Analysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems

Transcription

1 Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.-Ig. Georg Carle Char for Network Archtectures ad Servces Departmet of Computer Scece echsche Uverstät Müche

2 No Markov Systems Cotet: No Memory-less systems Embedded markov cha Geeral dstrbuted servce tmes Watg system M/GI/-s Ifte umber of sources State probabltes rasto probabltes Watg tme dstrbuto Impact of varace of the servce process o system performace Watg system GI/M/-s Ifte umber of sources State probabltes rasto probabltes Watg tme dstrbuto Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 2

3 No Markov Systems Markov Systems: Arrval process s memory-less. Servce process s memory-less. System s memory less at ay gve pot tme. No Markov Systems: Have oe compoet whch s memory-less AND oe compoet whch s NO memory-less. System becomes memory-less ether at the tme of a arrval or at the tme a servce s completed. M / GI / ad GI / M / Idea: Aalyze the system whe t s memory-less. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 3

4 Embedded Markov Cha Assumpto: State dscrete stochastc process { X t, t markova at tme {, 0,, }. t 0} whch s memory-less P P X t x X t x,..., X t0 x0 X t x X t x, t t... t t. 0 X tme Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 4

5 Embedded Markov Cha Characterstcs: he future developmet of the process oly depeds o ts curret state. Kowledge about the curret state X t at tme t s suffcet to calculate ts cosecutve states X t, X t, 2 Due to the state dscrete ature of the process, the states represet a cha. { X t} he cha s a markov cha accordg to our assumpto whch states that the process s memory-less at tme {, 0,, }. t Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 5

6 System state Embedded Markov Cha Pots of regeerato: Xt Arrvals 2 t P P P P Embedded pots tme whe the system s memory-less markova. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 6

7 Embedded Markov Cha he state probabltes at the embedded pots ca be descrbed by a state probablty vector. t { x,, 0,, } x, P{ X t } State trasto probablty matrx P whch descrbes the relato betwee ay cosecutve state probablty vectors ad at embedded pots ad s gve by P { p j } t t p j P{ X t j X t },, j 0,,2, P Relato of cosecutve state trasto vectors. P Probablty vector statoary state. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 7

8 Embedded Markov Cha Characterstcs: he probablty vector of the markov cha steady state s gve by the left egevector of the trasto probablty matrx P of egevalue. Embedded markov cha s usually appled f oly oe compoet s omemory less. Embedded pots are located where ths compoet becomes memory less. M / GI / Servce process s the oly o-memory less compoet. State process becomes memory less after servce completo. Embedded pots are located drectly after a servce completo. GI / M / Arrval process s the oly o-memory less compoet. State process becomes memory less after a arrval. Embedded pots are located drectly before a arrval occurs. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 8

9 M / GI / Watg system Model: Posso arrval M, Watg queue wth ulmted capacty GI Geeral depedet dstrbuted servce tme Model ad parameter descrpto: M / GI / No jobs are blocked! Arrval process s a Posso process wth a expoetal dstrbuted terarrval tme A. Servce tme B s geeral depedet GI dstrbuted. Jobs that arrve at a pot tme whe all servce uts are busy, are queued ad served FIFO order as soo as a free servg ut s avalable. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 9

10 M / GI / Watg system Arrval process: Arrval rate λ Average umber of arrvg jobs per tme ut. Servce process: Servce rate μ A t P A t e, E[ A] t Average umber of servce completos. assumg the servce ut oly has two states dle or busy B t P B t, E[ B] System: Watg system Watg queue wth ulmted capacty Queug strategy Frst I Frst Out FIFO Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 0

11 System state M / GI / Watg system State space: X t Radom varable descrbes the umber of watg ad curretly served jobs the system. State process s state dscrete ad tme cotuous stochastc process Xt Arrvals Watg jobs Busy servg ut t arrvals servce tme evets Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202

12 System state M / GI / Watg system State space: X t Radom varable descrbes the umber of watg ad curretly served jobs the system. State process s state dscrete ad tme cotuous stochastc process Xt Arrvals W t- B Watg jobs t t arrvals servce tme evets P P P P Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 2

13 M / GI / Watg system Embedded pots: State process becomes memory less at the tme of a servce completo. Embedded pots of the markov cha are located drectly after servce completo. Embedded Markov Cha: he pot tme of the th embedded pot correspods to the th servce completo. he sequece of the process states { X t0, X t,, X t, X t, } at these pots represet the embedded markov cha. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 3

14 M / GI / Watg system Aalyss: Itroduce a radom varable whch descrbes the umber of arrvals durg a servce durato. P wth GF 0 z z Geerato Fucto dgf z E[ ] dz z E[ B] Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 4

