Mathematical model of service quality performance s computing of multiservice network

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1 Mathematca mode of erce quaty performance computng of muterce networ Tuemoa Gufarda, Internatona academy of bune, maty, Kazahtan btract Th artce deoted to effectene of muterce networ functonng n order to prode the quaty of thee networ. Probem of the probabty-tme charactertc (PTC) of the networ cacuaton repreent the degner of mot nteret. Souton of th probem ge you opportunty to: gnfcanty mproe networ performance, preent networ faure durng oeroad traffc nformaton, to determne the optma drecton of bypa traffc; cacuate the optmum quaty of erce parameter of ntegrated networ. Deeopment of technooge for networ management coey ned to the mathematca modeng of procee and networ management eement. For ISDN muaton to determne t threhod or actuay parameter of the networ, wth whch t become neceary to achee a partcuar effect on the contro networ eement. The artce preent the mathematca mode cacuaton PTC, a new method for the formaton of noda oad ubnet wtchng channe, a ytem of equaton wth repect to the probabty of noda oad oe communcaton channe, obtaned a new method for cacuatng ca bocng probabte between each par of node, we hae proed the dependence of thee probabte of bocng probabte oe for a outgong drecton from the nta node, obtaned by the method of cacuatng the fow channe. Keyword: mut-erce networ (MSN), mathematca mode of probabtc-tme charactertc, method of hybrd and adapte wtchng, adapte routng, detour traffc, nter-zone connecton graph.. Introducton Today, the popuarty of the Internet o at that the technca capabte of the Word Networ do not hae tme to prode fu growng wth each pang day the fow of nformaton. Therefore, the ta of the quatate approach to networng reduced to the ta of eectng the topoogca tructure of the networ, determnng the optma number of node and n of the networ to the cacuaton of t capacty, fndng the hortet-path oad tranfer cacuaton probabty-tme charactertc of the networ, etc. In the proce of anay and ynthe of networ, the probem of cacuatng t probabty-tme charactertc for degner of greatet nteret a a outon to th probem, you can: gnfcanty mproe networ performance, preent networ faure durng oeroad traffc nformaton, to determne the optma drecton of bypa traffc; cacuate the optma parameter ntegrated networ quaty of erce. In muterce networ deery of deo and oce mut be carred out n rea tme wth the need to prortze n the cae of tranport networ congeton. Howeer, the networ ndutry neer focued on rea-tme networ, data deered n accordance wth the capabte of the networ n a pecfc perod of tme []. n mportant functon of the ISDN the etabhment of bandwdth on demand. Radca extenon of bandwdth oca area networ probem can be oed wth QoS, but from an economc pont of ew n the regona networ not feabe. Therefore, the regona communcaton networ degned to meet the optmzaton of reource uage for a partcuar type of traffc. When degnng and budng a networ wdey ued technoogy, whch reguate the dtrbuton of cot of arou erce, proded by the networ to tranmt nformaton. In uch networ n the frt pace there the concept of quaty of erce networ[].. Quatate probem of muterce networ. Quaty of Serce technoogy prode the patform neceary for adanced appcaton, hang much more trngent requrement for the wdth of the proded bandwdth and deay the paage of nformaton n the networ. Newer technooge prodng quaty erce depended prmary on arou mpementaton queung agorthm, whch ncude agorthm "frt n - frt out" prorty queue, far queue, etc. Thee agorthm are ued by router and other networ dece. But t wa mpobe to contro the contnuou fow of traffc. The technoogy abe to prode the quaty of erce traffc dtrbuton by category to aow the paage of hgherprorty traffc on the networ wth oftware ettng and regarde of competton from other traffc. Determnng when the ue of technoogy the quaty of erce to prode protecton for the mot prorty traffc from arou "attac" by the ower prorty traffc, and not ut "to ue the meda on a networ". 3. The probem of fndng probabtytme charactertc of erce quaty ISDN Th artce dcue fndng the foowng tattca parameter:

