Cost Effective Multi-Period Spraying for Routing in Delay Tolerant Networks

Transcription

1 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 Co Effcv Mul-Pro Sprayng for Roung n Dlay Tolran Nwor Eyuphan Bulu Mmbr IEEE Zjan Wang an Bollaw K. Szyman Fllow IEEE Abrac In h papr w prn a novl mul-pro prayng algorhm for roung n Dlay Tolran Nwor DTN. Th goal o mnmz h avrag copy coun u pr mag unl h lvry whl mananng h prfn mag lvry ra by h gvn aln. In ach pro om numbr of aonal cop ar pray no h nwor follow by h wa for mag lvry. A any m nanc h oal numbr of mag cop rbu o h nwor pn on h urgncy of achvng h lvry ra by h gvn aln for ha mag. Wang for arly lvry n h nal pro wh mall numbr of cop n nc cra h avrag numbr of cop pray n h nwor ll lvry. W fr cu - an -pro varan of our algorhm an hn w alo gv an a how h prn approach can b n o mor pro. W prn an n-ph analy of h algorhm an vala h analycal rul wh mulaon. Th rul monra ha our mul-pro prayng algorhm ouprform h algorhm wh ngl prayng pro. In Trm Dlay Tolran Nwor Roung Co Effcncy I. INTRODUCTION EAY Tolran Nwor DTN [] ar wrl Dnwor n whch a any gvn m nanc h probably ha hr an n-o-n pah from h ourc o h naon low. Thr ar many ampl of uch nwor n ral lf nclung cology monorng [ ] popln [4] ocan nor nwor [5 6] vhcular a hoc nwor [7] an mlary nwor [8]. Snc h anar roung algorhm aum ha h nwor connc mo of h m hy fal whn appl o roung of mag n DTN. Th rann nwor conncvy n o b of prmary Manucrp rcv Novmbr Rarch wa ponor by US Army Rarch aboraory an h UK Mnry of Dfnc an wa accomplh unr Agrmn Numbr W9NF Th vw an concluon conan n h ocumn ar ho of h auhor an houl no b nrpr a rprnng h offcal polc hr pr or mpl of h US Army Rarch aboraory h U.S. Govrnmn h UK Mnry of Dfnc or h UK Govrnmn. Th US an UK Govrnmn ar auhorz o rprouc an rbu rprn for Govrnmn purpo nowhanng any copyrgh noaon hron. Eyuphan Bulu. Auhor wh h Dparmn of Compur Scnc Rnlar Polychnc Inu Troy NY 80 USA. -mal: bulu@c.rp.u. Zjan Wang an Bollaw Szyman. Co-auhor ar wh h Dparmn of Compur Scnc Rnlar Polychnc Inu Troy NY 80 USA. -mal: {wangz zyman}@c.rp.u. concrn n h gn of roung algorhm for DTN. Hnc n rcn yar nw algorhm ung buffrng an conac m chul hav bn propo. Snc mo of h no n a DTN ar mobl h conncvy of h nwor manan only hrough mobl no whn hy com no ranmon rang of ohr no. Roung of mag ba on or-carry-an-forwar paragm. Tha f a no ha a mag copy bu no connc o anohr no or h mag unl an appropra communcaon opporuny ar. Th mporan conraon n uch a gn ar h numbr of mag cop ha ar rbu o h nwor an h lcon of no o whch h mag rplca. In h papr w uy how o rbu h cop of a mag among h ponal rlay no n uch a way ha h prfn prcnag r lvry ra of all mag ar lvr by h gvn lvry aln wh h mnmum numbr of cop u. Unl h prvou algorhm ba on mag prayng w nrouc a m pnn copyng chm whch bacally conr h m rmanng o h gvn lvry aln whn mang copyng con. Th a of our algorhm a follow. I fr pray h numbr of cop mallr han ncary o guaran h r lvry ra of mag o h naon bfor h gvn lvry aln. If h lvry o no happn for a cran pro of m hn h algorhm pray om aonal cop of h mag o ncra h probably of lvry. A a rul h algorhm paron h m o h prfn aln no vral varabl-lngh pro ach compo of prayng pha follow by h wa for lvry. In prayng pha carfully chon numbr of cop of h mag ar pa o no ha o no po on y. Thn h wang pha ar n whch h lvry of h mag wh avalabl cop amp npnnly by ach copy holr. I mporan o no ha onc h allow numbr of cop for h gvn pro ha bn rbu no mor cop wll b rbu o any no unl h bgnnng of n pro. If h mag lvr n h arly pro of uch mul-pro prayng frqunly nough h avrag numbr of cop u pr mag wll b ruc compar o h ngl pro prayng n whch all cop of h mag ar rbu a h bgnnng of h roung. W call h TT valu agn o mag whn hy ar gnra a ourc no a h lvry aln of h mag.

