VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. hp://www.ejouralofsciece.org Moiorig of Newor Traffic base o Queuig Theory S. Saha Ray,. Sahoo Naioal Isiue of Techology Deparme of Mahemaics Rourela 7698, Iia Email: saausaharay@yahoo.com, saharays@irl.ac.i ABSTRAT Newor raffic moiorig is a impora way for ewor performace aalysis a moior. The prese aricle explores how o buil he basic moel of ewor raffic aalysis base o Queuig Theory. I he prese wor, wo ueuig moels M/M/: +/FFS a M/M/: +/FFS have bee applie o eermie he forecas way for he sable cogesio rae of he ewor raffic. Usig his we ca obai he ewor raffic forecasig ways a he sable cogesio rae formula. ombiig he geeral ewor raffic moior parameers, we ca realize he esimaio a moior process for he ewor raffic raioally. Keywors: Newor raffic, Queuig Theory, sable cogesio rae. INTRODUTION Newor raffic moiorig is a impora way for ewor performace aalysis a moior. The prese aalysis sees o explore how o buil he basic moel of ewor raffic aalysis base o Queuig Theory []. Usig his, we ca obai he ewor raffic forecasig ways a he sable cogesio rae formula, combiig he geeral ewor raffic moior parameers. oseuely we ca realize he esimaio a moiio process for he ewor raffic raioally. Queuig Theory, also calle raom service heory, is a brach of Operaio Research i he fiel of Applie Mahemaics. I is a subjec which aalyze he raom regulaio of ueuig pheomeo, a buils up he mahemaical moel by aalyzig he ae of he ewor. Through he preicio of he sysem, we ca reveal he regulaio abou he ueuig probabiliy a choose he opimal meho for he sysem. Aopig Queuig Theory o esimae he ewor raffic, i becomes he impora ways of ewor performace preicio, aalysis a esimaio a, hrough his way, we ca imiae he rue ewor, i is useful a reliable for orgaizig, moiorig a efeig he ewor.. THE MATHEMATIAL MODEL OF THE QUEUING THEORY I ewor commuicaio, from seig, rasferrig o receivig aa a he proceeig of he aa coig, ecoig a seig o he higher layer, i all hese process, we ca fi a simple ueuig moel. Accorig o he Queuig Theory, his correspo proceure ca be absrace as Queuig heory moel [], lie fig.. osierig his i of simple aa rasmiig sysem saisfies he ueue moel []. N T s λ' T N λ T J T D T Figure : The absrac moel of commuicaio process
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. From he above fig., : Seig rae of he seer. T N : Trasporaio elay ime. : Arrivig spee of he aa paces N : Quaiy of aa paces sore i he buffer emporary sorage. : aces rae which have misae i seig from receiver i.e., los rae of he receiver. T s : Service ime of aa paces i he server where T s =T J +T D +T T J : Decoig ime T D : Dispachig ime T : alculaig ime or evaluaig ime or halig ime.. Moel-: The Queuig moel wih oe server M/M/:+/FFS hp://www.ejouralofsciece.org I moel M/M/, he wo M represe he seig process of he seer a he receivig process of he receiver separaely. They boh follow he Marov rocess [], also eep o oisso Disribuio, while he umber sas for he chael. Le N= be he legh of he ueue a he mome of. So he probabiliy of he ueue whose legh is, be =rob [N= ] I his moel, = Rae of arrival io he sae µ =Rae of eparure from he sae We have he rasiio rae iagram as follows The sysem of iffereial ifferece euaio is Figure : Sae rasiio iagram Here, λ is cosiere as he arrival rae while μ as he service rae. { } I he seay sae euaio, for L A for a L { } = I moel M/M/, we le Hece, from es. a whe we ge A Where λ a µ are cosas. for 5 The es. a reuces o a This implies for From e.5 whe =, we ge A for Therefore, =
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. I geeral, or, where Here, is calle server uilizaio facor or raffic hp://www.ejouralofsciece.org N Usig he Lile s law we have Also T s a iesiy. We ow, Therefore, Also, or, oseuely,, where < Hece,, =,,,. Suppose, L sas for he legh of he ueue uer he seay sae coiio. I iclues he average volume of all he aa paces which eer he processig moule a sore i he buffer. If L Also Hece L 7 L Sice, N eoes he average volume of he buffers aa paces he L N 8 9 Usig e.9, e.8 reuces o N This implies Ts N Ts Or, T T N N, ' The above euaio e. provies he relaio bewee followig parameers T s = Service ime Seig rae N 6 Quaiy of aa paces sore i he buffer If we ow ay wo