Seion 1. Statitic Method and Thei Alication Poceeding of the 11 th Intenational Confeence eliability and Statitic in Tanotation and Communication (elstat 11), 19 22 Octobe 2011, iga, Latvia,. 82-87. ISBN 978-9984-818-46-7 Tanot and Telecommunication Intitute, Lomonoova 1, LV-1019, iga, Latvia NETWOK TAFFIC PIOITIZATION USING MAP OF AIVALS Jelena evzina Tanot and Telecommunication Intitute Lomonoova t., 1, iga, LV-1019, Latvia Ph.: (371) 67100584. E-mail: lena_evzina@ti.lv ecent advance in telecommunication and the olifeation of netwoking technologie dictate the need fo thei deee and moe accuate efomance evaluation. Pactically, both emeging and exiting telecommunication evice, including voice, video and data, demand the mechanim fo netwok taffic ioitie allocation. In thi tudy we conide Makov-Benoulli ecoding of MAP of aival, fo which the econday oce tun out to be a MAP of aival. Seconday ecoding of an aival oce i a method that fom an oiginal aival oce geneate a econday new aival oce. In thi ae we eent the method uing Makov-Benoulli ecoding of MAP of Aival which can be ued fo taffic ioitization modeling. Key wod: Makov additive oce (MAP) of aival, Makov-Benoulli ecoding, netwok taffic, ioitization 1. Alication of MAP and thei Deivative MAP ha been widely ued fo modeling of many tye of queueing ytem. Thu, diffeent tye of MAP-baed queueing model have been develoed, including MAP/M/1 queue, MAP/G/1 queue and MAP/PH/1 queue. [1] One majo advantage of BMAP [2] ove MAP i that batche aociated with BMAP add to the modeling owe and flexibility of MAP. Thi fact ha been exloited by Klemm (2003) to model IP taffic. BMAP-baed queueing ytem have been extenively analyzed by many autho with eect to ATM taffic. Lucantoni (1993) ovide a uvey of the analyi of the BMAP/G/1 queue. A it i ointed out in Mauyama (2003), queue with batch Makovian aival ae o flexible that they can eeent mot of the queue tudied in the at a ecial cae. Many autho, including Heffe and Lucantoni (1986), Baiocchi (1991), Yamada and Sumita (1991), and Li and Hwang (1993), have ued MMPP(2) to model the ueoed ATM taffic. Thei analyi deal with MMPP/M/1 o MMPP/G/1 queueing ytem. Fiche AND Meie-Hellten (1992) dicu othe alication of MMPP. Zhou and Gan (1999) conide an M/MMPP/1 queue, which can be ued to model a ytem that ocee job fom diffeent ouce. The time to oce job fom each ouce (o job tye) i exonentially ditibuted, but each job tye ha a diffeent mean evice time. Moeove, afte a job comletion, the choice of next job to be oceed i govened by a Makov chain. Mucaiello (2005) ue a hieachical MMPP taffic model that vey cloely aoximate the long-ange deendence (LD) chaacteitic of the Intenet taffic tace ove elevant time cale. The LD oety of the Intenet taffic mean that value at any intant tend to be oitively coelated with value at all futue intant. Thi mean that it ha ome ot of memoy. Howeve, long-tem coelation oetie, heavy tail ditibution, and othe chaacteitic ae meaningful only ove a limited time cale. The oveflow taffic ha been modeled uing an inteuted Poion oce (IPP) by Kuczua (1973) and Meie-Hellten (1989). Min (2001) analyzed the adative wom-hole-outed tou netwok with IPP taffic inut. [3] A uvey of thee taffic model i given in Bae and Suda (1991), Fot and Melamed (1994), Michiel and Laeven (1997) and Ada (1997). 2. MAP of Aival We define the MAP (Makov additive oce) of aival to be oely adated to the multivaiate etting. We ue the definition and notation of Pacheco, Tang and Pabhu. [4] Fitly, we denote = (, ), a oitive intege, and E a countable et. 82
