Infinite Petri Nets. Dmitry A. Zaitsev.

Transcription

1 Infnte Petr Nets Dmtry A. Zatsev htt://aze.ho.a htt://cl.t.e

2 Verfcaton of rotocols Interacton of two systems BGP TCP Interacton of a efnte nmber of systems IOTP Blng Petr net moel LBS analyss of the moel Comostonal analyss (clans)

3 Moel of rotocol TCP

4 Decomoston of TCP moel

5 Problem of Marsan Verfcaton of common bs Ethernet rotocols Petr net moel of a worstaton How to comose the networ moel? How to analyze the networ moel? Varable an a ror nnown nmber of evces on the bs

6 Moel of a worstaton

7 Common bs Ethernet scheme 2 2 Z 2 Z

8 An examle of a bs wth for evces

9 Р-nvarants of Ethernet moel ) ) (( ) ( ) ) (( ); ) (( ); ) (( ); ) (( ); ) (( ; ) (

10 Strctres of nfnte nets Strctre System Moel Analyss Lnear Common bs Ethernet Tree-le Swtche Ethernet Rectanglar Comtng Gr Tranglar hexagonal Raobroacastng celllar networs Hyercbe Comtng Gr Hyertors Comtng Gr

11 Moel of rectanglar gr evce

12 Comoston of rectanglar gr x x 2 x (2x ) 4x R R 2 R 4x + (2x ) (3x ) x 2 4x 2 4x 2 R 2 R 22 R 4x 2+ (2x 2 ) 4x R R 2 R 4x 2+ (2x 2 ) x + (3x ) x +2 (3x 2 ) (3x ) x +

13 Oen gr 2 х 2

14 Parametrc secfcaton of a evce to : b ol o bl n t bl b l v n v v : v

15 Parametrc secfcaton of oen gr

16 Ege contons Oen gr Termnal (cstomer) evce Connecton of (ooste) eges Trncate commncaton evce

17 Eges Termnal evce Connecton of eges 4 2

18 Close gr 2 х 2

19 Trncate evce on eges

20 Tree-le strctres

21 Devces: tranglar rectanglar an hexagonal 4 t43 2o 4l t42 t2o 2ol b3 bl b 0 4o t23 2 4ol t4o b2 t24 2l t32 t34 t3o 3 3l 3o 3ol

22 Grs R R 2 R 3 R R 2 R 32 R 22 R R 23 R R 2 R 22 R 23 R 2.4 R 25 R 3 R 32 R 33 R 34 R 35 R 36 7 R 3 R 42 R 33 R 44 R R 4 R 4 R 42 R 43 R 44 R 45 R 46 R 47 R 4 R 43 R 45 R 47 5 R 5 R 52 R 53 R 54 R 55 R 56 R 5 R 52 R 53 R 54 R R R 57 R 58 R 59 6 R 6 R 62 R 63 R 64 R 65 R 6 R 62 R 63 R 64 R 65 R 66 R 67 R 68 R 69 R 60 R 6 7 R 7 R 72 R 73 R 74 x x 2 x (2x ) 4x R R 2 R 4x + (2x ) (3x ) x 2 4x 2 4x 2 R 2 R 22 R 4x 2+ (2x 2 ) 4x R R 2 R 4x 2+ (2x 2 ) x + (3x ) x +2 (3x 2 ) x + (3x )

23 Parametrc secfcaton of tors

24 Infnte system for -nvarants

25 Р-nvarants of tors

26 Invarants of 3 х 3 tors

27 Metho of grs analyss Solvng nfnte systems of lnear algebrac eqatons for - an t- nvarants Imlct constrcton of cyclc seqences of transtons frng Axlary grahs of acets transmsson an ossble blocngs of evces

28 Grah of transmssons o 2 o 2 4o 4o 2 4o 3 o o 2 2 o 2 22 o o 4o 2 4o 23 4o 22 3o o 3 32 o 32

29 Grah of ossble blocngs

30 Detors of ossble blocngs grahs

31 Comlex ealocs Crcle of blocngs Isolaton of a vertex Chan of blocngs wth ts en on an earler bloce chan Avalanche-le growth of the bloce noes nmber Possblty of blocng a networ by llntentone traffc

32 Examle of ealoc

33 Cbc evce ort 32 X 2 ort 22 ort ort 3 ort 2 X 3 ort 2 X

34 Comoston of a cbe X 2 X 3 R 32 X

35 Neghbors of a cbe

36 Hyercbe comoston 2 : ; 2 : n bl o ol b to n n n l b bl t n n n n n n n n n ol l ol o o l ol l o : : : : l ol o ol l o : : : :

37 Р-nvarants of hyercbe. 2 : : : : n n n xol xb xbl xo t xbl x xl xb to xl xb xbl x t xbl xo xol xb to n n n n )) ) (( ) ) ) (( (( )) ) (( ) ) (( ) 2 (( ; 2)) ( ( ; ) ( ) ( ; ) ( ) ( ol l ol l bl o o n b n b bl ol o l n n

