Stochastic Design of Enhanced Network Management Architecture and Algorithmic Implementations

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1 Ameica Joual of Opeaios Reseach hp://dx.doi.og/.4236/ajo.23.3a8 Published Olie Jauay 23 (hp://.scip.og/joual/ajo) Sochasic Desig of Ehaced Neo Maageme Achiecue ad Algoihmic Implemeaios Sog-Kyoo Kim W. CySip Gaduae School of Busiess Asia Isiue of Maageme Maai Philippies SKim@aim.edu Received Ocobe 2 22; evised Novembe 5 22; acceped Novembe 6 22 ABSTRACT The pape is focused o available seve maageme i Iee coeced eo eviomes. The local bacup seves ae hooed up by LAN ad eplace boe mai seve immediaely ad seveal diffee ypes of bacup seves ae also cosideed. The emoe bacup seves ae hooed up by VPN (Viual Pivae Neo) ih high-speed opical eo. A Viual Pivae Neo (VPN) is a ay o use a public eo ifasucue ad hoos up log-disace seves ihi a sigle eo ifasucue. The emoe bacup seves also eplace boe mai seves immediaely ude he diffee codiios ih local bacups. Whe he sysem pefoms a madaoy ouie maieace of mai ad local bacup seves auxiliay seves fom ohe locaio ae beig used fo bacups duig idle peiods. Aalyically acable esuls ae obaied by usig seveal mahemaical echiques ad he esuls ae demosaed i he fameo of opimized eoed seve allocaio poblems. The opeaioal oflo give he guidelies fo he acual implemeaios. Keyods: Sochasic Neo Maageme; N-Policy; Closed Queue; Algoihmic Implemeaio; Sochasic Opimizaio. Ioducio I ligh of he ece acs of eoism ad cybeeoism i becomes impeaive o oly o povide a eo secuiy (ha has eve bee a full-poof) bu o offe a paadigm of a eo secuiy sysem hich ca be applied o eoig fo he busiess coiuiy such as soc mae posal offices uclea poe plas ad goveme offices. Availabiliy [] of eoed seves is a majo issue i secuiy especs because of apid goh of Iee. The pape is focused o ehaceme of eo availabiliy o suppo moe eliable sevices. To ypes of bacup seves ae cosideed. Local bacup seves ae locaed i same aea ad hooed up via LAN (Local Aea Neo) ad he emoe bacup seves ae geomeically sepaaed ih mai seves bu emoe bacups ae hooed up by a Viual Pivae Neo (VPN) ih high-speed opical Iee. A VPN is eoig beee emoe seves ad clies via usig a public elecommuicaio ifasucue ih secue access o hei ogaizaio s eo. Ulie a expesive sysem of oed o leased lies a VPN ca povide he ogaizaio ih he same capabiliies bu a a much loe cos. The m mai oig seves ih he local bacups ad S emoe bacup seves ae geomeically sepaaed ih mai seves. The emoe (bacup) seves ae hooed up by a Viual Pivae Neo (VPN) ad ca be used duig he maieace of ieal bacup seves o absece of he epai faciliy (see Figue ). The umbe of he emoe bacups has he cool level N ihi he oal umbe S of emoe bacups. Ulie pevious eseach fom auho [2] N-policy is applied o esic he quaiy of exeal esouces. A VPN is eoig beee emoe seves ad clie via usig eihe a public elecommuicaio ifasucue such as Iee ih secue access o pivae eo ad secued eo such as miliay sysem. I his aicle e sudy a class of closed queueig sysems ih he iiial quaiy of m mai ueliable machies eseve machies ad S auxiliay eseve machies also called supe-eseve machies [3-4]. Mai oig machies ae subjec o expoeial failues ad hei epais ae edeed (i he FIFO ode) by a sigle epai faciliy (efeed o as he epaima) ih geeally disibued epai imes o eplaceme imes o exchage as e machies. As soo as a mai oig machie beas do i is immediaely eplaced by a eseve machie if available. The oal quaiy of oig machies mus o exceed m. Occasioally he goup of eseve machies is bloced fo he sae of some ouie maieace ad duig his Copyigh 23 SciRes.

