Numerical Simulations for Dynamic Stochastic and Hybrid Models of Internet Networks

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1 G Mc M Nu A L Cudu D O Nucl Sulon fo Dync Sochc nd Hybd Modl of Inn Nwo G MIRCEA M NEAMTU A L CIURDARIU D OPRIS 4 Dn of Econoc Infoc nd Sc 4 Dn of Mhc 4 W Unvy of Tşo Polhnc Unvy of Tşo Plozz S n 6A 5 Tşo P Vco n 4 Tşo 4 Bd V Pvn n 4 Tşo ROMANIA gblc@fuvo hlnu@fuvo clodn4@yhooco 4 o@huvo Abc: - In h w cond Inn odl whch ond o congon gnl fo h nwo dcbd by ochc nd hybd dffnl quon W cond Inn nwo wh on ouc nd cc ln wll wh ouc nd on cc ln W nlyz h condon fo h nc of oluon nd h lgoh ndd o dn h oluon W cy ou nucl ulon fo cn vlu Ky-Wod: Inn odl nwo dync ochc odl ochc dly dffnl quon Eul hod nucl ulon Inoducon Congon conol chn nd cv quu ngn ch AQM fo h Inn hv bn nvly udd nc h wo of Klly ll [5] In [] h Hof bfucon h bn udd fo h odl of n Inn nwo wh > ln nd ngl ouc whch cn b fould : v f b f wh h ndng of h ouc ov gn h u of fowd nd un dly v g on b ov l nd h congon ndcon funcon f ncng nonngv whch chcz h congon Th dczng odl of h bn nlyzd n [9] nd h vlu of fo whch h N- Sc bfucon lc h bn dnd Th dync odl wh ngl ln nd ouc cn b dcbd by: b c wh h whch h ouc n d h nd b ov l nub h lo obbly funcon ound- dly fo ouc c h ccy gn Th dczd odl of h bn nlyzd n [] By loyng h dync dlyd fdbc conol w cn cond h conolld congon conol y wh councon dly follow: ISSN: Iu Volu 8 Dcb 9

2 G Mc M Nu A L Cudu D O 4 5 wh 5 > h whu fl conn whch gun h bly of conoll 4 > h fdbc gn Th odl w nlzd n [4] Th odl nd ld o dync odl dcbd by ochc dly dffnl quon by ndozng on of h Thu n con w wll cond ochc dly dffnl quon obnd hough h ndozon of n h c of h odl cvly of c n h c of h odl nd h n h c of h odl 4 5 Fo cn funcon whch dcb h ochc dly dffnl quon SDDE h nc nd unqun of h oluon ufd In con w dcb h lgoh whch o h oluon of h quon fo h ochc y wh ln nd ouc wh ngl ln nd ouc cvly In con w dcb h lgoh nuc fo h c In con 4 w oc h dn odl hybd y by ddng h coz n Io ngl Fo cn vlu w cy ou nucl ulon wh h Ml ofw fo h ulon of Wn nd Lu oc SDDE Modl fo Inn nwo L Ω F P b col obbly c wh flon F fyng h uul condon; h h flon F gh -connuou nd ch F wh conn ll P -null n F Fo gnl hoy w f o [] Wh E d Ω w y fo h L L Ω F P f E < nd w dfn E dno h con L T E H w w w b - dnonl Wn oc gvn on h fld obbly c Ω F P Th ochc dly dffnl quon SDDE wh on fd lg < T < gvn by: d g f d Ψ [ T ] [ ] wh ω ω Ω Ψ F - ubl wh vlu n C[ ] R o h E Ψ < nd f : R R R 4 g : R R R ω ω Ω Th y 4 cn hn fould quvlnly : g f d 5 fo [ T ] nd wh Ψ fo [ ] Th cond ngl n 5 ochc ngl whch cn b nd ccodng o Io ngl Fo h funcon f g nd Ψ :[ ] R w cond h followng of condon []: Th funcon f nd g connuou; Th funcon f nd g fy unfo Lchz condon; Th funcon Ψ Hold - connuou wh onn ϕ ; 4 Th funcon f nd g fy ln gowh condon; 5 Th l dvv of f Φ Ψ nd unfoly boundd Pooon [] Au h h funcon nd g fy h bov uon - Thn h unqu ong oluon of quon 4 Th ochc odl fo n Inn nwo wh > ln nd ngl ouc dcbd by SDDE gvn by: f ISSN: Iu Volu 8 Dcb 9

