Network Reliability Optimization via the Cross-Entropy Method

Transcription

1 Ntwork Rliility Optimiztion vi th Cross-Entropy Mtho Dirk P. Kros, Kin-Ping Hui, n Sho Nrii Astrt Consir ntwork o unrlil links, h o whih oms with rtin pri n rliility. Givn ix ugt, whih links shoul purhs in orr to mximiz th systm s rliility? W introu nw pproh, s on th Cross-Entropy mtho, whih n l tivly with th onstrints n nois (introu whn stimting th rliilitis vi simultion) in this iiult omintoril optimiztion prolm. Numril rsults monstrt th tivnss o th propos thniqu. Inx Trms Ntwork Rliility, Cross-Entropy Mtho, Mont Crlo Simultion, Noisy Optimiztion, Mrg Pross. NOTATION i i C mx purhs proility o link i ost o link i totl ugt m, n numr o [links, nos] N smpl siz or CE p i rliility o link i r ntwork rliility (r optiml) x ntwork topology (x optiml, X rnom) y systm stt (Y rnom) α smoothing prmtr or CE ϕ x strutur untion o topology x This rsrh ws support y th Austrlin Rsrh Counil, grnt numr DP089. D. P. Kros n S. Nrii r with Dprtmnt o Mthmtis, Th Univrsity o Qunsln, Brisn, QLD 0 Austrli (-mil: kros@mths.uq.u.u; sho@mths.uq.u.u). K.-P. Hui is with th Dn Sin n Thnology Orgnistion, Austrli (-mil: Kin-Ping.Hui@sto.n.gov.u). ρ rrity prmtr or CE I. INTRODUCTION RAPID vlopmnts n improvmnts in inormtion n ommunition thnologis in rnt yrs hv rsult in inrs pitis n highr onntrtion o tri in tlommunition ntworks. Oprting ilurs in suh high-pity ntworks n t th qulity o srvi o lrg numr o onsumrs. Consquntly, th rul plnning o ntwork s inrstrutur n th til nlysis o its rliility om mor n mor importnt, in orr to nsur tht onsumrs otin th st srvi possil. On o th most si n usul pprohs to ntwork rliility nlysis is to rprsnt th ntwork s n unirt grph with unrlil links. Th rliility o th ntwork is usully in s th proility tht rtin nos in th grph r onnt y untioning links. This ppr is onrn with ntwork plnning, whr th ojtiv is to mximiz th ntwork s rliility, sujt to ix ugt. Mor prisly: givn ix mount o mony, n strting with non-xistnt ntwork, th qustion is whih ntwork links shoul purhs in orr to mximiz th rliility o th inish ntwork. Eh link rris prspii pri n rliility. Thr r two rsons why this Ntwork Plnning Prolm (NPP) is iiult to solv. First, th NPP is onstrin intgr progrmming prolm known s th Minimum Stinr Prolm, whih is 0- knpsk prolm with non-linr

2 ojtiv. It is wll known tht suh prolm is NP hr [], n it is lso APX-Complt []. For smll ntworks xt mthos, suh s rnh-n-oun, ynmi progrmming or onvxiition, my sussul (s or xmpl []); ut, sin th omplxity o th prolm inrss xponntilly with th numr o links, suh mthos quikly om insil or mort n lrg-sl prolms. Son, or lrg ntworks th vlu o th ojtiv untion tht is, th ntwork rliility oms iiult or imprtil to vlut [], []. A vil option thn is to us simultion to stimt th ntwork rliility, or xmpl vi th Cru Mont Crlo (CMC) thniqu. This noisy vrsion o th prolm is not vn in NP, us th vlu o givn solution is hr to omput. Morovr, or highly rlil ntworks whih typilly our in ommunition ntworks CMC rquirs vry lrg simultion ort in orr to stimt th rliility urtly. A numr o simultion thniqus hv n vlop to rss th ntwork rliility stimtion prolm. For xmpl, Kummoto t l. [6] propos simpl thniqu ll Dggr Smpling to improv th iiny o CMC simultion. Fishmn [] introu Prour Q, whih n provi rliility stimts s wll s ouns. Colourn n Hrms [8] propos thniqu tht provis progrssiv ouns tht vntully onvrg to n xt rliility vlu. Elprin t l. [9], [0] vlop Evolution Mols or stimting th rliility o highly rlil ntworks. Hui t l. [], [] propos hyri shm tht provis ouns n n provi sp-up y svrl orrs o mgnitu in rtin lsss o ntworks. Thy lso propos nothr shm [] whih mploys th Cross-Entropy thniqu to sp-up th stimtion in gnrl lsss o ntworks. Othr rlvnt rrns on ntwork rliility inlu [], [], [6]. Th litrtur on ntwork plnning rthr thn rliility stimtion is not xtnsiv, n virtully ll stuis prtin to ntworks or whih th systm rliility n ithr vlut xtly, or shrp rliility ouns n stlish. Cnl n Urquhrt [] mploy Simult Annling shm to otin mor rlil ltrntiv ntwork, givn usr-in ntwork topology. Dngiz t l. us Gnti Algorithm to optimiz th sign o ommunition ntwork topologis sujt to th minimum rliility rquirmnt [8]. Yh t l. [9] propos mtho s on Gnti Algorithm to optimiz th k no st rliility sujt to spii pity onstrint. Rihlt t l. [0] us Gnti Algorithm in omintion with rpir huristi to minimiz th ntwork ost sujt to spii ntwork rliility onstrint. Othr huristis n oun in []. To our knowlg, no simpl lgorithm is known tht n tkl t th sm tim th omintoril, onstrint n noisy spts o th NPP, n th purpos o this ppr is to introu suh mtho, n provi nw n tiv pproh to ntwork plnning. Our pproh is s on th Cross-Entropy (CE) mtho [], whih ws introu in [] s n ptiv thniqu or stimting proilitis o rr vnts in omplx stohsti ntworks. It ws soon rliz [], [] tht it oul us not only or rr vnt simultion ut or solving iiult omintoril optimiztion prolms s wll. Morovr, (n this is spilly rlvnt or th NPP) th CE mtho is wll-suit to solving noisy optimiztion prolms; xmpls r th Bur Allotion Prolm [6], th Vhil Routing Prolm [], n th Stohsti Shortst Pth Prolm [8]. For th NPP w will onsir oth th trministi s, whr th ntwork rliility n omput xtly, n th noisy s whr it is stimt vi simultion. A tutoril on th CE mtho n oun in [9], whih is lso vill on-lin rom th CE hompg: Th rst o th ppr is orgniz s ollows: In Stion II w ormult th NPP in mthmtil trms. In Stion III w prsnt th CE pproh to th prolm. This is urthr