2 88 S.-K. KIM Figue. Mixue of he local ad he emoe bacups. peiod of ime supe-eseve machies ae ove he duies of eseve machies. The supe-eseve faciliy is acivaed heeve he mai ad eseve faciliies combied ae esoed o is oigial quaiy m ad he he sysem egeeaes. While all mai machies eep o oig i he eve of failues he sysem us o supe-eseve faciliy ad he epaima is uavailable. Defecive machies ae eplaced by auxiliay eseve machies hose oal umbe is S. Hoeve he sysem ies o exhaus his quaiy ad ses up a smalle cool umbe N. Duig his peiod of ime he sys- em is obseved oly upo some adom epochs of ime hile dopped machies lie up i he aiig oom. If a oe of hese obsevaio epochs he umbe of defecive machies eaches o exceeds N (afe some delay) he epaima eus o his duies a busy peiod begis ad heeby he busy cycle coiues. This is a moe ealisic sceaio of a eliabiliy sysem ha fucios ude esiced obsevaios a leas duig is maieace peiods. The sceaio ca be diecly applied fo eo maageme. The cool iege vaiable N (less ha o equal o S ) hose value amog ohe paamees is deemied i he fameo of a compehesive opimizaio. Opeaioal oflo gives he implemeaio guidelies fo eo maageme based o he mahemaical esuls. The mahemaical val- ues ae he iiial codiios fo eo maageme opeaios ad he deailed oflo ill be explaied i his pape. 2. Mahemaical Desig fo Ehace Neo Achiecue The Dualiy Piciple [5] is applied ad i icludes aohe eliabiliy model hich is moe simple ha he mai model (i.e. Model ) ad o hich e ill efe as o Model 2. Model 2 is simila o Model excep ha i does o have he supe-eseve faciliy ad idle peiods. Besides he oal umbe of eseve machies is (i.e. less by oe ha i Model. We ahe associae i ih epaima s vacaios hich ae disibued as egula epais. Hoeve upo his eu he epaima bigs a bad e machie hich eplaces ay oe ha beas do duig his vacaio ip if ay such available. Oheise he e machie he bigs i subsiues ay ohe machie ad i boh cases he old machie is disposed. Model 2 is diecly coeced ih ye aohe model hich e ill call Model 3. Model 3 is a mulichael queueig sysem ih buffe of capaciy ad sae depede aival pocess i oaio G G2 M m. Le be he successive momes of epai compleios ad le R R. be he successive 2 Copyigh 23 SciRes.

3 S.-K. KIM 89 epai duaios all duig a busy peiod. (Fo beviy of oaio e use R R2. as geeic adom vaiables fo evey busy peiod.) The adom vaiables R ae iid ih a commo pobabiliy disibuio fucio [6] A x: PR R x x (2.) ad mea a. Each of he mai machies beas do idepedely of each ohe ad of epais ad accodig o he expoeial disibuio ih paamee. Noice ha eed o equal R uless he coespodig epai belogs o a busy peiod. The pebusy peiod is icluded i he busy peiod; he easo fo disiguishig his ime fom he es of he busy peiod is fo he descipioal coveiece ad fo belo agumes egadig he dualiy piciple. We iepe he eie pebusy peiod as a pa of sae depede sevice ih he fis sevice iiiaig a busy peiod disibued as he covoluio ˆBx Bˆ x A x A x (2.2) hee deoes he PDF of he adom vaiable B. If a ime v (immediaely afe he h epai compleio) he oal quaiy of iac (i.e. mai oig ad eseve machies) is less ha m he busy peiod goes o. Model 3 descibes he umbe of cusomes i a G G2 M m queueig sysem ih sae depede aival seam. Moe specifically i is lie a mulichael queue GI M m (of Taacs [7]) excep fo he ipu is o a geeal idepede bu i vaies depede o he queue legh. If upo ay aival he oal umbe of cusomes (icludig hose i sevice) ae less ha m he PDF of he ex ie-aival ime is A x. Oheise he cusomes ges los ad he ex ie-aival ime is disibued as A x of (2.2). While Models 2 ad 3 seem o be ideical e call hem sochasically cogue. x Le π : lim P Z m be he limiig pobabiliies of he pocess Z. These pob- abiliies exis ude he same codiios as hose fo he embedded pocess [2]. π lim P Z m ad x m hee ad Pm a P m (2.3) π π π m (2.4) m m B. x Ad π : lim P Z is subjec o ou fuhe cosideaio. I he ohe had his model is coolled by he socalled fis excess level pocess fom flucuaio heoy. This is a maed hee-vaiae poi eeal pocess ih all depede compoes. This pocess by iself ca be applied he pacical applicaios such as oue desig 8. The pocess ill be emiaed a some of he adom obsevaio imes he oe of is acive compoes cosses N ad because is value ca be of ay magiude ih posiive pobabiliy he fis excess level ill be cuailed o is maximal umbe S should i fomally exceed S. The vacaio peiod eds ad he epaima esumes his usual duy. The peiod of ime fom uil may o may o iclude a vacaio peiod ad e heefoe call i he h sevice cycle. Duig epaima s vacaio peiod all eseve machies ae bloced ad he mai oig faciliy is baced up by supe-eseve machies hich he sysem boos fom a souce limied o S uis. While all S of hem ae available he sysem aemps o uilize o all supeeseve machies. Namely i ses up a heshold N S a specific efeece umbe (o be opimized) he sysem ies o o exceed. I is assumed ha fom he begiig of a vacaio peiod he saus of he sysem is obseved upo some adom epochs of ime. To simplify oaio ad ihou loss of geealiy e ill fomalize his pocess o he fis sevice cycle. Suppose ha a he all of m machies become iac he epaima leaves he mai eseve faciliy ad he sysem is obseved upo he imes 2. We ill begi ih hich is he aveage peiod of usig he supe-eseve machies. Le us assume ha he adom vaiables ae expoeially disibued ih commo mea. By he heoem by auho [3-4] e have N. m No e u o v ha is he aveage umbe of supe-eseve machie usage: N SN SN v S S hee is he aveage epai ime fo sigle machie. Sice e ge N SN B. hee e : m. Copyigh 23 SciRes.

