween he nodes). Usng he formulae (4) and (6) we oban Cu (... u) = CF ( ( n)... F( n)) = + ( ) h ( u ) < N+ r r r= = ( r P Nl h u ) r r l r= r= n + r. (7) Suppose ha we have no = such ha =. We ae he value of l beween and or n he conrar case ( l ) he proper s fulflled oo. l Denoe b γ l and δ l he real numbers such ha γ = and γ = and δ =. From he formula (5) we oban 0 l l l l δ l =. Obvousl
5 D. Cuu / Gordon and Newell Queueng Newors and Copulas 05 = + n + γ l ( ) l u l n + δl = ( ) l u +. (8) We have he followng proposon. Proposon. If here ess no = such ha = we have N + l γ l δ r r l Cu (... u ) = + ( ) ( ) l u r r l u r r = + + r= r=. h ( u ) < N+ r r r= Proof: For he formula from he enuncaon we use he formulae (7) and (8). I remans o prove ha C s a copula. The random varables can be consdered as connuous random varables on [0 N + ) wh he cumulave dens funcon havng he same values n he neger argumens. We denoe b N m = h( u) and we prove ha Cm (... m ) s ncreasng n each. Ths proper can be proved frs f we consder m Q 0 N + as follows. We consder for each m one raonal value ecep one for ha we wan o prove he monoon. For hs we consder wo dsnc raonal values. We reduce all he nvolved fracons o he same denomnaor p and we buld a Gordon and Newell m [ ) queueng newor wh N p cusomers nodes and he same. I resuls ha for hs = m p N m p newor we have Cm (... m) PN (... ). on Therefore we have proved he monoon on Q [0 N + ) and he monoon R [0 N + ) resuls b lm. If we have an = wh u = 0 we have h (0) = 0 and each erm from he enuncaon ha conans hs ( u + ) and wce whou hs facor. u appears wo mes wh opposes sgns: once wh N + l γl δl Ths s rue ncludng he erm ( ) l u + l u + = whch can be reduced wh l from he begnnng of he formula. I resuls ha Cu (... u ) = 0. If u r = for all r = wh r we oban N + l γl δl Cu (... u) = ( ) l u + l u +. We ae now l = and we oban N + n n + + N+ N+ Cu (... u ) = = = u. I resuls ha C s ndeed a copula and he proposon s proved.
6 06 D. Cuu / Gordon and Newell Queueng Newors and Copulas Remar. The copula from he above proposon s ndeed a connuous funcon because we can prove n he same wa as for he monoon ha f h ( u ) = N + he nvolvng erm becomes zero. From hs we oban he lef connu and he rgh connu s obvous because he nvolvng erm does no appear. Suppose now ha we can have = such ha = bu here ess l = such ha. In hs case we replace n he above proposon for each wh u he epresson l ( N+ ) u l l ( u + l b ) γ ( N+ ) u l r = and he epresson. B lm we can prove n hs case ha C s also a copula. r r = ( u + l b Fnall we consder he case wh = for an =. Because n fac s p onl proporonal o we can consder =. We use now he formulae (5 ) and (7) we oban ) δ Cu (... u) = + ( ) ur = r= u r < r= (9) B lm we can also prove n hs case ha C s a copula. 3. SOME PROPERTIES FOR THE COPULA THAT CONNECTS N We wll compue now for he copula from he prevous secon he value ρ of Spearman (see [8]): ρ = Cuvdudv ( ) 3. (0) 00 In he case = we have C = W hence ρ =. () oban In he case we use he proposon and he recurrence relaon (4 ). We Cu ( u) = u+ u + ( u + ) ( u+ )( u + ) () N + δ f h ( u ) + h ( u ) N + and n he conrar case Cu ( u) = u+ u. ( ) Suppose frs ha and.