3 vlop in Stion IV or th noisy s, in prtiulr with rspt to vrin rution thniqus suh s Prmuttion Mont Crlo n th Mrg Pross. Stion V ouss on implmnttion issus with rgr to sping up th lgorithm. W illustrt th tivnss o th CE pproh vi numr o numril xprimnts in Stion VI. Finlly, in Stion VII w prsnt our onlusion n irtions or utur work. II. PROBLEM DESCRIPTION ssigns to h stt vtor y th stt o th systm (working = or il = 0). Tht is, i ll th trminl nos r onnt, ϕ x (y) = () 0 othrwis. Now, onsir th sitution with rnom stts, whr h purhs link works with proility p. Lt Y rnom stt o link, n lt Y th orrsponing rnom stt vtor. Th rliility o th ntwork in y purhs vtor x is givn y Consir n unirt grph G(V, E), with st V o nos r(x) = E[ ϕ x (Y )] = y ϕ x (y)pr{y = y}. () (vrtis), n st E o links (gs). Suppos th numr o links is E = m. Without loss o gnrlity w my ll th links,...,m. Lt K V st o trminl nos. With h o th links is ssoit ost n rliility p. Th ojtiv is to purhs thos links tht optimiz th rliility o th ntwork in s th proility tht ll th trminl nos r onnt y untioning links sujt to totl ugt C mx. Lt = (,..., m ) not vtor o link osts, n p = (p,..., p m ) th vtor o link rliilitis. W introu th ollowing nottion. For h link lt x suh tht i link is purhs, x = 0 othrwis. W ll th vtor x = (x,..., x m ) th purhs vtor. Th st o ll possil purhs vtors is not y X. To intiy th oprtionl links, w in or h link th link stt y i link is untioning, y = 0 othrwis. Not tht or h link tht is not purhs, th stt y is pr inition qul to 0. Th vtor y = (y,..., y m ) is ll th stt vtor. For h purhs vtor x lt ϕ x th strutur untion o th purhs systm. This untion W ssum rom now on tht th links il s-inpnntly, tht is, Y is vtor o s-inpnnt Brnoulli rnom vrils, with suss proility p or h purhs link n 0 othrwis. Dining p x = (x p,..., x m p m ) s th vtor o proilitis o th omponnts o Y, or givn purhs vtor x, w writ Y Br(p x ). It ollows tht or h x, th rliility is omput s r(x) = m ϕ x (y) (x j p j ) yj ( x j p j ) yj, () y j= whr 0 0 :=. Our min purpos is to trmin mxr(x), () x X sujt to th onstrint on th totl ugt x C mx. () E Lt r := r(x ) not th optiml rliility o th ntwork, whr x is th optiml purhs vtor. III. CROSS-ENTROPY APPROACH In this stion w show how th CE mtho n us to solv th onstrin omintoril optimiztion prolm (), (). Th CE mtho onsists o two stps whih r itrt: ) gnrt rnom purhs vtors X,...,X N oring to som spii rnom mhnism, n ) upt th prmtrs o this mhnism in orr to otin ttr systm rliilitis in th nxt itrtion.