2 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 Th rmanr of h papr organz a follow. In Scon w prn h prvou wor on on h opc an cu om bac mobly a roung concp. W alo commn abou h ffrnc bwn our algorhm an h ohr. In Scon w crb our algorhm n al an prov an analy of ffrn varan. In Scon 4 w valua h prn algorhm ung mulaon an monra h achv mprovmn. W alo compar h rul of our analy wh h mulaon rul. Fnally w offr concluon an ouln h fuur wor n Scon 5. II. REATED WORK Thr ar varou clafcaon of roung algorhm for lay olran nwor [9 0]. Hr w v hm no wo cla: rplcaon ba algorhm an cong ba algorhm. In rplcaon ba algorhm mulpl or a ngl copy of h mag gnra an rbu o ohr no ofn rfrr o a rlay n h nwor. Thn all of h no npnnly of ohr ry o lvr h mag copy o h naon. In cong ba algorhm [] [] a mag convr no a larg of co bloc uch ha any uffcnly larg ub of h bloc can b u o rconruc h orgnal mag. Conqunly a conan ovrha manan an h nwor ma mor robu agan h pac rop whn h congon ar. Howvr h algorhm nrouc an ovrha of an ra wor n for cong forwarng an rconrucng co bloc. Epmc Roung [] an approach u by h rplcaon ba roung algorhm. Bacally urng ach conac bwn any wo no h no chang hr aa o ha hy boh hav h am cop. A h rul h fa pra of cop achv ylng h hor lvry m. Th prformanc analy of pmc roung wllu n many arcl nclung [4] [5]. Th man problm wh h approach h ovrha ncurr n banwh buffr pac an nrgy conumpon by h gry copyng an orng of mag. Hnc h approach nappropra for rourc conran nwor. To ar h wan of pmc roung h algorhm wh conroll rplcaon or prayng hav bn propo [6] [7] [8] [9] [] []. In h algorhm only a mall numbr of cop ar rbu o ohr no an ach copy lvr o h naon npnnly of ohr. Of cour uch an approach lm h aformnon ovrha an prov an ffcn ulzaon of nwor rourc. Th rplcaon ba chm wh conroll rplcaon ffr from ach ohr n rm of hr aumpon abou h nwor. Som of hm aum ha h rajcor of h mobl vc ar nown whl om ohr aum ha only h m an uraon of conac bwn no ar nown. Morovr n om of hm [0] aum ha vn h no movmn can b conroll. Ohr han h u whch aum om aonal faur hr ar alo om wor whch aum zro nowlg abou h nwor. Th algorhm whch fall n h la cagory m o b h mo rlvan o h applcaon bcau mo ofn nhr h conac m nor h rajcor ar nown for cran n h applcaon of lay olran nwor n ral lf. An ampl coul b a wl lf racng applcaon whr h no ar aach o anmal ha mov unprcably. Th algorhm whch aum zro nowlg abou h nwor nclu h on prn n [] MaProb [] SCAR [] an Spray an Wa [4]. In ach of h algorhm lm numbr of cop ar u o lvr a mag. Y h proc of choong h no for placng nw mag cop ffrn n ach of hm. In [] an MaProb ach no carr lvry probably whch upa n ach conac wh ohr no. If a no wh a mag copy m anohr no ha o no hav h copy rplca h mag o h no n conac only f ha no' lvry probably hghr han own probably. A mlar a u n SCAR. Each no manan a uly funcon whch fn h carrr qualy n rm of rachng h naon. Thn ach no r o lvr aa n bunl o a numbr of nghborng no whch hav h hgh carrr qualy. In [4] Spyropoulo al. propo a ngl pro prayng algorhm n whch all h mag cop ar gvn o h ohr no a h bgnnng hn h wang pha nr an h lvry of h mag by any of h cop pc. Hr no ha h algorhm a pcfc ca of our algorhm n whch h numbr of pro ju on. In ha papr auhor alo propo wo ffrn way for h rbuon of mag cop o h nwor: Sourc Spray an Wa an Bnary Spray an Wa. Whl n h formr only h ourc capabl of prayng cop o ohr no n h lar all no havng a copy of h mag ar alo allow o o o. In Bnary Spray an Wa whn a no cop a mag o anohr no alo pa h rgh of copyng half of rmanng copy coun o ha no. Th rul n rbu an far prayng compar o h ourc prayng bu onc h prayng on h pc lvry lay h am for boh. In [] an analy on h pc lvry lay of mag n h wo algorhm prov by h am auhor. Alhough hr ar many algorhm ulzng h conroll floong approach o h b of our nowlg h a prn n h papr complly nw an ha no bn u by any of h pror wor. Th man conrbuon of h wor h rbuon of mag copyng proc ovr many pro wh ncrang urgncy of mng h r lvry ra by h aln. Th rulng aapvy of h numbr of cop pray o h nwor complly ffrn from ohr aapv copyng rag [5] [6]. Th prn mul-pro prayng algorhm can b conr a a gnralzaon of prayng ba algorhm uch a h ngl pro pray an wa algorhm prn n [4]. W how unr wha conon uch gnralzaon ruc h avrag numbr of cop u pr mag whou crang h ra of mag lvr by h

