A hybrid method to find cumulative distribution function of completion time of GERT networks

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1 Jounal of Indusal Engneeng Inenaonal Sepembe 2005, Vol., No., - 9 Islamc Azad Uvesy, Tehan Souh Banch A hybd mehod o fnd cumulave dsbuon funcon of compleon me of GERT newos S. S. Hashemn * Depamen of Indusal Engneeng, Islamc Azad Uvesy, Scence and Reseach Banch, Tehan, Ian S.. T. Faem Ghom Depamen of Indusal Engneeng, Amab Uvesy of Technology, Tehan, Ian Absac Ths pape poposes a hybd mehod o fnd cumulave dsbuon funcon (CDF of compleon me of GERT-ype newos (GTN whch have no loop and have only exclusve-o nodes. Poposed mehod s ceaed by combng an analycal ansfomaon wh Gaussan quadaue fomula. Also he combned cude one Calo smulaon and combned condonal one Calo smulaon ae developed as alenave mehods of soluon pocedue. Then, hough a compaave sudy made fo dffeen soluon pocedues, he supeoy of hybd mehod s ndcaed. Compung me and accuacy ae consdeed as fundamenal facos fo compason puposes. Keywods: GERT newo; Compleon me; Dsbuon funcon; Hybd; Gaussan quadaue fomula; Condonal smulaon. Inoducon Gaphcal evaluaon and evew echque s a song ool fo he analyss of he ndusal engneeng poblems. Taylo and Davs [0] have exploed he use of GERT-ype newos (GTN analyss as a ool fo planng and deemng he expeced me and cos of sysem mplemenaon. Ineane and Begel [4] have descbed a modfed GERT newo whch has been developed fo auomacally acqung empoal nowledge o be used n an nellgen smulaon ang sysem. Dowson [2] has noduced a dynamc samplng echque fo he smulaon of pobablsc and genealzed acvy newos. Zmmemann [3], usng GERT newo pecedence consans, has examned me complexy of sngle-and decal paallel-machne schedulng. In hs sudy, he duaon and pecedence consans of acves ae assumed o be sochasc. Acquson of GTN compleon me can povde useful daa. Analycal mehods fo hs mae have been noduced n [6, 7, 8,, 2]. Fuhemoe Whehouse [] has noduced smulaon mehods. Kuhaa and Nshuch [5] have poposed effcen one Calo smulaon mehod o esmae GTN chaacescs such as pojec me, cos, ec. Shbanov [9] has developed an algohm o fulfll equvalen smplfyng ansfomaons of he sucue of GTN. Ths pape dscusses GTN whch have no loop and have exclusve-o nodes. The newo has one sa node and numeous end nodes. Acvy duaons ae assumed o be conuous andom vaables o consans. Ths pape also develops a hybd (analycalnumecal mehod ceaed by combng an analycal ansfomaon wh negaon hough Gaussan quadaue fomula. Ths pocedue behavo s moe exac n especve exsng smulaon mehods and even when hese mehods combne wh he poposed analycal ansfomaon of hs pape. In addon, he dffeen soluon pocedues ae capable o fnd he occuence pobably of end nodes of newos n evey abay gven me. The pape has he followng sucue: Secon 2 noduces noaons. Secon 3 ansfoms he above-meoned GTN o a GERT newo wh paallel pahs. Secons 4 and 5 combne cude one Calo smulaon and condonal one Calo smulaon wh analycal ansfomaon of secon *Coespondng auho. E-mal: s_s_hashemn@yahoo.com