15 System state System state M / GI / Watg system rasto behavor: Xt Arrvals j-+ arrvals durg the servce Xt Arrvals j arrvals durg the servce j t X t 0 B t+ X t t arrvals j j 2 =0 X t 0 t B t+ X t t arrvals j servce tme evets servce tme evets P State trasto j wth 0 P State trasto =0 j Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 5

16 M / GI / Watg system State trasto: State trasto betwee cosecutve embedded pots t ad. t p j P{ X t j X t } Case : System ot empty 0 at tme t At tme are jobs the system. he servce of the ext job starts drectly after j jobs rema the system at tme t. j-+ jobs have to arrve durg the terval correspods ths case to the servce durato. p j t. j,,2,, j,, t t ; whch Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 6

17 M / GI / Watg system Case 2: System s empty = 0 at tme t At tme are =0 jobs the system. he servce of the ext job starts drectly after the arrval of the job. j jobs rema the system at tme t. j jobs have to arrve durg the servce durato. t p j 0 j, j 0,, P { p j } Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 7

18 M / GI / Watg system he state probabltes at the embedded pots ca be descrbed by a state probablty vector. t { x0,, x,,, x j,, j 0,, } x j, P{ X t j} State trasto probablty matrx P whch descrbes the relato betwee ay cosecutve state probablty vectors ad at embedded pots ad s gve by t t P Relato of cosecutve state trasto vectors. 0 A start vector s suffcet to calculate the future state probablty vectors,,2,. hs method allows us to evaluate systems overload or durg traset phase whch are typcal ssues commucato etworks. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 8

19 M / GI / Watg system Statoary state equato: A system s called stable f t state probablty vector does ot further chage. c.f. Chapter 3 { x0, x,, x j, } P x j x0 j Probablty vector statoary state. j x j j 0,, State probabltes ca be calculated by usg the dstrbuto of RV ad the probablty vector. I geeral the process or system s a statoary state at the begg. he statoary state vector s typcally determed by umercal method. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

20 M / GI / Watg system Power method: Robust umercal method to determe the steady probablty state vector. Calculate the geeral state probablty equato a certa aborto crtera s reached. P utl It s assumed that the statstcal equlbrum s reached f the followg aborto crtera holds true: E[ X t ] E[ X ] 0 t 6 Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

21 M / GI / Watg system Complemetary watg tme dstrbuto fucto M / GI / : I the followg the complemetary watg dstrbuto fucto of a M / GI / Watg system depedg o st utlzato ad the coeffcet of varato cb of the servce tme dstrbuto s show. Coeffcet of varato Varatoskoeffzet he coeffcet of varato s a ormalzed measure of dsperso of a probablty dstrbuto It s a dmesoless umber whch does ot requre kowledge of the mea of the dstrbuto order to descrbe the dstrbuto c X X, E[ X ] E[ X ] 0 c 0 c c determstc costat servce durato varace lower tha expoetal dstrbuto varace hgher tha expoetal dstrbuto Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 Pcture take from ra-ga, Eführug de Lestugsbewertug ud Verkehrstheore 2

22 M / GI / Watg system Complemetary watg tme dstrbuto fucto M / GI / : Probablty that the system s busy. x0 x0 Watg durato creases wth hgher varace of the servce process! Varace of the servce process becomes the domatg factor. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 Pcture take from ra-ga, Eführug de Lestugsbewertug ud Verkehrstheore 22

23 M / GI / Watg system Average watg tme of watg jobs: Hgh utlzato should be avoded f the servce durato has a hgh coeffcet of varato! Best performace s acheved by systems wth costat servce tmes. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 Pcture take from ra-ga, Eführug de Lestugsbewertug ud Verkehrstheore 23

24 M / GI / Watg system State probabltes at radom observato pots: * x, 0,,, x A, 0,,, State probabltes at a radomly chose observato pot t*. State probabltes at pots of arrval. State probabltes of a M / G / Watg system are vald at ay radomly chose observato pot t*. x x * x A, 0,, Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

25 M / GI / Watg system Idea: Observe the state process over a terval of legth. [ X ] Focus o a sgle state ad cout the followg state trastos: Arrval evet: State trasto A, [ X ] [ X ] umber of these state trastos durg terval. Departure evet: State trasto D, [ X ] [ X ] umber of these state trastos durg terval. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

26 System state M / GI / Watg system State probabltes at radom observato pots: Number of arrvals durg terval whle the system was state. Xt Arrvals t arrvals A, Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

27 System state M / GI / Watg system State probabltes at radom observato pots: Number of departures durg terval whch chaged the system state from + to state. Xt Arrvals t departures D, Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

28 M / GI / Watg system Both evets are alteratg durg the process developmet. Durg a observato terval the followg equalty holds true: A,, D he total umber of arrval ad departure evets durg tme terval s gve by: A 0 A, otal umber of arrvals durg terval D 0 D, otal umber of departure durg terval Start state s gve by X0 ad the fal state s gve by X. D X 0 X A otal umber of departure evets durg terval Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