2 ) Capacty of branche (bandwdth) ubnet and manua CC compoed ISDN; ) The magntude and nature of the oad comng on each hybrd wtchng node n CC and manua mode and the tota oad determned by each group path ntegra ISDN; 3) The probabty of oe n the branche and between each par of node ubnet pacecraft, a we a the dtrbuton of thee oe on trant route and node n the networ, the aue of the aerage deay n the tranmon of meage compoed ubnet CP ISDN. of the aboe tattca parameter durng operaton ISDN, uuay not contant, and changng them often mpobe to predct. So oad change and graty between wtchng node caed the commonng of new unt, a we a other factor. In th regard, the modern communcaton ytem a ot of attenton pad to the choce of uch an agorthm n cutomer erce ISDN, whch woud tae nto account the emergng change n the tuaton on the networ and enure coneraton n a changng et pont generazed crteron of quaty of erce. 4. Method of aement erce quaty n ubnet CS Th ecton preent the mathematca mode for cacuatng the parameter of quaty of erce muterce networ[4]. It ue traffc type muterce networ ung the concept of oad tranfer detour. In determnng the parameter of quaty of erce on the ubnet QC made the foowng aumpton: ) Inta ca fow are Poon; ) Poon character tored a tream for redundant and mng for oad; 3) The ytem n a tate of tattca equbrum; 4) Sytem wth obou oe; 5) Doe not tae nto account oe n the wtchng and contro dece; 6) Setup tme zero. Baene data to determne the parameter of quaty of erce are: ) Structure (ocaton and capacty of the branche); ) nput oad for erce n BH between the node of each par of node; 3) a pan for fow on the networ. 4. Probabty of oe on branche In ummary, you can get a number of QoS parameter, whch were obtaned: ) The magntude of the tota oad on each branch; ) The probabty of oe n the branche; 3) The probabty of oe between the node of each par of node (the rato of ot oad to receed); 4) The probabty of oe n the mdde on the networ (the rato of oad ot on the networ, receed a erce); 5) The aue of the oad ered and ot n each trant node on the networ a a whoe. The man parameter here the probabty of oe n the branche, and other parameter can be eay cacuated through thee aue. Cacuaton of the o probabte on the branche n networ wth crcutou drecton compcated by the fact that the probabty of oe n each branch n genera depend on the o probabte for a other branche. Score predcted bandwdth and quaty of erce a ery mportant tage of degnng muterce networ. Choong an approprate traffc mode defned eecton crtera that ue the foowng parameter: Mode Incomng ca ca bocng, Number of ource Hod Tme. Thee deeopment n the Erang mode ued for trun group wthout repeated ca nce caer are redrected or expected ow bocng factor. Cacuaton formua n Erang traffc mode B( c, a) c c a c! a 0! Let reform t for more comfortabe uage: P( ; )! 0! Th recurrence reaton formua Erang o probabty cacuaton and preentaton of th formua n ntegra form. Ge proof of Lemma, theorem, coroare. nd the formua () tranformed nto formua (3), (4) and (5). P(, ) (3) P(, 0 d ( d ) ) P(, ) () () (4) e (, (5) y P ) Thee equaent formua much eaer to cacuate the probabte of bocng ubcrber meage[3]. e y dy 4. Search for mng and exce burden The magntude of mng or exce burden depend on the probabty of o t ( ) of traffc dtrbuted to