4 9 S.-K. KIM Model 3 as meioed is he G G2 M m ad Q z of (4.) is he geeaig fucio cove- (muli-chael) queue ih sae depede aivals m ge i he ope disc ceeed a zeo. By usig he Kolpaallel chaels ad a buffe o aiig oom of mogoov diffeeial equaio ad he semi-egeeaive capaciy [7]. A cusome ees a fee chael avail- echiques [2-48-9] his sysem has bee solved by able ih his sevice demad disibued expoeially Dshalalo [6]. The limiig disibuio ih paamee. Model 2 as e see i is cogue o π π π π m is: Model 3 hile Model is dual ih Model 2 (See Dshalalo [5] ad Kim [4] ). The saioay pob- m π π abiliies P P P Pm fo he embedded pocess = ae o o saisfy he folloig fomulas: P π m (2.8) m PA m U A P P π m m Qm m m m PA hee hee PA ap Pm apm A U Qj Qz j Fo he pocess Z (2.7) he coespodig fomulas yield mzmzz Q a Pm mzz π π m (2.9) a P u A u e d e u Ad u m W U a a a s i s 2 as : i i s s i s 2 as : i i s m W Q Q S m m j j j S j m j ji m m i m i m S j m m i m m m i m i m S m m mx m e d Ax m! e m x m! dax m alog ih (2.8). m P m π m a Pm 3. Neoed Seve ih Coolled Bacup Opimizaio (2.) The sochasic opimizaio echiques ae used fo he sample illusaio of he opimizaio ad he sochasic opimizaio echiques by iself ca be applied o ealold poblems such as compue-eoig huma esouces ad maufacuig pocess. Le a saegy say specify ahead of he ime a se of acs e impose o he sysem ad he sysem ca be subjec o a se C of cos fucios. The geeal fomula of sochasic opimizaio is [2-4]: C: lim C (3.) No e u o covegece heoems fo semiegeeaive semi-maov ad Maov eeal pocesses [] i i lim R PM i iilim P Zs ds π i PM iiilim P Yu du PM o aive a he objecive fucio C hich gives he oal expeced ae of all pocesses ove a ifiie hoizo. As a easoable pefomace measue le us Copyigh 23 SciRes.

