5.1 Properties of Random Numbers

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1 UNIT - 5 : RANDOM-NUMBER GENERATION, RANDOM-VARIATE GENERATION: Propertes of radom umbers; Geerato of pseudo-radom umbers; Techques for geeratg radom umbers; Tests for Radom Numbers. Radom-Varate Geerato: Iverse trasform techque; Acceptace-Rejecto techque; Specal propertes.8 Hours RANDOM-NUMBER GENERATION Radom umbers are a ecessary basc gredet the smulato of almost all dscrete systems. Most computer laguages have a subroute, object, or fucto that wll geerate a radom umber. Smlarly smulato laguages geerate radom umbers that arc used to geerate evet tmes ad other radom varables. 5. Propertes of Radom Numbers A sequece of radom umbers, R, R... must have two mportat statstcal propertes, uformty ad depedece. Each radom umber R, s a depedet sample draw from a cotuous uform dstrbuto betwee zero ad. That s, the pdf s gve by The desty fucto s show below: The expected value of R, s E( R) The varace s gve by V ( R) 3 [ x /3] / xdx [ x x dx [ E( R)] (/ ) / ] / /3 / 4 Some cosequeces of the uformty ad depedece propertes are the followg:. If the terval (, ) s dvded to classes, or subtervals of equal legth, the expected umber of observatos m each terval N/ where A' s the total umber of observatos.. The probablty of observg a value a partcular terval s of the prevous values draw. 5. Geerato of Pseudo-Radom Numbers Pseudo meas false, so false radom umbers are beg geerated. The goal of ay geerato scheme, s to produce a sequece of umbers betwee zero ad whch smulates, or tates, the deal propertes of uform dstrbuto ad depedece as closely as possble. Whe geeratg pseudo-radom umbers, certa problems or errors ca occur. These errors, or departures from deal radomess, are all related to the propertes stated prevously. Some examples clude the followg ) The geerated umbers may ot be uformly dstrbuted. ) The geerated umbers may be dscrete -valued stead cotuous valued 3) The mea of the geerated umbers may be too hgh or too low. 4) The varace of the geerated umbers may be too hgh or low 5) There may be depedece. The followg are examples: a) Autocorrelato betwee umbers. b) Numbers successvely hgher or lower tha adjacet umbers. c) Several umbers above the mea followed by several umbers below the mea. Usually, radom umbers are geerated by a dgtal computer as part of the smulato. Numerous methods ca be used to geerate the values. I selectg amog these methods, or routes, there are a umber of mportat cosderatos. ackvk@gmal.com