3 the branch (). Dtrbuton of the nput oad for a ubequent branche of the tree baed on the probabty of erce, cacuated on a the preou branche of the tree path. In turn, med branch oad ) (mn ( ) mutaneouy nput to the oad node. Fndng trant oad at each node of the tree performed ung the foowng formua: L u t u (, ) = t u (, ) = = t u (, ) = r u ( ) (6) Where r u ( ) - nput oad comng to the te and degned a node - the node number of outbound drecton. Note that n th cae, the nput oad and topoer are not fundamentay dfferent from each other. In other word, the trant oad nput to the proce to excude the cycc tranfer path. Input burden r u ( ) for any node dtrbuted oer a outgong u drecton ( ) L ( ) from t. boe a, nce any node n the tree path, except the frt, ncuded ony one branch, then th node, regarde of the choce of outgong drecton there from, the aue of trant oad are the ame. In accordance wth the crteron of optmaty for each branch, a member of one or the other route determned by ome of t weght (cot). The route wth the mnmum or the maxmum weght whch a near um of the weght of the branche optma accordng to th crteron, or hortet route. If there are no aaabe machne or free any one branch of the route - the path condered oced. If a route et boced, the oad t ( ) a dena of erce. When ntroduced aumpton ubnet crcut wtchng (CS) a Maro ytem wth a fnte - dmenona phae pace E, whch tate change occur at dcrete pont n tme correpondng to the tme of recept meage to the ubnet CS. We aume that the networ management U = {Ud, d = 0,,,...} maro type, then operaton of the networ can be repreented a a controed maro type proce. 4. Woraround drecton of oad tranfer ubnet CS For more effcent ue of tme channe tract modern automatc wtchng ytem ocated on the node, n addton to aow the man way of communcaton (the way the frt choce) to ue woraround (path next eecton). For carty, the aboe-decrbed ue of bypa area of oad tranfer CS ubnet networ utrated by the fragment contng of the -th wtchng node and out of t to neghborng node,, 3 drecton oad tranfer t (Fgure ). Fgure - Ung detour tranmon noda oad Orderng of the eement of the et K ( ) made n accordance wth the choce of node outgong drecton prorty route n the matrx M. Order gen by a matrx of oad dtrbuton route: M ( ( ( ( 3 4 ) ) ) ) 3 4 In accordance wth the eement of th matrx prmary ued branch path of frt choce (3). When t formed oeroad exce fow, whch ered by a equence of branche (4), () and (). Then the et ( ) = {3, 4,, }[5]. K Fgure - Count four ntermedate networ node. Probabty of empoyment erce area (, ) cacuated by the formua ( ) ( ) p ( ) p ( ) (7) K ( ) Mutpcaton the exce burden on the branch (). ( ) * p ( ) the hare of

4 Probabty of oe П equa to ( ) wth oad t ( ) n node п ( ) p ( ), п ( ) (0;). (8) When cacuatng the tota oad (nput at each branch, med each branch and redundant for th branch ) we retrct ouree to ony fndng mng branch tota oad. In turn, through the aue p ( ) can eay determne the tota oad how each ncomng branch and excee for t. To eect a method for aeng the quaty of erce on the ubnet pacecraft ue hghy compex adapte ytem, networ management, nformaton fow and procee ca erce, ubcrber to hare the nformaton. Management ytem, fndng quaty deteroraton charactertc of arou channe, rearrange the order of channe o that thee channe w be engaged n the at turn. Lewe wa but adapte contro ytem, whch aow reducng any ehood of o of ca, the deay tme to connect. Quaty of erce requet for the tranfer of nformaton communcaton ytem depend on a number of parameter, the man of whch uuay ncude: the amount of nformaton to be tranmtted, the bandwdth networ tructura reabty and networ urabty, etc. In fact, a between each par of node and are excuded cycc route, any node for a gen addre the uptream node reate to. Th mean that eery path from node to, pang through the node, dffer from a the path between two node,, branch ony (). In that cae oe P ( ) w be proportonay decreaed on amount of branch oe (), t mean P ( ) >P ( ). Bandwdth t ( ) * P ( ) repreent the exce oad, and the aue g -med oad throughout the path between node and. To decrbe the decentrazed agorthm for cacuatng the man charactertc of data networ ung crcutou drecton formazed method of optma aocaton of channe reource muterce networ. 