Cost-Benefit Analysis for Network Reliability Improvement

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1 Cost-Benefit Anlysis fo Netwok elibility Impovement huming FANG, Hioshi WAKABAYAHI tudent Membe of JCE, Fculty of Ubn cience, Meijo Univesity (Nijigok --, Kni, Gifu 09-06, Jpn) Membe of JCE, Pofesso, Fculty of Ubn cience, Meijo Univesity (Nijigok --, Kni, Gifu 09-06, Jpn) A highly elible tffic netwok is vey impotnt fo both bnoml nd noml peiod. Netwok elibility cn be impoved effectively by impoving the most impotnt key link in the netwok. When such impotnt link is once found, netwok elibility cn be impoved nd mintined efficiently. Thus, some indexes hve been poposed fo finding the most impotnt key link in the netwok. Howeve, these indexes hve thei defects. Theefoe, good solution cnnot be obtined by these indexes fo evluting netwok elibility impovement. In ddition, the point of view of cost / benefit is lso impotnt. This ppe pesents the following contents: Fistly, peviously poposed indexes e summized. econdly, since the clcultion wok fo netwok elibility inceses exponentilly with the numbe of links of the netwok, n enomous mount of CPU time nd memoy size should be needed (NP-hd poblem). Theefoe n efficient Clcultion Algoithm fo Boolen Absoption (CABA) with ptil diffeentil is poposed. It enbles to clculte elibility nd impotnce utomticlly even fo vey lge scle netwok. Using CABA, the pocesses of netwok impovement with some indexes e comped fo smll netwok. Then the fetues nd defects of these indexes e pointed out. Thidly, cost-benefit nlysis method is poposed in ode to impove the peviously poposed indexes. eies, pllel nd simple bidge netwoks e discussed. Depending elibility-cost function, the behvio of netwok impovement pocess will be found diffeent. A genel conclusion fo effective nd efficient netwok impovement is pesented. Key Wods: Teminl elibility; Pobbility Impotnce; Citiclity Impotnce; Cost-Benefit Anlysis; Boolen Absoption.. INTODUCTION It is impotnt to keep highly elible tffic netwok fo bnoml peiod especilly disste peiod. Howeve, the tffic system my be dmged seiously, nd wht dmged od should be selected is vey difficult fo econstucting to mintin o impove the tffic netwok elibility. In ddition, teminl elibility of the tffic netwok is often clculted by the combintion of pth sets nd cut sets. Fo exmple, since pth sets e set of links, the elibility of one pth set is combintion of link elibility. Theefoe, when key link fo impoving the teminl elibility is once found, the netwok elibility cn be impoved nd mintined efficiently by impoving the link elibility of the key link. Thus, some indexes such s pobbility impotnce nd citiclity impotnce hve been poposed. Howeve, these indexes hve own demeits fo finding the most impotnt key link in the netwok. In ddition, the cost fo epiing the tffic system is lso impotnt fo impoving the netwok elibility. Theefoe, the point of view of cost/benefit fo impoving the tffic netwok elibility should be discussed.. CUENT EIABIITY INDICE The concept of impotnce hs been poposed fo long peiod in the system engineeing field, but hs ppeed in only some ppes in the tnspottion field ). Impotnce is defined s the degee of mgnitude tht impovement in elibility of link contibutes fo system elibility. In this ppe, impotnce is bsed on the connectivity elibility. () elibility Impotnce The teminl elibility of the highwy netwok is defined s the pobbility tht two given nodes ove the netwok e connected with cetin sevice level of tffic fo given time peiod 7),0). imilly, link elibility in the netwok is defined s the pobbility tht the tffic is in cetin sevice level fo given