2 2 S. S. Hashemn and S.. T. Faem Ghom 3. Secon 6 pesens a hybd mehod. Secon 7 gves an example o llusae he capables of hybd mehod n compason wh combned cude one Calo smulaon and combned condonal one Calo smulaon. Fnally secon 8 s devoed o he concludng emas and ecommendaons fo fuue sudes. F ˆ ( : Esmaon (appoxmaon of ( f ( : Pobably densy funcon of -h acvy duaon ( q ( f ( : f q ( n q-h smulaon un F 2. Noaons The followng noaons have been used n hs pape: N : Numbe of acves (acs : Numbe of end nodes Q : Numbe of smulaon uns n : Numbe of pahs whch sa fom sa node and emnae n -h end node S : Acvy se of j-h pah whch emnaes n -h end node P : Accomplshmen pobably of -h acvy, gven ha sa node of hs acvy has occued P : Occuence pobably of j-h pah whch emnaes n -h end node P ( : Occuence pobably of -h end node n Pˆ ( : Esmaon (appoxmaon of ( P : Occuence pobably of -h end node when + : Duaon andom vaable of -h acvy : Duaon of -h acvy n q-h smulaon un T : Compleon me of j-h pah whch emnaes n -h end node T : Compleon me of j-h pah whch emnaes n -h end node n q-h smulaon un L : Coune of T F ( F q F P whch smalle han o equal o ( : CDF of -h acvy duaon ( : ( n q-h smulaon un F ( : CDF of j-h pah whch emnaes n -h end node ( F q ( : F ( n q-h smulaon un Fˆ ( : Esmaon (appoxmaon of ( L : Save of sum of F (, S, F ( : CDF of occuence me of -h end node, gven ha hs node has occued F 3. Tansfomaon of GERT newos In GTN whch have no loop and have only exclusve-o nodes we can we: ( j= P F ( F = =,2,..., P j= When +, he lm of above equaly would be: lm + Based on P PF( = lm F ( = P + j = j = = lm + whee n = P P, we can we: n j = j= P F ( P =. P S Fo shoe me (when s emaably smalle han + we have: P ( = j= P F ( and lm P ( = P s evden. Consequenly we + can ansfom he above-meoned GTN o he n GERT newos wh paallel pahs such ha = j= j-h pah whch emnaes n -h end node consss of

3 A hybd mehod o fnd cumulave dsbuon funcon of 3 acves whch belong o S. So f we can compue F (, F ( can be defned. To compue F (, we mus defne he CDF of sum of acvy duaons whch belong o S ( =. Ths as can be done S by numeous mehods. These mehods have been noduced n he nex secons. 4. Combned cude one Calo smulaon If we geneae he andom value fo each, we can compue he compleon me of all pahs usng: T = q =, 2,...,Q S An esmaon of F ( ( ˆ ( s obaned by dvd- ng he numbe of T whch s smalle han o equal o ove Q. Then Fˆ ( and P ˆ ( can be asceaned. The followng algohm s poposed fo he abovemeoned compuaons: F Sep : q = Sep 2: Fo j-h pah whch emnaes n -h end node se L = 0. Sep 3: Geneae andom values, 2,..., N, hen compue T fo =,2,...,, j =,2,...,. ( Sep 4: If T q, hen L L +. Sep 5: Se q q +. If q Q, go o sep 3; ohewse go o sep 6. L Sep 6: Fs compue F ( =, hen compue he Q followng esmaons: 5. Combned condonal one Calo smulaon Condonal one Calo smulaon has been poposed by Bu and Gaman [] fo sochasc newo analyss. Based on T = we can we: P ( T = P( If S, hen S P(T = P( S S Usng he condonal pobably P(T, S, = P( S, S, Rgh hand sde of he above equaly epesens he condonal CDF of -h acvy and lef hand sde of he above equaly epesens he condonal CDF of compleon me of j-h pah whch emnaes n -h end node. So we can we: F (, S, = F ( S, S, The followng algohm s poposed fo combned condonal one Calo smulaon mplemenaon: Sep : q = Sep 2: Fo j-h pah whch emnaes n -h end node se L = 0. Sep 3: Geneae andom values, 2,..., N Sep 4: Compue: n ( P F ( j= S F ( = =,2,..., ( P j= S P ( ( P F ( =,2,..., = j= S Sep 7: Sop. ( (2 F (, S, = F ( S, S fo,2,..., j =,2,..., =. Then L L + F (, S,., go o sep 3; ohe- Sep 5: Se q q +. If q Q wse go o sep 6.,

