Dstrbutons 8/03/06 /06 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
outlne densty, dstrbuton, moments unform dstrbuton Posson process, eponental dstrbuton Pareto functon densty and dstrbuton resdual watng tme tal of dstrbuton, fttng hypoeponental dstrbuton (Erlang) smulaton hypereponental dstrbuton smulaton fttng G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
moments G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 3
G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 4 varance and second order moment varance and second order moment varance σ of a dscrete random var. wth mean E[] ] [ ] [ ] [ ] [ ] [ ) ( E E E E E n n n n σ ] [ ] [ ) ( ) ( ) ( ) ( ) ( ] [ ] ) [( µ µ µµ µ µ µ µ µ µ µ σ E E d f d f d f d f E E varance σ of a contnuous random var. wth mean μ
unform dstrbuton Posson process eponental dstrbuton G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 5
unform dstrb. over the nterval (0,ma f ( ) ma) f( ) 0< < 0 0 ma ma ma ma ma F( ) dt 0 < E [ ] d ma 0 ma ma 0 ma ma ma ma F( ) densty 0 ma / ma dstrbuton 0 ma G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 6
Posson process random pont process whose assocated countng process N(t) satsfes: ndependent ncrements statonary ncrements P[one event n (t,th)] ho(h) P[two or more events n (t,th)] o(h) events occur sngly at a rate unform n tme the number of arrvals n any tme nterval t has a Posson dstrbuton wth mean t t P() t e P () t e 0 ( t) t G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 7!
Posson process when multple Posson streams are merged the resultng stream s a Posson stream wth ntensty equal to the sum of the ntenstes 3 3 a sngle Posson stream can be splt nto ndependent Posson streams p p 3 p p p p 3 3 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 8 p p p 3 3
eponental dstrbuton f () mean / e densty standard dev. / coeff.of varaton σ/µ 0 F() P [ ] 0 e a da e dstrbuton 0 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 9
ep. dstr. - memoryless property (random) P[ > t h > t] P[( > t h) ( P[ > t] > t)] e ( t h) e t e h P[ t > h] h tme last event arrval new event arrval : nterarrval tme, t tme unts elapsed snce last event, the dstrbuton of the remanng watng tme h s ndependent of t, the system s memoryless G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 0
eponental dstrbuton: percentles r th percentle P[ ( r) ] r 00 90% of values are less than 90-th percentle P[ ( 90)] 0. 9 e ( 90) 0. ( 90) log0. e ( 90) ( 90) log0. log0. 3 ( r) log ( 00 ) 00 r G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
eponentally dstrb.. sequence we can obtan a random devate of an EP() random varable by frst generatng a random number y from a unform dstrbuton over (0,) and then usng the relaton F - (y) y e log y e y y log y log( e [0 y ) log( random y) ] nput random ep. dstrb. 0 output G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
Pareto functon G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 3
Web performance ndces ces typcal phenomenon observed on the Internet for several performance ndces (download tmes, connecton tmes, fle szes, thn tme of web browser, ) etreme varablty of traffc characterstcs (arrval tmes, number and szes of downloaded fles, ) etreme varablty of performance ndces (download tmes, connecton tmes, ) teletraffc dstrbutons follow an eponental decay, Web traffc dstrbutons follow a power decay very hgh values of varables occur wth non neglgble probablty (heavy tal) G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 4
eponental vs heavy tal decrease esponenz. epon. 3 - coda heavy lunga tal -3 00 0 5 4 0, 0,0 3 0,00 0 3 4 5 6 0 0 3 4 5 6 effetto heavy coda tal effect lunga G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 5
Pareto functon Pareto eponental α α p() α α > 0 0< e α α F () P [ ] α h dh α [ h ] e α α α > mean value α α α α α f α nfnte varance f α nfnte mean G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 6