3 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 aln. Th al of h novl approach ar gvn n h n con. Whl gnng a roung algorhm for mobl nwor an mporan u ha mu b conr h mol of mobly of no n h nwor. Thr ar many mobly mol propo for mobl no [7] [8] [9]. Howvr ranom rcon ranom wal an ranom waypon mobly mol ar h on u mo ofn by h publh roung algorhm. In gnral no ncounr n a mobly mol coul b characrz by a paramr call pc nrmng m EM. In many mol aum ha h m lapng bwn wo concuv ncounr of a gvn par of no ponnally rbu wh h man EM. Howvr h rbuon of h nrmng m of h nwor no pcfc o ach mobly mol o h paramr can b rv whn h nwor paramr an h aum mobly mol ar nown [0]. III. MUTI-PERIOD SPRAYING In h con w fr l h aumpon of our mol an hn crb our roung algorhm n al. Morovr w alo prn h analy of h propo algorhm wh varan. A. Nwor Mol an Aumpon W aum ha hr ar M no movng on a N N D oru accorng o a ranom mobly mol. Each no ha a ranmon rang R an all no ar ncal. Th mng m of no ar aum o b npnn an ncally rbu IID ponnal ranom varabl. Furhrmor w alo aum ha h buffr pac n a no unlm h aumpon no crucal nc h prn algorhm u h prfn numbr of cop wh h mamum numbr comparabl o h ngl pro prayng algorhm. W alo aum ha h communcaon bwn no prfcly parabl ha any communcang par of no o no nrfr wh any ohr mulanou communcaon. To b conn wh prvou rarch by w no h numbr of cop ha ach mag rbu o h nwor. In Spray an Wa algorhm [] h lvry of a mag can happn boh n pray an wa pha. Th probably of mag lvry a or bfor m whn hr ar cop of h mag n h nwor p - - whr /EM h nvr of h pc nrmng m bwn wo concuv ncounr of any par of no. Durng wang pha nc conan p grow wh h am valu. Howvr nc h numbr of cop ncra urng h prayng pha p funcon chang ach m a nw copy rbu o ohr no. To mplfy h analy of mag lvry probably w aum n h papr ha M >> whch ofn ru n DTN an whch w nforc by lmng prmbl valu of. Morovr for DTN o b of praccal u h lvry probably p mu b clo o o w aum alo ha p 0.9. W wll how blow ha from h wo aumpon follow ha h formula p - - a goo appromaon of h lvry probably a m. Fg. Th cumulav rbuon funcon of lvry probably n h Spray an Wa algorhm for ffrn valu of λ/em whr λ < λ < λ. A h h ncounr wh anohr no h prayng no lvr h mag o h naon wh probably /M o h oal probably ha h mag lvr urng prayng bwn /M an /M- an nc M>> /M a goo appromaon of h probably. Bnary prayng u log p ach wh avrag m abou EM/M n h p - no pray a mag copy o - ohr no o h oal prayng lay logem/m. Th approma formula achv h am lvry probably a h arlr m EM/M an from ha m on mach h bhavor of h algorhm prfcly. Hnc h avrag ffrnc bwn m a whch algorhm an formula achv h am lvry probably log-em/m. Thu h rlav rror p of ung h approma formula for p : p p log. M Snc p 0.9 for <048 o much byon h rang of uful valu of h rlav rror of appromaon mallr han /M whch a mall fracon for M>>. Fg. how h cumulav rbuon funcon cf of h lvry probably n a ngl pro pray an wa algorhm for ffrn valu. Clarly whn ncra h man valu /λ EM/ of ponnal cf cra an h pc lay oghr wh h m n o rach h r lvry probably hrn. Our man conrbuon o nrouc an analyz h mulpro prayng algorhm an o how unr wha conon mor ffcv han h ngl pro prayng. In our algorhm prayng of mag cop fn by h urgncy of mng h r lvry probably by h gvn lvry aln. Mor prcly h algorhm ar wh prayng fwr mag cop han h mnmum n by h ngl prayng algorhm an hn wa for a cran pro of m o f h mag lvr. Whn h lvry o no happn h algorhm pray om aonal cop of a mag an agan wa for h lvry. Th proc rpa unl hr h mag lvr or h lvry aln pa. Hnc a h m rmanng o h lvry aln cra an lvry ha A h n of Scon w plan how h lvr mag ar acnowlg o ohr no.

4 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 4 no y happn h numbr of no carryng h mag copy ncra. To h b of our nowlg h a ha no bn u by any of h prvouly publh algorhm for DTN roung. Fg. Th cumulav rbuon funcon of lvry probably of a mag whn ffrn cop ar pray n wo ffrn pro. Fg. ummarz wha our algorhm gn o achv. In h pcfc vron of h algorhm w allow wo ffrn prayng pha. Th fr on ar whou lay an h con on ar a m. Th man objcv of h algorhm o amp lvry wh mall numbr of cop an u h larg numbr of cop only whn h amp unuccful. Wh propr ng h avrag numbr of cop pray unl h lvry m can b lowr han n h ca of prayng all mag whou lay whl h lvry probably by h aln rman h am. To analyz h prformanc of our algorhm analycally w n o rv wo formula; on for h avrag numbr of cop u by h algorhm an h con on for h cumulav rbuon of h probably of lvry wh h ncrang numbr of cop an hrfor wh h ncrang λ valu. In our chm h rm pro rfr o h m uraon from h bgnnng of on prayng pha o h bgnnng of h n prayng pha. Thr may b mulpl prayng pha an h corrponng pro bwn hm ach of ffrn lngh. In h n con w ar wh h analy of h wo pro ca o fn h opmal pro lngh an h corrponng copy coun for ach pro. In ubqun con w analyz h hr an mulpl pro ca. B. Two Pro Ca If hr ar wo pro unl h mag lvry aln h quon ha n o b anwr ar whn o fnh h fr pro an ar h con on? an how many cop houl b allow n ach?. In ohr wor wha houl b h valu of n Fg. o mnmz h avrag numbr of cop u by h algorhm an how many cop houl b pray n ach pro? ' aum ha h ngl pro pray an wa algorhm u cop nclung h copy n h ourc no of a mag o achv h r lvry probably p by h aln. ' furhr aum ha h Two Pro Sprayng algorhm pray cop o h nwor a h bgnnng of cuon an aonal - cop a m h bgnnng of h con pro. Thn h cumulav rbuon funcon of h probably of mag lvry a m : f cf f > whr h lay wh whch h prayng wh cop woul n o ar o mach h prformanc of our algorhm n h con pro S Fg.. Th valu of h can b foun from h qualy of rpcv cf funcon a m : Th pc lvry probably whn cop ar u n h ngl pro pray an wa algorhm by fnon p. Our objcv o mach h lvry probably whl crang h avrag numbr of cop blow. Hnc by h lvry aln h followng nqualy mu b af: W can u h nqualy o boun a. A g largr h avrag copy coun cra whn an valu rman conan. Snc our algorhm am a crang h avrag copy coun whl mananng h lvry probably of h ngl pro prayng algorhm a m h opmal mu b h larg pobl an hrfor: W wan o mnmz h avrag numbr of cop c fn a: c No ha f h mag no lvr n h fr pro hn h co w fn co a h numbr of cop u pr mag bcom cop. Subung n h abov w g: c Tang rvav of c w oban: c W ar only nr n h gn of h rvav o w can gnor alway pov facor. For h am raon w can alo mulply h rul by alway pov