2 time peiod. Teminl elibility,, is given by the miniml pth sets expession s following 7) ; N ( ) E[ ( )] (Eq.) P Whee Ps is the -th miniml pth set, nd N is the totl numbe of miniml pth set, nd is biny indicto vible fo link (nd equls if link suvives o povides the cetin suppot, nd equls 0 othewise), nd link elibility = E [ ]. The teminl elibility of tffic netwok depends on the netwok stuctue nd the link elibilities. Thee e, theefoe, two bsic ppoches to impoving netwok elibility, nmely, to impove the netwok stuctue o to impove the elibility of the links. The focus hee is identifying which links should be impoved, to mximize the impovement in netwok elibility. In ode to find out the key link to impove the teminl elibility most efficiently, the Binbum s stuctul impotnce ) hs been poposed. The Binbum s stuctul impotnce is shown s Eq.; I = ( ) / (nd0 I ) (Eq.) Although Binbum s stuctul impotnce hs potentility in impoving netwok elibility, it hs demeit to be stted in this section. Fo the cse of two links in seies netwok, the teminl elibility AB follows fom Eq. is: (Eq.) AB Fo the cse of two links in pllel netwok, the teminl elibility AB follows fom Eq. is: AB ( )( ) (Eq.) The pobbility impotnce fo these two links in pllel netwok, I nd I, e obtined fom Eq. nd Eq. s I = nd I =. If >, I > I is hold. The esult indictes tht in cse of pllel typed netwok, impoving the moe elible link will be moe effective fo impoving teminl elibility. Accoding to common sense, it is difficult to impove moe elible link whees it is the esy to impove less elible link. This esult is ctully itionl fo impoving, nd mintining netwok. () Citiclity Impotnce Becuse of the demeit of Binbum s stuctul impotnce, Citiclity Impotnce ws poposed which is the tio of the popotionl impovement in the netwok elibility to the popotionl impovement in the link elibility ). CI / / I (Eq.) Bsed on the defect of Eq., Wkbyshi lso poposed the citiclity impotnce s the popotion of the mginl chnge in teminl elibility ginst the mginl chnge in the elibility engineeing, nd Citiclity Impotnce is intoduced s Eq.6. ( ) / ( ) = lim = I (Eq.6) q 0 q / q Whee q =- is used fo the unelibility of link. Fo the cse of two links nd in seies netwok, it follows fomeq. nd Eq. tht: CI = = CI (Eq.7) Fo the cse of two links nd in seies netwok, it follows fom Eq. nd Eq.6 tht: = nd (Eq.8) If >, < is hold fom Eq.8. Fo the cse of two links nd in pllel, it follows fom Eq. nd Eq.6 tht: = ( )( ) = (Eq.9) Fom Eq.7 nd Eq.9, the Citiclity impotnce index is sme fo both links in some type netwok, so it does not help distinguish between them in tems of impoving netwok elibility. The fomul of the citiclity impotnce which Henley nd Kummoto poposed o Wkbyshi poposed cn t mke n expected esult The elibility impotnce nd the citiclity impotnce mentioned bove cnnot explin explicitly fo selecting the most impotnt key link of tffic netwok becuse of thei own demeits. Theefoe, good solution cnnot be obtined by these indexes fo evluting netwok elibility impovement. In ddition, lthough the point of view of cost-benefit is lso impotnt 8), those indexes cnnot explin explicitly the incese in cost of impoving link elibility when link elibility inceses. Thus the tffic netwok elibility incese in ccodnce with the viety of the cost should be discussed.. COT-EIABIITY FUNCTION Accoding to the index in Chpte, the less elible link in seies netwok should be impoved in ccodnce with Eq.8. Howeve, the esult fom Eq.9 does not povide distinguishble infomtion which link should be impoved fistly in pllel netwok. Thus the impovement of tffic netwok elibility in ccodnce with the viety of the cost will be poposed in this Chpte.

3 Tble Viety of cost incese with elibility incese elibility incese Cost incese of Cse Cost incese of Cse The needed cost sttegies to impove the link elibility e ssumed to be thee cses: Cse : The cost incese is constnt mount when the link elibility inceses unde the sme incese degee of link elibility nd is shown in cse l of Tble (constnt mount equls 00 (unit is 0,000 Yen)). Cse : The cost incese is pogessive incese when the moe elible link is impoved unde the sme incese degee of link elibility nd is shown in cse of Tble (pogessive incese equls 00). Cse : The cost to incese the link elibility is fixed unde the sme incese degee of link elibility (fixed cost is 000). The effect of impovement of netwok elibility cused by cost incese my be not obvious in the shot time, thus simple cost-benefit function s the impovement of the netwok elibility ginst the cost incese fo long time is defined s Eq.0. The effect of cost incese to impove the link elibility my be not obvious in the shot time, so the cost-effect function fo long time is defined s Eq.0, whee Y shows the numbe