7 D. Cuu / Gordon and Newell Queueng Newors and Copulas 07 Because ( u + u ) du du =0 resuls ha 00 N + I C( u u) dudu 00 = = γ δ ( u ) dudu ( u )( u ) dudu h( u) + h( u) N+ h( u) + h( u) N We use now he subsuons δ ( u + ) = ( ). We oban u = du ln d = ( u + ) = and γ I = d d d d N+ N+ N+ N+ N + ln ln ln ln N+ N+ N + ln N+ ln N+ 0 0 ( ) d ( ) d = N + N + ln 0 d N+ N+ N + ln ( ) ln ( ) d d If we have also and we oban N+ N+ ln N+ ln N+ (ln + ln ln ln ) (ln + ln ln ) I = ( ) ( ) + ln N + ( ln + ln + ln ln ) ( ) and from here = N + 6 ln N + (ln + ln ln ln ) ρ = ( ) N + ln N + (ln + ln ln ) ( ) + 6ln N + (ln + ln + ln ln ) In he same wa we oban ( ) 3 N+ N+ ln. (3) 6( N + ) ln (9 3) ln ρ = 3 f = (4) 6( + )ln ( N + ) ln ρ = 3f = and (5) N+ N+ ln N ln 3( + 3) ln 6( N + ) ln ρ = + 3 f =. (6) Suppose now ha and =. I resuls ha = and
8 08 D. Cuu / Gordon and Newell Queueng Newors and Copulas N+ N+ N+ N+ N+ ( N+ ) 0 N+ 0 N+ 0 I = d d + d. If we have also and we oban N+ N+ N+ N+ ( N+ ) (ln ln ln ) ( N+ ) (ln ln ) I = ( ) ( ) + ( N+ ) (ln + ln ln ) N + ( ) + and from here N+ N+ 6 N+ N+ ( N+ ) (ln ln ln ) ( N+ ) (ln ln ) ρ = ( ) ( ) + 6 ( N+ ) (ln + ln ln ) N + ( ) 3. (3 ) In he same wa we oban 6 6 ρ = 3 f = and ( N + )ln N + (4 ) N 3( + 3) 6 + ρ = + 3 f ( N ) ln = (6 ) If = and we swch he ndees l and n (3 ) (4 ) and (6 ). If we ae can consder p = p and. I resuls ha = we oban = = and =. We N + N+ h( u) + h( u) N I = ( u u uu ) du du = ( ) = N+ N+ N+ N+ ln + ln ln + ln + dd( 3 ) dd 3 dd ln + ln 3 N + We consder now 0 + = N + N + ln N + ( N + ) ( ) d 3 ( ) d = 0 N + N + N + ln N + ( ) 3 and we use l' Hôpal and we oban I = and from here ρ =. (7) We wll compue now for he copula from he prevous secon he value τ of Kendall (see [8]): τ = P(( X X )( Y Y ) 0) P(( X X )( Y Y ) < 0) = C C = C u v u v C( u v) dudv 4 dudv. As for Spearman's ρ we consder frs he case. 6 (8)
9 D. Cuu / Gordon and Newell Queueng Newors and Copulas 09 Usng he subsuons u + + C C τ = 4 d d = and (8) we oban N N. (8 ) 0 0 Bu for h ( u ) + h ( u ) N + we have δ C ( ) = u u( u + ) = ( ). C C (ln ln ln ) ln Therefore + = ( ) n hs case. If h ( u ) + h ( u ) N + we have also N ( ) N + δ γ u + + C ( ) = u ( u + ) + = + and from here N + C C (ln + ln ln ) ln (ln ln ) ln =. I resuls ha N+ N+ N+ N+ N+ N+ C C C C C C N+ I= dd = dd+ dd = N + N+ (ln + ln ln )ln d 0 0 (ln + ln ln ) ln d + d d N+ N+ N + (ln ln ) 0 N+ If d d. and here and we oban N+ N+ 0 N+ N+ N+ N+ ln N+ ln N+ ln I = d d + d N+ N+ N+ N+ ln N+ ln N+ ln d d + d. We noce ha he above condon we oban can be avoded b lm. If ln N+ N+ ln N+ N+ (ln ln ) (ln+ ln ln ) I = ( ) + ( ( ) ) and from 4ln N+ N+ 4ln N+ N+ (ln ln ) (ln + ln ln ) τ = ( ) ( ) 3. (9) In he same wa we oban 4N ln 4( N + ) ln τ = + 3 f = and (9 ) N+ N+
10 0 D. Cuu / Gordon and Newell Queueng Newors and Copulas N + 4 N + 4 N + ln τ = 4 + f = (9 ) If = and we have 4 4 N+ 4 ( N+ ) (ln ln ) ( N+ ) (ln ln ) ( N+ )(ln ln ) τ = N+ 3 (9 ) and n he case and = we change he ndees n he above formula. Because ρ = n he case = for an we have (see [8]) τ =. (0) Now we chec when he wo from (3) become =. For W we have from he recurrence formula (4 ) Cu ( u) = u+ u + PN ( n N n). I resuls ha l N + l γ l γ l l Cu ( u) = u+ u + ( u+ ) ( u + ) δ l δ l ( u + ) ( u + ) f h ( u ) + h ( u ) < N + and () Cu ( u) = u+ u fh( u) + h( u) N+. ( ) I resuls ha n he case = we have C = W f h ( u ) + h ( u ) N +. Because he Boole nequal s proved b nducon f here ess and such ha h ( u ) + h ( u ) < N + we have Cu (... u ) Wu (... u ). If h ( u ) + h ( u ) N + for an and we oban b usng he proposon and he resul for = ha Cu (... u ) = Wu (... u ). In he case of he upper Fréche bound mn and = here ess for an and n 0 he number u such ha u u < < and h ( u ) + h ( u ) N +. I resuls ha Cu ( u) Cu ( u) = u + u < u and analogousl Cu ( u) < u. Therefore Cu ( u) = mn( u u) ff a leas one of he argumens s O or l. In he case u u we denoe b u = mn( u... u ). If u u (0) and here ess anoher such ha u (0) we oban Cu (... u) Cu ( u ) < mn( u... u ). In hs wa we have proved ha Cu (... u ) = mn( u... u ) f here ess u = 0 or n he conrar case u = for ndees. 4. CONCLUSIONS As we can noce when we deduce he analcal form from he copula ha connecs N we can see ha we have o consder wo cases of he Gordon and Newell queueng newor: he newor wh wo nodes and he newor wh a leas hree