5 S.-K. KIM 9 coside he eliabiliy faco hich epeses he pobabiliy of he umbe of iac machies a ay mome of ime i equilibium: m m C π. (3.2) This is o oly a eliabiliy measue of he sysem bu i ca also seve as a cosai o a opimally fucioig he sysem. We aive a he folloig expessio fo he sample objecive fucio [2]: N cπ m GB Z m (3.3) H G mπ m Bm PM Tae he oal umbe of mai eoed seves as 2 ad he oal umbe of local bacups is 4. We ae seig up he maximal availabiliy of emoe bacup seves o 5. Hece m 2 4 ad S 5. No e calculae N C ad N ha gives a miimum fo N C. I ohe ods he cool level N sads fo he excess level of emoe bacup hich miimizes he oal cos of his sysem. Belo is a plo of NC fo N2 S5. Ou calculaio yields ha N 9 fo hich he miimal cos equals I meas ha e allocae ou ieal esouces o 2 mai m ad 4 ieal eoed seves ad obai he heshold value N 9 hich gives us he decisio poi ha is he umbe of emoe bacups hich e eed fom exeal esouces o miimize he cos of he bacup sysem. Usig he above example of ou model e aive a he eliabiliy faco is.267. I ells us ha he lielihood of havig a leas m iac mai eoed seve is Algoihmic Implemeaios fo he Ehaced Neo Achiecue The eo achiecue ha has meioed i he pevious is he mahemaical ad heoeical appoach o aalyze he sochasic model. The opeaioal mehod is he guidelie fo acual implemeaio. The oflo of opeaig he ehace eo maageme ca be easily adaped fo sofae pogammig ad simulaio. All of he mahemaical esuls fom he pevious secios ae applied io he opeaioal mehod as he iiial codiios. The vaiables eed o be defied fo usig he esuls fom he mahemaical model. Numbe of ieaio umbe of mai ad bacup seves he saus of he epai faciliy ae some of ey facos fo implemeaio. The vaiables fo opeaios of ehaced e- o maageme ae as follo: : umbe of ieaios m : umbe of mai seves a ieaio : umbe of local bacup seves a ieaio S : umbe of emoe bacup seves a ieaio Q : m L : cool level of emoe bacup seves c : couig umbe of emoe bacup usage G : umbe of seves ha have bee fixed ihi Ieaios Repaiema : ON epai faciliy is available oig OFF epai faciliy is o vacaio. The values i he mahemaical model ae applied as he iiial codiios i he opeaioal oflo bu he oaios ae diffee. The dela of oaios beee mahemaical model ad opeaioal mehod is sho i Table. The opeaioal oflo ca be peseed afe defiig he iiial codiio (see Figue 2) The oflo is he depicio of a sequece of opeaios fo ehaced eo maageme ha is focused o sevice availabiliy. If he opeaios is applied i he example case i Secio 3 he acual values of he iiial codiio ae give: m 2 4 S 5 L 9 based o he dela lis (see Table ) The eo maageme based o above opeaio oflo gives he opimal pefomace i seve availabiliy pespecive. 5. Coclusio I his aicle heoeical appoaches of he eo defese model is peseed. Ulie simulaed model e ca fid he explici fomulas ha is he ey elemes of he complex model. I addiio his model ca be also applied vaious eal-old applicaios such as eo sysem desig 8 ad sofae achiecue []. Aalyically acable esuls ae obaied by usig a dualiy piciple Table. Dela lis of he iiial codiio. Op. Mah. Descipios m m S L N S umbe of mai seves umbe of local bacup seves umbe of emoe bacup seves fis exceed level of emoe seves Copyigh 23 SciRes.

6 92 S.-K. KIM Figue 2. Opeaios oflo of ehace eo. Copyigh 23 SciRes.

7 S.-K. KIM 93 (hich eables us o ea a moe udimeay sysem) semi-egeeaive aalysis ad he heoy of flucuaios of mulivaiae maed eeal pocesses. The esuls ae applied i he fameo of opimizaio poblems. REFERENCES [] D. Russell ad G. T. Gaemi S. Compue Secuiy Basics O Reilly ad Asso. Ic. Sebasopol 26. [2] S.-K. Kim Ehaced Neoed Seve mg. ih Radom Remoe Bacups Poceedigs of SPIE 5244 Olado 7 Sepembe 23 pp doi:.7/2.542 [3] S.-K. Kim Ehaced Maageme Mehod of Soage Aea Neo (SAN) Seve ih Radom Remoe Bacups Mahemaical ad Compue Modellig Vol. 42 No pp doi:.6/j.mcm [4] S.-K. Kim Ehaced Sochasic Mehodology fo Combied Achiecue of e-commece ad Secuiy Neos Mahemaical Poblems i Egieeig Vol Aicle ID: [5] J. H. Dshalalo O a Dualiy Piciple i Pocesses of Sevicig Machies ih Double Cool Joual of Applied Mahemaics Vo. No pp [6] J. H. Dshalalo Queueig Sysems ih Sae Depede Paamees I: J. H. Dshalalo Ed. Foies i Queueig CRC Pess Boca Rao 997 pp [7] L. Taacs Some Pobabiliy Quesios i he Theoy of Telephoe Taffic Magya Tudomáyos Aadémia. Maemaiai és Fiziai Oszály Közleméyei Vol pp [8] S.-K. Kim Desig of Sochasic Hiless-Pedicio Roue by Usig he Fis Exceed Level Theoy Mahemaical Mehods i he Applied Scieces Vol. 28 No pp doi:.2/mma.626 [9] J. H. Dshalalo O he Muliseve Queue ih Fiie Waiig Room ad Coolled Ipu Advaced Applied Pobabiliy Vol. 7 No pp doi:.237/42748 [] E. Cila Ioducio o Sochasic Pocesses Peice Hall Egleood Cliffs 975. [] S.-K. Kim Desig of Ehaced Sofae Poecio Achiecue by Usig Theoy of Iveive Poblem Solvig IEEE Poceedigs of IEEM Hog Kog 8- Decembe 29 pp Copyigh 23 SciRes.