2 . The route should be fast. The total cost ca be maaged by selectg a computatoally effcet method of radom-umber geerato.. The route should be portable to dfferet computers, ad deally to dfferet programmg laguages.ths s desrable so that the smulato program produces the same results wherever t s executed. 3. The route should have a suffcetly log cycle. The cycle legth, or perod, represets the legth of the radom-umber sequece before prevous umbers beg to repeat themselves a earler order. Thus, f, evets are to be geerated, the perod should be may tmes that log. A specal case cyclg s degeeratg. A route degeerates whe the same radom umbers appear repeatedly. Such a occurrece s certaly uacceptable. Ths ca happe rapdly wth some methods. 4. The radom umbers should be replcable. Gve the startg pot (or codtos), t should be possble to geerate the same set of radom umbers, completely depedet of the system that s beg smulated. Ths s helpful for debuggg purpose ad s a meas of facltatg comparsos betwee systems. 5. Most mportat, ad as dcated prevously, the geerated radom umbers should closely approxmate the deal statstcal propertes of uformty ad depedeces 5.3 Techques for Geeratg Radom Numbers The lear cogruetal method It wdely used techque, tally proposed by Lehmer [95], produces a sequece of tegers, X, X,... betwee zero ad m accordg to the followg recursve relatoshp: X + (ax + c) mod m,,,... (7.) The tal value X s called the seed, a s called the costat multpler, c s the cremet, ad m s the modulus. If c Equato (7.), the form s called the mxed cogruetal method. Whe c, the form s kow as the multplcatve cogruetal method. The selecto of the values for a, c, m ad X drastcally affects the statstcal propertes ad the cycle legth. A example wll llustrate how ths techque operates. EXAMPLE Use the lear cogruetal method to geerate a sequece of radom umbers wth X 7, a 7, c 43, ad m. Here, the teger values geerated wll all be betwee zero ad 99 because of the value of the modulus. These radom tegers should appear to be uformly dstrbuted the tegers zero to 99. Radom umbers betwee zero ad ca be geerated by R X /m,,, (7.) The sequece of X ad subsequet R values s computed as follows: X 7 X (7*7 + 43) mod 5 mod R /. X (7* + 43) mod 77 mod 77 R 77 /. 77 X 3 (7*77+ 43) mod 35 mod 5 R 3 5 /. 5 Secod, to help acheve maxmum desty, ad to avod cyclg (.e., recurrece of the same sequece of geerated umbers) practcal applcatos, the geerator should have the largest possble perod. Maxmal perod ca be acheved by the proper choce of a, c, m, ad X. The max perod (P) s: For m a power of, say m b, ad c ¹, the logest possble perod s P m b, whch s acheved provded that c s relatvely prme to m (that s, the greatest commo factor of c ad m s ), ad a + 4k, where k s a teger. ackvk@gmal.com

3 For m a power of, say m b, ad c, the logest possble perod s P m / 4 b-, whch s acheved provded that the seed X s odd ad the multpler, a, s gve by a 3 + 8k or a 5 + 8k, for some k,,... For m a prme umber ad c, the logest possble perod s P m -, whch s acheved provded that the multpler, a, has the property that the smallest teger k such that a k - s dvsble by m s k m. 3 Multplcatve Cogruetal Method: Basc Relatoshp: X + a X (mod m), where a ad m (7.3) Most atural choce for m s oe that equals to the capacty of a computer word. m b (bary mache), where b s the umber of bts the computer word. m d (decmal mache), where d s the umber of dgts the computer word. Example : Usg the multplcatve cogruetal method, fd the perod of the geerator for a 3, m 6 64, ad X,, 3, ad 4. Whe the seed s ad 3, the sequece has perod 6. However, a perod of legth eght s acheved whe the seed s ad a perod of legth four occurs whe the seed s 4. Perod Determato Usg Varous seeds X X X X EXAMPLE 3 Let m, a 9, c, ad X 63, ad geerate a sequece c radom tegers usg Equato (7.). X 63 X (9)(63) mod 97 mod 97 X (9) (97) mod 843 mod 43 X3 (9) (43) mod 87 mod Whe m s a power of, say m b, the modulo operato s accomplshed by savg the b rghtmost (decmal) dgts. EXAMPLE 4 Let a 7 5 6,87, m 3 -,47,483,647 (a prme umber), ad c. These choces satsfy the codtos that sure a perod of P m-. Further, specfy a seed, X 3,457. The frst few umbers geerated are as follows: X 7 5 (3,457) mod ( 3 - ),74,94,799 mod ( 3 - ) X,74,94,799 R X / 3 ackvk@gmal.com