5 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 5 facor -. A w conr only valu > hn w oban: c gn gn W conclu ha gn of rvav chang only onc a: > an chang from ngav o pov o h co funcon ha h unqu mnmum a h pon. Hnc an hrfor: c [ Agan by ang h rvav of c n rgar of an comparng o zro w can oban h opmum valu of. z an A hn: z c z z A A z z Sng f z c z w g f z z A A z z Tang rvav of h funcon n rgar of z w oban: z f z Az z z Thn by mulplyng by an alway pov facor w oban: z z g z f z z Az W noc ha bcau h numbr an yp of rm pon of a funcon ar fn by h numbr of zro an h rvav gn nar h zro funcon g z an c hav h am numbr an yp of rm pon rla by quaon z. Th fr zro of funcon g z a z 0 an corrpon o mamum whn A > bcau nar zro g z z Az z A o pov for z < 0 an ngav for z > 0. If A < hn of cour h pon z 0 a h mnmum an corrpon o whch man ha h on-m prayng opmal n uch a ca. For A h pon z 0 nhr bu hn h rvav non-ngav for z > 0. Thrfor on-m prayng opmal agan. Howvr h conon ha A > af n ralc applcaon bcau A p > 6%. Uually h raonabl valu for p ar n hgh 90 prcn whch man ha h nrouc algorhm wll prform br whn raonabl lvry probably by h aln rqur. Mor formally for 0 < 4z < mn A 4/7 w hav: z z z z z < z 0! z z z < z z z ] z an z Az < z o g z < z z 0. Smlarly for < z < 0 w hav z > z o g z > z z Az z z A > 0. Th alo man ha hr a la on mor zro pon a g z a connuou funcon ngav n clo pov nghborhoo of 0 an pov n. Morovr conrng h qualy of wo funcon: z z Az W noc ha z z a conv funcon whch hav a mo wo nrcon pon wh any ragh ln nclung h ln Az. Hnc z z an Az nrc a z 0 an for A > acly onc mor a h pon corrponng o h mnmum whch h pon of nr o u hr. In l z op > 0 no h nar o 0 nrcon pon of h wo funcon. From h conv propry of h fr funcon follow ha o h rgh of z h fr funcon alway abov h ragh ln Az. op z > zop. z op < A a for z A w hav Hnc h wo funcon canno nrc for Furhrmor mu b ha z z > A Az > Az o h co funcon alray growng. W conclu ha hr a unqu opmum pon a z op > 0 f an only f A > or n ohr wor f an only f h rqur lvry probably by h aln grar han or uccncly p > 6% a vry raonabl conon for praccal oluon. Th pon can b foun n log A p by bcng h nrval 0 A unl w g h rang of h oluon whn wo concuv ngr. Thn w can u h floor an clng of h appromaon o fn ngr oluon ha w ar nr n. Comply of h algorhm low O log A an bcau A ln p A h naural logarhm of h nvr of h probably of nonlvry of a mag by h n of h aln. Hnc h comply h polylogarhmc funcon of h nvr. W can alo fn h opmal valu of an by a mplr mho whch gnralz ncly o ca wh mor pro o w wll prn hr. From h quaon fnng c clar ha h avrag numbr of cop pray

6 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 6 by our algorhm largr han o for our algorhm o b abl o cra h avrag numbr of cop blow mu b mallr han. A a rul h followng bounar for mu hol: < < 0 Snc h pobl valu for all varabl ar ngr w can u numraon mho a plan n Algorhm an oban h opmal valu rlavly qucly n O p. Wh conan valu of EM an h a logarhmc funcon of h nvr of nonlvry probably hnc grow far han comply of fnng a oluon va h rvav of h co funcon. C. Thr Pro Ca In h con w aum ha hr ar hr pray an wa pro unl h lvry aln. In h ca w n o fn wo ffrn bounary pon whch para h hr pro. an no h bounary pon. Whl h formr an a h bounary bwn h fr an h con pro h lar mar h bounary bwn h con an h hr pro. Th cumulav rbuon funcon of h probably of mag lvry by h m : ] ] ] [0 cf whr an ar h lay wh whch h con an h hr prayng woul hav o ar o qual h cf of our algorhm ovr h con an hr prayng pro rpcvly. A bfor ung h qualy of h funcon a m an w can oban h valu of an : an analogouly: Fg. Th cumulav rbuon funcon of lvry probably wh cop pray n hr ffrn pro. Fg. llura our approach wh hr pro. Smlar o h wo pro ca w wan o achv h am or hghr lvry probably p a h gvn aln whl mnmzng h avrag numbr of cop u. Tha w n o afy h followng nqualy: Ung h nqualy w can lmna bcau a g largr h avrag copy coun g mallr whn all ohr paramr ar p conan. Thrfor rplacng h abov nqualy wh an quaon w oban: Furhrmor h avrag copy coun u n h hr pro prayng can b fn a: c Whn w ubu an n c an a h rvav c w oban: c Afr gnorng h alway pov facor h gn of h rvav bcom: gn c gn W ha f < h gn alway pov no ha 0 by fnon ha h co funcon alway growng wh ncrang. Thrfor mnmum co oban a 0. Ohrw w noc ha h gn of h rvav chang from ngav o pov only onc a: / ln For boh ca w oban h opmum valu of. Thn w can aly oban formula c by