of yes to invest, F shows the benefit of ll kinds of tffic consumption, b0 mens the oiginl netwok elibility, nd Cost b shows the cost incese to impove the netwok elibility fom b0 to b. b b0 Eff ( Y, F) Y F (Eq.0) Cost b. CACUATION AGOITHM OF BOOEAN ABOPTION FO TEMINA EIABIITY AND I Boolen bsoption is used to clculte the exct vlue of teminl elibility 6). Howeve, the mnul wok, including n expnsion nd Boolen bsoption, is vey complicte nd impcticl s the size of the netwok expnds. In ddition, it tends to led to misclcultion. Theefoe, n lgoithm fo pocessing this clcultion by compute nmed s CABA hs been developed. The min point of this lgoithm is to expnd Eq. diectly. In ddition, it is designed to genete nd to unify ech tem by tuns fo efficiency. Futhemoe, only one bit of the memoy of compute is used to memoize ech ndom vible of evey link of the netwok. In ddition, the elibility impotnce ( I ) cn be obtined by Eq.. Thus, the clcultion fo I of link is lso obtined by this lgoithm. The lgoithm is s following: tep : et p be the numbe of miniml pth sets to be used in this clcultion. Memoize these miniml pth sets. Hee, evey miniml pth set tht is composed of links expessed s biny numbe is memoized s deciml numbe. Fo exmple, the miniml pth set, 0, tht is, {,,, 0}, is expessed s the biny numbe (ed this figue fom the ight). At this step, the numbe is tnslted into deciml numbe then memoized; the biny numbe is memoized s the deciml numbe (=0+++9). This pocedue pemits eduction of the memoy egion size used in the compute. tep : et m. m is the numbe of miniml pth sets in evey itetion. tep : Any poduct composed of m miniml pth sets (obtined in the expnsion of Eq. into p m tems) is expessed s: ( ). n Fig. Pocess of Boolen bsoption fo Teminl elibility Fig. Pocess of Boolen bsoption fo elibility impotnce

4 Ange this poduct by Boolen bsoption in tems of links. Fo exmple, the poduct of the miniml pth sets {,,, 0}, {,, 9, }, {, 8,, } is tnslted into the memoy vible 999 which indictes This pocedue is demonstted in Fig.. Bsed on the memoy vible of poduct fo teminl elibility, the memoy vible fo I of ll links cn be clculted nd stoed in othe stoe egions. If the coesponding bit of does not exist in the memoy vible of poduct fo teminl elibility, the memoy vible fo I of link tnsltes into 0, othewise, the coesponding bit of in memoy vible of poduct fo teminl elibility is tnslted into 0 nd the new memoy vible is stoed in othe stoe egions s the memoy vible fo I of link. Fo exmple, the poduct of I is bsed on the memoy vible 999 fo teminl elibility, thus the memoy vible fo I of link is 998. Howeve, the memoy vible fo I of link 6 is 0. This pocedue id demonstted in Fig.. tep : Combine like tems. The poducts geneted in step e checked whethe o not the sme poduct hs been geneted in the peceding pocess. Fo the bove exmples, the numbe 999 nd 998 e checked whethe o not the sme numbe exists in the own stoe egions. When the sme poduct exists, the coefficient of the poduct is updted; when not, it is newly stoed. tep : Itete step nd step fo ll combintions of m ( ). The numbe n p of itetions is. m tep 6: Itete step to step fo m,,, p. tep 7: Ech numbe in the stoe egions coesponds to ech tem in the polynomil expession of, fo which Boolen bsoption hs ledy been cied out. If the numbe 999 emins in the stoe egion fo teminl elibility, the coesponding tem, exists in the polynomil expession fo teminl elibility. imilly, if the numbe 998 emins in the stoe egion fo elibility impotnce, the coesponding tem, 890 exists in the polynomil expession fo elibility impotnce. Theefoe, the vlue of teminl elibility nd elibility impotnce e obtined by substituting the vlue fo the link elibility in the coesponding tems.. NUMEICA EAMPE The Cost-benefit function fo impovement of tffic netwok elibility ws poposed in Chpte, nd thee cost sttegies wee poposed in ode to find the most impotnt key link to impove the netwok elibility mostly. Accoding to the index poposed by Wkbyshi, the lest elible link in seies type netwok should be selected s the most impotnt key link to impove the netwok elibility mostly, thus, in this section, only simple pllel netwok nd simple bidge netwok e selected to nlyze the cost benefit function fo impovement of netwok elibility. In this Chpte, two sttegies fo selecting the most impotnt key link to impove the netwok elibility e discussed: The most elible link will be selected s the most impotnt key link ccoding to I nd CI ; The lest elible link will be selected s the most impotnt key link ccoding to common sense. () The Cost-benefit Anlysis fo imple Pllel Netwok The effect of the cost-benefit nlysis will be Fig. the effect of thee cost sttegies fo impoving the link elibility of simple pllel netwok