4 X 7 5 (,74,94,799) mod ( 3 - ) 559,87,6 R X / 3.67 X (559,87,6) mod ( 3 - ),645,535,63 R 3 X 3 / Notce that ths route dvdes by m + stead of m; however, for such a large value of m, the effect s eglgble. Combed Lear Cogruetal Geerators As computg power has creased, the complexty of the systems that we are able to smulate has also creased. Oe frutful approach s to combe two or more multplcatve cogruetal geerators such a way that the combed geerator has good statstcal propertes ad a loger perod. The followg result from L'Ecuyer [988] suggests how ths ca be doe: If W,, W,,..., W,k are ay depedet, dscrete-valued radom varables (ot ecessarly detcally dstrbuted), but oe of them, say W,, s uformly dstrbuted o the tegers to m, the k j W ( ) W, j modm j s uformly dstrbuted o the tegers to m. To see how ths result ca be used to form combed geerators, let X,, X,,..., X,k be the th output from k dfferet multplcatve cogruetal geerators, where the j th geerator has prme modulus m j, ad the multpler a j s chose so that the perod s m j. The the j'th geerator s producg tegers X,j that are approxmately uformly dstrbuted o to m j -, ad W,j X,j s approxmately uformly dstrbuted o to m j -. L'Ecuyer [988] therefore suggests combed geerators of the form k j X ( ) X, j modm j X, X > m Wth R m, X m Notce that the (-) j- coeffcet mplctly performs the subtracto X, -; for example, f k, the j - X X X ( ) ( ) ( ) ( ) ( ),, j, j The maxmum possble perod for such a geerator s ( m )( m )...( m ) p k k EXAMPLE 5 For 3-bt computers, L'Ecuyer [988] suggests combg k geerators wth m , a 44, m , ad a 469. Ths leads to the followg algorthm: ) Select seed X, the rage [,,4,74,83,56] for the frst geerator, ad seed X, the rage [,,47,483,399]. Set j. ) Evaluate each dvdual geerator. ) X, j+ 44 X,j mod,47,483,563 ) X,j+ 469 X,j mod,47,483,399 3) Set X j+ (X, j+ - X, j+ ) mod,47,483,56. 4) Retur 4 ackvk@gmal.com

5 X j +, X j + >,47,483,563 R,47,483,56, X j +,47,483,563 5) Set j j + ad go to step. 5 Tests For Radom Numbers. Frequecy test. Uses the Kolmogorov-Smrov or the ch-square test to compare the dstrbuto of the set of umbers geerated to a uform dstrbuto.. Rus test. Tests the rus up ad dow or the rus above ad below the mea by comparg the actual values to expected values. The statstc for comparso s the ch-square. 3. Autocorrelato test. Tests the correlato betwee umbers ad compares the sample correlato to the expected correlato of zero. 4. Gap test. Couts the umber of dgts that appear betwee repettos of a partcular dgt ad the uses the Kolmogorov-Smrov test to compare wth the expected umber of gaps. 5. Poker test. Treats umbers grouped together as a poker had. The the hads obtaed are compared to what s expected usg the ch-square test. I testg for uformty, the hypotheses are as follows: H : R ~ U[,] H : R ɫ U[,] The ull hypothess, H, reads that the umbers are dstrbuted uformly o the terval [, ]. I testg for depedece, the hypotheses are as follows; H : R ~ depedetly H : R ɫ depedetly Ths ull hypothess, H, reads that the umbers are depedet. Falure to reject the ull hypothess meas that o evdece of depedece has bee detected o the bass of ths test. Ths does ot mply that further testg of the geerator for depedece s uecessary. Level of sgfcace a a P (reject H H true) Frequetly, a s set to. or.5 (Hypothess) Actually True Actually False Accept - α β (Type II error) Reject α (Type I error) - β Frequecy Tests A basc test that should always be performed to valdate a ew geerator s the test of uformty. Two dfferet methods of testg are avalable.. Kolmogorov-Smrov ad. Ch-square test. Both of these tests measure the degree of agreemet betwee the dstrbuto of a sample of geerated radom umbers ad the theoretcal uform dstrbuto. Both tests are o the ull hypothess of o sgfcat dfferece betwee the sample dstrbuto ad the theoretcal dstrbuto.. The Kolmogorov-Smrov test. Ths test compares the cotuous cdf, F(X), of the uform dstrbuto to the emprcal cdf, SN(x), of the sample of N observatos. By defto, F(x) x, x ackvk@gmal.com