7 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 7 ubung wh h opmum valu n corrponng conon. Snc < < an an all h valu ar ngr by numraon plan n Algorhm w can mply fn h copy coun ha gv h mnmum copy coun for a gvn. Howvr o u numraon w n o ablh boun on boh an. Ung nqualy ha mu b < af for h con pro o ar bfor h aln for mag lvry w can calcula h uppr boun for no a a follow: Boun > c > < Boun Now w hav h rang for an. Thu for a gvn w wll ry o oban h opmum valu ha ma h co funcon mnmum. Whn w a h rvav c w oban: c m whr m Snc w ar nr n h gn of h rvav w can gnor h alway pov facor. Thn w hav: c gn gn A w conr h valu > w conclu ha h gn of h rvav chang from ngav o pov only onc a: [ ] Whn w ubu n h quaon wh h opmum n Eq. w hav h followng quaon: ln ln Snc o fn h valu of n h abov funcon no op o ay w can na fn a rang for hn by op bcng whn h rang w can rach h opmum ngr valu of. I obvou ha af h followng: R op ln op < R On h ohr han op alo bggr han ln R bcau R ln > op op R ln R R Thrfor l whn h rang R ]. Snc a op [ R ncra h valu of ln ncra w can fn h ngr opmum valu of by bcng n h rang. On can aly ha h comply of h bcng arch O log. In Algorhm for ach par w fr fn h opmum mnmzng h co funcon hn w compar wh h currn opmum co. Hr no ha f < hn c oban by ung h opmum 0. Ohrw c compu ung h opmum valu gvn n Eq.. To a comply of Algorhm w obrv ha can b approma a follow: Boun < < ma p bcau funcon f ha rvav an hrfor mamum a. Hnc h comply :

8 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 8 p O log In concluon h comply of numraon n h ca log p O o nvrly proporonal o p m ra m logarhm of h nvr of m ra. D. Incrang h Numbr of Pro by Rcurv Paronng In h con w how ha by applyng rcurv paronng of ach pro mor prayng pro can b cra n uch a way ha h oal co of prayng can b cra vn mor. An ampl gvn n Fg. 4. From Two Pro Ca con w now how o achv h opmum paronng of h nr m nrval from h ar o h lvry aln no wo pro. Howvr alo pobl o paron ach of h wo pro nvually o cra h co of prayng vn furhr. Alhough h may no b h opmal paronng n h rulng numbr of pro ll cra h prayng co. If w wan o hav hr pro unl h mag lvry aln w can paron hr h fr pro wh paramr λ or h con pro wh λ an lc h on whch achv h lowr co. In ohr wor w n o lc hr λ λ 4 λ or λ λ 5 λ 6 a h ponnal facor n h corrponng hr ponnal funcon. Furhrmor afr obanng h hr pro prayng w can run h am algorhm o fn a lowr co prayng wh four pro. Howvr w n o paron ach pro carfully conrng h bounar of pobl valu. Aum ha w currnly hav pro of prayng. no h copy coun afr prayng n h pro an no h n m of ha pro. Thn h cumulav rbuon funcon of h probably of mag lvry by h m bcom: [0 ] ] cf... ] whr h lay wh whch prayng wh cop woul hav o ar o qual h cf of our algorhm ovr h h prayng pro. Obvouly 0 an for > w hav: Fg.4 Rcurv paronng algorhm o fn mor pro of prayng an furhr cra h oal co of prayng. Th pron ay o rv from h followng mpl rav fnon of for > rulng from h qualy of h rpcv ponnal funcon a pon : W wan o ncra h numbr of pro o whl crang h oal co for prayng wh h am lvry probably a h lvry aln. Algorhm an 4 ummarz h p o achv h goal. Bacally w paron ach pro no wo pro on by on o fn h nw co for h currn paronng. Thn from h pobl paron w lc h on ha achv h low co. For ach pro w n o fn nw numbr of cop o agn o ach of h wo nwly cra pro no whch h orgnal pro pl. Th lvry probably a h n of h boh pro n o ay unchang bu h avrag co houl b mallr han h orgnal avrag co of pro. For ach pro bng pl cp h la on hr ar h followng boun on ho wo numbr: < < < W can alo fn an uppr boun for h la pro whch w wll no for convnnc a. pl no h bounary pon n whch h con nnr pro ar. h ar of pro for prayng aonal - cop. Th valu of pl can b foun from h qualy of h probably of mag lvry by h n of h orgnal an h pl pro: Subung an by h formula n Eq. whch clarly an mu alo oby w oban: j pl j j j For h la pro w n o fn an uppr boun for h valu of wh gvn. Th co of h la pro n rm of avrag numbr of cop u lghly ffrn

9 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 9 han h co of ohr pro. p no h probably of mag lvry bfor h pro ar. Smlarly l p pl no probably of mag lvry bfor h con a pro ar. Of cour p p pl p whr p no h probably of lvry of h mag by h aln. Th co of h orgnal pro can b mply wrn a: Co p whra h co of h pl pro : Co p p pl pl p p Snc w wan Co > Co pl hn h followng nqualy mu hol: p p < p whch yl h followng uppr boun for fabl valu of : p < p Algorhm how how h opmal paronng of a ngl pro whr 0<< foun. For convnnc w no 0 0. For ach par of numbr uch ha h co of prayng foun an < opmal par whch gv h mnmum co lc. Clarly h comply of h algorhm O. E. Acnowlgmn of Dlvry Th crpon of h mo of h publh roung proocol for lay olran nwor o no conan al of how h no n h nwor larn abou h lvry of a mag o h naon o avo prayng afr h mag lvry. Y h a crucal u n our algorhm bcau rcly affc h co of copyng of mag. If a mag lvr o naon bu a pcfc no no nof abou h lvry h no wll connu prayng h mag ncrang h avrag co of copyng. In h papr w uy wo yp of acnowlgmn for nofyng h no abou h lvry of h mag. TYPE I: Whn naon rcv a mag fr cra an acnowlgmn for ha mag an n o ohr no whn rang whch aum o b am for all h no n h ca. Thn ung pmc roung h acnowlgmn pra o all ohr no whnvr hr a conac bwn a no carryng h acnowlgmn an a no whou. No ha ofn h acnowlgmn pac whch carry only acnowlg mag ar much mallr han aa mag. In uch ca h co of h acnowlgmn pmc roung mall compar o h co of roung h aa pac. Mor coly h lay wh whch all no n h nwor larn abou h lvry of h mag. Durng h lay hr may b ul prayng of h alray lvr mag ncrang h oal co of copyng. TYPE II: In h yp of acnowlgmn w aum ha h naon u on m broaca ovr h mor powrful rao han h ohr no h aumpon ofn af n pracc o h broaca rach all h no n h nwor. n h prvou ca h acnowlgmn mag hor o broaca npnv. Howvr o ma h chm mor ffcn w u h followng pmology npr a. W conr an nvronmn n whch a ffrn m nvual nfc by ffrn pahogn. Each pahogn ha an ncubaon pro urng whch h nfc nvual no conagou. Afr h ncubaon pro h c nvual conguou an abl o nfc ohr. W aum ha hr ar ffcv vaccn for all pahogn an w wan o vaccna h nr populaon wh h propr m of vaccn n h mo ffcn way. Th b way o achv h goal o wa unl h clo n of an ncubaon pro of any nfc nvual an o apply h vaccn for all obrv nfcon o h nr populaon a ha m. Such lay vaccnaon campagn allow mrgnc of nw nfcon pobly wh nw yp of pahogn bfor lng c nvual nfc ohr. Th approach mnmz h numbr of ncary vaccnaon campagn ach wh all vaccn ncary o op alray ar pmc. Inpr by h a w u h followng ffcn acnowlgmn chm. A h naon rcv mag wa unl h clo pro chang m of any of h rcv mag. A ha m h naon broaca an acnowlgmn of all o far rcv