4.3 Zona prncpe herarchca addreng and routng Herarchca routng agorthm nformaton fow baed on the appcaton of the prncpe of zona herarchca addreng and routng. When ung a herarchca addreng zona addre of any node may be repreented a a ector, I (,...,, ) (9) Where I - the number of ee of the herarchca addre, - the node addre n the area of -th ee. When degnng herarchca routng agorthm ha adopted the foowng aumpton: ) The traffc between the MC ee n a nge zone at any ee ung ony the nterna path of th zone; ) The traffc between dfferent zone CC - th ee ( =, m - ) but beongng to the ame zone ( + )-th ee, ent to the centra zone to the CC ee, then the path zone ( + )-th ee to the Crmna Code, whch the centra area n the -th ee, n whch the recpent node, and further aong the path zone - ee node to the recpent; 3) May aow a drect n between adacent CC adacent zone at any ee. Ta decompoton networ nto zone regarded a a goa graph partton G ( X, Y ) X X, I ={,,, }, Y Y, (0) Where, I - number of pece nto whch the graph (number of zone -ee). Partton graph G can be defned by anaogy wth the partton et. Coecton of pece of P (G ) caed a partton of the graph G = (X, У ), G Р (G ) [ G ], I; G Р (G ) [ G G ] = Y = Y Y = Y =[Y ], X In other word, the et pece of Р (G ) = { X = G =G. () G,..., G } a partton of the graph G, f any pece of th et not empty, f for any two pece of Р (G ) the nterecton of the et of ertce empty and the nterecton of the et of edge may not be empty, and f the unon of a the pece exacty equa to the graph G. The expreon () defne a ubet Y of the et of edge Y Y, fang n between the pece and cut the graph G and G, or n term of herarchca addreng, et Y defne a et of drect n between zone G and G. Each of the peceg,..., G, you mut eect the

5 et of ertce correpondng to X ; S () where the aue of S determned by requrement to networ connectty. (When S = there ony one way out of the zone n the Crmna other area of the -t ee, when S = - two path, etc.) Next, we form a graph ubnet ee :, G =(X,Y ), (3) where X =, Y =, Y. Ioaton et,..., X, houd be baed on the requrement of the connected ubnet ee. Graph G need brea apart G = X, Y, I ={,,, } and o on up unt the next partton to I =. Thu, the reutng partton m obtan the foowng reaton, pecfe the dentty of node and edge ytem-orented graph (SOG) zone of dfferent ee: {( Y ),... ( Y ),( Y ), (X m,y m )} (4) Let ca Thu, K = r pq, (p,q) Y (5) K equa to the tota ength of a the pece and the connectng edge G and G of the graph Gˢ. The connectng edge of a the pece of the graph on the SIO -m ee K = (6) Tota ength of a edge of mutee parttonng m K= Tota ength of a edge of mutee parttonng K (7) obect of m-ee partton graph G = (X, Y ) to fnd a et of pece, the tota ength of the connectng rb at a ee meet the pecfed crtera K mn. Let ee graph Gˢ broen nto pece G,..., G. In accordance wth th partton the et of edge Yˢ of the graph Gˢ can be wrtten a Y = Y Then each ubet Y repreented a foow: Y = Y Y Y (8) Y, (9) Where Y - a ubet of a the edge X ncdent to ertce peceg ; Y - a ubet of edge X connectng ertce ubet pece G together; Y - a ubet of edge connectng pece G and G We ca the rato of the tota ength of the nner rb (rb ubety ) to the tota ength of connectng rb (rb ubety ) of the partton coeffcent (G ) graph G : (G )= r / K (0) Mutee partton coeffcent defned a m (G )= r / K. () Th coeffcent, a we a the aue of K may ere a crteron for eauatng the tered partton graph. The ta refer to the probem of combnatora and ogca type, obtanng an optma outon whch aocated wth a ot tryng dfferent partton. In order to mpfy and reduce the reut of the number of opton offered to oe the probem ung the own agorthm. In th cae, each of the (m - ) tep neceary to oe the probem of parttonng the graph n a tradtona ettng. Howeer, n th cae the number of choce at each ee of the partton arge enough. So, for a graph G = (X, Y), X = n when pttng nto pece G,, G the ame dmenon n =... = n = p the number of opton p p N Cn Cn p... C! p p n!!( p!) () Een for n = 9, = 3, p = 3, we get N = 680. Soe th probem by ortng opton are not pobe, o t neceary to ue heurtc agorthm.