5 discussed in the cse of two links nd in pllel netwok, nd the oiginl elibility of two links e shown s =0. nd =0.. Y equls 0 yes nd F equls hunded million evey ye in ode to shot cut clcultion. Fig. shows thee cost sttegies fo impoving the link elibility of the simple pllel netwok, nd the left bnch of evey cse in Fig. shows tht the moe elible link should be impoved, nd the ight bnch shows tht the less elible link should be impoved. () Cost sttegy of Cse The cost incese is constnt mount when the link elibility is impoved with the sme degee in cse. The esult Eff ( 0, ) fom the left bnch of cse is 8 by using Eq., nd the esult Eff ( 0, ) fom the ight bnch of cse is 7.. It mens tht Eff ( 0, ) > Eff ( 0, ). These esults suggest tht the moe elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. (b) Cost sttegy of Cse The cost incese is pogessive when the link elibility is impoved with the sme degee in cse. The esult Eff ( 0, ) is.8 fom the left bnch of cse by using Eq., nd the esult Eff ( 0, ) fom the ight bnch of cse is.. It mens tht Eff ( 0, ) < Eff ( 0, ). These esults suggest tht the less elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. (c) Cost sttegy of Cse The cost is fixed when the link elibility is impoved with the sme degee in cse. The esult Eff ( 0, ) fom the left bnch of cse is 0 by using Eq., nd the esult Eff ( 0, ) fom the ight bnch of cse is.. It mens tht Eff ( 0, ) > Eff ( 0, ). These esults suggest tht the moe elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. Bsed on the cost sttegies of cse nd cse, the moe elible link should be selected s the most impotnt key link; on the conty, the less elible link should be selected s the most impotnt key link to be impoved bsed on the cost sttegy of cse. Theefoe, the diffeent link should be selected s the most impotnt key link ccoding to the diffeent cost sttegies by using the cost-benefit function Eq.0 in pllel type netwok. () Cost-benefit Anlysis fo imple Bidge Netwok It is esy to clculte the exct vlue of teminl elibility nd elibility impotnce of the bove-mentioned pllel netwok. Howeve, it is vey complicte nd impcticl to clculte the exct vlue of teminl elibility s the size of the netwok expnds. In this section, thee is simple bidge netwok tht hs fou notes nd five links shown in Fig.. The miniml pth sets of this bidge e P = {, }, P = {, }, P = {,, }, independent miniml pth set is seies netwok system 9), thus, the elibility of miniml pth set is shown s following; ( P), ( P ), ( P ), ( P ) (Eq.) Teminl elibility of this bidge netwok is given by using Clcultion Algoithm of Boolen Absoption (CABA) s Eq.; P = {,, }. The ( ). (Eq.) elibility impotnce of this bidge netwok is given by using CABA s followings; PI, (Eq.) PI, (Eq.) PI, (Eq.) PI, (Eq.6) PI. (Eq.7) The oiginl dt of the simple bidge netwok is shown s =0., =0., =0., =0., =0., Y = 0 nd F =00,000,000 yen /ye. () Cost sttegy of Cse The pocess of cost-benefit nlysis is simil s the pevious desciption of ection.(), nd the esult Eff ( 0, ) bout the simple bidge netwok fom the left bnch of cse is. by using Eq.0, nd the esult Eff ( 0, ) fom the ight bnch of cse is 7.. It mens tht Eff ( 0, ) < Eff ( 0, ). These esults suggest tht the lest elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. (b) Cost sttegy of Cse The esult Eff ( 0, ) is.0 fom the left bnch of Fig. A simple bidge netwok