10 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 0 mag. Hnc h naon broaca acnowlgmn rlavly nfrqunly proporonally o a ubanal fracon of h whch aum larg. Evn hough acnowlgmn of om mag ar lay prayng of any rcv mag afr h lvry m ar uppr. I clar ha Typ II acnowlgmn rul n br prformanc han Typ I acnowlgmn n rm of h oal numbr of cop u pr mag. Howvr may rqur hghr nrgy conumpon. In mulaon w compar h prformanc of boh yp of acnowlgmn by howng how hy affc h rul of our algorhm. IV. SIMUATION MODE AND RESUTS To valua our mul-pro algorhm w hav vlop a cr vn-rvn mulaor n Java. W prform nv mulaon wh ffrn paramr ha may affc h prformanc of h propo algorhm. Fr of all w compar h rul of mulaon wh h analycal rul ha w hav oban n prvou con. Morovr w alo loo a h ffc of wo ffrn mobly mol on h rul. W ploy M00 mobl no nclung h n ono a oru of z 00 m by 00 m. All no cp h n ha ha hgh rang of acnowlgmn broaca n TYPE II ca ar aum o b ncal an hr ranmon rang a R 0 m no ha h paramr gnra a par lay olran nwor whch h mo common ca n pracc. Th movmn of no ar c accorng o wo ffrn mobly mol [0]: Ranom Wal Mol: Th p of a no ranomly lc from h rang [4 ]m/ an rcon alo ranomly chon. Thn ach no go n h lc ranom rcon wh h lc p unl h poch la. Each poch' uraon agan ranomly lc from h rang [8 5]. Ranom Waypon Mol: Fr a nw naon n h nwor ara chon ranomly. Thn h no mov owar ha naon wh a ranomly lc p from h rang [4 ]m/. Whn no mov accorng o h abov mol wh gvn paramr h valu of EM n h formr an lar bcom 480 an 50 rpcvly w boh compu h valu from h gvn paramr an vala h rul by mulaon. Aumng ha h r p by h gvn aln 0.99 fr w hav foun h opmum copy coun for boh wo pro p an hr pro p ca ung Algorhm an Algorhm. Tabl how h valu of h opmum ' for ffrn valu a wll a h mnmum valu ha achv h r p n ngl pro p pray an wa algorhm. Clarly a h aln cra mn mnmum achvng p by n p ncra bcau mor cop ar n o m h r p by h aln. Such an W hav lc a hgh r lvry probably bcau h mo lly ca n ral applcaon. Howvr w alo loo a h ffc of ffrn p valu n lar mulaon. ncra alo obrv for valu u n boh p an p algorhm. I alo mporan o rmar ha h opmum valu ar ffrn for ranom wal an ranom waypon mol bcau h EM valu gnra n h wo ffrn ng ar ffrn. Alhough w mnon n Scon ha our algorhm ar gn for h nvronmn n whch h aln no o gh wh rpc o EM valu n h mulaon w alo our algorhm wh gh aln uch a 00 an 50 4 o how hy prform n h ca. Morovr for h opmum valu of hr pro w alo ran Algorhm an Algorhm 4 ovr h rul ha w oban wh Algorhm an obrv ha h rul cloly mach h opmum valu ha w oban ung Algorhm. TABE I OPTIMUM COPY COUNTS THAT MINIMIZE THE AVERAGE NUMBER OF COPIES WHIE PRESERVING THE DESIRED PROBABIITY OF DEIVERY. mn n p Ranom Wal p Opmum p Opmum mn n p Ranom Waypon p Opmum p Opmum W ar by compung an h opmum valu from hory. Thn w prform mulaon o fn h avrag copy coun u pr mag whn h compu valu ar u. W hav gnra mag from ranomly lc no o h n no who nal locaon wa alo chon ranomly. Furhrmor w u bnary prayng whl rbung h allow copy coun n ach pro. All rul ar h avrag of 000 run. In Fg. 5 an Fg. 6 w how h avrag copy coun oban whn h opmum valu ar u n p an p vron of our algorhm an whn h prfn ranom wal mol u. Our analy fn h co funcon a h avrag copy coun u pr mag a h ac lvry m an compu h opmum valu whch mnmz h co funcon. Hnc o compar hory wh mulaon w oban h avrag copy coun n mulaon ung Typ II acnowlgmn. Howvr w alo nclu h avrag copy coun oban n mulaon whn Typ I acnowlgmn u. From h rul n boh fgur w obrv ha analy rul ar vry clo o Typ II rul bu a h aln g gh hy bcom an uppr boun for Typ II rul. Th bcau for h mallr valu of h numbr of cop pray o h nwor 4 Th valu can urly b conr a gh aln bcau no ha rc lvry n ngl prayng can achv p 0.99 a 0 an 60 n h gvn ranom wal an ranom waypon mol rpcvly.