3 ] [ ] [ Ψ T f d f f v 6 Th ochc odl fo h Inn nwo wh ngl ln nd ouc cn b dcbd by: ] [ - - ] [ Ψ g d c T g d b g 7 Th ochc odl fo Inn ocd o cn dcbd by: 5 4 d d d 8 If funcon f g nd Ψ fy condon -5 hn SDDE gvn 6 7 cvly 8 h unqu oluon Nucl ulon fo h SDDE quon 6 7 nd 8 Th obl of olvng n SDDE ducd o on of olvng qunc of y of SDDE of ncng dnon on uccv nvl ] [ n n Ung h Eul hod of f od fo [] [] [] fo 6 w obn: h N G h f f b f v h > 9 wh ]] h ndo[nold[ h G Ung ul fo [] [] [] w hv h h lgoh gvn by 9 convgn Fo 8 b v 5 5 n h f wh Ml w obn n Fg h ob n Fg h ob nd n Fg h ob Fg Th ob WSEAS TRANSACTIONS on MATHEMATICS G Mc M Nu A L Cudu D O ISSN: Iu Volu 8 Dcb 9

4 G Mc M Nu A L Cudu D O Fg 8 h ob nd n Fg 9 h ob Fg Th ob Fg4 Th ob Fg Th ob Ung h Eul hod of f od fo 7 w obn: h g b g G h h c G h Fg5 Th ob Fo h 5 c 65 9 b 5 b 5 4 g wh Ml w obn h ob: n Fg 4 h ob n Fg 5 h ob n Fg 6 h ob n Fg 7 h ob n Fg6 Th ob ISSN: Iu Volu 8 Dcb 9

5 G Mc M Nu A L Cudu D O Ung h Eul hod of f od fo 8 w obn: h G h h G h h G h 4 5 Fg7 Th ob Fo h wh Ml w obn h ob: In Fg h ob n Fg h ob n Fg h ob n Fg h ob n Fg4 h ob nd n Fg5 h ob Fg8 Th ob Fg Th ob Fg9 Th ob ISSN: Iu Volu 8 Dcb 9

6 G Mc M Nu A L Cudu D O Fg Th ob Fg4 Th ob Fg Th ob Fg5 Th ob Fg Th ob 4 Hybd dffnl quon ocd fo h SDDE quon nd Rndon n bc y of obcv uncnly nd obbly hoy bnch of hc fo udyng h bhvo of ndo hnon Th conc of fuzzy w nd by Zdh [4] v h bh funcon n 965 In od o u fuzzy vn Lu B [6] noducd h conc of cdbly u Th cdbly hoy w foundd bnch of hc fo udyng h bhvo of fuzzy hnon Fuzzn nd ndon wo bc y of uncny In ny c fuzzn nd ndon ulnouly n y ISSN: Iu Volu 8 Dcb 9

7 G Mc M Nu A L Cudu D O L Θ b nony nd l Ρ b h ow of Θ ll ub of Θ Ech ln n Ρ clld n vn In od o n n oc dfnon of cdbly w cc h followng fou o: C Θ ; C A C B whnv A < B ; { } { } { } { } c C A C A fo ny A Ρ ; 4 C U A uc{ A } fo ny vn { } u C A < { } 5 A wh Th funcon C clld cdbly u nd Θ ΡC cdbly c A fuzzy vbl funcon fo h cdbly c Θ ΡC o h of l nub If fuzzy vbl ξ dfnd funcon on cdbly c hn w y g bh funcon v: { } µ C ξ R Suo h Θ ΡC cdbly c nd Ω FP obbly c Th oduc Θ Ρ C Ω FP clld chnc c A hybd vbl ubl funcon fo chnc c Θ Ρ C Ω FP o h of l nub fo ny Bol B of l nub θ ω Θ Ω ξ θ ω B n vn h { } L T b n nd nd l ΡC Θ b cdbly c A fuzzy oc funcon fo T ΘΡC o h of l nub A fuzzy oc C d o b C oc f: C ; C h ony nd ndndn ncn; vy ncn C C nolly dbud fuzzy vbl wh cd vlu nd vnc σ who bh funcon : π µ R 6σ L C oc b ndd f nd σ Th C oc ly h ol of Bownn oon L T b n nd nd Ρ C Ω FP Θ chnc c A hybd oc ubl funcon fo T Θ Ρ C Ω FP o h of l nub fo ch T nd ny Bol B of l nub h θ ω Θ Ω X θ ω B n vn { } Suo w ndd Bownn oon C ndd C oc nd f g h o gvn funcon Thn: d f d g h dc 4 clld hybd dffnl quon A oluon hybd oc h f 4 dnclly n Hybd dffnl quon ocd o : d vf b f d dc Ψ [ ] > { } wh h oluon of quon v f Th nucl ulon of 5 gvn by: h vf b f N h > { } z > 5 G h L z 6 S L z h G h ndo[ nold[ πz 6σS h]] 7 Fo 8 b v 5 5 n 5 h f wh Ml 5 w obn n Fg 6 h ob n ISSN: Iu Volu 8 Dcb 9