29 Network Securty, WS 2008/09, Chapter 9 29 IN2072 Aalyss of System Performace, SS M / GI / Watg system State probablty at embedded pots: 0,,,, lm, lm X X x A A D A D D State trasto at arrvals 0,,,, lm x A A A 0,, 0,,, lm X X A A A D A A 0,, lm A A D wth ad

30 M / GI / Watg system State probablty at embedded pots: Statoary system lm X 0 X A 0 x x, 0,, A he arrval process s memory less. hus, a arrval sees the system from the same perspectve state tha a depedet observer. PASA x * x A, 0,, q.e.d. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

31 Questos What s a embedded cha? What s a embedded markov cha? Whe does a system become stable? What s the power method? Does the start probablty vector have a mpact o the power method? How ca you aalyze a M/GI/ watg system? What mpact does the varato coeffcet of the servce dstrbuto have o the system performace? What s the dfferece betwee watg tme of all jobs ad watg tme of watg jobs? How ca you proof that arrval see a M/GI/ system from the same perspectve tha a depedet observer? Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 3

32 Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems GI / M / Watg system

33 GI / M / Watg system Model: Geeral depedet dstrbuted GI, Watg queue wth ulmted capacty Model ad parameter descrpto: GI / M / No jobs are blocked! M Posso process Arrval process s geeral depedet GI dstrbuted. Servce tme B s a Posso process wth a expoetal dstrbuted ter-arrval tme A. Jobs that arrve at a pot tme whe the servce ut s busy, are queued ad served FIFO order as soo as the servg ut has served the curret job. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

34 GI / M / Watg system Arrval process: Arrval rate λ Average umber of arrvg jobs per tme ut. Servce process: Servce rate μ A t P A t, E[ A] Average umber of servce completos. assumg the servce ut oly has two states dle or busy B t P B t e, E[ B] t System: Watg system Watg queue wth ulmted capacty Queug strategy Frst I Frst Out FIFO Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

35 System state GI / M / Watg system State space: X t Radom varable descrbes the umber of watg ad curretly served jobs the system. State process s state dscrete ad tme cotuous stochastc process. Xt Arrvals W t- B Watg jobs t t arrvals servce tme evets P P P P P P P Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

36 GI / M / Watg system Embedded pots: State process becomes memory less at a arrval. Embedded pots of the markov cha are located drectly before a arrval. Embedded Markov Cha: he pot tme of the th embedded pot correspods to the th servce completo. he sequece of the process states { X t0, X t,, X t, X t, } at these pots represet the embedded markov cha. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

37 GI / M / Watg system Aalyss: Itroduce a radom varable whch descrbes the umber of served jobs wth a ter-arrval terval. P wth GF 0 z z Geerato Fucto E[ ] dgf z dz z E[ A] Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

38 GI / M / Watg system State trasto: t. State trasto betwee cosecutve embedded pots ad t p j P{ X t j X t } Case : System ot empty j 0 at tme t. jobs the system rght before the th arrval. + jobs are the system the embedded pot t. j jobs rema the system at the ext embedded pot. +-j jobs have to be served durg the terval t ; t. p j j, 0,,, j,2,, Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

39 Network Securty, WS 2008/09, Chapter 9 39 IN2072 Aalyss of System Performace, SS GI / M / Watg system Case 2: System s empty j = 0 at tme At tme are + jobs the system. o jobs rema the system at tme t t t. + jobs have to be served durg 0,,, 0 0 k k p k k } { k k k j k k k p P. ; t t

40 GI / M / Watg system he state probabltes at the embedded pots ca be descrbed by a state probablty vector. t { x0,, x,,, x j,, j 0,, } x j, P{ X t j} State trasto probablty matrx P whch descrbes the relato betwee ay cosecutve state probablty vectors ad at embedded pots ad s gve by t t P Relato of cosecutve state trasto vectors. 0 A start vector s suffcet to calculate the future state probablty vectors,,2,. hs method allows us to evaluate systems overload or durg traset phase whch are typcal ssues commucato etworks. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

41 Network Securty, WS 2008/09, Chapter 9 4 IN2072 Aalyss of System Performace, SS GI / M / Watg system Statoary state equato: A system s called stable f t state probablty vector does ot further chage. c.f. Chapter 3 P Probablty vector statoary state. },,, 0, { j x x x k k k x k x x,2,, 0 j j x j x j x j

42 Questos How ca you aalyze a GI/M/ watg system? What s the utlzato of a GI/M/ watg system? What mpact has the varato of the arrval process o the watg tme of a GI/M/ watg system? Whe does the system become memory-less? Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