4 4 S. S. Hashemn and S.. T. Faem Ghom L Sep 6: Compue F ( =. Also compue F ˆ ( and Q P ˆ ( usng fomulae ( and (2. Sep 7: Sop. 6. A hybd (analycal-numecal mehod F In he pevous secon, has been shown ha: F (, S, = F ( S, S, To avod smulaon eo, we can compue ( exacly by he followng elaon. F ( = S... F ( S S Excep fo specal cases, he above analycal compuaon s no so much easy job. So, now applcaon of Gaussan quadaue fomula ha genealzed fo sochasc newos by Faem Ghom and Hashemn [3] s beng poposed. In ulzaon of poposed numecal mehod, he deemnaon of negal s bounds s no an easy as. We have: F ( F ( S S 0 = 0 f f (f S S ( d mn{ } < mn{ } Then negaon nevals fo all negals wll be [ n{ }, ] fo S,. So we can we F ( S F ( = mn{ } mn{ } mn{ l} ( f ( d S Whee {,...,,..., l} = S { }. By compung F ( fo =,2,...,, j =,2,...,, F ( and P ( can be appoxmaed accuaely usng fomulae ( and (2. 7. Example On a poducon lne a pa s manufacued a he begnng of he lne. The manufacung opeaon s assumed o ae 4 hous. Befoe he fshng ouches ae pu on he pa, s nspeced, wh 25% of pas falng he nspecon and equng ewo. The nspecon me (ncludng wang fo nspecon s assumed o be dsbued accodng o he exponeal dsbuon, wh a mean of hou. Rewong aes 3 hous, and 30% of he pas ewoed fal he nex nspecon. The nspecon of he 3 ewoed ems s assumed o ae hou. Pas 4 whch fal hs nspecon, ae scapped. If he pa passes ehe of he above nspecons, s sen o he fnal fshng opeaon, whose me s dsbued accodng o he exponeal dsbuon wh a mean of hou n 40% of he me and hou n 60% of 2 me. A fnal nspecon, whch aes hou, ejecs 5% of he pas; hese ae scapped. The manufacue nends o now wha ae he pobables of havng non-defecve and scapped pas and he coespondng cumulave dsbuon funcon of he mes wll ae fo ecep of nondefecve and scapped pas. The GERT newo fo he above poducon lne s llusaed n Fg.. The example s desgned n such a way ha he exac analycal soluon can be found. The am has been he esablshmen of possbly fo a saghfowad compason beween he acual soluon and he soluon ganed fom he ohe mehods. The ansfomed GERT newo s shown n Fg. 2. We should fs compue P fo =,2,...,, j =,2,...,. n P = PPP 2 4P 6 = ((0.25( (0.3 = P 2 = P P2 P4 P5 P7 P0 = ((0.25((0.7(0.4(0.5 = P 3 = P P3 P7 P0 = ((0.75(0.4(0.05 = 0.05 P4 = P P2 P4 P5 P8 P0 = ((0.25((0.7(0.6(0.05 = P 5 = P P3 P8 P0 = ((0.75(0.6(0.05 = P 2 = P P2 P4 P5 P7 P9 = ((0.25((0.7(0.4(0.95 = P 22 = P P2 P4 P5 P8 P9 = ((0.25((0.7(0.6(0.95 =

5 A hybd mehod o fnd cumulave dsbuon funcon of 5 P 23 = P P3 P7 P9 = ((0.75(0.4(0.95 = P 24 = P P3 P8 P9 = ((0.75(0.6(0.95 = F ( and hen F (, P ( can be obaned exacly by analycal mehod. Fˆ( and hen F ˆ (, P ˆ ( can be obaned by: I Combned cude one Calo smulaon, II Combned condonal oe Calo smulaon, III Hybd mehod. lm P ( = s evden. Fo smalle = = + P ( <, and he pobably of none of end nodes o occu n s P (.In ohe wod fo each = poduc pobably of beng on poducon lne s P ( n. F ˆ ( and P ˆ ( ae compued fo 53 = values of hough he above mehods. Obaned esuls fo some values of ae shown n ables hough 8. Table 2. Analycal mehod esuls fo end node 2. P 2 ( F 2 ( Table. Analycal mehod esuls fo end node. P ( F ( Table 3. Combned cude one Calo smulaon esuls fo end node. Pˆ ( Fˆ(

6 6 S. S. Hashemn and S.. T. Faem Ghom Table 4. Combned cude one Calo smulaon esuls fo end node 2. Pˆ 2 ( Fˆ 2 ( Table 6. Combned condonal one Calo smulaon esuls fo end node 2. Pˆ 2 ( Fˆ 2 ( Table 5. Combned condonal one Calo smulaon esuls fo end node. Pˆ ( Fˆ ( Table 7. Hybd (analycal-numecal mehod esuls fo end node. Pˆ ( Fˆ (

