Iteratoal Mathematcal Forum,, 2006, o. 40, 2003 207 A Mult-Etry Smulated ad Iversed Fucto Approach for Alteratve Solutos Kev Wag a, Che Chag b ad Chug Pg Lu b a Computg ad Mathematcs School Joh Moores Uversty, Lverpool, U.K trust236@ms5.het.et b Departmet of Electrcal Egeerg Yua Ze Uversty, Tawa, R.O.C cchag@mal.tt.edu.tw Abstract Usually, a smulated ad versed fucto assocated wth ts geerated radom varables s used as a smulato put supportg alteratve solutos. Sgle-etry smulated ad versed fucto s popularly used dustry maagemet for may years. But most of cases, mult-etry compoets problems exsted such as the mult-meda traffc (voce, data & fax) telecommucato etwork ad the mult-szed cotaer (cotaer szed wth 20, 30 & 40 feet) mare trasportato maagemet. The purpose of ths compoud versed fucto approach s to solve the mult-etry alteratve problems. Furthermore, a case study of etwork mult-servce path traffc evet by MATLAB geetc module s demostrated to verfy the model we propose. Keywords: Mult-etry versed fucto; smulated; ARIMA; K-S test; geetc
2004 Kev Wag, Che Chag ad Chug Pg Lu. Itroducto Usually, a smulated ad versed fucto assocated wth ts geerated radom varables s used as a smulato put supportg alteratve solutos. Sgle-etry smulated ad versed fucto s popularly used dustry maagemet for may years. But most of cases, mult-etry compoets problems exsted such as the mult-meda traffc (voce, data & fax) telecommucato etwork ad the mult-szed cotaer (cotaer szed wth 20, 30 & 40 feet) mare trasportato maagemet. The purpose of ths compoud versed fucto approach s to solve the mult-etry alteratve problems [] [2]. Furthermore, a case study of etwork mult-servce path traffc evet by MATLAB geetc module s demostrated to verfy the model we propose. Ths approach maly cludes several processes: () Buldg a mult-etry versed model; (2) Verfy the selected dstrbuto; (3) Alteratve dex calculato & decso makg. 2. Mathematcal Modelg The appled mathematcal model defto ad formulato of ths study are explaed as follows. 2. Formula A: A sgle-etry versed fucto Let x be a varable wth cumulatve dstrbuto of F j (x), j deotes each data samplg evet. For F j (x) s a o-decreasg fucto, the verse smulato fucto of F j (y) could be defed as F (y)=f{x;f j (x) y} 0 y () j Assume that samplg data ca be grouped to adjacet tervals [b 0, b ], [b,
A Mult-Etry Smulated ad Iversed Fucto Approach 2005 b 2 ),.,[b -, b ], ad defe the pecewse-lear fucto: F(x) j 0 F(b j ) + x b b b [ F(b ) F(b )],x < b = j j Where F j (b 0 ) = 0, F (b ) r j = k k = / r,, b 0,x > b x < b =,2,.., r s the umber of observatos the th terval (b -, b ) ad r = for somel < r = (2) Fd the mmum postve teger k(0 k -), such that Y F(X k ) ad retur X= F (Y)=bk+[Y-F j (b k )] (b k+ -b k )/[F j (b k+ )-F j (b k )] (3) j 2.2 Formula B: A mult-etry versed fucto Defe a compoud fucto as a put of mult-etry versed formula (3) Fˆ (x) F (x)p (x (4) = j= j j ) P = where 0< P j, ad Pj deotes the Percetage Weght (PW) value of j j= servce etry j. The summato of the percetage parameters of voce, data ad fax s equal to value. Usually, there are two cases for the calculato of PW values: a. Case : If the mult-servce traffc s a stable stage, the the costat PW values of each servce wll be derved by samplg data statstcs drectly