11 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 ncra opmum valu n p an p ar larg u o larg mn n p o ha prayng pro a longr. B h alo ncra h ffrnc bwn h avrag copy coun n whn Typ I an Typ II acnowlgmn ar u bcau a valu g largr mor no carryng mag cop n o b acnowlg abou h lvry whn Typ I acnowlgmn u. Avrag Numbr of Cop Ranom Waypon p-thory p-sm Typ II p-sm Typ I Ranom Wal Dlvry aln Avrag Numbr of Cop p-thory p-sm Typ II p-sm Typ I Dlvry aln Fg.5 Th comparon of h avrag numbr of cop oban va analy an mulaon for h wo pro ca whn ranom wal mol u. Ranom Wal Fg.7 Th comparon of h avrag numbr of cop oban va analy an mulaon for h wo pro ca whn ranom waypon mol u. Avrag Numbr of Cop Ranom Waypon p-thory p-sm Typ II p-sm Typ I Dlvry aln Avrag Numbr of Cop p-thory p-sm Typ II p-sm Typ I Dlvry aln Fg.8 Th comparon of h avrag numbr of cop oban va analy an mulaon for h hr pro ca whn ranom waypon mol u. TABE II AVERAGE NUMBER OF COPIES USED IN SINGE P TWO-PERIOD P AND THREE-PERIOD P SPRAYING AGORITHMS WITH DIFFERENT ACKNOWEDGMENT TYPES AND DEADINES Typ I Typ II Fg.6 Th comparon of h avrag numbr of cop oban va analy an mulaon for h hr pro ca whn ranom wal mol u. W alo compar h rul whn ranom waypon mobly mol u. Fg. 7 an Fg. 8 how h comparon of avrag copy coun oban n mulaon wh ho compu analycally. Th concluon ar mlar o ho ma abov for h ranom wal mol vn hough mn valu ar ffrn from ho u n h ranom wal mol nc h ng n h mol gnra an EM of 50. Th how ha our analy hol for ffrn mobly mol. I only rl on h EM h avrag nrmng m bwn no for h appl mobly mol. To compar h prformanc of h propo algorhm wh h ngl pro p prayng algorhm whch a pcal ca of our algorhm w fr compar h avrag numbr of cop u n boh algorhm whn ffrn yp of acnowlgmn mchanm ar u. mn p p p p p p In Tabl II w prn h avrag copy coun u n hr compar algorhm whn ranom wal mol u w no nclu h rul whn ranom waypon mol u bcau hy ar mlar o h rul prn hr. From h abl w obrv ha n boh acnowlgmn yp p algorhm u fwr cop on avrag han hr p or p prayng algorhm o. Howvr whn Typ I acnowlgmn u h avng n h numbr of cop oban by p algorhm cra. Morovr n om ca 00 prformanc bcom wor han p algorhm. Th bcau whn h aln g gh h numbr of cop ha ar pray o h nwor ncra o ha h numbr of no carryng h mag cop ncra an h uraon of pmc l acnowlgmn longr. Conqunly mor runan cop ar pray by h

12 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 no havng mag cop bfor hy ar nform abou h lvry. Morovr w alo noc ha ung h propo algorhm vn wh Typ I acnowlgmn rul n lowr avrag cop u han whn ung h ngl pro prayng algorhm wh Typ II acnowlgmn. I houl alo b no ha n ngl pro prayng algorhm wh copy coun h avrag numbr of mag cop pray o h nwor l han. Th mply bcau vn n ngl pro prayng whch o all prayng a h bgnnng hr non-zro chanc ha h mag wll b lvr bfor all cop ar ma. To furhr compar h prformanc of h propo algorhm wh h ngl pro prayng algorhm w hav maur om aonal mrc. Fg. 9 an Fg. 0 how h comparon of avrag mag lvry lay an h avrag m of prayng complon 5 m by whch h la copy pray n h algorhm rpcvly. Inpcng h wo graph w obrv ha h propo p an p algorhm ncur hghr avrag lay han p algorhm bu hy achv h am lvry probably 6 bfor h aln compar o h p prayng algorhm. Morovr nc h propo algorhm popon h prayng of all cop o lar m hy fnh prayng lar han h ngl pro pray an wa algorhm o. Th rul n lowr mmory uag avrag ovr cuon m of our algorhm whn compar wh uch uag ncurr by h ngl pro prayng algorhm. W alo compu h prcnag of h avng achv n h numbr of copy coun wh h propo mul-pro algorhm. Fg. char h fracon - avg / wh h gvn. Hr h avrag copy coun u n ngl pro prayng an avg h avrag copy coun achv n h mul-pro prayng algorhm. Th m w prn h rul whn ranom waypon mol u h rul wh ranom wal mol ar mlar. From h rul hown n Fg. w obrv ha p algorhm prov hghr avng han p algorhm. Morovr clar ha h avng wh Typ II acnowlgmn ar hghr han h avng wh Typ I acnowlgmn n boh p an p algorhm. Th ffrnc bwn h avng of Typ I an Typ II acnowlgmn g mallr a h aln ncra. Th bcau largr cra h numbr of copy coun pray o h nwor rulng n acnowlgmn rachng all no carryng mag cop arlr. On h ohr han w alo obrv flucuaon vn n h avng of a ngl algorhm wh ffrn lvry aln. Th 5 Th valu n Fg.0 ar compu ovr ca n whch h mag lvr afr all ponal cop ar pray. 6 In mulaon w aum ha collon or collon avoanc o no mpac mag lvry. In hy can only forc mng no o communca qunally layng om parw no communcaon. Y h avrag rqur communcaon m abou 0. wh Mb/ banwh an 00Kb pac mall compar o h avrag mng m of wo no.7 n ranom wal ng. Morovr mng of four or mor no vry unlly blow % n our ng. Thu unlly blow 0.05% n our ng ha a communcaon lay u o collon or collon avoanc wll c mng m jufyng our aumpon. A pc n mulaon all hr algorhm achv h r p by h aln for h a of brvy h rlvan plo om hr. bcau for om concuv valu mn valu n p algorhm whch achv h r p h am.. mn whl valu n mul-pro algorhm ar ffrn. In h ca mul-pro algorhm a h avanag of prayng n mulpl pro an lay h prayng furhr whn h aln largr for ampl n p algorhm whn 600 hn 400 an h opmum 5 bu whn 700 hn 55 an h opmum 6. Hnc mul-pro algorhm can prov mor avng ovr ngl pro algorhm n uch ca. Fg.9 Th comparon of h avrag lay for h ngl pro an mulplpro algorhm ranom wal mol. Fg.0 Th comparon of avrag n of prayng m n h ngl pro an mulpl-pro prayng algorhm ranom wal mol. Fg. Th prcnag of avng achv by h propo algorhm wh wo ffrn acnowlgmn chm ranom waypon mol. W alo loo a h ffc of h r p on avng achv by h propo algorhm. A an ampl w plo h prcnag of avng oban n p-typ II algorhm wh hr ffrn p valu n Fg.. Hr w prform mulaon n a ffrn way o how alo h fla bhavor of prcnag of avng wh rpc o na of. Wh h gvn an p valu w fr foun h mnmum valu ha achv h gvn p n ranom waypon mol an hn oban h avng prov by p-typ II algorhm whn opmum valu ar u a ha valu whl mananng h gvn p.

