Joural of Appled Mathematcs ad Computatoal Mechacs 16, 15(), 77-84 www.amcm.pcz.pl p-issn 99-9965 DOI: 1.1751/jamcm.16..1 e-issn 353-588 DIFFUSION APPROXIMATION OF THE NETWORK WITH LIMITED NUMBER OF SAME TYPE CUSTOMERS AND TIME DEPENDENT SERVICE PARAMETERS Mkhal Matalytsk 1, Dmtry Kopats 1 Isttute of Mathematcs, Czestochowa Uversty of Techology Częstochowa, Polad Faculty of Mathematcs ad Computer Scece, Grodo State Uversty Grodo, Belarus m.matalytsk@gmal.com, dk8395@mal.ru Abstract. The artcle presets research of a ope queueg etwork (QN) wth the same types of customers, whch the total umber of customers s lmted. Servce parameters are depedet o tme, ad the route of customers s determed by a arbtrary stochastc trasto probablty matrx, whch s also depedet o tme. Servce tmes of customers each le of the system s expoetally dstrbuted. Customers are selected o the servce accordg to FIFO dscple. It s assumed that the umber of customers oe of the systems s determed by the process of brth ad death. It geerates ad destroys customers wth certa servce tmes of rates. The etwork state s descrbed by the radom vector, whch s a Markov radom process. The purpose of the research s a asymptotc aalyss of ts process wth a bg umber of customers, obtag a system of dfferetal equatos (DE) to fd the mea relatve umber of customers the etwork systems at ay tme. A specfc model example was calculated usg the computer. The results ca be used for modellg processes of customer servce the surace compaes, baks, logstcs compaes ad other orgazatos. Keywords: queueg etwork, brth ad death process, asymptotc aalyss 1. Itroducto Exact results for fdg state probabltes of Markov chas the ostatoary regme (trastoal regme) was obtaed oly certa specal cases [1, ] because of the large dmeso of systems of dfferece-dfferetal equatos, whch they satsfy. The dffuso approxmato method for fdg them wth a large umber of customers has bee vestgated [3-5]. Its essece s to approxmate a dscrete stochastc process that descrbes the umber of customers etwork systems, a cotuous dffuso process. I ths paper, ths method s used to aalyze a ope Markov etwork wth a umber of features that were ot prevously cosdered other works.
78 M. Matalytsk, D. Kopats Queueg etworks (QN) are used the mathematcal modellg of varous ecoomc ad techcal systems related to the servcg of clet requests, the umber of whch s actually lmted. Ofte, however, the total umber of customers studed systems chages over tme. It predetermes to use at ther smulato a ope QN wth a lmted umber of customers servced them. Cosder a ope QN, cossts of 1 queueg systems (QS) S, S 1,..., S. Supposg that servced parameters of ts etwork deped o tme t, let the umber of servce les the system S tme t descrbe a fucto of tme m (t), that takes teger values, =,. Servce probablty of customer each servce le of the system S o the tme terval [t t] equal to µ (t) t o( t), =,. Customers are selected o the servce accordg to FIFO dscple. Customer ad servce of whch the QS S was completed, wth probablty p j (t) move the queue of QS S j,, j =,. Trasto matrx, P(t)= p j (t) s the matrx of trasto probabltes of a rreducble Markov cha ad depeds o tme p j (t) 1. I addto, we assume that the umber of customers the system S, except for fuctos µ (t), p (t), m (t) determed by the brth ad death process, whch geerates ew customers wth the testy () λ t. λ t ad destroys the exstg wth testy Hece, the object uder study s a ope QN, the total umber of customers whch s lmted ad vares accordace wth the process of brth ad death, occurrg the system S. The etwork state s determed by the vector where k ( 1 ) k t = k t,k t,...,k t, (1) t - cout of customers the system S tme t, =,. Vector (1) vew of the above, s a Markov radom process wth cotuous tme ad a fte umber of states. Obvously, the total umber of customers servced the etwork at tme t equals K( t ) = k ( t). We carry out the asymptotc aalyss of Markov = process (1) wth a bg umber of customers usg the techque proposed [6, 7]. Note that the aalytcal results, whe the parameters of servce customers ad trasto probabltes of customers ot depedet o the tme, have bee obtaed [8]. Supposg that QN operate uder a hgh load regme of customers,.e. value K t. It s suffcetly bg, but ot lmted: K( t) K. Ths artcle derves a partal dfferetal equato of the secod order, ad s the equato of the Kolmogorov-Fokker-Plack equato for the probablty desty of the vestgated process. A system of o-homogeeous ordary dfferetal equatos of the frst order for the average values of the compoets of the vector of the etwork state was obtaed. The soluto of ths system allows us to fd the average relatve umber of customers each QS terested tme.