7 A hybd mehod o fnd cumulave dsbuon funcon of 7 Table 8. Hybd (analycal-numecal mehod esuls fo end node Conclusons Pˆ 2 ( Fˆ 2 ( a Table 9 pesens compung me and mean absolue eo fo dffeen poposed mehods. b Hybd mehod pesens moe accuae soluons n shoe me. Fo moe complex newos, dmenson of negals and compung me wll ncease. Smulaon me wll be longe fo moe complex newos oo. c Combned condonal one Calo smulaon s moe accuae han combned cude one Calo smulaon, bu compung me of combned condonal one Calo smulaon s longe. d Wh ncease n he value of, he negaon neval wll be wde and consequenly he negaon eo nceases. Wh he dvson of negaon nevals no he smalle nevals and he compuaon of negal fo each small subnevals, he accuacy of soluons can be nceased. In he poposed example, alhough hs opeaon has no been pefomed, he soluons ganed by Gaussan quadaue fomula have been moe accuae han he soluons of smulaon mehods. e The smla eseach can be done on he GERT newos wh loop. f The smla eseach can be pefomed on he GERT newos wh nclusve-o and AND nodes. g Ths pape assumed ha he andom vaables of acves ae of addve ype. The smla eseach can be done unde he assumpon of mulplcave ype fo andom vaables of acves. h The ceaon of a powe ool has been one of he ams of hs eseach, so ha can help us n he nex sages of eseach whee he equed esouces o accomplsh acves ae lmed. Table 9. Compung me and mean absolue eo of dffeen poposed mehods. ehods Compason cea Compung me (Second ean absolue eo of compuaon fo ( P ean absolue eo of compuaon fo ( F Hybd (analycal-numecal ~ Combned cude one Calo smulaon Combned condonal one Calo smulaon

8 8 S. S. Hashemn and S.. T. Faem Ghom Fgue. The GERT newo of ypcal example. Fgue 2. Tansfomed GERT newo of ypcal example.

9 A hybd mehod o fnd cumulave dsbuon funcon of 9 Refeences [] Bu, J.. and Gaman,. B., 97, Condonal one Calo: a smulaon echque fo sochasc newo analyss. anagemen Scence, 8, [2] Dawson, C. W., 995, A dynamc samplng echque fo he smulaon of pobablsc and genealzed acvy newos. Omega, 23, [3] Faem Ghom, S.. T. and Hashemn, S. S., 999, A new analycal algohm and geneaon of Gausson quadaue fomula fo sochasc newo. Euopean Jounal of Opeaonal Reseach, 4, [4] Ineane, L. D. and Begel, J. E., 99, A modfed GERT newo fo auomac acquson of empoal nowledge. Compues and Indusal Engneeng, 2, [5] Kuhaa, K. and Nshuch, N., 2002, Effcen one Calo smulaon mehod of GERT-ype newo fo pojec managemen. Compues and Indusal Engneeng, 42, [6] Pse, A. A. B. and Happ, W. W., 966, GERT: gaphcal evaluaon and evew echque, Pa I, Fundamenals. Jounal of Indusal Engneeng, 7. [7] Pse, A. A. B. and Whehouse, G. E., 966, GERT: gaphcal evaluaon and evew echque, Pa II, Pobablsc and ndusal engneeng applcaons. Jounal of Indusal Engneeng, 7. [8] Pse, A. A. B., 966, GERT: gaphcal evaluaon and evew echque. Rand emoandum R-4973-NASA. [9] Shbanov, A.P., 2003, Fndng he dsbuon densy of he me aen o fulfll he GERT newo on he bass of equvalen smplfyng ansfomaon. Auomaon and Remoe Conol, 64, [0] Taylo, B. W. and Davs, K. R., 978, Evaluang me/cos facos of mplemenaon va GERT smulaon. Omega, 6, [] Whe house, G. A., 973, Sysem Analyss and Desgn Usng Newo Techques. Pece- Hall, Inc., Englewood Clffs, New Jesey. [2] Whehouse, G. E. and Pse, A. A. B. 969, GERT-geneang funcons, condonal dsbuons, counes, enewal mes and coelaons. AIIE Tansacon. [3] Zmmemann, J., 999, Tme complexy of sngle-and decal paallel-machne schedulng wh GERT newo pecedence consans. ahemacal ehods of Opeaons Reseach, 49,

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Name of the Student: Engneeng Mahemacs 05 SUBJEC NAME : Pobably & Random Pocess SUBJEC CODE : MA645 MAERIAL NAME : Fomula Maeal MAERIAL CODE : JM08AM007 REGULAION : R03 UPDAED ON : Febuay 05 (Scan he above QR code fo he dec