R i c h a r d (2. 0gic.r ant1 L l a t l Rutenburg Sli I liit,cwiat,iorial Menlo Park, ('A 94025 Abstract 'I'his paper considers tlie problem of finding n routing st,rategy t.hat# minimizes the expected delay from c'veiy sonrce t,o a single tlestinat,ion in a network in whicli m r l i link fails and recovers according to a Markov chain. \'\:e assume Ohat each node knows the current st,a.t,e of it.s own out,goiiig links and the state-transition probnhilitries for every link of t.he network. We show t,hat, t l i e gcweral prohlem is #P-coniplet*e and consider t,wo s p -cia1 cases: (lase I assullies t,lie network is a clirect,cd acyclic graph ( D A G ) orient,etl toward t.lie dvstiii:ilioir, a t i t l (-:asp 2 assunies t.hat, the link states a.re iiitlrpeiitlclit, froin slot, t,o slot. For each case. we prove t,h;it t,hc 01)t.iiiial rorit.ing st.rabegy has a simple st,at.e-intlc,pcirtl~~lit rc~presrntation and we present, an efficient algorit h i r i for finding t he opt.iinal strategy. IC f0 11 OW i Ilg 1Jf /,I1 2171 11 112-e3j"Pf 1 cr/d C 7'0 It f Zn g 1) TO h1C 1Jt : INFOCOM '92 Theorem 2 Strategy plc zs optzmal, 2.e.. the best niiaoiig all slrategzes Namely, for each znztzal slate .c Of the OUlg(JZng /ZllkS, ,J(p{<)(X') < , l ( U ) ( . l ' ) f07 fill POIZczes U Proof To prove the t,heoreiii, it. suf1icc.s t,o sliow t,liat J ( @ ) ( z ) 5 J ( $ " ) ( z ) for all admissible decision rules $, since we know there exist.s a. st,at,ionary opt.inial policy. (H.ecall t,lia.t p~ = q5F.) Clea.rly, we only need to consider decision rules I ) siicli that., for any stmate .I:? .$ either c,hooses the best, link tliat i s up or waits. Let, d) be such a decision rule. By Theorem 1,
J ( d , E ) ( z ) < J ( $ P ) ( r ) for all P and all k , which by t,lieCompa.rison Lemma implies J ( d , F ) ( . r ) 5 , J ( + k r $ g ) ( z ) for a.ll z and all IC. Now for a.ny stsate 3: t,lierc exists a k such that 4 k performs exactly the same action a.s li,. (If $ chooses t,he best. link that is u p , let, IC he greakr tlian t,he niimber of t,Iiis link; ot,herwise IC(. k I)r Icss ~~I i a i r tIir niiniI~rr ol' t.Iiis IiiiI~.)-'I~Iif,rc,forr, J ( @ ) ( x ) 5 J(li,qhg)(x) for all ; E , wliicli by bliv Coniparison Lemma implies J(q5g)(x) < J ( $ " ) ( z ) for all 2. which proves t,he theorem.
Algorithms for the Global ProblemIn this sect,ion, we apply tjhe solnt,ion to t,he local problem t.o solve t,he minimum-expect.ed-delay rout,ing prohl e i n for (:ases 1 and 2 defined in Sectmioil I . Recall (,hat, t h r soliit,ioii t,o t,lir loci11 prol)lt~iii assuiiictl t,tiat. t,lir 1,iiric for a 1)acltet 1.0 rracli x? from a.ny nc,iglil)or j is iiitlepc'ndent, of the cnrrent, stat.e of t,lie orkgoing links at, node i . 'l'his assnmpt,ioii does not. hold for t,he general 5A.2.5 5A.2.9