6 If the sample from the radom-umber geerator s R R,,..., RN, the the emprcal cdf, SN(x), s defed by umber of R,R,, R whch are x S ( x) N The Kolmogorov-Smrov test s based o the largest absolute devato betwee F(x) ad SN(X) over the rage of the radom varable. That s. t s based o the statstc D max F(x) -S N (x) For testg agast a uform cdf, the test procedure follows these steps: Step : Rak the data from smallest to largest. Let R () deote the th smallest observato, so that R () R () R (N) Step : Compute D D + max max N R ( ) R ( ) N Step 3: Compute D max (D+, D-). Step 4: Determe the crtcal value, D α, from Table A.8 for the specfed sgfcace level α ad the gve sample sze N. Step 5: D D Accept: No Dfferece betwee S N (x) ad F(x) D > α D α Reject: No Dfferece betwee S N (x) ad F(x) We coclude that o dfferece has bee detected betwee the true dstrbuto of {R, R,... R N } ad the uform dstrbuto. 6 EXAMPLE 6: Suppose that the fve umbers.44,.8,.4,.5,.93 were geerated, ad t s desred to perform a test for uformty usg the Kolmogorov-Smrov test wth a level of sgfcace α of.5. Step : Rak the data from smallest to largest..5,.4,.44,.8,.93 Step : Compute D + ad D - R N D + ( ) max R D max R( ) N N ~ ~ Step3: Compute D max (D+, D-). Dmax (.6,.).6 Step 4: Determe the crtcal value, D α, from Table A.8 for the specfed sgfcace level α ad the gve sample sze N. Here α.5, N5 the value of D α.565 Step 5: Sce the computed value,.6 s less tha the tabulated crtcal value,.565, the hypothess of o dfferece betwee the dstrbuto of the geerated umbers ad the uform dstrbuto s ot rejected. ackvk@gmal.com

7 . The ch-square test. The ch-square test uses the sample statstc χ ( O E ) E Where, O s observed umber the th class E s expected umber the th class, N E N No. of observato No. of classes Note: samplg dstrbuto of χ s approxmately the ch square has - degrees of freedom Example 7: Use the ch-square test wth α.5 to test whether the data show below are uformly dstrbuted. The test uses tervals of equal legth, amely [,.), [.,.)... [.9,.). (REFER TABLE A.6) 7 Computatos for ch square test Iterval Rage O E O E ( ) O ( O E ) E E The value of tabulated value of χ χ s 3.4. Ths s compared wth the crtcal value.5,9 χ.5,9 6.9.Sce, the ull hypothess of a uform dstrbuto s ot rejected. χ s much smaller tha the ackvk@gmal.com

8 Ru Tests (Up ad Dow) The rus test exames the arragemet of umbers a sequece to test the hypothess of depedece. A ru s defed as a successo of smlar evets preceded ad followed by a dfferet evet. E.g. a sequece of tosses of a co, we may have H T T H H T T T H T The frst toss s preceded ad the last toss s followed by a "o evet". Ths sequece has sx rus, frst wth a legth of oe, secod ad thrd wth legth two, fourth legth three, ffth ad sxth legth oe. A few features of a ru two characterstcs:. umber of rus ad the legth of ru. a up ru s a sequece of umbers each of whch s succeeded by a larger umber; a dow ru s a squece of umbers each of whch s succeeded by a smaller umber Cosder the 4 umbers; both the Kolmogorov-Smrov ad Ch-square would dcate that the umbers are uformly dstrbuted. But, ot so Now, rearrage ad there s less reaso to doubt depedece Cocers: Number of rus Legth of rus Note: If N s the umber of umbers a sequece, the maxmum umber of rus s N-, ad the mmum umber of rus s oe. If a s the total umber of rus a sequece, the mea ad varace of a s gve by µ a (N - ) / 3 σ a (6N - 9) / 9 For N >, the dstrbuto of a approxmated by a ormal dstrbuto, N (µ a, σ ). Ths approxmato ca be used to test the depedece of umbers from a geerator. Substtutg for µ a ad σ a > Z (a - µ a ) / σ Z a a { a [( N ) 3] } { (6N 9) /9} where ~ N(,) Acceptace rego for hypothess of depedece Zα Z Zα Z a a 8 α / α / -Z α / Z α / ackvk@gmal.com