mean α α mean µ f() d α d α [ ] α α α α d α 0 < α α α α f α > mean α α α f α mean snce α [ ] G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 7
varance varanza varance σ : E[( E [ ]) ] E [ ] ( E [ ]) α α α α f () d f () d α d α d α α α α α α α > 0 0< α α se f se f α α α α α α > σ α α α α α α σ α pochè because [ ] G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 8
shape parameter α decreasng the value of α ncreases the porton of probablty mass that s present n the tal of the dstrbuton a random varable dstrbuted accordng to a Pareto functon wth α can gve rse to very large values wth a non neglgble probablty G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 9
Pareto - dstrb.. mean 0.95 dstrbuzone F() F() 0.9 0.85 0.8 eponental 0.75 0 5 0 5 0 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 0
Pareto dstrb. mean dstrbuzone F() F() 0.995 0.99 0.985 0.98 0.975 0.97 alfa. 0.09 alfa.5 0.33 alfa.9 0.47 alfa 0.5 alfa3 0.66 alfa4 0.75 eponental esponenzale 0.965 0 5 0 5 0 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
Pareto dens. mean 0 8 alfa. 0.09 denstà f() f() 6 4 alfa.5 0.33 alfa.9 0.47 alfa 0.5 alfa3 0.66 alfa4 0.75 esponenzale eponental 0 0 3 4 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb -
Pareto dens. mean 0.0 0.008 denstà f() f() 0.006 0.004 eponental 0.00 0 5 6 7 8 9 0 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 3
log dens.. of Pareto funct. mean 0 0 α. 0.09 0 - α.5 0.33 - F() f() 0-4 0-6 α4 0.75 α3 0.66 α 0.5 eponental 0-8 0 5 0 5 0 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 4
dstrbuton tal relablty R(t) of a system: probablty that the system survves untl tme t (F(t) unrelablty functon) R ( t) P( > t) F( t) F( t) hgh values are the most crtcal for performance (tal dstrbuton s very mportant) ep.dstrb. F( ) e Pareto dstrb. F( ) α 0< α 0< G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 5
tal of the dstrbutons, mean esp. F () e F () 0 0 e Pareto 3 LogLog complementary plot α.5 0 - -F() - F() 0-0 -3 eponental eponental esponenzale Pareto coda lunga 0-4 0 0 0 0 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 6
fractals (self smlarty) G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 7
self-smlarty smlarty of web traffc from Wllnger-Pason Pason, Where Mathematcs meets the Internet, Notces of the Amercan Mathematcal Socety, 45(8), pp.96-970, 970, 998 Posson Pacet arrvals measured mean tme scale 00ms 6sec tme scale s 60sec tme scale 0s 600sec tme scale 60s G.Serazz 05/06 Dmensonamento h degl Impant Informatc dstrb - 8
Unform resdual watng tme resdual watng tme Pareto Eponental last event tme G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 9
resdual watng tme Y : tme between two consecutve events gven that t tme unts are elapsed snce the last event occurred, we want to compute the dstrbuton of Y, the resdual watng tme Y - t t Y tme arrval of last event arrval of new event φ ( ) lm y 0 P [ t y > t] y G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 30
resdual watng tme PA ( B) PAB ( ) PB ( ) Pt [ < t y] f( ) ( ) lm y 0 y P [ > t] F ( ) φ G Y (y t): condtonal probablty of the event Y y gven that the event >t has occurred GY ( y t) P[ Y y > t] P[ t y > t] P[( t y) ( > t)] P[ t < t y] P [ t y > t] P [ > t] P [ > t] t y Pt [ < t y] f( ) d t F( t) F( t) G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 3
resdual watng tme unform dstrb. ep. dstrb. Pareto dstrb. ma T ( t) T ( t) t T ( t) α t G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 3
resdual wat. tme: unform and ep. dstrbutons ma Y ma ma f() t ( t Y) dt ( t Y) f( t) dt FY ( ) FY ( ) Y ma Y ma ( ma ) ma z z z f( Y z) dz Y 0 ma Y ma f() t ( t Y) dt z f ( Y z) dz FY ( ) FY ( ) 0 0 Y Y z e z Y e 0 0 z e e dz z e dz Y e G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 33