4 An iint mtho to gnrt rnom purhs vtors tht stisy () is s ollows: First, gnrt uniorm prmuttion (,..., m ) o (,...,m), y s-inpnntly rwing m numrs rom th uniorm istriution on [0, ] n ltting,..., m orrspon to th inis o th orr osrvtions. Son, givn suh prmuttion, lip oin with suss proility to i whthr to purhs link or not. I sussul n i thr is nough mony vill to purhs link, st X =, tht is, link is purhs; othrwis st X = 0. W rpt th ov prour or links,, t. For h link i w hk whthr th rmining ugt llows us to purhs th link, n i so, w purhs th link with proility i. Th min lgorithm or gnrting rnom purhs vtor using uniorm prmuttion is thus summriz s ollows: Algorithm [Gnrtion Algorithm]. ) Gnrt uniorm rnom prmuttion (,..., m ). St k =. ) Clult C = k + k i= X i i. ) I C C mx, rw X k Br( k ). Othrwis st X k = 0. ) I k = m, thn stop; othrwis st k = k + n ritrt rom stp. Th usul CE prour [] is to onstrut squn o rrn vtors { t, t 0} (i.., purhs proility vtors), suh tht { t, t 0} onvrgs to th gnrt (i.., inry) proility vtor tht orrspons to th optiml purhs vtor x =. Th squn o rrn vtors is otin vi two-stp prour, involving n uxiliry squn o rliility lvls {γ t, t 0} tht tns to th optiml rliility γ = r t th sm tim s th { t } tn to. At h itrtion t, or givn t, γ t is th ( ρ)-quntil o prormns (rliilitis). Typilly ρ is hosn twn 0.0 n 0.. An stimtor γ t o γ t is th orrsponing smpl ( ρ)-quntil. Tht is, gnrt rnom smpl X,...,X N using th gnrtion lgorithm ov; omput th prormns r(x i ), i =,...,N n lt γ t = r ( ( ρ)n ), (6) whr r ()... r (N) r th orr sttistis o th prormns. Th rrn vtor is upt vi CE minimiztion, whih (s []) rus to th ollowing: For givn ix t n γ t, lt t,j = E t [X j r(x) γ t ]. An stimtor â t o t is omput vi N i= â t,j = I {r(x X i) γ t} ij N i= I, j =,...,m, () {r(x i) γ t} whr w us th sm rnom smpl s in (6), n whr X ij is th j-th oorint o X i. Th min CE lgorithm or optimizing () using th ov gnrtion lgorithm is thus summriz s ollows: Algorithm [Min CE Algorithm]. ) Initiliz â 0. St t= (itrtion ountr). ) Gnrt rnom smpl X,...,X N using Algorithm, with = â t. Comput th smpl ( ρ)- quntil o prormns γ t using (6). ) Us th sm smpl to upt â t, using (). ) I mx(min(â t, â t )) β (8) or som smll ix β, thn stop (lt T th inl itrtion); othrwis st t = t + n ritrt rom stp. Not tht th ost vtor, rliility o links p, th initil rrn vtor â 0, th smpl siz N, totl ugt C mx, th rrity prmtr ρ, n th stopping prmtr β n to spii in vn. Rmrk [Smooth Upting]. Inst o upting irtly using (), on my hoos to us smooth upting prour â t = α ã t + ( α) â t (9)

5 TABLE I LINK COSTS AND RELIABILITIES i i p i i i p i i i p i Fig.. 6-no omplt grph whr ã t is th prmtr vtor otin vi () n α is ll th smoothing prmtr. It is sily sn tht or α = th originl upting prour is otin. By stting th smoothing prmtr twn 0 < α < w tk th pst into ount whn upting th prmtr vtor. Rmrk [Unrliility]. In mny pplitions th link n ntwork rliilitis r los to. Th pproprit quntity to onsir is thn th ntwork unrliility r = r. In suh s Algorithm n rily moii to minimiz th unrliility, rthr thn to mximiz th rliility. Th only irns r tht ˆγ t now rprsnts th smpl ρ-quntil o th unrliilitis, n tht r(x i ) ˆγ t in () is rpl with r(x i ) ˆγ t. Exmpl. On or is 6-no ully onnt grph givn in Figur. Th two lk nos in th grph rprsnt th trminl nos. Th ntwork is untioning i th two trminl nos r onnt y oprtionl links. Th link osts n rliilitis r givn in Tl I. Th totl ugt C mx is qul to 00. Th optiml purhs vtor n lult (y totl numrtion) to x = (, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0,, 0), whih givs ntwork unrliility o r = r = Th our lk links orm th optiml ntwork. Tl II isplys th volution o th purhs proility vtor or this prolm. W us th ollowing CE prmtrs: N = 0, ρ = 0., α = 0., β = 0.0 n w took â 0 = (0.,...,0.). W s tht s t, γ t n â t quikly pprohs r n x rsptivly. IV. NOISY OPTIMIZATION As mntion in th introution, or ntworks involving lrg numr o links th xt vlution o th ntwork rliility is in gnrl not sil, n simultion oms vil option. In th orrsponing simultion-s optimiztion prolm th ojtiv untion (th ntwork rliility) is thus orrupt y nois. In this stion w show how th CE mtho n sily moii to tkl suh noisy NPPs. In orr to pt Algorithm w gin, t itrtion t, gnrt rnom smpl X,..., X N oring th Br(â t )-istriution. Howvr, th orrsponing prormns (ntwork rliilitis) r now not omput xtly, ut stimt. For xmpl, stimtion vi CMC involvs, or h vtor X i, rwing rnom smpl o stt vtors Y,..., Y K, h oring to Br(p Xi )-istriution, n stimting th prormn s r(x i ) = K ϕ Xi (Y j ), i =,...,N. (0) K j= Th upting ormul is similr to (). Th only irn is tht r(x i ) is rpl with r(x i ). Thror th upting ormul t t-th itrtion is givn y N i= â t,j = I { r(x X i) γ t} ij N i= I, j =,...,m. () { r(x i) γ t} Th min CE lgorithm or stimting th rliility o ntwork is summriz s ollows: Algorithm [Noisy Vrsion o th CE Algorithm]. ) Initiliz â 0. St t = (itrtion ountr). ) Gnrt rnom smpl X,...,X N using Algorithm. Lt r (),..., r (N) th orr sttistis o