variables, i is easy o gai he umerical value of he hir oe. So, hese hree variables are ey parameers for measurig he performace of he rasmissio sysem.. QUEUING THEORY AND THE NETWORK TRAFFI MONITOR..Forecasig he ewor raffic usig Queuig Theory The ewor raffic is very commo [5]. The sysem will be i worse coiio, whe he raffic becomes uer exreme siuaio, i which leas o he ewor cogesio [6]. There are a grea eal of research abou moiorig he cogesio a prese,besies, he ocumes which mae use of Queuig Theory o research he raffic rae appear more a more. For forecasig he raffic rae, we ofe es he aa isposal fucio of he rouer use i he ewor. osierig a rouer s arrival rae of aa flow i groups is, a he average ime which he rouers use o ispose each group is, he buffer of he rouers is, if a cerai group arrives, he waiig legh of he ueue i groups has alreay reache, so he group has o be los. Whe he arrivig ime of group imeous, he group has o rese. Suppose, he group s average s s
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. waiig ime is. We ieify i o be he arrival probabiliy of he ueue legh for he rouers group a he mome of, supposig he ueue legh is i: =,,...,, i =,,...,+. hp://www.ejouralofsciece.org The he ueuig sysem of he rouer s ae groups saisfies simple Marov rocess [7], accorig o Marov rocess, we ca fi he iversio sregh of marix of moel as follow: Q. Newor ogesio Rae Newor cogesio rae is chagig all he ime [8]. The isaaeous cogesio rae a he sable cogesio rae are ofe use o aalysis he ewor raffic i ewor moior. The isaaeous rae A is he cogesio rae a he mome of. The A ca be obaie by solvig he sysem legh of he ueue s probabiliy isribuig, which is calle. Le, =,,...,+ o be he arrival probabiliy of he ueue legh for he rouers group a he mome of by cosierig he ueue legh is. The, he ueuig sysem of he rouer s ae groups saisfies simple Marov rocess. Accorig o Marov rocess, saisfies he followig sysem of iffereial ifferece euaios. Le, = rob { umber of aa paces prese i he sysem i ime } a = rob { umber of aa paces prese i he sysem i ime + } ase : For = rob { umber of aa paces prese i he sysem a ime } rob { o aa pace arrival i ime } rob { o aa pace eparure i ime } + rob { - umber of aa paces prese i he sysem a ime } rob { aa pace arrival i ime } rob { o aa pace eparure i ime } + rob { + umber of aa paces prese i he sysem a ime } rob {o aa pace arrival i ime } rob { aa pace eparure i ime }+ { o }{ o } { o }{ o } + o { o }{ o } + o Diviig boh sies by a aig limi as { } o, sice lim
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. Here, i sae, aa paces arrival is i.e. Also, i sae, aa pace eparure is i.e. Hece, e. reuces o { } { } where =,,, hp://www.ejouralofsciece.org = rob { + o. of aa paces prese i he sysem a ime + } = rob { o. of aa paces prese i ime } prob { aa pace arrival i ime } rob { o aa pace eparure i ime } + rob { + o of aa paces prese i ime } rob { o aa pace eparure i ime }+ { o }{ o } { o } o o ase : For =, we have + = rob { o aa pace prese i he sysem a ime + } = rob { o aa pace prese i ime } + rob { o aa pace arrival i ime } + rob {oe aa pace prese i ime } rob { o aa pace arrival i ime } rob { oe aa pace eparure i ime } Diviig boh sies by we ge sice a aig limi as { } { }, By solvig his iffereial euaio sysem, we ge he isaaeous cogesio rae A as = { o } { o }{ o } o o Diviig boh sies by a aig limi as, we obai A e The isaaeous cogesio rae ca o be use o measure he sable operaig coiio of he sysem, so we mus obai he sable cogesio rae of he sysem. The so-calle sable cogesio rae meas, i will o chage wih he ime chagig, whe he sysem wors i a sable operaig coiio. The efiiio of he sable cogesio rae is A lim A } { ase : For =+, we have sice, a osierig, lim as he isribuig of he sable legh of he ueue a as he buffer of he rouer, he sable cogesio rae ca be obaie i wo ways: firsly, we obai he isaaeous cogesio rae, he fi is limi. Accorig o is efiiio, i ca be obaie wih he isribuig of he legh of he ueue. Secoly, accorig o he Marov rocess, we ow ha he isribuig of he sable legh of ueue 5