9 Example 8: Based o rus up ad rus dow, determe whether the followg sequece of 4 umbers s such that the hypothess of depedece ca be rejected where a The sequece of rus up ad dow s as follows: There are 6 rus ths sequece. Wth N4 ad a6, 9 µ a {(4) - } / ad σ a {6(4) - 9} / The, Z ( ) Now, the crtcal value s Z.5.96 (Ht: Z a/ Z.5/ Z.5 ) Refer Table A.3, so the depedece of the umbers caot be rejected o the bass of ths test. Poker Test It based o the frequecy wth whch certa dgts are repeated. Example 9: Note: Par of lke dgts appear each umber geerated. I 3-dgt umbers, there are oly 3 possbltes.. The dvdual dgts ca be all dfferet. Case.. The dvdual dgts ca all be the same. Case. 3. There ca be oe par of lke dgts. Case 3. P(3 dfferet dgts) (d dff. from st) * P(3rd dff. from st & d) (.9) (.8).7 P(3 lke dgts) (d dgt same as st) * P(3rd dgt same as st) (.) (.). P(exactly oe par) Example : A sequece of three-dgt umbers has bee geerated ad a aalyss dcates that 68 have three dfferet dgts, 89 cota exactly oe par of lke dgts, ad 3 cota three lke dgts. Based o the poker test, are these umbers depedet? Let a.5. The test s summarzed ext table. Combato, Observed Frequecy, O Expected Frequecy, E ( O E ) E Three dfferet dgts Three lke dgts Exactly oe par The approprate degrees of freedom are oe less tha the umber of class tervals. Sce χ.5, 5.99 < 47.65(Table A.6), the depedece of the umbers s rejected o the bass of ths test. ackvk@gmal.com

10 The Gap Test It measures the umber of dgts betwee successve occurreces of the same dgt. A gap of legth x occurs betwee the recurrece of some dgt. Steps volved the test. Step : Specfy the cdf for the theoretcal frequecy dstrbuto gve by Equato below based o the selected class terval wdth. F x x+ ( x). (.9).9 Step : Arrage the observed sample of gaps a cumulatve dstrbuto wth these same classes. Step 3: Fd D, the maxmum devato betwee F(x) ad S N (x) as Equato. D max F(x) -S N (x) Step 4: Determe the crtcal value, D α, from Table A.8 for the specfed value of α ad the sample sze N. Step 5: If the calculated value of D s greater tha the tabulated value of D α, the ull hypothess of depedece s rejected. Example: legth of gaps assocated wth the dgt 3. 4,, 3, 5,, 7,, 8,,, 7, 9,, 3, 5,, 7, 9, 4,, 6, 3 3, 9, 6, 3, 4, 8,, 3,, 9, 4, 4, 6, 8, 4,, 3, 8, 9, 5, 5, 7 3, 9, 5, 9, 8, 5, 3,,, 3, 7, 4, 7,, 3, 6, 3, 5, 9, 9, 5, 5 5,, 4, 6, 8,, 4, 7,, 3, 3,, 9, 5, 7, 9, 5,, 6, 6, 3, 8 8, 8, 9,, 9,, 8, 5, 4, 4, 5,,, 3, 9, 7,,,, 3, 6, 3 Note: eghtee 3 s lst > 7 gaps, the frst gap s of legth We are terested the frequecy of gaps. P (gap of ) P (ot 3) P (ot 3) P (3), Note: there are terms of the type P (ot 3) (.9) (.) The theoretcal frequecy dstrbuto for radomly ordered dgt s gve by F x x+ ( x). (.9).9 Note: observed frequeces for all dgts are compared to the theoretcal frequecy usg the Kolmogorov-Smrov test. Example : Based o the frequecy wth whch gaps occur, aalyze the dgts above to test whether they are depedet. Use a.5. The umber of gaps s gve by the umber of dgts mus, or. The umbers of gaps assocated wth the varous dgts are as follows: Dgt # of Gaps ackvk@gmal.com