6 6 TABLE II THE EVOLUTION OF THE CE ALGORITHM WITH C mx = 00, N = 0, ρ = 0., α = 0., AND β = 0.0 t γ t â t r(x ),..., r(x N ). Lt γ t = r ( ( ρ)n ). ) Us th sm smpl to upt â t using (). ) Stop i (8) hols or som smll ix β (lt T th inl itrtion); othrwis st t = t + n ritrt rom stp. W oul tk â T s our inl purhs vtor, i th lttr wr inry. Sin â T is los to, ut not xtly inry, w roun â T to th nrst inry vtor, not th solution y â, n tk this roun vtor s our solution or th NPP. I, in ition, w wish to otin n stimt o th optiml rliility r, w gnrt (lrgr) smpl Y,..., Y N rom Br(â ), n stimt r vi r = N ϕâ (Y i ). () N i= Rmrk [Choi o K]. Not tht th simultion tim n ru y hoosing rltivly smll K in (0), sy K = 00 in Exmpl. Howvr, this orrspons to rltivly lrg vrin o th rliility stimtor, whih in turn oul l to suoptiml onvrgn o th lgorithm. On th othr hn, hoosing lrgr K, sy K = 0000 in Exmpl, n inrs th ury n hn o loting n optiml solution ut t ost o inrs simultion tim. To ovrom this, on my hoos K ptivly y stting K = min{ K min N/N uniqu, K mx }, () whr N uniqu is th numr o uniqu ntworks gnrt with no rptitions n K min n K mx r th minimum n mximum vlus o K llow in th simultion. Th i o using () is simpl: In th rly stgs o th simultion, whn N uniqu N, w tk K K min to quikly nrrow th srh sp. During th ours o th simultion th srh sp oms smllr n th lgorithm strts to gnrt ntworks with similr topologis n prormns, so tht N uniqu rss. Whn this strts to hppn, K is inrs utomtilly to urtly istinguish twn ntworks with similr rliilitis. It is importnt to rliz tht CMC only works stistory whn th ntwork rliility is nithr too smll nor too lrg. For xmpl, onsir highly rlil ntwork. For ny givn purhs vtor x th CMC stimtor o th (vry smll) unrliility r(x) = r(x), whih is th pproprit quntity to onsir hr, is givn y ǫ = r(x) := N ( ϕ x (Y i )), N i= n th orrsponing s-rltiv rror is Vr( r(x)) E[ r(x)] = r(x) r(x)n r(x)n, () whih shows tht or ǫ = 0.0, sy, w n smpl siz o t lst N 0 /r(x), whih n prohiitivly lrg. Prmuttion Mont Crlo A mor iint wy o stimting th ntwork unrliility in highly rlil ntworks is Prmuttion Mont Crlo (PMC) [9]. Th i is s ollows. Consir ntwork with strutur untion s in (), n rliility r = r(x) s in (). Lt l th numr o purhs links n lt E x th st o

7 purhs links. W ssum hr tht ϕ x (x) =, so tht th purhs ntwork untions i ll its links work. Now, osrv ynmi ntwork G(V, E) in whih h purhs link hs n xponntil rpir tim with rpir rt λ() = log( p ). Th rpir rt or h link tht is not purhs is st ollows tht Pr{T > } = xp{log( p )} = p. Thror, th proility o link ing oprtionl t tim t = is p, n hn th proility tht th ntwork is untioning t tim t = is prisly th ntwork rliility. By onitioning on Π w hv to λ() =, so tht th link os not om oprtionl in init tim. At tim t = 0 ll links r il. Assum tht ll rpir tims r s-inpnnt o h othr. Th stt o t tim t is not y Y (t) n th stt o th link st E t n r = E[ϕ(Y ())] = Pr{Π = π} Pr{ϕ(Y ()) = Π = π}, π (6) tim t is givn y th vtor Y (t), in in similr wy s or. Thn, (Y (t)) is Mrkov pross with stt sp r = r = Pr{Π = π} Pr{ϕ(Y ()) = 0 Π = π}. π () {0, } m. This pross is ll th Constrution Pross (CP) o th ntwork. Lt Π not th orr in whih th links r onstrut (om oprtionl) in init tim, n lt A 0, A 0 + A,...,A A l th tims t whih thos links r onstrut. Hn th {A i } r sojourn tims o (Y (t)). Π is rnom vril whih tks vlus in th sp {(,..., l ) {,...,m} l : i j, j i}. For ny suh π = (,..., l ) in Lt E 0 = E x, E i = E i \ { i }, i l, λ(e i ) = E i λ(). (π) = min i {ϕ(e x \ E i ) = } th ritil numr o π, tht is, th numr o rpirs rquir to ring th systm up in th orr spii y π. From th gnrl thory o Mrkov prosss it is not iiult to s tht Pr{Π = π} = l j= λ( j ) λ(e j ). () Morovr, onitionl on {Π = π}, th sojourn tims A 0,...,A l r s-inpnnt n h A i is xponntilly istriut with prmtr λ(e i ), i = 0,...,l. Rll tht h link hs n xponntil rpir rt λ() = log( p ). Lt T th orrsponing rpir tim. It Using th initions o A i n (π), w n writ th lst proility in trms o onvolutions o xponntil istriution untions. Nmly, or ny t 0 w hv Lt Pr{ϕ(Y (t)) = 0 Π = π} = Pr{A A (π) > t Π = π} = Conv 0 i (π) { xp[ λ(e i)t]}. (8) G(π) = Pr{ϕ(Y ()) = 0 Π = π}, (9) s givn in (8). Eqution () n thn rwrittn s r = E[G(Π)], (0) n this shows how th CP simultion shm works. Nmly, lt Π (),..., Π (K) s-inpnnt intilly istriut rnom vtors, h istriut oring to Π. Thn r = K K G(Π (i) ) () i= is n unis stimtor or r, whr h G(Π (i) ) n lult s th onvolution o xponntil untions. Mrg Pross A rul stuy o th volution o th CP shows tht mny o th rsults rmin vli whn w omin vrious stts to orm supr-stts t vrious stgs o th pross. Th mthmtil ormultion is s ollows (s [9], [0] or mor

