LOAD TOOL TRAFFIC GENERATION FOR LARGE USER POPULATIONS (CMG 12 Presentation Follow-up)

Transcription

1 LOAD TOOL TRAIC GENERATION OR LARGE USER POPULATIONS (CMG 2 Psnaion ollow-up) Jams Bady Capaciy Plann fo h Sa of Nvada jfbady@admin.nv.gov This is a follow-up o my CMG 2 psnaion ild Whn Load Tsing Lag Us Populaion Wb Applicaions h Dvil is in h (Viual) Us Dails, [BRAD22]. Duing my alk I mniond ha mos load ools do no off a Ngaiv-Exponnial disibuion im opion, vn hough i is h logical on o daw hink ims fom whn amping o mimic h qus pan of a lag us populaion. I followd ha commn wih a dfiniion of h simpl fomula usd o gna his yp of affic and som ands xpssd an ins in h fomula s oigin and h pincipls suounding is divaion. Th following is inndd o saisfy ha qus by, poviding h ason o poduc h affic pan,saing h im fomula, dscibing h mhodology usd o ca i, illusaing is implmnaion wih an xampl, and dailing is divaion mahmaically. Moivaion Moivaion fo gnaing Ngaiv-Exponnially disibud ims bwn quss suls fom h fac ha ansacions poducd by a lag populaion of uss, who hav no cocion bwn hm whn pssing h En Ky, a indpndn fom ach oh. This qus iming pan is fd o in h liau as a Poisson pocss [WIKI], o andom aivals, and is chaacizd by ims bwn aivals bing Ngaiv- Exponnially disibud and h numb of aivals in consan lngh invals possssing Poisson disibuion aibus. If you a unfamilia wih hs affic flow concps, an inuiiv illusaion of hm using a on m ul and wo ss of numbd chips is conaind in [BRAD29]. omula Th fomula which poducs h dsid Ngaiv-Exponnially disibud ims bwn quss is psnd by ih Eq. o Eq. 2. Using Eq. 2, h im unil h nx qus,, is obaind by muliplying minus h man im bwn quss,, by h naual logaihm of a unifomly disibud andom numb bwn zo and on, ln ( ). Eq. 2 is usually usd ov Eq. o obain ln( ) bcaus ln( ) = and = yild symmical suls and Eq. 2 quis on lss aihmic opaion han dos Eq.. Wh: = im o nx qus ln = naual log (ln = ) ( ) ( ) = ln (Eq. ) = ln (Eq. 2) = man im bwn quss = andom numb Mhodology Eq. suls fom applying a chniqu calld h Invs Tansfomaion Mhod [WIKI2] o h Ngaiv- Exponnial disibuion. This mhod, usd xnsivly in load ools and simulaion packags, poducs sampls fom h chosn pobabiliy disibuion by xploiing h fac ha h disibuion s Cumulaiv Disibuion uncion [WIKI3] posssss h sam ang of valus as a (,) andom numb gnao. Eq. 3 illusas h laionships bwn h unifomly disibud (,) andom numb,, h Cumulaiv

2 Disibuion uncion, ( ), and h ingal of h Pobabiliy Dnsiy uncion [WIKI4], f ( ) of h Invs Tansfomaion Mhod. Wh: = = andom numb Ng Exp Cum Disibuion = im o nx qus ( ) = f ( ) d (Eq. 3), wihin h conx ( ) = unc(cd) f ( ) = Ng Exp Pob Dnsiy unc(pd) Th xpssion ha yilds h dsid valus of is divd by pfoming h ingal in, valuaing i a h limis shown and solving h quaion fo. Eq. 4 is h Ngaiv-Exponnial disibuion implmnaion of Eq. 3. = ( ) = d (Eq. 4) Whn h ingal is pfomd, valuad a is limis, and h xpssion solvd fo Divaion scion of his documn conains h sp by sp mahmaical dails of h pocdu. Illusaion = f ( ) d, h sul is Eq.. Th Th gaph and associad abl in igu povid an inuiiv illusaion of h Invs Tansfomaion Mhod fo h Ngaiv-Exponnial disibuion using = 5 as h man im bwn quss. As shown, a unifomly disibud (, ) andom numb is dawn and is locaion found on h ( ) a hoizonal lin is dawn fom h ( ) axis. Thn, axis unil i inscs h Eq. cuv. Th im o dlay unil making h nx qus is h axis valu a ha inscion poin. igu : Ng-Exp Tim o Nx Rqus,