6 Let u now determne the ze of the zone ue, the choce of metrc for determnng the ength of the edge of the centra node and eect zone SOG. In th cae, nce we are conderng the probem of eparatng the contro zone SOG on a topoogca ee, we ue the reut obtaned for ymmetrc networ and ymmetrc networ wth centrazaton. Seectng an agorthm adapte routng contro wthn the zone houd be baed on the coeffcent of centrazaton zone = Nс / (N ), where Nс - number of commutaton pont (CP), reated to the centra node; N - tota number of node n the ytem. The maxmum aowabe ze of the area houd be determned by the eected method of adapte routng. erage pacet tranfer deay tme n the manua mode can be repreented a a um 4.4 Determnng the ze of the zone The ery choce of adapte routng contro agorthm nde the zone houd be baed on the coeffcent of centrazaton zone t = Nc / (N - ), where Nc - number of the authorzed capta, adacent to the centra node; N - tota number of node n the ytem. The maxmum aowabe ze of the zone houd be determned by the eected method of adapte routng. erage pacet tranfer deay tme n the manua mode can be repreented a a um of T a = T 0 + T c (3) Where T0 - pacet deay tme at a fxed routng; Tc - addtona pacet deay tme, dependng on the erce traffc. Gen that Tc ncreae wth the ze of the zone, t adabe to a a certan threhod aue and determne the ze of the zone baed on the reate N3: N3 Т с. (4) Moreoer, nce t depend on the choce of contro method, the aue w depend on the N3 N N 3 = N д () () ;, (5) () Where - the boundary aue of the coeffcent of centrazaton, defned for a gen networ parameter and. Dependng on the boundary aue of the coeffcent of centrazaton eected centrazed or decentrazed area. o adapte routng agorthm n dgta networ houd be mpemented agorthm for adapte contro contrant ntenty of nformaton fow. The contro agorthm a nge et of procedure that contro routng and oume (ntenty) of the ncomng tream of nformaton. Management nformaton exchange can be reduced ony to the procedure of fow dtrbuton (routng) n a reatey ma traffc, e, at ow ntente n the networ fow nformaton. 4.5 Аgorthm mt the ntenty of fow In accordance wth the decon of the aboe ta and the bandwdth aocaton by crcutou path ISDN oad tranfer mode commutaton pont (CP) agorthm retrcton fux ntenty can be wrtten a foow: Step. Data Entry: ca wth mut-channe ca (MCC)c, tract ntegra group(tig); Step. Ca defnton wth(mcc)c, Step 3. Seect (TIG) accordng to the routng matrx; Step 4. If the number of (MCC)c n buffer (TIG) (c) e than the threhod <, then taen to oad the nformaton erce,.e. ecured her an approprate buffer. If the condton <, not atfed, then the oad a dena of erce. Th agorthm to determne the fow of nformaton (DFI) qute mpe to mpement, but n certan condton of ISDN, uch a a rapd bud-up of the oad ntenty of the upper cae n the commutaton pont, the retrcton mpoed by them, may not be enough. In order to more qucy and effectey mt the fux ntenty n ISDN conder a more compcated agorthm IPR wth the exchange of nformaton oeroad between adacent CC. The bac dea of the DFI that when the number of t own MCC buffer - th TIG -th CC threhod a neghborng -th Commutaton pont reatng to the channe wtchng (CS) tranmtted meage about bocng -th CS. fter that, the -th CS own MCC who were to be ent to the - th CS are boced (or forwarded to the woraround). Wth decreang aue beow the threhod a the -th CS meage are ent to wthdraw -th boc of the commutaton pont[6] DFI cont of two part. Upon admon to the unt CC, foow thee tep: Step. Data Entry: ca wth MCC, TIG ; Step. Ca defnton wth MCC; Step 3. Seect TIG n accordance wth the matrce route, Step 4. If the number of MCC n buffer TIG doe not exceed the threhod of (c) < L (c), then go to tep 6, otherwe - go to tep 9. () () Step 5. If < L and TIG not boced, go to tep 6, otherwe - go to tep 9. Step 6. Tae a oad to the erce. Step 7 Enarge counter MCC n buffer TIG per unt : < L +, Step 8. If TIG boced, then fnh; f not, end a meage about bocng the -th CC and fnh. Step 9. Boc CS and fnh. By, the end of erce n the -th CC, CS perform the