11 D. Cuu / Gordon and Newell Queueng Newors and Copulas nodes. Ths paron no cases can be eplaned b he fac ha for he frs case he copula s he lower Fréche bound W whch s onl a copula no an n copula for n 3. For he queueng newor wh nodes we have C = W n he general case no onl for he Gordon and Newell queueng newor. Ths can be eplan b he fac ha N+ N = N hence he varables are srongl anhec as we now for C = W. For he queueng newor wh a leas 3 nodes we have no C = W even f =. Ths s because N and N can ncrease ogeher b means of decreasng anoher N. Bu for = C can end o W f he servces n he oher nodes end o nfn (he are ver fas reporng o hose of he wo nodes) because he N cusomers end o sa n he wo nodes. For < we se N and we buld a Jacson queueng newor wh he frs nodes (he las node of he Gordon and Newell queueng newor can be consdered as an ousde newor). The node has he same servces and he ranson probables and he arrvals from ousde he newor ep( P ). If from () s less han he obaned Jacson queueng newor s sable and an copula from hs paper nvolvng < from he frs nodes ends o he copula Prod ( N ends o be ndependen). An open problem s o fnd an analcal form for he copula ha connecs N for a more general queueng newor hen he Gordon and Newell queueng newor f 3. We can sar wh he Buzen queueng newor where he servce n he node depends on he number N of cusomers n ha node: s dsrbuon s ep( a ( n ) ) where 0 and a s a gven funcon (see []). As we can noce for = and he correspondng lm case for 3 he copula C = W s Archmedean and he same hng we can sa abou he correspondng lm case for C=Prod. Anoher open problem s o sud f n he oher cases he obaned copula C s Archmedean and f no o oban anoher copula ha s Archmedean (n he dscree case we now ha he copula s no unque). Λ REFERENCES [] Dall' Aglo G. "Fréche classes: he begnnng" n: G. Dall' Aglo S. Koz and G. Salne (eds.) Advances n Probabl Dsrbuons wh Gven Margnals. Beond he Copulas Kluwer Academc Publshers [] Buzen J. "Compuaonal algorhms for closed queueng newors wh eponenal servers" Communcaons of he ACM 6 (9) (973) [3] Cuu D. "Smulaon epermens on he servce ssems usng arrvals and servces dependng hrough copulas" The Annals of Buchares Unvers (005) [4] Garza M. Garza R. Kemele M. and Lochar C. Newor Modelng Smulaon and Analss Marcel Decer New Yor Bassel 990. [5] Gordon W.J. and Newell G.F. "Closed queueng ssems wh eponenal servers" Operaons Research 5 () (967)
12 D. Cuu / Gordon and Newell Queueng Newors and Copulas [6] Iosfescu M. Fne Marov Chans and Applcaons Techncal Edure Buchares/ Romanan 977. [7] Klenroc L. Queueng Ssems John Wle and Sons Inc. New Yor Torono London Sdne 975. [8] Nelsen R. "Copulas and assocaon" n: G. Dall' Aglo S. Koz and G. Salne (eds.) Advances n Probabl Dsrbuons wh Gven Margnals. Beond he Copulas Kluwer Academc Publshers [9] Schwezer B. "Thr ears of copulas" n: Advances n Probabl Dsrbuons wh Gven Margnals. Beond he Copulas (eds.) G. Dall' Aglo S. Koz and G. Salne Kluwer Academc Publshers [0] Sungur E. and Tuncer Y. "The use of copulas o generae new mulvarae dsrbuons. The froners of sascal compuaon smulaon & modelng" Proceedngs of he ICOSCO-I Conference (The Frs Inernaonal Conference on Sascal Compung Çeşme Izmr Ture) I (987) 97-. [] Văduva I. Fas Algorhms for Compuer Generaon of Random Vecors used n Relabl and Applcaons Preprn no. 603 Jan. 994 TH-Darmsad. [] Văduva I. "Smulaon of some Mulvarae Dsrbuons" The Annals of he Buchares Unvers (003) 7-40.
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