resdual watng tme: Pareto dstrb. Y f() t ( t Y) dt z f( Y z) dz FY ( ) FY ( ) α α z[ α ( Y z) ] dz 0 ( α ) Y 0 α α α α ( Y z)[ ( Y z) ] dz Y ( Y z) dz ( α α α ) Y 0 0 α α α α α α Y α Y Y α αy Y ( α ) α Y α Y αy Y α α α α( α ) ( α ) Y α G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 34
hypereponental dstrbuton G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 35
hypereponental dstrbuton a process conssts of alternate phases that have eponental dstrbutons durng a sngle vst the process eperences one and only one of the many alternate phases α S α - - S - α S α α < 0 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 36
G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 37 hypereponental dstrbuton hypereponental dstrbuton ) ( ] [ ) ( 0,,,,... ) ( t t e t P T t F t e t f α α α α >
G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 38 hypereponental dstrbuton hypereponental dstrbuton. ] [ ] [ ] [ > varaton of coeff Var E E mean α α α α
hypoeponental dstrbuton (Erlang) G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 39
G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 40 Erlang (hypoeponental) dstrbuton Erlang (hypoeponental) dstrbuton a process wth K sequental phases wth dentcal eponental dstrbutons ( stageerlang) > 0! ) ( ] [ ) ( 0,,,3,... )! ( ) ( ) ( r r t t r t e t P T t F t t e t f - S/ S/ S/ S/
eample of Erlang dstrbuton (mean) 3.5 3 60.5 densty f().5 0.5 0 0 5 0 0 0.5.5.5 3 varable G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 4
G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 4 Erlang (hypoeponental) dstrbuton Erlang (hypoeponental) dstrbuton decreases varance the ncreases as S S varaton of coeff S S Var Var S S E E mean /. ] [ ] [ ] [ ] [ <
problem: eecuton tme of a web applcaton consder the global eecuton tme T of a transacton of a web applcaton on an ntranet web browser web server web applcaton db server each one of the three software components (mutually ndependent) have an average eecuton tme (response tme) of 0 ms, eponentally dstrbuted a complete eecuton of a command requre the sequental eecuton of 5 sw components (web server, web appl., db server, web appl., web server) compute the probablty that the complete eecuton tme requres more than 60 ms, more than 90 ms G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 43
problem: eecuton tme of a web applcaton the dstrbuton of the global eecuton tme T s an Erlang-5 mean E[ T ] 50 ms 50 0.0 Var[ ] E[ T ] 500 5 500ms F( t) e t /0 t 0 t 0 3 t 6 0 4 t 0 4 P[ T 60] F(60) 0.749 P[ T > 60] F(60) 0.85 P[ T 90] F(90) 0.945 P[ T > 90] F(90) 0.055 G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 44
fttng: Erlang- dstrbuton the payload of the messages of an applcaton have the followng dstrbuton lenght (m ) crt 5 50 70 00 40 frequency (f ) 0.4 0.3 0. 0.5 0.05 mean E[ ] 5 m f 5 0.4 50 0.3... 54crt Var[ ] 5 ( m E[ ]) (5 54) 0.4... 054crt E[ ] E[ ].77 Var[ ] Erlang wth E[ ] 54crt Var[ ] 54 458crt G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 45
fttng: Q-Q Q Q plot graphcal technque for determnng f two data sets come from populatons that have the same dstrbuton quantles of data set o oooo o o o o o o o o o o o o oo ooo o o oo o quantles of data set G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 46
fttng: Q-Q Q Q plot provde nsght nto the nature of the dfference between two samples the ponts of two data sets that come from two populatons wth the same dstrbuton should fall appromatvely along the reference lne the sample szes do not need to be equal probablty plot: the values of one data set are replaced wth the ones of a theoretcal dstrbuton e.g.: two data sets come from populatons whose dstrbutons dffer by a shft n locaton, the ponts should le along a straght lne that s dsplaced up/down from the reference lne G.Serazz 05/06 Dmensonamento degl Impant Informatc dstrb - 47