6 cse by using Eq.0, nd the esult Eff ( 0, ) fom the ight bnch of cse is.6. It mens tht Eff ( 0, ) < Eff ( 0, ). These esults suggest tht the lest elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. (c) Cost sttegy of Cse The esult Eff ( 0, ) fom the left bnch of cse is 9. by using Eq.0, nd the esult Eff ( 0, ) fom the ight bnch of cse is 7.9. It mens tht Eff ( 0, ) > Eff ( 0, ). These esults suggest tht the most elible link should be selected s the most impotnt key link bsed on cost sttegy of Cse. Bsed on cost sttegies of cse, the most elible link should be selected s the most impotnt key link; on the conty, the lest elible link should be selected s the most impotnt key link to be impoved bsed on cost sttegy of cse nd cse. In pctice, the cost sttegies of cse nd cse e tionl, in ccodnce with the well-known lws of diminishing etuns 8). Futhemoe, the conclusion, tht the lest elible link in the simple bidge netwok should be selected s the most impotnt key link fom cse nd cse bsed on the cost-benefit nlysis, is lso tionl, in ccodnce with the well-known it is difficult to impove highly elible link whees it is the esy to impove lowe elible link ). Theefoe, the lest elible link in genel tffic netwok should be selected s the most impotnt key link to be impoved fo incesing the netwok elibility most effectively. 6. CONCUION In this ppe, fistly, in ode to discuss the impovement of the tffic netwok elibility, the cuent indexes of elibility including PI, CI nd, nd wee intoduced nd the demeits of these indexes wee lso pointed out. And then, the Cost-elibility function ws poposed fo impovement of the tffic netwok elibility bsed on the simple cost-benefit nlysis nd the Clcultion Algoithm of Boolen Absoption fo teminl elibility nd pobbility impotnce hs been developed. stly, two numeicl exmples fo the pllel netwok nd bidge netwok wee simulted bsed on the Clcultion Algoithm of Boolen Absoption nd the cost-elibility function. Fom these simultions, genel conclusion cn be obtined s following: In the vey simple netwok, the diffeent link should be selected s the most impotnt key link ccoding to the diffeent cost sttegies fo impovement of the tffic netwok elibility. In genel netwok, the lest elible link should be selected s the most impotnt key link fo netwok elibility impovement. Howeve, thee e only thee types of tffic netwok hve been used to cetificte this conclusion. Moe types of tffic netwok should be used fo finding the most impotnt key link in some typicl netwoks in the futue studies. EFEENCE ) Blow,.E. nd Poschn, F. (97). ttisticl Theoy of elibility nd ife Testing: Pobbility Models. Holt, ineht nd Winston, New Yok, UA. ) Binbum, Z.W. (969). On the Impotnce of Diffeent Components in Multi-Component ystem. Multivite Anlysis II., Acdemic Pess, New Yok, UA. ) Fng,.M. nd Wkbyshi, H. (00). Impovement of Netwok elibility bsed on Govenment uppot. Poceedings of Fouth Intentionl ymposium on Tnspot Netwok elibility, Minesod Univesity, UA. ) Henley, E.J. nd Kummoto, H. (98). elibility Engineeing nd isk Assessment [M]. Pentice-Hll, Inc., Englewood Cliffs, UA. ) Henley, E.J. nd Kummoto, H. (99). Pobbilistic isk Assessment: elibility Engineeing, Design nd Anlysis. Institute of Electicl nd Electonics Enginee, New Yok, UA. 6) Iid, Y. nd Wkbyshi, H. (988). An Efficient Clcultion Method to Obtin Uppe nd owe bounds of Teminl elibility of od Netwoks using Boolen lgeb. Poceeding of JCE, No.9/IV-9, pp ) Iid, Y. nd Wkbyshi, H. (989). An Appoximtion Method of Teminl elibility of od Netwok Using Ptil Miniml Pth nd Cut ets. Poceeding of the Fifth WCT, Yokohm.Vol., pp ) Nicholson, A. (007). Optimizing Netwok Teminl elibility. Poceedings of Thid Intentionl ymposium on Tnspot Netwok elibility. Delft Univesity, Nethelnds. 9) Wkbyshi, H. nd Iid, Y. (99). An Efficient Evlution Method fo od Netwok elibility in Disste. Intentionl ymposium on Ntul Disste eduction nd Civil Engineeing, JCE, pp ) Wkbyshi, H. nd Iid, Y. (99). Uppe nd owe Bounds of Teminl elibility of od Netwoks: An Efficient Method with Boolen lgeb [J]. Jounl of Ntul Disste cience, Volume, No., pp. 9-. ) Wkbyshi, H. nd Iid, Y. (99). Impovement of od Netwok elibility with Tffic Mngement (Edited by B.iu nd J.M.Blosseville). Tnspottion ystems: Theoy nd Applictions of Advnced Technology. Pegmum Elsevie cience, United Kingdom, pp ) Wkbyshi, H. (00). Netwok elibility Impovement: Pobbility Impotnce. Poceedings of econd Intentionl ymposium on Tnspot Netwok elibility. Univesity of Cntebuy, Chistchuch, New Zelnd. 6