7 foowng tep: Step 0. Decrement counter CS n buffer TIG = -; Step. If < L - d and - th TIG boced, then go to tep 3, otherwe fnh (parameter d ntroduced to enure the tabty of the agorthm ); Step. Send to a the -th CC adacent to the -th of the commutaton pont, the notce of wthdrawa of the oc -ro CC. The agorthm DFI deeoped effectey contro the ntenty of fow n ISDN proded a contant rato ( c ) between dfferent cae of traffc n CC. = cont. ( c ) I term of arabe DFI need, aowng optma () (4) rato between the quantte L - L to redtrbute - a otherwe the traffc w be boced by the ower cae f there uffcent pare capacty buffer CC reered for the upper cae of traffc. The conequence of th unutfed decne n performance ISDN. Let t - update ntera threhod; P ~ - Etmate of the aerage bocng probabty MCC -th ntera of the ~ ( ) commutaton pont on [ t t, t] c ; P - Etmate of the aerage bocng probabty MCC ca c n the ntera n the -th CC. The dea of the propoed agorthm to conert threhod that, under ow oad ISDN P ~ P doe not prode any adantage MCC of cae, by ncreang a oad, e P ~ P recee hgher prorty traffc cae. Th agorthm can be ncuded a part of the agorthm decrbed aboe n the TIG; whe perodcay at ntera tep. t, perform the foowng Step. Detect mode ISDN, for whch to cacuate P ~ the ntera t t, ] - the current tme. [ t Step. Secure another traffc ca Tc and perform pp 3-5. ~ ( c ) Step 3. If P - P ~ <, where a certan threhod aue, then go to tep 4, otherwe - to tep 5. ~ ( c ) Step 4. If P - P ~ - then ncreae the (c) threhod L : L = L + 0. Step 5.If P ~ ~ ( c ) - P then reduce the threhod: (c) L : L = L - 0 ( 0 - ncrement threhod). Step 6. If condered a traffc cae, then fnh, otherwe - go to p.. IPR deeoped contro agorthm aow for the retrcton n the fow ntenty ISDN, worng n a arety of condton and can be ued to mt the ntenty of fow when acceng the contro zone, o; and to mt the ntenty of the fow when acceng GM wthn the contro area. Seecton of a partcuar agorthm IPR defned comprome between mpementaton compexty and effcency, whch n turn depend on the nature of the nterna and mxed-zone traffc parameter IHT and GM, dmenon, zone contro, and other factor, whch may tae nto account when ung a muaton mode of ISDN. Numerca mpementaton of the agorthm demontrated by a pecfc exampe. 5. Сoncuon Song the probem of accuratey cacuate networ performance ead to a gnfcant reducton n mean - tme deery of data pacet to a uer and reduce o n the ca. Thu, due to the effecte bandwdth redtrbuton ntegra path networ can achee the optma parameter aue n cutomer erce ISDN. thee aboe factor aocated wth the probem of optma aocaton of bandwdth between ubnet CC and CP, n the end reut, gnfcanty ncreae the effectene of mut- networ communcaton n genera and thereby actuay a tranton to a more cot-effecte networ. 6. Reference [] Budng a Reabe and Secure Muterce Networ for Homeand Securty and Defene, Cco Sytem, Inc, [] 5th Progre Report on the Snge European Eectronc Communcaton Maret 009 [3] Tuemoa, G.E. Some mathematca repreentaton of Erang Formua / Tuemoa G.E. // Izetya uzo magazne, #5. Bhe: Scentfc magazne and chdren terature. 0. Page 3-5. [4] Tuemoa, G.E. mathematca mode for cacuatng the parameter of the aynchronou data networ wth crcutou drecton / Tuemoa G.E. Izetya uzo magazne, #5. Bhe: Scentfc magazne and chdren terature. 0. Page 4-8. [5] Tuemoa, G.E. method of optma aocaton of channe reource n muterce networ. / Tuemoa G.E. // Scence and new technooge magazne, #. Bhe: Scentfc magazne and chdren terature. 0. Page [6] Tuemoa, G.E. gorthm for cacuatng the parameter of quaty of erce dgta networ wth ntegrated erce / Tuemoa G.E., Аhgaye D.U., Mutafn S.. // Matera of Scence Forum, dedcated to the 75th annerary of Prof., cademcan MIN Byahe R.G. maty, Page 38-4