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. ca be obaie hrough sysem of seay sae euaios. From e., e. a e., we have he sysem of iffereial ifferece euaios as follows { } { } 5 for =,,,, { } for = 6 { } for =+ 7 Accorig o some properies of Marov process, we ow ha i i=,,,,+ saisfies he above iffereial euaio. Here, [,,..., ],,,..., [,,..., ] For seay sae euaio, lim a lim Uer seay sae coiio, es.5,6 a 7 rasform o followig balace euaios. { } for =,,,, 8 for = 9 for =+ The above sysem of seay sae euaios ca be wrie i marix from as hp://www.ejouralofsciece.org where,,..., a i a Q For =, From e.9, we have i Q Also, Solvig a we ge Hece, A For = Also, From e., we ge Therefore,, From e.5, we have, sice, 5 6 7 6
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. hp://www.ejouralofsciece.org 7 Usig e.6, we obai ] [ From e.7 yiels.. Hece, A For =, 8 9 Also, From e.8, we obai a From e.9, we have From e., we have From e., we have Hece, A For =, yielig 5 6 7 From e.6, we obai a From e. From e. From e.7, we have Also, Hece, A
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. O he aalogy of his, we coclue ha, he sable cogesio rae is A A, for { } A hp://www.ejouralofsciece.org A 5. THE QUEUING MODEL WITH ADDITIONAL ONE SERVER M/M/ : +/FFS There will be o ueue. Therefore - server will remai ile a he combie service rae will be, ase- For The, all he servers will be busy. So, maximum - umber of aa paces prese i he ueue. The combie service rae will be, I his moel, umber of servers or chaels is wo a hese are arrage i parallel. Here, arrival isribuio is oisso isribuio wih mea rae per ui ime. The service ime is expoeioal wih mea rae per ui ime. Each server is ieical i.e. each server gives ieically service wih mea rae per ui ime. The overall service rae ca be obaie i wo siuaios. If here are umbers of aa paces are prese i he sysem. Hece, combiig ase- a ase-, we have for all for,,, ase- For < Figure : Sae rasiio rae iagram The seay sae euaios are for = 8 for 9 { } for for The above sysem of seay sae balace euaios ca be wrie i marix form as Q 8
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. hp://www.ejouralofsciece.org 9 a i i where,...,, a Q From e.9, we obai sice,, From e.5, we ge, Usig he value of Sice, or, ] [ Hece A For = 5 5 55 56 57 From e.5, we have a From e.5 From e.55, we ge
VOL., NO., November ISSN XXXX-XXXX ARN Joural of Sciece a Techology - ARN Jourals. All righs reserve. [ ] From e.57 8 hp://www.ejouralofsciece.org he ewor raffic hrough ueuig heory moels. I he prese wor wo ueuig moels M/M/: +/FFS a M/M/:+/FFS have bee applie. These wo moels are use o eermie he forecas way for he sable cogesio rae of he ewor raffic. Usig he Queuig Theory moels, i is coveie a simple way for calculaig a moiorig he ewor raffic properly i he ewor commuicaio sysem. We ca moior he ewor efficiely, i he view of he ormal, opimal a or eve for he high overhea ewor maageme, by moiorig a aalyzig he ewor raffic rae. Fially, we ca say ha ewor raffic rae ca have a impora role i he ewor commuicaio sysem. REFERENES Also, [ ] 8 8 8 O he aalogy of his, we coclue ha, he sable cogesio rae is A A { } A A, for 6. ONLUSION This research paper cies he aalysis of he ewor raffic moel hrough Queuig Theory. I he prese aalysis, we escribe ha how we ca mae a ueuig moel o he basis of ueuig heory a subseuely we erive he esimaio afer aalyzig [] Joh N. Daigle, 5, Queueig Theory wih Applicaios o ace Telecommuicaio, ISBN: -87-857-8, Spriger, Boso, USA. [] Ver axso, Sally Floy, 997, Why We Do Kow How To Simulae The Iere. I roceeigs of he 997 Wier Simulaio oferece, e. S. Araóir, K. J. Healy, D. H. Wihers, a B. L. Nelso, USA:AM. [] Re Xiagcai, Xiog Qibag,, A Applicaio of Mobile Age for I Newor Traffic Maageme, ompuer Egieerig, -. [] Li Da-Qi, She Ju-Yi, a Zhou Jiag-liag, 7, Hece, Queuig Theory Supervisig K-Meas luserig Algorihm a ITS Appllicaio i Opimize A Desig of TT Newor, Joural of Asroauics, 8 8, pp. 75-756. [5] Wag ei-fa, Zhag Shi-wei, Li Ju, 5, The Applicaio a Achieveme of SVG i Newor Neflow Moior Fiel, hiese Joural of Semicoucors,, pp. 6-65. [6] Wag Tig, Wag Yu, 7, Survey o a Queue Theory Base Haover Scheme for UAVS ommuicaio Newor, hiese Joural of Sesors a Acuaors, 7,. [7] Guher, N., 998, The racical erformace Aalys, McGraw-Hill Ic., New Yor. [8] Ha Jig, Guo Fag, Shi Ji-Hua, 7, Research o he raffic moiorig of he isribue ewor base o huma immue algorihm, Microcompuer Iformaio, 7-8.