2006 Kev Wag, Che Chag ad Chug Pg Lu b. Case 2: Otherwse, the mult-servce traffc s a bg varato stage, a statstcal ARIMA (Auto-Regresso Itegrated Movg Average) tme-seres estmato formula for each telecommucato servce (voce, data, fax) PW value wll be used. Costruct a statstcal ARIMA (Auto-Regresso Itegrated Movg Average) tme-seres estmato formula for each telecommucato servce (voce, data, fax) as Formula C [3] [4]. 2.3 Formula C: A ARIMA formula of order (p, d, q) Auto-regresso model Y t = + φ Yt + φ Yt + K+ 2 2 φpyt p et (5) where φ : Idex of auto-regresso ; : Servce observato of tme φ Y t seres t- term ; φ p Y t p : Servce observato of tme seres t-p term; et : Partal error Movg average model: Yt = et θ et θ 2 et 2 K θqet q (6) where θ :Servce observato of tme seres t term ; θ et :Partal error of tme seres t- term; θ q et p :Partal error of tme seres t-p term Itegrated model: Yt = φ Yt + φ Yt 2+ L+ φ Yt p+ et θ et L θ qet 2 p q (7)
A Mult-Etry Smulated ad Iversed Fucto Approach 2007 I ARIMA (p, d, q) model, p deotes the order of Auto-Regresso, d deotes the order of dfferece, q deotes the order of Movg Average. 2.4 Formula D: K-S model verfcato test The authors use K-S testg to verfy the verse fucto model [5] [6]. It s defed as the follows: F(x) = x, 0 x (8) R,R,...,R S x = N N ( ) 2 N x (9) Where N s the largest observato umber ad S N (x) s the optmal approxmato value of F(x). Step : Rak the data from smallest to largest, let deote the th smallest observato, so that () ( 2) L ( N ) R R R Step 2: Compute D + = max R () N N (0); = max () D N R N () Step 3:Calculate D=max ( + D, D ) (2) Step 4: Determe the crtcal value D, for the specfed sgfcace level α α ad the gve sample sze N. Step 5: If the sample statstc D s greater tha the crtcal value D,the ull hypothess that the data a sample from a uform dstrbuto s rejected If D D α, coclude that o dfferece has bee detected betwee the true dstrbuto of { R, R2,..., R N }ad the uform dstrbuto. α
2008 Kev Wag, Che Chag ad Chug Pg Lu 2.5 Formula F: A queue ad related score fucto The queue modelg M/M/K ca be used as a bass to calculate the related performace parameter values. Further, the related explaatos are provded the followg part. ρ = λ k µ (3) L = kρ + ( kρ) k(k! )( k + p 0 ρ) 2 (4) L ω = (5) λ where λ deotes ter-arrval rate; µ deotes servce rate; ρ deotes system utlzato ;L deotes servce umber system; ω deotes delay tme (servce tme); ad P o deotes the probablty dle status. For smplcty of computg processes, a lear trasformato fucto of score fucto ragg from to 0 was employed supportg the alteratve dexes calculato. 2.6 Formula E: A object fucto By usg the geetc aalyss [7] [8] of MATLAB software package, the g X, Y supportg alteratve authors defe a geeral object fucto ( ) soluto: g (, ) MAX X Y (6) = c k, ST α X k=, 2,, l (7) s= k,s s β Y d, k=, 2,, l (8) s = k,s s k where g (X,Y ) deotes the score fucto path performace parameters; X s, Y s deote the path performace parameters; α k,s, β k,s deote the coeffcets to acheve the studed path ; c k, d k deote the costraed upper boud values ; l deotes the umber of costraed codtos.