11 Gap Legth Frequecy Relatve Frequecy Cumulatve Relatve Frequecy F(x) F(x) - S N (x) The crtcal value of D s gve by D Sce D max F(x) - S N (x).4 s less tha D.5, do ot reject the hypothess of depedece o the bass of ths test. Tests for Auto-correlato The tests for auto-correlato are cocered wth the depedece betwee umbers a sequece. The lst of the 3 umbers appears to have the effect that every 5th umber has a very large value. If ths s a regular patter, we ca't really say the sequece s radom. The test computes the auto-correlato betwee every m umbers (m s also kow as the lag) startg wth the th umber. Thus the autocorrelato Form the test statstc Z ρm betwee the followg umbers would be of terest. R, R, m R,.., + + m R + ( M + ) m ρm ˆ whch s dstrbuted ormally wth a mea of zero ad a varace of oe. σ ρ m ˆ M The actual formula for ρ ad the stadard devato s ρ îm ˆ ( ). 5 R + kmr k + m + ad m M k 3M + 7 σ ρ m ˆ ( M + ) After computg Z, do ot reject the ull hypothess of depedece f z where α s the level of sgfcace. Z z α α EXAMPLE : Test whether the 3rd, 8th, 3th, ad so o, umbers the sequece at the begg of ths secto are auto correlated. (Use a.5.) Here, 3 (begg wth the thrd umber), m 5 (every fve umbers), N 3 (3 umbers the sequece), ad M 4 (largest teger such that 3 + (M +)5 < 3). ackvk@gmal.com

12 Ad σ SYSTEM MODELING AND SIMULATION UNIT-5 VIK 3(4) + 7 ( 4 + ) ρ m ˆ ρ m ˆ [(.3)(.8) + (.8)(.33) + (.33)(.7) + (.7)(.5) + (.5)(.36) ] The, test for statstc assumes the value.945 Z.56.8 Now the crtcal value from Table A.3 s Z.5.96 Therefore, the hypothess of depedece ca t be rejected o the bass of ths test..5 Radom Varate Geerato TECHNIQUES: INVERSE TRANSFORMATION TECHNIQUE ACCEPTANCE-REJECTION TECHNIQUE All these techques assume that a source of uform (,) radom umbers s avalable R,R.. where each R has probablty desty fucto ad cumulatve dstrbuto fucto. Note: The radom varable may be ether dscrete or cotuous. Iverse Trasform Techque The verse trasform techque ca be used to sample from expoetal, the uform, the Webull ad the tragle dstrbutos. Expoetal Dstrbuto The expoetal dstrbuto, has probablty desty fucto (pdf) gve by ad cumulatve dstrbuto fucto (cdf) gve by The parameter λ ca be terpreted as the mea umber of occurreces per tme ut. For example, f terarrval tmes X, X, X 3... had a expoetal dstrbuto wth rate, ad the could be terpreted as the mea umber of arrvals per tme ut, or the arrval rate. For ay, E(X ) /λ ackvk@gmal.com