Dffuso approxmato of the etwork wth lmted umber of same type customers 79. The dervato of system of dfferetal equatos for the average relatve umber of customers the etwork system We troduce I ( 1) -vector, all compoets are equal to zero except -th, whch equals 1, =,. Cosder all possble trastos to the state = ( k,t t) of process ki I,t k t durg tme t: from the state ( j ) µ t m( m t,k t 1) pj( t) t ο( t) k t t = ca get to ( k,t t ) wth probablty from the state ( k I,t) we ca get to λ ( t) k ( t ) tο t ; from the state ( I,t) we ca get to 1 k,, j =, ; k,t t wth probablty k,t t wth probablty λ ( t) K k( t ) 1 t ο( t) ; from the state ( k,t ) ( t t) = k, wth probablty 1 µ ( t) m m ( t ),k( t ) λ t k t λ t K k t tο t ; = = from other states ( k,t t ) wth probablty ο( t ). From the formula of total probablty, we obta a system of dfferece equatos for the state probabltes P( k,t ): ( j ) m( 1) j P k,t t = P k I I,t µ t m t,k t p t, j= P k I,t λ t k t t 1 P( k I,t) λ ( t) K k( t ) 1 t P( k,t) = 1 µ ( t) ( m m( t ),k( t )) λ ( t) k( t ) λ ( t) K k( t) t ο( t). = = Therefore, the system of dfferece-dfferetal Kolmogorov equatos for these probabltes s: dp k,t dt,j= m ( ) = µ t m t,k t p t P k I I,t P k,t j j ( ( ) ) µ t m m t,k t 1 µ t m (m t,k t p t P( k I I,t), j= j j λ t k t P k I,t P k,t λ t P k I,t λ ( t) K k( t) ( P( k I,t) P( k,t )) λ ( t) P( k I,t). () =
8 M. Matalytsk, D. Kopats The soluto of system () a aalytc form s a dffcult task. We shall therefore cosder the asymptotc case of a bg umber of customers o the etwork, that s, we assume that K 1. To fd the probablty dstrbuto of the radom k t, we move o to the relatve varables ad cosder the vector ξ ( t). vector Possble values of ths vector at a fxed t belog to a bouded closed set G = x = ( x, x 1,...,x ) : x, =,, x 1, (3) = whch they are located odes ( 1) - dmesoal lattce at a dstace 1 ε = K from each other. By creasg K fllg desty of set G possble vector compoets ξ( t ) creases, ad t becomes possble to cosder that t has a cotuous dstrbuto wth probablty desty fucto p( x,t ), whch satsfes the asymptotc relato K 1 P( k,t) p( x,t). We use the followg approxmato fucto k 1 1 P k,t : K P( k,t ) = K P( xk,t ) = p( x,t ), x G. Rewrtg the system of equatos () for the desty p( x,t ), we obta p x,t t, j= m ( ) = K µ t p t l t,x p xe e,t p x,t j j ( l( t ),x) m µ t p t p x e e,t ( ) j j, j= x ( ) Kλ t x p xe,t p x,t λ t p xe,t Kλ ( t) 1 x ( p( x e,t) p( x,t )) λ ( t) p( x e,t), = 1 where e= I, =,. If p( x,t ) twce cotuously dfferetable x, the K vald the followg expaso ε p( x,t) p x,t p( x ± e,t) = p( x,t ) ± ε o ε x x p( x,t) p x,t p( xe e j,t) = p( x,t ) ε x x j ε p( x,t) p( x, t) p( x, t) o ( ε ) x x xj x j,,, j =,.