A Mult-Etry Smulated ad Iversed Fucto Approach 2009 Iter-Arrval Tme Relatve Cumulatve Iverse Frequecy (Mll Sec.) Frequecy Frequecy referece pot 0.5 x<2.0 0 0.0 0.0 (0.0,2.00) 2.0 x<4.0 5 0.5 0.25 (0.35,4.00) 4.0 x<6.0 20 0.20 0.45 (0.75,6.00) 6.0 x<8.0 40 0.40 0.85 (0.85,8.00) 8.0 x 0.0 5 0.5.00 (.00,0.00) Table. Voce pack servce samplg frequecy 3. Appled Example 3.. Samplg & Model Verfcato below: Frstly, ths study takes a group of etwork samplg data as above ad Table 2. Data pack servce samplg frequecy Iter-Arrval Tme (Mll Sec.) Frequecy Relatve Frequecy Cumulatve Frequecy Iverse referece pot 0.5 x<2.0 25 0.25 0.25 (0.25,2.00) 2.0 x<4.0 40 0.40 0.65 (0.65,4.00) 4.0 x<6.0 20 0.20 0.85 (0.85,6.00) 6.0 x<8.0 0 0.0 0.95 (0.95,8.00) 8.0 x 0.0 5 0.05.00 (.00,0.00)
200 Kev Wag, Che Chag ad Chug Pg Lu F - (y) Iter-Arrval Tme (Mll Secods) Servce Tme(M Secods) 0.0 8.0 6.0 4.0 2.0 0.5 0.0 (.00, 0.0) (0.95, 8.0) (0.85, 6.0) (0.65, 4.0) (0.25, 2.0) 0.2 0.4 0.6 0.8.0 Fg Voce pack verse fucto Fg 2 Compoud verse fucto
A Mult-Etry Smulated ad Iversed Fucto Approach 20 Fg 3 ARIMA ( p=0, d=, q=) voce servce PW values estmato Table 3 A example of K-S test model verfcato Terms K-S 2. 99 00 Calculatos R() 0.02 0.05.. 0.969.000 / N 0.025 0.05.. 0.975.000 / N R() 0.003 -.. 0.005 0.000 R / () ( ) N 0.02 0.026. 0.09 0.025 A example of Table 3 s the Calculato parameters of each term for K-S test, here K-S Calculato deotes the parameters. Terms deote the dexes we make for ths statstcs. From ths table, the authors ca fd ad D = max R() N N D + = max R () N N values, ad fally, we ca get D=max ( + D, D ) for the decso model fttg. H 0 : the radom varable of falure ter-arrval tme s expoetally dstrbuted; H : ot expoetally dstrbuted. A example of D=0.063, α = 0. 05, Dα =0.36 ; where D value derved from emprcal test of Table 3 ad Dα value derved from statstcal referece table; If D, accept the H0: the radom varable expoetally dstrbuted. Oce the hypothess D α
202 Kev Wag, Che Chag ad Chug Pg Lu model of K-S test s accepted, the authors ca use the selected dstrbuto parameters to calculate the queue performace dexes; otherwse, whle model test rejected, the classcal statstcal mmum varace method wll be the, used to estmate the performace dex. 3.2 Alteratve Study Secodly, for smplcty, a MATLAB software geetc program assocated wth a object fucto s used to support the followg research [9][0]. (Usg path -2-4-7-9 as a example Fg 4) Fg 4 Network archtecture of ths study Table 4 Factor scores of each lk Lk 2 2 4 4 7 7 9 X 2.6500.8838 3.046 4.300 Y 6.6000 7.265 6.866 7.7000 Table 4 has factor scores for each lk. For each lk, there are two factor scores X ad Y, X s a lear trasformato of delay tme, ad Y s a lear trasformato of utlzato. There are some of geetc parameter deftos: () Objectve
A Mult-Etry Smulated ad Iversed Fucto Approach 203 Fucto: X 2 + Y 2 - XY ; (2) No. of populato: 500; (3) Crossover rate: 0.9; (4) Mutato probablty: 0.05; (5) Geerato maxmum evoluto: 30 Table 5. A geetc example for alteratve (3: p=0, d=, q=) No. of geeratos Optmal values of ftess Average values of ftess Total values of ftess No. of populato 0 763.29 370.26 853.33 500 02 763.29 435.40 27702.53 500 03 77.38 478.05 239028.68 500 04 77.38 525.05 262525.93 500 05 77.38 543.9 27954.89 500 06 77.38 565.50 282753.60 500 07 777.84 58.9 290595.83 500 08 778.35 599.84 299920. 500 :: :: :: :: :: 9 778.35 675.4 337705.5 500 20 778.35 670.36 33583.42 500 2 779.63 676.73 338365.87 500 22 779.63 676.58 338293.90 500 23 779.63 682.49 34247.6 500 :: :: :: :: :: 29 779.63 686.62 34333.84 500 30 779.63 696.66 348334.08 500