13 Ad so /λ s mea terarrval tme. The goal here s to develop a procedure for geeratg values X, X, X 3... whch have a expoetal dstrbuto. The verse trasform techque ca be utlzed, at least prcple, for ay dstrbuto. But t s most useful whe the cdf. F(x), s of such smple form that ts verse, F -, ca be easly computed. A step-by-step procedure for the verse trasform techque llustrated by me expoetal dstrbuto, s as follows: Step : Compute the cdf of the desred radom varable X. For the expoetal dstrbuto, the cdf s F(x) -e -λx, x. Step : Set F(X) R o the rage of X. For the expoetal dstrbuto, t becomes e -λx R o the rage x. Sce X s a radom varable (wth the expoetal dstrbuto ths case), so -e -λx s also a radom varable, here called R. As wll be show later, R has a uform dstrbuto over the terval (,)., Step 3: Solve the equato F(X) R for X terms of R. For the expoetal dstrbuto, the soluto proceeds as follows: e -λx R e -λx R -λx l( - R) x-/λ l( R) ( 5. ) Equato (5.) s called a radom-varate geerator for the expoetal dstrbuto. I geeral, Equato (5.) s wrtte as XF - (R). Geeratg a sequece of values s accomplshed through steps 4. 3 Step 4: Geerate (as eeded) uform radom umbers R, R, R3,... ad compute the desred radom varates by X F - (R ) For the expoetal case, F - (R) (-/λ)l(- R) by Equato (5.), so that X -/λ l ( R ) ( 5. ) for,,3,... Oe smplfcato that s usually employed Equato (5.) s to replace R by R to yeld X -/λ l R ( 5.3 ) whch s justfed sce both R ad - R are uformly dstrbuted o (,). Table 5. Geerato of Expoetal Varates X, wth Mea, Gve Radom Numbers R, I R X Table 5. gves a sequece of radom umbers from Table A. ad the computed expoetal varates, X, gve by Equato (5.) wth a value of. Fgure 5.(a) s a hstogram of values, R, R, R from the uform dstrbuto ad fgure 5.(b) ackvk@gmal.com

14 4 (a) (b) Fgure 5.: (a) Emprcal hstogram of uform radom umbers; (b) emprcal hstogram of expoetal varates; (c) theoretcal uform desty o (, ); (d) theoretcal expoetal desty wth mea. Fgure 5.. Graphcal vew of the verse-trasform techque Acceptace-Rejecto techque Useful partcularly whe verse cdf does ot exst closed form Illustrato: To geerate radom varates, X ~ U(/4, ) Procedures: Step : Geerate a radom umber R ~ U [, ] ackvk@gmal.com

15 Step a: If R ¼, accept XR. Step b: If R < ¼, reject R, retur to Step R does ot have the desred dstrbuto, but R codtoed (R ) o the evet {R ³ ¼} does. Effcecy: Depeds heavly o the ablty to mmze the umber of rejectos. 5 Posso Dstrbuto A Posso radom varable, N, wth mea a > has pmf α e α p( ) P( N ),,,,...! N ca be terpreted as umber of arrvals from a Posso arrval process durg oe ut of tme The tme betwee the arrvals the process are expoetally dstrbuted wth rate α. Thus there s a relatoshp betwee the (dscrete) Posso dstrbuto ad the (cotuous) expoetal dstrbuto, amely N + A < A A < + A l R α R e α < > + + R l R α The procedure for geeratg a Posso radom varate, N, s gve by the followg steps: Step : Set, ad P Step : Geerate a radom umber R + ad let P P. R + Step 3: If P < e -α, the accept N. Otherwse, reject curret, crease by oe, ad retur to step Nostatoary Posso Process (NSPP): A Posso arrval process whose arrval rate (l ) chages over tme. Thk fast food. Arrval rates at the luch ad der hour much greater tha arrval rates durg off hours. Thg Process: Geerates Posso arrvals at the fastest rate, but accept oly a porto of the arrvals, effect thg out just eough to get the desred tme-varyg rate. Nostatoary Posso Process (NSPP) Thg Algorthm: To geerate successve arrval tme (T) whe rates vary: Step Let λ* max t T λ (t) be the maxmum arrval rate, ad set t ad. Step Geerate E from the expoetal dstrbuto wth rate λ*, ad let t t+e (the arrval tme of the ext arrval usg max rate). Step 3 Geerate radom umber R from U [, ). If R < λ (t) / λ* the T t ad +. Step 4 Go to step. Note: Some part of radom varate geerato chapters are ot covered please refer to text book. ackvk@gmal.com

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