8 8 Fig.. A rig ntwork, th propr prtition σ = {{}, {,, }}, n its orrsponing losur F σ = {,, }. tils): Givn th grph G(V, E) n sust F E o gs, prtition σ = {V,..., V k } o V is si to propr (with rspt to F) i h inu sugrph G(V i ) o th sugrph G(V, F) is onnt. Lt F i th g-st o th inu sugrph G(V i ). Th st F σ = k i= F i o gs is th losur o F. Th sist wy to visuliz th supr-stt σ is to intiy it with th grph G(V, F σ ), s in Figur. Hr F = {, }, whih inus th propr prtition σ = {{}, {,, }}, so tht F = n F = {,, } n F σ = {,, }. Lt L(G) th olltion o ll propr prtitions o V, orr y th rltion σ τ F σ F τ, whr τ is otin y mrging omponnts o σ. Rll th CP (Y (t)) o th ntwork in y x. Th CP inus Mrkov pross (Y(t)) on L(G), ll th Mrg Pross (MP). Initilly, this pross strts in th supr-stt σ 0 in whih ll nos r isolt rom h othr, n it ns in th supr-stt σ ω orrsponing to th originl ntwork with ll gs untioning. Furthrmor, t ny tim t 0, ϕ(y (t)) = ϕ(y(t)). For h σ L(G), th sojourn tim in σ hs n xponntil istriution with prmtr λ(σ) := E σ λ(), s- inpnnt o othr prtitions, whr E σ = E F σ n th trnsition rom urrnt prtition σ to on o its sussors τ ours with proility λ(σ) λ(τ) λ(σ) W in trjtory o (Y(t)) s squn θ = (σ 0,..., σ (θ) ), whr (θ) is qul to th numr o trnsitions rquir in orr or th ntwork to om oprtionl.. Fig.. () σ 0 = {{},{},{},{}}. () σ = {{,}, {,}}. () σ = {{,}, {},{}}. () σ = {{,,,}}. A possil trnsition squn o th MP (Y(t)). For xmpl, onsir th rig ntwork givn in Figur. Th ntwork is oprtionl whn th two lk nos r onnt y oprtionl links. A possil trnsition squn rom th initil stt is shown in Figur. Th ntwork oms oprtionl upon th trnsition rom σ to σ. Thror (θ) =. Sin ϕ(y (t)) = ϕ(y(t)), th proility tht th ntwork is not untioning t tim t = is qul to th ntwork unrliility. Thus th ntwork unrliility n writtn s r = θ Pr{Θ = θ}pr{ϕ(y()) = 0 Θ = θ}. () For ny t 0, w n writ th lst proility s Lt Pr{ϕ(Y(t)) = 0 Θ = θ} = Conv 0 j (θ) { xp[ λ(σ j)t]}. () G MP (θ) = Pr{ϕY() = 0 Θ = θ}. () s givn in (). Eqution () n rwrittn s r = E[G MP (Θ)] () whr Θ is th rnom trjtory in (Y(t)). Lt Θ,...,Θ K th s-inpnnt n intilly istriut rnom tr-

9 9 jtoris, h istriut oring to Θ. Thn r = K G MP (Θ i ) (6) K i= is n unis stimtor or r. A simpl wy o gnrting suh trjtoris is y irst gnrting rnom vtor Π xtly s in th CP. Th squn o istint prtitions otin rom th orr in whih th links om up (givn Fig.. Exmpl ntwork. y Π) trmins uniqu trjtory Θ V. IMPLEMENTATION ISSUES Algorithm is sign to in singl optiml solution. Howvr, thr r situtions whr th purhs proility () Ntwork A. () Ntwork B. o rtin link i, â t,i, i {,...,m}, oul osillt or vry long tim or onvrging to ithr 0 or, nmly whn ) mny o th nit ntworks hv rliilitis tht () Ntwork C. () Ntwork D. r vry los to th optiml ntwork rliility; Fig.. Cnit ntworks. ) thr r suprluous links in th ntwork, tht is, links tht o not t th ovrll ntwork rliility. Th rsult o suh osilltory hvior is tht th simultion tim inrss signiintly. In this stion, w introu two lgorithms, th hyri CE lgorithm n th suprluous link rmovl lgorithm, whih n ru th omputtionl ort. A. Hyri CE Mtho In th sitution whr only on optiml ntwork xists ut whr mny ntworks n purhs whos ntwork (un)rliilitis r vry los to th optiml on, th lmnts o th purhs proility vtor oul osillt, n this woul inrs th omputtionl ort. In suh ss, on my trmint th lgorithm on th numr o purhs proilitis tht li twn [β, β] lls low rtin thrshol, sy δ, n thn gnrt ll nit ntworks oring to th purhs proility vtor. W vlut th prormns o ths ntworks n tk th ntwork with th st prormn s our inl solution to th prolm. As n xmpl, onsir th 6-no ully onnt grph givn in Figur. Th lk n gry soli lins init tht th purhs proilitis o ths links li in th rngs [ β, ] n [0, β] rsptivly. Th two sh links init tht th orrsponing purhs proilitis li in th intrvl [β, β]. In suh s, w trmint th min CE loop (stps n o th lgorithm) n gnrt ll nit ntworks, nmly th our ntworks shown in Figur. W thn vlut th ntwork unrliilitis to trmin whih ntwork topology is optiml. Th min hyri CE lgorithm or stimting th rliility o ntwork is thus summriz s ollows: Algorithm [Hyri CE Mtho]. ) Initiliz â 0. St t =. ) Gnrt rnom smpl X,...,X N using Algorithm, with = â t. Comput th smpl ( ρ)- quntil o prormns γ t using (6).