38 Generators of moels Sqare gr (grahcal form) Hyercbe gr Hyertors gr Canvas of generalze negborhoo Comters of oble exonent

39 Generator of an oen gr man( nt argc char * argv[] ) { nt = ato( argv[] ); for(=; <=; ++) for(=; <=; ++) { rntf("tr {to_^%%} {ol_^%%} {b_^%%} -> {o_^%%} {bl^%%}\n" ); rntf("tr {t_2^%%} {_^%%} {bl^%%} -> {l_^%%} {b_2^%%}\n" ); rntf("tr {t_3^%%} {_^%%} {bl^%%} -> {l_^%%} {b_3^%%}\n" ); rntf("tr {t_4^%%} {_^%%} {bl^%%} -> {l_^%%} {b_4^%%}\n" ); rntf("tr {to_4^%%} {ol_4^%%} {b_4^%%} -> {o_4^%%} {bl^%%}\n" ); rntf("tr {t_4^%%} {_4^%%} {bl^%%} -> {l_4^%%} {b_^%%}\n" ); rntf("tr {t_42^%%} {_4^%%} {bl^%%} -> {l_4^%%} {b_2^%%}\n" ); rntf("tr {t_43^%%} {_4^%%} {bl^%%} -> {l_4^%%} {b_3^%%}\n" ); rntf("tr {to_2^%%} {l_4^%%} {b_2^%%} -> {_4^%%} {bl^%%}\n" + + ); rntf("tr {t_2^%%} {o_4^%%} {bl^%%} -> {ol_4^%%} {b_^%%}\n" + + ); rntf("tr {t_23^%.%} {o_4^%%} {bl^%%} -> {ol_4^%%} {b_3^%%}\n" + + ); rntf("tr {t_24^%%} {o_4^%%} {bl^%%} -> {ol_4^%%} {b_4^%%}\n" + + ); rntf("tr {to_3^%%} {l_^%%} {b_3^%%} -> {_^%%} {bl^%%}\n" + + ); rntf("tr {t_3^%%} {o_^%^%} {bl^%%} -> {ol_^%%} {b_^%%}\n" + + ); rntf("tr {t_32^%%} {o_^%%} {bl^%%} -> {ol_^%%} {b_2^%%}\n" + + ); rntf("tr {t_34^%%} {o_^%%} {bl^%%} -> {ol_^%%} {b_4^%%}\n" + + ); } rntf("net n2o%\n" ); }

40 Atomatc vsalzaton

41 Generate grahcal format

42 Generalze neghborhoos 2-negborhoo for 3 mensons

43 Generalze neghborhoos 2-negborhoo for 3 mensons on eges

44 Moore neghborhoo n a tors

45 Theorems an Hyothess Theorem. Moel of a hyercbe s a -nvarant Petr net for arbtrary natral nmbers. Theorem 2. Moel of a hyercbe s a t-nvarant Petr net for arbtrary natral nmbers. Statement. Moel of a hyercbe s not a lve Petr net (t contans ealocs). Hyothess. Aearng ealocs n the gr moels ncrease robablty of aearng new ealocs. Hyothess 2. Process of ealocs creaton can be controlle by generators of (erlos) traffc.

46 Basc blcatons Zatsev D.A. Zatsev I.D. an Shmeleva T.R. Infnte Petr Nets: Part Moelng Sqare Gr Strctres Comlex Systems 26(2) Zatsev D.A. Zatsev I.D. an Shmeleva T.R. Infnte Petr Nets: Part 2 Moelng Tranglar Hexagonal Hyercbe an Hyertors Strctres Comlex Systems 26(4) 207 forthcomng Zatsev D.A. A generalze neghborhoo for celllar atomata Theoretcal Comter Scence 666 (207) Zatsev D.A. Verfcaton of Comtng Grs wth Secal Ege Contons by Infnte Petr Nets Atomatc Control an Comter Scences 203 Vol. 47 No Zatsev D.A. Shmeleva T.R. Verfcaton of hyercbe commncaton strctres va arametrc Petr nets Cybernetcs an Systems Analyss Volme 46 Nmber (200) 05-4 Shmeleva T.R. Zatsev D.A. Zatsev I.D. Verfcaton of sqare commncaton gr rotocols va nfnte Petr nets MESM th Mle Eastern Smlaton Mltconference

47 Conclsons Fnte arametrc secfcaton of nfnte Petr nets Constrcton an solvng nfnte systems n arametrc form Proof of nfnte Petr net nvarance Reresentaton of comlex ealocs Analyss of comtng grs (lane mltangle hyercbe)

48 Research rectons Formal methos for solvng nfnte systems Methos for lveness nvestgaton Methos of boneness conservatveness lveness enforcng Behavoral roertes of moels Comoston of a few basc elements on a gven attern

49 Infnte Petr Nets