Dffuso approxmato of the etwork wth lmted umber of same type customers 81 By usg them, ad that εk = 1, we obta the followg represetato: p x,t p x,t p x,t = µ ( t) m( l ( t),x) p j( t) t, j= x x j ε p( x,t) p( x,t) p( x,t) x x x j x j m( l( t ), x) p( x,t) p( x,t) µ ( t) pj( t) p( x,t) ε, j= x x x j ε p( x,t) p( x,t) p( x,t) x x x j x j ε ε p x,t p x,t p x,t p x,t λ ( t) x λ ( t) p( x,t ) ε x x x x p( x,t) ε p( x,t) λ ( t) 1 x = x x We troduce otatos: ε p x,t p x,t λ ( t) p(x, t) ε O ε x x A x, t = µ l,x p λ x λ x j= = j( t) m j( t) j j t t t 1, (4) A x,t = µ t m l t,x p t, = 1,, (5) j j j j j= j= = j( t) m j( t) j j 1, (6) B x,t = µ l, x p t λ t x λ t x B x,t = µ t m l t,x p t, = 1,, (7) j j j j j= j 1 p t, j, pj t = p j t, = j, ( t) m( ) pj( t ), j, pj( t) = j() p ( t ) 1, = j, B x,t = µ l t, x p t, j, (8) j j j. mj() t l t =, j =,. (9) K Cosderg them, t turs out that the case of asymptotc for a suffcetly bg K dstrbuto desty p(x,t) of vector relatve varables
8 M. Matalytsk, D. Kopats k t ξ( t ) = ( ξ( t ),ξ1( t ),...,ξ( t )) = = K a O( ε ), where k t k1 t k t,,..., K K K 1 ε = the Kolmogorov-Fokker-Plack equato: K satsfes up to p x, t ε = A x, t p x, t B x, t p x, t t x x x ( ) j, (1) at pots of exstece of dervatves. =, j= j The, accordg to [9], expectatos ( ) t = M ξ t, =,, accurate to terms order of magtude Ο( ε ) determed from the system of DE d ( t) dt ( ) = A t, =,. (11) From (3), (4) t s obvous that the rght-had sde of (11) are cotuous pecewse lear fuctos. Such systems are approprately addressed by dvdg the phase space ad fd solutos the areas of the learty of the rght Ω t =,1,,..., the compoets of the dex set () t. We dvde parts. Let { } Ω( t ) to two dsjot sets Ω( t ) ad Ω1( t ) : { 1} Ω t j ( t) l ( t) { } Ω t = : l t < t, 1 = : j j. For fxed t the umber of parttos 1 of the type equals. Each partto wll be defed the set G( t ) = ( t ) : ( t ), ( t) 1 of dsjot regos G τ () t such that, = Gτ( t ) = ( t ) : l( t ) < ( t ) 1, Ω( t ); j( t) l j( t ), j Ω1( t ); c( t) 1 c= 1,,..., 1 G t = G t. τ =, τ You ca wrte the system of equatos (11) explctly for each of the areas of A: Ω t =,Ω t = 1,,...,, phase space subdvso. Cosder the feld { } { } 1 whch accordg to o queues systems S1, S,..., S average ad the presece of queues the system S. The system of dfferetal equatos (11) ths feld s of the form: ' ( t ) = λ ( t) ( t ) λ ( t) 1 ( t ) µ j( t) j( t) pjo( t ) µ ( t) l( t) p( t ), = (1) ' ( t ) = µ ( t) l( t) pj( t ) µ j( t) j( t) pj( t ), = 1,.