10 0 6 9 h B B 8 g Fig.. B A grph G with 8 loks. Fig. 6. Systm with thr trminl nos not y lk nos. ) Us th sm smpl to upt â t, using (). ) I th numr o purhs proilitis in th rng [β, β] is lss thn or qul to rtin thrshol, pro to th nxt stp (lt T th inl itrtion); othrwis st t = t + n ritrt rom stp. ) Gnrt ll nit ntworks oring to th purhs proility vtor â t n vlut th ntwork unrliilitis. Output th ntwork with th smllst unrliility s solution to th prolm. B. Suprluous Links W nxt isuss th rol o suprluous links in purhs ntwork. Ths r links tht o not t th ovrll ntwork rliility, whthr thy r untioning or not. As n xmpl, onsir th 8-no ntwork with trminl nos in Figur 6. In this ntwork, thr r iv suprluous links in totl, nmly links,,, 8, n 9. Sin ths links o not t th rliility o th ntwork, th CE lgorithm oul, unnssrily, hv iiulty iing whthr or not to purhs thm. In prtiulr, th purhs proilitis o ths links, â t,i with i {,,, 8, 9}, oul osillt or long tim or thy onvrg to ithr 0 or. This inrss th omputtionl ort signiintly. Not tht w only n to onsir rmoving suprluous links whn ϕ x (x) = ; tht is, i ll trminl nos r onnt givn tht ll purhs links r in untioning. Fortuntly, th intiition o suprluous links n B B B Fig. 8. Th lok-utvrtx tr o G. rlt to wll-unrstoo prolm in grph thory, nmly tht o intiying th loks in grph []. To illustrt th onpts, onsir Figur. Th opn irls in th grph r th so-ll utvrtis. Whn ny o ths is rmov, th grph sprts into isjoint omponnts. A similr onpt is tht o rig: n g whos ltion sprts th grph. A sugrph is ll lok i it is () rig inluing its nvrtis, () omplt grph with vrtis, or () grph with or mor vrtis suh tht no singl vrtx sprts it. Exmpls o h o th ss (), () n () r givn in th igur s B, B n B, rsptivly. Th nxt stp in th nlysis is to orm lok-utvrtx tr whos vrtis r intii with th loks n utvrtis o th grph, n whos gs join utvrtis to loks. Figur 8 givs th lok-utvrtx o th grph G o Figur. Th lrg lk n gry vrtis orrspon to loks; th smll whit vrtis orrspon to th utvrtis. Th thr lk vrtis r th trminl loks whih n to onnt in orr or th systm to untion. Th prour or rmoving suprluous links now oms lr: I lok is ithr () n isolt lok ( lok o

11 gr 0), () n n lok whih ontins no trminl no, or () n n no whih ontins xtly on trminl no n its jnt lok ontins th sm trminl no, lok n rmov. W rpt this prour until ll suprluous loks r rmov rom th lok-utvrtx tr. Thr xist iint lgorithms or ining ll th loks in grph,.g., Algorithm. o []. Th ollowing lgorithm n ppli to rmov suprluous links rom th grph. Algorithm [Suprluous Link Rmovl]. ) Fin ll isolt loks n rmov rom th lokutvrtx tr. ) Fin ll singl gr loks tht o not ontin ny trminl no n rmov rom th lok-utvrtx tr. ) Lt G n unmrk singl gr lok in th lokutvrtx tr tht ontins xtly on trminl no n pro to stp. Trmint i no suh lok xists. ) Lt G th lok to whih G is onnt. I G ontins th sm trminl no s G, prun G ; othrwis mrk G to kp in th lok-utvrtx tr. Rturn to stp. Not tht suprluous link in ntwork in y x os not mn tht th link is lso suprluous in x. By rmoving suprluous link with proility, th omputtionl ort rss, us th orrsponing purhs proility no longr osillts. Howvr, lik in th hyri CE mtho, th ury oul ngtivly t i th purhs proility onvrgs too quikly to 0, whih might l to suoptiml solution. To ovrom this iiulty, on oul hoos to rmov th link with proility lp, 0 lp. In Stion VI, w invstigt t wht proility suprluous link shoul rmov in orr to otin th mximum prormn o th lgorithms with minimum omputtionl ort. VI. NUMERICAL EXPERIMENTS In this stion, w prsnt thr tst ss. For ll tst ss w ompr th prormn o th trministi CE lgorithm (CE-DET) n th hyri CE lgorithm (CE-HYBRID) with thos o th noisy vrsions using CMC simultion (CE-CMC) n MP simultion (CE-MP). Thus, in th ormr s th systm rliilitis or h purhs vtor r stimt vi Cru Mont Crlo simultion, n in th lttr s vi th Mrg Pross. In stp o th hyri CE lgorithm, w rnk th nit ntworks using ithr trministi pproh (CE-HYBRID-DET) or MP simultion (CE-HYBRID-MP). All xprimnts wr rpt tims in orr to ssss th vriility n ury o th sttistis. Tst Cs : For th irst xprimnt w rturn to Exmpl, whr th optiml purhs vtor is givn y x = (, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0,, 0), whih givs minimum ntwork unrliility o r = Th orrsponing optiml ntwork is pit in Figur. W tk th sm CE prmtrs s in Exmpl n st K min = 000, K mx = 000, n δ =. In CE- HYBRID-MP, 0000 smpls wr us to rnk nit solutions. Tls III n IV show typil volutions o CE- CMC n CE-MP rsptivly. As ws th s with th trministi vrsion (s Tl II), oth mthos oun th optiml purhs vtor vry quikly. Th ntry in Tl III mns tht th orrsponing ntwork unrliility ws stimt (vi CMC) s 0. Not tht this n our whn th smpl siz K is rltivly smll. Tl V summrizs th sttistis ovr inpnnt rplitions. Hr Mtho rprsnts th mtho us in simultion; worst is th worst ntwork unrliility otin ovr th rplitions; st is th st unrliility otin in simultion; mn(t ) is th vrg numr o itrtions; n how otn is th numr o tims th mtho otin th optiml solution. Eh mtho prorm xptionlly wll, otining th