The 11 th Intenational Confeence ELIABILITY and STATISTICS in TANSPOTATION and COMMUNICATION -2011 A oce ( X, J ) = {( X ( t), J ( t)), t 0} on the tate ace E i an MAP if it atifie the following two condition: (a) (X,J) i a Makov oce; (b) Fo, t 0, the conditional ditibution of (X(t) X() J(t)) given (X(),J()) deend only on J(). In thi cae, a MAP of aival i imly a MAP with the additive comonent taking value on the nonnegative (may be multidimenional) intege, i.e., N with N = {0,1,2,...}. Fo claity we conide time-homogeneou MAP, fo which the conditional ditibution of (X(t) X(),J(t)) given J() deend only on t. A MAP of aival (X,J) i imly a MAP taking value on N E and, a a conequence, the incement of X may then be aociated with aival event, with the tandad examle being that of diffeent clae of aival into a queueing ytem. Accodingly, without lo of geneality, we ue the tem aival to denote the event tudied and make the inteetation X i (t) = total numbe of cla i aival in (0,t] (1) fo i=1,2,,. We than call X, the additive comonent in the teminology fo MAP, the aival comonent of (X,J). It follow fom the definition of MAP that J i a Makov chain and, following the teminology fo MAP, we call it the Makov comonent of (X,J). Since a MAP of aival (X,J) i a Makov ubodinato on N E, it i a Makov chain. Moeove, ince (X,J) i time-homogeneou it i chaacteized by it tanition ate, which ae invaiant tanlation in the aival comonent X. Thu, it i ufficient to give, fo j, k E and m, n N, the tanition ate fom (m,j) to (mn, k), which we denote imly (ince the ate doe not deend on m) by λ ( n ). Wheneve the Makov comonent J i in the tate j, the following thee tye of tanition in (X,J) may occu with the eective ate: (a) Aival without change of tate in J occu at ate λ jj ( n ), n > 0 ; (b) Change of tate in J without aival occu at ate λ ( 0 ),k E,k j ; (c) Aival with change of tate in J occu at ate λ ( n ),k E,k j, n > 0. We denote Λ (λ ( )), n = n n N, and ay that (X,J) i a imle MAP of aival if Λ (n,...n ) = 0 if n > 1 1 l fo ome l. In cae X (=X) ha a tate ace N we ay that (X,J) i a univaiate MAP of aival, an examle of which i the Makov-Poion oce, which i known a MMPP (Makov-modulated Poion oce). The identification of meaning of each of the above tanition i vey imotant fo alication. In aticula, aival ae clealy identified by tanition of tye (a) and (c), o that we may talk about aival eoch, inteaival time, and define the comlex oeation like thinning of MAP of aival. Moeove, the aameteization of thee ocee i imle and, by being Makov chain, they have nice tuctual oetie. Thee ae eecially imotant comutationally ince it i a imle tak to imulate Makov chain. 3. Makov-Benoulli ecoding of MAP of Aival Seconday ecoding of an aival oce i a mechanim that geneate a econday aival oce fom the oiginal aival oce. A claical examle of econday ecoding i the Benoulli thinning which ecod each aival in the oiginal oce, indeendently of all othe, with a given obability o emove it with obability 1. Fo MAP of aival (X,J) we conide econday ecoding that leave J unaffected. A imle examle fo which the obability of an aival being ecoded vaie with time i the cae when thee i a ecoding tation (o contol oce) which i on and off fom time to time, o that aival in the oiginal oce ae ecoded in the econday oce duing eiod in which the tation i 83
Seion 1. Statitic Method and Thei Alication oeational (on). The contol oce may be intenal o extenal to the oiginal oce and may be of vaiable comlexity (e.g. ule fo acce of cutome aival into a queueing netwok may be imle o comlicated and may deend on the tate of the netwok at aival eoch o not). Anothe examle i the one in which the oiginal oce count aival of batche of cutome into a queueing ytem, while the econday oce count the numbe of individual cutome, which i moe imotant than the oiginal aival oce in cae evice i offeed to cutome one at a time (thi i a way in which the comound Poion oce may be obtained fom the imle Poion oce). Hee the econday oce uually count moe aival then the oiginal oce, wheea in the eviou examle the econday oce alway ecod fewe aival than the oiginal oce doe (which coeond to thinning). Seconday ecoding i elated to what i called making the oiginal oce. Suoe that each aival in the oiginal oce i given a mak fom a ace M, indeendently of the mak given to othe aival (e.g. fo Benoulli thinning with obability each oint i maked ecoded with obability o emoved with obability 1 ). If we conide the oce that account only the aival which have mak on a ubet C of M, it may be viewed a a econday ecoding of the oiginal oce. A natual mak aociated with cell moving in a netwok i the ai of oigin and detination node of the cell. If the oiginal oce account fo the taffic geneated in the netwok, then the econday oce may eeent the taffic geneated between a given ai of node, o between two et of node. The mak may eeent ioitie. Thee ae commonly ued in modeling acce egulato to communication netwok ytem. A imle examle of an acce egulato i the one of which cell (aival) ae judged in violation of the contact between the netwok and the ue ae maked and may be doed fom evice unde the ecific ituation of congetion in the netwok; othe cell ae not maked and caied though the netwok. The econday ocee of aival of maked and unmaked cell ae of obviou inteet. 