Dffuso approxmato of the etwork wth lmted umber of same type customers 83 The system (1) s a system of ordary homogeeous DE. Its soluto of a system for a bg s aalytcally dffcult, so the evet of a etwork of a bg dmeso, t s approprate to use umercal methods. 3. Example I the computer system Mathematca, a mathematcal programmg procedure has bee developed that mplemets calculato examples. It shows oe example of the calculato of the average relatve umber of customers the system etwork, whch s a mathematcal model of the processg of customer requests for a surace compay. Cosder the QN, cosstg of 6 QS S, S 1, S, S 3, S 4, S 5, where K = 1. Defe the followg trasto probabltes betwee QS: p 5 (t) =.cos (3t); p 4 (t) =.s (3t); p 3 (t) =.4cos t; p (t) =.4s t; p 1 (t) =.s (t); p (t) = =.cos 5s(5t) 1 (t); p 1 (t) = 1; p j = other cases. l () t=. N () t= 17; 1 N () t 13 ; N () t 5 ; N () t 3 ; N () t 1 ; N () t 1. 1 = = 3 = 4 = = Let's preted that () =, = 1,5, ad cosder perod where there are o queues systems S 1, S, S 3, S 4, S 5 average. The (1) takes the form 5 5 ' ( t) = µ j( t) j( t) p j p( t) l( t) µ ( t) λ ( t) ( t) λ ( t) 1 ( t ), = ' 1( t ) = µ j( t) j( t) p j( t ) µ ( t) l( t) p j( t) µ 1( t) 1( t ), ' ( t ) = µ j( t) j( t) p j( t ) µ ( t) l( t) p j( t ) µ ( t) ( t ), (13) ' 3( t ) = µ j( t) j( t) p j( t ) µ ( t) l( t) p j( t ) µ 3( t) 3( t ), ' 4( t ) = µ j( t) j( t) p j( t ) µ ( t) l( t) p j( t ) µ 4( t) 4( t ), ' 5( t ) = µ j( t) j( t) p j( t ) µ ( t) l( t) p j( t ) µ 5( t) 5( t ). Let µ (t) = t 1.3 t ; µ 1 (t) = t 1 1. t ; µ (t) = t 3.5 t ; µ 3 (t) =.1t t ; µ 4 (t) = =.5t 1.5 t ; µ 5 (t) = t 3 t, () t. t, [ 5s ( 5t) 1] λ = 7 λ() t = t, l (t)=, where 1 [.] - teger part, paretheses. Solvg the system (13) the package Mathematca, we obta
84 M. Matalytsk, D. Kopats N (t) = (1.3 t 1. t )(t 5t) 17; N 1 (t) = (.5 t 1. -t )( t 5t) 13; N (t) = (.5 t.5 t )( t 3 7t 8t) 5; N 3 (t) = (.7 t 1.3 t )(t.7t) 3; N 4 (t) = (.9 t 3 t )(t.7t) 1; N 5 (t) = (.9 t 3 t.5 t )(t 1.1t) 1. 4. Coclusos I ths paper, Markov QN wth a lmted umber of the same type customers was vestgated. The umber of customers of systems vares accordace wth the process of brth ad death. For obtag a system of DE for a average umber of customers ts systems, the method of dffuso approxmato was appled, allowg oe to fd them wth hgh accuracy for a bg umber of customers. The results may be useful modellg ad optmzato of customer servce the surace compaes, baks, logstcs compaes ad other orgazatos [1-1]. Refereces [1] Kelly F.P., Wllams R.J., Stochastc Networks. The IMA Volumes Mathematcs ad ts Applcatos, Sprger-Verlag, New York 1995. [] Nykowska M., Model tademowego systemu obsług, Przegląd Statystyczy 1984, 9(3), 531- -54. [3] Kobayash H., Applcato of the dffuso approxmato to queueg etworks, Joural of ACM 1974, 1(-3), 316-38, 456-469. [4] Gelebe E., Probablstc models of computer systems. Dffuso approxmato watg tmes ad batch arrvals, Acta Iformatca 1979, 1, 85-33. [5] Lebedev E.A., Checheltsky A.A., Dffuso approxmato of queueg et wth a sem- Markov put rate, Cyberetcs 1991,, 1-13. [6] Medvedev G.A., O the optmzato of the closed queug system. Proceedgs of the Academy of Sceces of the USSR, Techcal Cyberetcs 1975, 6, 65-73. [7] Medvedev G.A., Closed queug systems ad ther optmzato. Proceedgs of the Academy of Sceces of the USSR, Techcal Cyberetcs 1978, 6, 199-3. [8] Ruslko T.V., The asymptotc aalyss of closed queug etworks wth tme-depedet parameters ad prorty applcato, Vestk of GrSU. Seres, Mathematcs. Physcs. Computer Scece, Computer Facltes ad Maagemet 15,, 117-13. [9] Paraev Y.I., Itroducto to Statstcal Process Cotrol Dyamcs ad Flterg, Sovet Rado, 1976. [1] Matalytsk M.A., Ruslko T.V., Mathematcal Aalyss of Stochastc Models of Clams Processg Isurace Compaes, GRSU, Grodo 7. [11] Ruslko T.V., Matalytsk M.A. Queug Network Models of Clams Processg Isurace Compaes, LAP LAMBERT Academc Publshg, Saarbrucke 1. [1] Matalytsk M.A., Kturko O.M., Mathematcal Aalyss of HM-etworks ad ts Applcato Trasport Logstcs, GrSU, Grodo 15.