13 IEEE/ACM Tranacon On Nworng 85:50-4 Ocobr 00 Fg. Th prcnag of avng achv by p-typ II algorhm wh hr ffrn p valu ranom waypon mol. Fg. Th ffc of numbr of no on h ffrnc bwn h analy an mulaon rul. Fg.4 Th avrag numbr of cop u pr mag n h mulaon of ral rac from RollrN. From Fg. w fr obrv ha h avng ar almo h am whn plo accorng o h valu whr h mnmum m ha prayng of cop achv h gvn p provng h propry analycally h ubjc of our fuur wor. Aonaly w obrv ha a h gvn p valu cra h avng prov by mul-pro algorhm cra. Th bcau a p cra mnmum valu achvng p wh h gvn cra an h cf of lvry probably g mor vrcal aroun h valu. Bcau of h wo raon h chanc of avng n mul-pro algorhm cra wh lowr valu of h r p. In h abov mulaon w alway aum a conan numbr of no M00 n h nwor. Howvr h valu of M affc h prformanc of h algorhm a wll. For ampl n Fg. w plo h mulaon an analy rul n ranom wal mol for p algorhm wh hr ffrn M valu whr p I clar ha a M ncra h ffrnc bwn p-sm Typ II an analy g mallr. Th h rul of fa prayng wh ncrang M 7. 7 I houl b no ha EM o no chang wh ncrang M. Only h ra of mng wh nw no ncra whch rul n h fa prayng of mag. Morovr h ffrnc bwn p-sm Typ I an p- Sm Typ II rul cra bcau largr M valu nabl far acnowlgmn proc. In aon o h valuaon of h propo proocol wh ranom mobly mol w hav alo loo a prformanc on ral DTN rac. From h vral aa rla o far w hav lc RollrN [] rac han o ay uably for ampl all h mng bwn no ar muually rcor. RollrN rac nclu h opporunc ghng of Bluooh vc by group of rollrblar carryng Mo n h rollr our n Par. Snc h no ar no ncal h gnra nrmng m bwn par of no vary gnfcanly. Alhough our proocol no gn for nwor wh hrognou nrmng m bwn no w mply appl our mul-pro prayng a ung h rul from our analy an lf h gn of an algorhm pcfcally for hrognou nwor a a fuur wor. Th RollrN rac ar a an n a In ach 0 arng from h bgnnng unl h m afr whch h la mag wll no hav nough m o b lvr w hav gnra a mag from a ranom ourc o a ranom naon n h nwor. In ngl pro roung for ach numbr of cop allow w foun h lvry m of ach mag an alo h m w call covr a whch 99% p of all mag ar lvr nc hr gnraon a h ourc no. Thn w ran h wo pro roung on h am rac wh h am of mag. In h fr pro w allow h prayng of / cop whch h mo frqun ca accorng o h rul of our analy. In h con pro w r ffrn copy coun an foun h ncary copy coun ha achv h am lvry ra of all mag by h covr. Snc h nrmng m bwn no an alo h lvry m of mag ar ffrn from ach ohr w compu h ar of con pro nvually for ach mag. Tha for ach mag w u lvry m n h ngl pro roung wh mag a h mag own an compu accorngly. In Fg. 4 w how h avrag copy coun u pr mag n p an p prayng algorhm. Clarly mul-pro prayng a can ruc h avrag copy coun u n ral DTN rac vn whn h frqunc of no mng how hrognou bhavor. Th avng n h ca ar n h rang of 6%-8%. Howvr w blv ha a mor carful gn of mul-pro a can ncra h avng vn furhr. Th gn of a mul-pro prayng ba roung algorhm for hrognou nwor wll b h ubjc of our fuur wor. V. CONCUSION AND FUTURE WORK In h papr w nrouc a gnral mul-pro prayng algorhm for DTN whch rbu h mag cop pnng on h rmanng m o lvry aln an hn ung formal analy an mulaon w valua prformanc. W fr how analycally how o paron m unl aln n a ngl pro prayng algorhm no wo

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