12 TABLE III A TYPICAL EVOLUTION OF CE-CMC FOR TEST PROBLEM t γ t â t TABLE IV A TYPICAL EVOLUTION OF CE-MP FOR TEST PROBLEM t γ t â t TABLE V NUMERICAL RESULTS FOR TEST CASE Mtho worst st mn(t ) how otn CE-DET CE-CMC CE-MP CE-HYBRID-DET CE-HYBRID-MP optiml purhs vtor in most o rplitions. This suggsts tht th propos noisy CE lgorithms using th CMC simultion n MP simultion n th hyri CE lgorithm n prorm s tivly n rlily s th trministi vrsion. Th s-rltiv rror in th rliility stimtion (or K = 00) ws roun 0 to 0% or CMC n % or CE- MP. Thus vn with this mount o unrtinty ll lgorithms n tivly lot th optiml ntwork topology or this prolm h j i 9 g Fig. 9. Ntwork topology or tst s. Tst Cs : Tst s is onrn with th 0-no ully onnt ntwork with trminl nos pit in Figur 9. This tst s is lot hrr thn th prvious on, not only us thr r nit ntworks, ut lso us mny o ths nit ntworks hv unrliilitis tht r vry los to th st oun solution (lss thn 0 0 wy).

13 TABLE VI THE PSEUDO CODE FOR GENERATING THE COST AND PROBABILITY VECTORS FOR TEST CASE i 0, j whil j m i i + 6 (mo 98 + i)/0 i > 0.99 n p k, or ll k =,..., j o p j thn j 0/ xp(8 ) j j + Fig.. l 6 8 s m 6 v h 9 p t k 6 n o 8 0 i x r u 8 w 0 q j Ntwork with nos n links. g g Fig. 0. Th st oun ntwork or tst s. h 0 9 j i TABLE VIII PSEUDO CODE FOR GENERATING THE LINK COST AND RELIABILITIES FOR CASE i 0, j whil j m i i + 6 (mo i)/0 i > 0.99 n p k, or ll k =,..., j o p j thn j 0/ xp(8 ) j j + Th totl ugt is C mx = 0, n w tk th ollowing CE prmtrs: N = 0, K min = 000, K mx = 0000, ρ = 0.0, α = 0., β = 0.0, n δ =. In CE- HYBRID-MP, 0000 smpls wr us to rnk nit solutions. Tl VI givs th psuo o or gnrting th link osts n rliilitis. Bs on inpnnt trils, th st oun solution or tst s is givn in Figur 0, with ntwork unrliility o Th rsults o rplitions or h mtho r summriz in Tl VII, with th xption o CE-CMC, whih il to prou solution th rson is tht CMC simultion rquir too lrg omputtionl ort (tht is, lrg smpl siz) to urtly stimt th vry smll ntwork unrliility;. (). It is nourging tht CE-HYBRID-DET otin th st oun ntwork in ll trils. CE-MP n CE-HYBRID-MP prorm rlily, otining n tims rsptivly. W oun tht CE-DET i not prorm s wll s othr mthos. On rson is tht th sp o onvrgn ws too st. Exprimnts show tht y hoosing smllr α, sy α = 0., CE-DET prorm s wll s its noisy ountrprt. Tst Cs : Th ntwork topology or tst s, tkn rom th survivl ix tlommunition ntwork sign lirry wsit ompriss nos, links n trminl nos, s Figur. In this prolm, w invstigt how suprluous link rmovl (Algorithm ) ts th ovrll prormn o th mthos, using irnt vlus or lp. For this tst s, w us th ollowing CE prmtrs: N = 000, K min = 000, K mx = 0000, ρ = 0.0, α = 0., β = 0.0, n δ =. Th totl ugt C mx = 000. In CE-HYBRID-MP, 0 smpls wr us to rnk nit solutions. Th psuo o or gnrting n p is givn low. Th st oun solution or tst s is givn in Figur,