4. Conideed Cae Conide a witch which eceive the inut fom a numbe of ouce and uoe that the oiginal oce account fo inut fom thoe ouce. If we ae inteeted in tudying one of the ouce in aticula, we hould obeve a econday oce which account only the inut fom thi ouce; thi coeond to a magin of the oiginal oce. In cae all ouce geneate the ame time of inut, what i elevant to the tudy of efomance of the ytem it i a total inut aiving to the witch; thi i a econday oce which coeond to the um of the coodinate of the oiginal aival oce. Thi how that ome of the tanfomation of aival oce may alo be viewed a ecial cae of econday ecoding of an aival oce. Suoe that (X,J) i an MAP of aival on N E with ( Q,{ Λ n } n> 0 ) ouce. We conide a ecoding mechanim that indeendently ecod with obability ( n, m) an aival in X of a batch of ize n aociated with a tanition fom j to k in J a an aival of ize m in the econday oce (with m N fo ome 1 ). The oeation i identified by the et of ecoding obabilitie = { (, ) ( (, )) :, n m = n m n N m N }. (2) Whee ( n, ) i a obability function on N, and ( 0, m) = δ m0. We call thi ecoding Makov-Benoulli ecoding with obabilitie and denote the eulting econday oce a ( X, J ). Thu ( X, J ) i a oce on N E which i non-deceaing X, and which i inceaed only when X inceae, i.e. ( 1 X ( t) = X ( T ) T t T ). (3) Moeove, fo n>0 n, = ( m) P( X ( T 1) X ( T ) = m A ( n)} (4) ( = 1 1 1 = k with A n ) { X( T ) X( T ) = n, J ( T ) = j, J ( T ) }. 84
The 11 th Intenational Confeence ELIABILITY and STATISTICS in TANSPOTATION and COMMUNICATION -2011 Theoem. Suoe that(x,j) i a MAP of aival on N E with ( Q,{ Λ n } n> 0 ) ouce and = { (, ),, n m n N m N } i a et of ecoding obabilitie. Then: (a) The oce ( X, X, J ) i a MAP of aival on N E with ouce Q,{ Λ n (n,m) }( n, m) > ) (5) ( 0 (b) The oce ( X, J ) i a Ma of aival on N E with ouce ( Q,{ Λ m ) m> 0 ) = ( Q,{ Λ n (n,m)} m> 0 ). (6) n> 0 The oof of the theoem adduced in Pacheco, Tang and Pabhu. The ecial cae of Makov-Benoulli making with the dicete ace of mak M i conideed too. Secifically, each aival of a batch of ize n aociated with a tanition fom tate j to tate k in J i given a mak m M with (making) obability c ( n, m), indeendently of the mak given to othe aival. If we aume without lo of geneality that M = { 0,1,..., K} N, Makov-Benoulli making become a ecial cae of Makov-Benoulli ecoding fo which the maked oce ( X, X, J ) i uch that ( X, J ) i a Makov-Benoulli ecoding of (X,J) with ecoding obabilitie ( n, m) = ( ( n, m)) = ( 1{ m M } c ( n, m)) fo n>0 and m N. Suoe that X i a Poion oce with ate λ, J i a table finite Makov chain with geneato matix Q, and X and J ae indeendent. We view J a a tate of a ecoding tation, and aume that each aival in X i ecoded, indeendently of all othe aival, with obability j o non-ecoded with obability 1 j, wheneve the tation i in tate j. We let X (t) be the numbe of ecoded aival in (0,t]. Uing theoem fom [4] we may conclude that ( X, J ) i an MMPP with ( Q,( λ j δ )) ouce. In the ecial cae whee the obabilitie j ae eithe 1 o 0, the tation i on and off fom time to time, with the ditibution of the on and off eiod being Makov deendent. 5. Examle We conide now two examle of Makov-Benoulli ecoding. In the fit examle we ee the oveflow oce fom a tate deendent M/M/1/K ytem a a ecial cae of econday ecoding of the aival oce of cutome to the ytem. In the econd examle we ee a moe elaboated cae of econday ecoding. 5.1. Examle 1 We conide a Makov-modulated M/M/1/k ytem with batch aival with (indeendent) ize ditibution{ n} n> 0. When thee i an aival of a batch with n cutome and only m<n oition ae available, only m cutome fom the batch ente the ytem. Aume that the evice ate i μ j and the aival ate of batche iα j, wheneve the numbe of cutome in the ytem i j. We let J(t) be the numbe of cutome aival in (0,t]. The oce (X,J) i a MAP of aival on N { 0,1,..., K} with ate α j λ ( n) = μ j 0 n k = min( j n, K) k = j 1, n = 0 othewie 85