3 Th shadd xampl shows ha if =.5 is dawn h im o wai unil making h nx qus is = 3.5. Divaion Wha follows is a daild divaion of Eq. and Eq. 2 pfomd by applying h Invs Tansfomaion Mhod o h Ngaiv-Exponnial disibuion. Th fis sp in his pocss is o inga h Ngaiv-Exponnial Pobabiliy Dnsiy uncion (PD), Eq. 5 hough Eq. 9, and valua h suling Cumulaiv Disibuion uncion (CD) ov h ang ( ) hough Eq. 3.,, Eq. Th scond sp, shown in Eq. 4 hough Eq. 2, is o s h suls of his ingal qual o a (,) andom numb,, and solv fo, h im o wai bfo making h nx qus. Sp : Inga h PD ov h ang (, ): f ( ) = (Eq. 5) = 9 ( ) f ( ) d (Eq. 6) ( ) = d ( ) = d (Eq. 7) (Eq. 8) om Calculus: Evalua h CD a is limis (, ): a a d = (Eq. 9) a ( ) = (Eq. )

4 = ( ) ( ) (Eq. ) ( ) = (Eq. 2) Sp 2: s ( ) = and solv fo : ( ) = (Eq. 3) ( ) = = (Eq. 4) = (Eq. 5) (Eq. 6) = ln = ln ( ) (Eq.7) ( ) = ln (Eq. 8) ( ) = ln (Eq. 9) and : Givn h symmy of ( ) ( ) = ln (Eq.2) Summay Whn load sing lag us populaion applicaions Ngaiv-Exponnially disibud hink ims a h logical choic bu ha hink im opion is ofn no on h load ool s lis of availabl ims. Eq. 2 claly illusas is absnc isn du o mahmaical complxiy. Th fomula is simpl and, as h pvious scion dmonsas, is divaion is saigh fowad. Thos who andd my CMG 2 psnaion had ha oh hink im pobabiliy disibuions can ffcivly mimic a lag us populaion qus pan if a sufficin numb of load ool hads a implmnd. Th affic qualiy valuaion ffo associad wih his appoach, I dscibd in my alk, can b miigad by ingaing Eq. 2 ino h load ool s famwok bcaus his im algoihm poducs hink ims ha a indpndn fom had coun.

5 Rfncs [BRAD29] J.. Bady, Th Rosa Son of Taffic Concps and Is Load Tsing Implicaions, CMG MasuIT, (Spmb 29). hp:// [BRAD22] J.. Bady, Whn Load Tsing Lag Us Populaion Wb Applicaions Th Dvil Is In h (Viual) Us Dails, CMG 2 Innaional Confnc, (Dcmb 22). [WIKI2] Wikipdia, Poisson pocss, (2) hp://n.wikipdia.og/wiki/poisson_pocss. [WIKI22] Wikipdia, Invs ansfom sampling, (23) hp://n.wikipdia.og/wiki/invs_ansfom_sampling. [WIKI23] Wikipdia, Cumulaiv disibuion funcion, (23) hp://n.wikipdia.og/wiki/cumulaiv_disibuion_funcion [WIKI24] Wikipdia, Pobabiliy Dnsiy funcion, (22) hp://n.wikipdia.og/wiki/pobabiliy_dnsiy_funcion Copyighs and Tadmaks All bands and poducs fncd in his documn a acknowldgd o b h admaks o gisd admaks of hi spciv holds.