43 Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems M / GI[,K] / S Batch Servce wth Start hreshold

44 M / GI[,K] / S Loss system Batch servce system wth start threshold: Model: Actvato Posso arrval process M, S 2 K GI GI GI Batch servce process wth GI dstrbuted servce durato Model ad parameter descrpto: M / GI[,K] / S System wth S watg slots. Arrval process s a Posso process. Servce durato s GI dstrbuted. Up to K jobs ca be served parallel. Servce ut s actvated f or more jobs are watg. All jobs of a batch experece exactly the same servce tme. Arrvg jobs have to wat utl the curret batch s served. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

45 M / GI[,K] / S Loss system Model ad parameter descrpto: System has S watg slots. Jobs are blocked f S jobs are watg the queue. At the ed of a batch servce, ew jobs are loaded to the servers f at least jobs are watg. Up to K jobs are loaded to the system after a batch servce. If less tha jobs are watg the queue at the ed of a batch servce, the servers rema dle utl jobs are the queue ad a ew batch servce starts. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

46 System state M / GI[,K] / S Loss system State space: X t Radom varable descrbes the umber of watg the system. State process s state dscrete ad tme cotuous stochastc process. Xt Arrvals K 2 t t2 servce process t3 t4 t Embedded pots U Case U Case 3 U Case 2 Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

47 M / GI[,K] / S Loss system Embedded pots: State process becomes memory less at the tme of a servce completo. Embedded pots of the markov cha are located drectly after servce completo. Embedded Markov Cha: he pot tme of the th embedded pot correspods to the th batch servce completo. he sequece of the process states { X t0, X t,, X t, X t, } at these pots represet the embedded markov cha. Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

48 M / GI[,K] / S Loss system Aalyss: Itroduce a radom varable whch descrbes the umber of arrvals durg a servce durato. P he state probabltes at the embedded pots ad ca be descrbed by a state probablty vector. t t p j P{ X t j X t } Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

49 M / GI[,K] / S Loss system State trasto: t. State trasto betwee cosecutve embedded pots ad t p j P{ X t j X t } Case : Less tha jobs the queue at tme - jobs have to arrve utl the servce process s started. hs watg perod s Erlag - dstrbuted Servce ut s actvated as soo as jobs are watg. j jobs arrve durg the servce. rasto tme U s gve by: E. t. U E B p j j, j 0,,, S p S ks k, j S Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

50 M / GI[,K] / S Loss system State trasto: t. State trasto betwee cosecutve embedded pots ad t p j P{ X t j X t } Case 2: k Number of jobs the queue hgher tha the threshold. Servce process s started mmedately ad the queue s empted. rasto tme U s detcal wth the servce durato. State trasto probablty s detcal to that case. j jobs arrve durg the servce. U B p j j, j 0,,, S p S ks k, j S Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS

51 M / GI[,K] / S Loss system Case 3: K S Number of jobs the queue s hgher tha the umber of servers. Servce process s started mmedately ad K jobs are served. -k jobs rema the queue after the servce s started. rasto tme U s detcal wth the servce durato. j jobs are the queue after the batch servce s completed. j-+k jobs arrve durg the servce durato. U B p j j k, j 0,,, S p S ksk k, j S Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 5

52 Network Securty, WS 2008/09, Chapter 9 52 IN2072 Aalyss of System Performace, SS M / GI[,K] / S Loss system State trasto matrx: K k S k S k S k S k S k j k K k S k S k S k S k S p P } { S- S 0 K K+ K+2 S

53 Average watg tme E[W] M / GI[,K] / S Loss system Average watg tme: Start threshold raffc load Varace of the servce process Characterstcs: Hgh watg tme for systems wth low traffc load ad hgh start threshold. Start threshold becomes domatg for systems wth low traffc load sce t takes a log tme utl the start threshold s reached. E [ B] / k M / GI[,K] / S wth K=32 servce uts ad S=64 watg slots Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 Pcture take from ra-ga, Eführug de Lestugsbewertug ud Verkehrstheore 53

54 Average watg tme E[W] M / GI[,K] / S Loss system Average watg tme: Start threshold raffc load Varace of the servce process Characterstcs: Impact of varace of the servce process decreases wth hgher start thresholds. Optmum terms of average watg tme s hghly parameter sestve. Start threshold M / GI[,K] / S wth K=32 servce uts ad S=64 watg slots Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS 202 Pcture take from ra-ga, Eführug de Lestugsbewertug ud Verkehrstheore 54

55 Questos Descrbe the M/GI[,K]/-S loss system ad ts parameter. What mpact has the start threshold o the watg durato? How ca you aalyze the M/GI[,K]/-S loss system? Whch state chages are possble betwee to cosecutve embedded pots? Network IN2072 Securty, Aalyss WS of System 2008/09, Performace, Chapter 9 SS