204 Kev Wag, Che Chag ad Chug Pg Lu I Table 5, the geetc expermet result of alteratve s derved by umerc tred aalyss. Ths aalyss shows that f the teratve of geerato over 2, the optmal ftess value wll be rather smoothg. Therefore, geeral, ths ftess value of geerato 2 ca cosdered as optmal ftess value for alteratve. Table 6 Maaged alteratves defto Alteratves of Strateges Traffc Status Attrbute Itesty Order (p, d, q) of ARIMA A < 75% & B+C >25% L p=0,d=0,q= 2 A < 85% & B+C >5% M p=0,d=,q=0 3 A < 85% & B+C >5% M p=0,d=,q= 4 A < 85% & B+C >5% M p=,d=0,q=0 5 A > 85% & B+C < 5% H p=,d=0,q= 6 A > 85% & B+C < 5% H p=,d=,q=0 7 A > 85% & B+C < 5% H p=,d=,q= (Here, A deotes the PW value of voce servce, B deotes the PW value of data servce, C deotes the PW value of voce servce, H deotes Hgher performace requremet, M deotes Mddle performace requremet, L deotes Lower performace requremet)
A Mult-Etry Smulated ad Iversed Fucto Approach 205 Table 7 Geetc ftess dexes of alteratve expermets Smulato Data Numbers Alteratves of PW Values of Dfferet ARIMA Order (p, d, q)for Mult-etry Iversed Fuctos Sum of Averaged Square Error Mult-etry Iversed Fucto for (Voce, Data, Fax) 500 000 2000 Averaged Geetc Ftess Idexes (AGFI) : p=0,d=0,q= A 0.0260 (0.72,0.5,0.3) 532.63 552.39 577.59 554.20 2: p=0,d=,q=0 B 0.02352 (0.77,0.3,0.0) 750.5 802.43 843.27 798.74 3: p=0,d=,q= B 0.03570 (0.79,0.2,0.09) 779.63 830.30 887.59 832.5 4: p=,d=0,q=0 B 0.0360 (0.8,0.,0.08) 052.87 27.82 95.52 25.40 5: p=,d=0,q= C 0.03792 (0.86,0.09,0.05) 285.98 369.56 464.06 373.20 6: p=,d=,q=0 C 0.03876 (0.88,0.08,0.04) 375.63 463.67 563.9 467.50 7: p=,d=,q= C 0,04329 (0.9,0.06,0.03) 498.65 595.56 703.79 599.33 Fally, Table 7, the authors also show the possble decsos/actos of Network Servce Provder (NSP). Usually, the etwork users have dfferet performace requremets accordg to how much trasmsso fee they wat to pay, therefore, NSP defes servce strateges wth the costraed codto of Sum of Averaged Square Error less tha 0.05 as: A. Supportg lower performace requremet of path desg (AGFI less tha 600) B. Supportg mddle performace requremet of path desg (AGFI greater tha 600 ad less tha 200) C. Supportg hgher performace requremet of path desg (AGFI greater tha 200)
206 Kev Wag, Che Chag ad Chug Pg Lu Fg 5. AGFI lst of dfferet alteratves 4. CONCLUSION. I ths paper, the authors preseted a smulated aalyss patter to support mult-servce etwork path alteratves. I specfc, a seres of smulated procedures are provded for the readershp to follow up by ths artcle. At the same tme, ths evaluato model ca also be used to meet the requremet of the maagemet level s decso makg actvtes may other dustres. 5. REFERENCE [] Jerry Baks etc., Dscrete-evet system smulato, Pretce Hall Iteratoal Edto, pp. 355-390, 996. [2] Reuve Y. Rubste Moder Smulato ad Modelg, A Wley Iter-scece Publcato, 998, pp. -60. [3] Tra, N.; Reed, D.A., Automatc ARIMA tme seres modelg for adaptve I/O, Parallel ad Dstrbuted Systems, IEEE Trasactos o, Volume: 5 Issue: 4 Aprl 2004, Page(s): 362-377. [4] Papagaak, K.; Taft, N.; Zh-L Zhag; Dot, C., Log-term forecastg of Iteret backboe traffc, Neural Networks, IEEE Trasactos o, Volume: 6 Issue: 5 Sept. 2005, Page(s): 0-24.
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