8 G Mc M Nu A L Cudu D O Fg7 h ob nd n Fg8 h ob d b d dc d c d dc 8 > > { } wh h qulbu on of h Th nucl ulon of 7 gvn by: Fg6 Th ob h b Gh L z h c Gh L z 9 N h > { } z > wh G h L z gvn by 7 Fg7 Th ob Fo n 5 h 9 5 b 5 b 5 c z 5 z z 4 wh Ml w obn n Fg9 h ob n Fg h ob nd n Fg h ob Fg8 Th ob Th hybd dffnl quon ocd o : Fg9 Th ob ISSN: Iu Volu 8 Dcb 9

9 Fg Th ob Fg Th ob Th hybd dffnl quon ocd o : 5 4 dc d d dc d d dc d d wh h qulbu on coodn Th nucl ulon of gvn by: 5 4 z L G h h z L Gh h z L G h h Fo z 8 z 5 z 4 h n wh Ml w obn n Fg h ob n Fg h ob nd n Fg4 h ob Fg Th ob Fg Th ob WSEAS TRANSACTIONS on MATHEMATICS G Mc M Nu A L Cudu D O ISSN: Iu Volu 8 Dcb 9

10 G Mc M Nu A L Cudu D O Fg4 Th ob 5 Concluon Th h noducd SDDE fo dync ochc nd hybd odl of Inn Nwo Th h hown h h quon blong o h cgoy of quon h cc unqu oluon W hv dcbd nucl lgoh n od o dn h o oluon Th oluon hv bn vulzd wh h hl of og n Ml ung h Bo-Mull hod fo h ulon of Wn nd Lu oc A l udy wll b conducd fo c n whch oh confdnc wll b ndozd nd fuzzn Alo w wll nlyz h bly lly o wh nlyzd n [7] Th odl fo h cn b ndd condng h fconl ngl [] [4] Acnowldgn: Th ch w uod by h Gn wh h l Th qulv nly nd nucl ulon fo o conoc odl whch conn von nd couon Th Nonl Unvy Rch Councl fo Mny of Educon nd Rch of Ron gn No 5/9 Rfnc: [] B CTH Bucw E - Nucl nly of lc on hod fo ochc dly dffnl quon LHS JCou Mh London 5-5; [] Chungung L Gunong C Xofng L Jubng Y - Hof bfucon n n Inn congon conol odl Cho Solon nd Fcl Vol ; [] Go S F F Dlf F M To - Ncy Oly Condon fo Fconl Acon-L Pobl wh Innc nd Obv T WSEAS Tncon on Mhc vol 7 n 8 6-; [4] Guo S Fng G Lo X Lu Q- Hof bfucon conol n congon conol odl v dync dlyd fdbc Cho ; [5] Klly FP Mulloo A Tn DKH - R conol n councon nwo: hdow c ooonl fn nd bly J O R Soc ; [6] Lu B Uncny Thoy d Edon c 9 by UTLAB [7] Lu JC Mcy MC -Sochc Dffnl Dly Equon Mon Sbly And Alcon To Hooc S Cll Rgulon Sy SIAM J APPL MATH Socy fo Indul nd Ald Mhc Vol 67 No ; [8] Mc G Nţu M Oş D - Hof bfucon fo dyncl y wh dly nd lcon n Ronn Mon Publhng Hou Tşo 4; [9] Mc G O D- Inn odl wh n cc ln nd fdbc dly Econoy Infoc Vol III Nub / ISSN Publhd by Econoy Infoc Dn nd INFOREC Aocon wh h uo of Educon nd Rch Mny Econoc Publhng Hou Buch 78-8; [] Mc G - Inn congon conol odl Pocdng of h 9 h WSEAS Innonl Confnc on Mhc nd Cou n Bun nd Econoc Buch Ron ; [] Mc G O D - N-Sc bfucon n dc dync y WSEAS Tncon on Mhc Vol 8 N 9 6-7; [] Mohd SEA - Sochc ch fo ochc dffnl y wh oy Thoy l nd lcon Pog n Pobbly Bhu 998; [] Tudo M - Aoon ch fo ochc quon wh hdy gun Sud Cc M ; [4] Ud C O D Eul-Lgng-Hlon Dync wh Fconl Acon WSEAS Tncon on Mhc vol 7 n 8 9-; [5] Zdh LA Fuzzy b fo hoy of obly Fuzzy S nd Sy ISSN: Iu Volu 8 Dcb 9