14 TABLE VII NUMERICAL RESULTS FOR TEST CASE Mtho worst st mn(t ) how otn CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP l 6 8 s m v 8 9 h p t k n 6 o 8 0 i x r u 8 w 0 q j g Fig.. Th st oun ntwork or tst s. with ntwork unrliility o Tl IX shows th numril rsults or tst s, vrg ovr inpnnt rplitions or irnt vlus o lp. Hr how otn is th numr o tims th mtho otin th st oun solution or nr-st solutions, tht is, solutions whos ntwork unrliilitis r lss thn % wy rom th st oun solution. CPU nots th vrg simultion tim in sons. W s tht pplying Algorithm hivs signiint rution in omputtionl tim. Howvr or vlus o lp 0.8, ll our mthos il to otin th st oun or nr-st solutions. Th rson is tht som o th links in th ntwork my hv n onsir s suprluous links in rly itrtions. This us th purhs proilitis to onvrg to 0 vry quikly. This n voi y tking smllr lp. It is intrsting to not tht CE-MP prorm wors with suprluous link rmovl thn without. A likly rson is tht th s-rltiv rror in th ntwork rliility stimtor ws quit lrg, so tht th ntworks oul not urtly rnk. Howvr, to rs th rltiv rror to out 0.0 smpl siz o t lst K = 0000 woul rquir, whih woul signiintly inrs th omputtionl ort. In ontrst to CE-MP, th othr thr mthos prorm s wll with 0. lp 0. s with lp = 0, otining th st oun or nr-st solutions roun th sm numr o tims. This shows tht y rmoving suprluous links with proility 0. lp 0., w wr l to ru th omputtionl tim up to % without ting th ury o th mthos. Morovr, oth hyri CE lgorithms, with or without suprluous link rmovl, signiintly ru th omputtion tim, without ting signiintly th ury. VII. CONCLUSIONS AND FUTURE RESEARCH In this ppr, w propos CE pproh or solving th Ntwork Plnning Prolm, whih is iiult onstrin omintoril optimiztion prolm with noisy ojtiv untion. This nois is introu whn th ojtiv untion (th systm rliility) is stimt rthr thn vlut xtly. W rviw svrl thniqus or ruing this nois. W xplin how th trministi vrsion o th CE lgorithm (whn th ojtiv untion is known) oul rily moii to hnl th noisy sitution. Th numril rsults show tht th noisy CE lgorithm n prorm s tivly n rlily s its trministi ountrprt. W propos two thniqus to sp up th onvrgn o th lgorithm: proilisti suprluous links rmovl n hyri CE. Exprimnts show tht with n pproprit hoi o prmtrs ths thniqus n sp up th optimiztion

15 TABLE IX NUMERICAL RESULTS FOR TEST CASE WITH DIFFERENT VALUES FOR DELP lp Mtho worst st mn(t ) how otn CPU CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP CE-DET CE-MP CE-HYBRID-DET CE-HYBRID-MP pross y ovr 00% with vry littl impt on th ury o th lgorithm. W hv tkn prgmti pproh rgring onvrgn, osrving tht numrilly th CE lgorithms tns to onvrg in glol rthr thn lol sns. Although svrl CE onvrgn rsults r isuss in [], [], this topi rmins n intrsting hllng or utur rsrh. Anothr possil irtion is to invstigt whthr rliility rnking [] n urthr improv th prormn o th lgorithms. Hr only th rltiv rnkings o th ntworks r importnt, rthr thn th xt vlus o thir rliilitis. In som ss th ormr r sir to stimt thn th lttr. This oul l to n improvmnt o th prormn o th CE lgorithm. A urthr improvmnt oul hiv y xmining t wht stg in th simultion on shoul strt using suprluous link rmovl, n wht vlu o lp shoul tkn. Finlly, vloping n iint rr vnt prigm tht n ru th nois in th ntwork rliility stimtion with rltivly low omputtionl ort is nothr importnt r or utur rsrh.

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17 [] L. Mrgolin, On th onvrgn o th ross-ntropy mtho, Annls o Oprtions Rsrh, vol., no., pp. 0, 00. [] A. Cost, O. Jons, n D. P. Kros, Convrgn proprtis o th ross-ntropy mtho or isrt optimiztion, Oprtions Rsrh Lttrs, 00, to ppr. [] K.-P. Hui, Mont Crlo ntwork rliility rnking stimtion, IEEE Trnstions on Rliility, April 00. Dirk P. Kros hs wi rng o pulitions in ppli proility n simultion. H is pionr o th wll-known Cross-Entropy mtho n outhor (with R.Y. Ruinstin) o th irst monogrph on this mtho. H is ssoit itor o Mthoology n Computing in Appli Proility n gust itor o Annls o Oprtions Rsrh. H hs hl rsrh n thing positions t Printon Univrsity n th Univrsity o Mlourn, n is urrntly working t th Dprtmnt o Mthmtis o th Univrsity o Qunsln. Kin-Ping Hui is urrntly rsrh nginr t th Dn Sin n Thnology Orgniztion in Austrli. His rsrh intrsts inlu ntwork rliility stimtion, ntwork optimiztion, ntwork survivility, n ntwork rovry. Sho Nrii is PhD stunt in th Dprtmnt o Mthmtis t th Univrsity o Qunsln. His rsrh intrsts inlu pplitions o th Cross-Entropy mtho to ntwork sign prolms, ontinuous optimiztion, n noisy optimiztion.