Seion 1. Statitic Method and Thei Alication It ouce ( Q,{ Λ n } n> 0 ), whee Q i obtained fom the matice { Λ n } n 0 though () and (). If we define X (t) a the oveflow fom the ytem in (0,t], then it i eadily een that ( X, J ) i a Makov- Benoulli ecoding of (X,J) with ecoding obabilitie ( n, m) = 1 { m= n ( k j)} fo n,m>0. Thu ( X, J ) i a MAP of aival on N { 0,1,..., K} with ( Q,{ Λ m} m> 0 ) ouce, whee λ m) = α δ fo m>0. ( j kk m ( K j) 5.2. Examle 2 Suoe X = ( X 1,..., X ), Y = ( Y 1,..., Y ), and ( X, Y, J 2 ) i a MAP of aival on N E2 with ouce. We ae inteeted in keeing only the aival in X which ae eceded by an aival in Y without a imultaneou aival in X. The aival counting oce which we obtain by thi oeation i denoted by X. Fo imlicity, we aume that E 2 i finite and Λ ( n, m ) = 0 if n>0 and m>0. We let { T X } > 0 and { T Y } > 0 be the ucceive aival eoch in X and Y, eectively, denote J = J 1, J ) with ( 2 J 1 ( 1 t) = 0 max{ T Y : T Y t} > max{ T othewie X : T X t}. It can be checked in a outine fahion that ((X,Y),J) i a MAP of aival on N { 0,1} E2 * * ( ( n,m) ( n, m) > 0 with Q,{ Λ } ) ouce (with the tate of J, ) odeed in lexicogahic ode), whee Λ * ( n, m) =0 if n>0 and m>0, and ( 1 J 2 Q Λ Λ * m> 0 (0, m) m> 0 (0, m) Λ Λ * ( n,0) (0, m) Q =, Λ ( n, m) = Λ Λ n> 0 ( n,0) Q n> 0 ( n, o) Λ ( n,0) Λ (0, m) X if eithe n=0 o m=0. It i eay to ee that (, J ) i a Makov-Benoulli ecoding of ((X,Y),J) with ecoding obabilitie (, j )( q, k ) ( n, m), l) = 1{ = 1, n= 1}, fo (n,m)>0. Thu (, J ) aival on N { 0,1} E2 with ( Q,{ Λ n} n> 0 ) -ouce, whee Λ n 0 = Δ (n,0) 0 0. X i an MAP of A aticula cae of the eviou examle with =1 and Y being a Poion oce wa conideed biefly by Neut. In addition, He and Neut conideed an intance of the ame examle in which X, Y, J ) = ( X,Y,( K, )) eult fom atching togethe the indeendent MAP X, K ), a geneal ( 2 1 K 2 ( Y, K 2 ( 1 MAP of aival, and ), a imle univaiate MAP of aival. Thi lead to ecoding of the aival in X that ae immediately eceded by an aival on the imle univaiate MAP of aival X. 6. Concluion In thi ae we define a tye of econday ecoding which include imotant ecial cae of making and thinning. We conide in aticula Makov-Benoulli ecoding of MAP of aival, fo which the econday oce tun out to be a MAP of aival. We invetigate two examle of behavio of the queueing ytem. In the fit examle we conide the oveflow oce fom the tate deendent M/M/1/K ytem a a ecial cae of econday ecoding of the aival oce of cutome to the ytem. In the econd examle we ee a moe comlicated cae of econday ecoding. 86
The 11 th Intenational Confeence ELIABILITY and STATISTICS in TANSPOTATION and COMMUNICATION -2011 Thi analyi of behavio of uch ytem hel the telecommunication netwok to be maintained table egadle of congetion uing ioitization mechanim. [5,6] We omit the numeical comutation without infomation fo the olution. The main uoe of thi tudy i to how the uefulne and the flexibility of the ooed method. efeence 1. Olive C. Ibe. Makov. Pocee fo Stochatic Modeling. Academic Pe, 2009, 490. 2. Dudin A., Klimenok V. Ma evice ytem with coelated flow. Mink, BGU, 2000 (in uian). 3. G.Bolch, S.Geine, H.de Mee, K.S.Tivedi. Queueing Netwok and Makov Chain: Modeling and Pefomance Evaluation with Comute Science Alication. A.JohnWiley&Son, 2006, 878. 4. Pacheco A., Tang L.C., Pabhu N.U. Makov-Modulated Pocee & Semiegeneative Phenomena. Wold Scientific, 2009, 224. 5. evzina J. Stochatic Model of Data Flow in the Telecommunication Netwok, Jounal Comute Modelling and New Technologie. No 2, Vol. 14. iga, 2010. 6. Scott S.L., Smyth P. The Makov Modulated Poion Poce and Makov Poion Cacade with Alication to Web Taffic Modeling. Bayeian Statitic 7, Oxfod Univeity Pe, 2003. 87