The Monitoring of The Network Traffic Based on Queuing Theory


 Gilbert Fletcher
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1 The Moitorig of The Networ Traffic Based o Queuig Theory A roject Thesis Submitted by alash Sahoo Roll No: 49MA7 I partial fulfillmet of the requiremets For the award of the degree Of MASTER OF SIENE IN MATHEMATIS Uder the supervisio of rof. S. Saha Ray DEARTMENT OF MATHEMATIS NATIONAL INSTITUTE OF TEHNOLOGY, ROURKELA, ORISSA7698
2 ERTIFIATE This is to certify that the roject Thesis etitled The Moitorig Of The Networ Traffic Based O Queuig Theory" submitted by alash Sahoo, Roll o: 49MA7 for the partial fulfilmet of the requiremets of M.Sc. degree i Mathematics from Natioal Istitute of Techology, Rourela, is a boafide record of review wor carried out by him uder my supervisio ad guidace. The cotet of this dissertatio, i full or i parts, has ot bee submitted to ay other Istitute or Uiversity for the award of ay degree or diploma. Dr. S. Saha Ray Associate rofessor Departmet of Mathematics Natioal Istitute of Techology, Rourela Rourela Orissa, Idia
3 DELARATION I declare that the topic `The Moitorig of the Networ Traffic Based o Queuig Theory' for my M.Sc. degree has ot bee submitted i ay other istitutio or uiversity for the award of ay other degree or diploma. lace: Date: alash Sahoo Roll.No. 49MA7 Departmet of Mathematics Natioal Istitute of Techology Rourela7698 Orissa, Idia
4 AKNOWLEDGEMENT I would lie to warmly acowledge ad express my deep sese of gratitude ad idebtedess to my supervisor Dr. S. Saha Ray, Associate rofessor, Departmet of Mathematics, Natioal istitute of Techology, Rourela, Orissa, for his ee guidace, costat ecouragemet ad prudet suggestios durig the course of my study ad preparatio of the fial mauscript of this roject. I would lie to tha the faculty members of Departmet of Mathematics for allowig me to wor for this roject i the computer laboratory ad for their cooperatio. My heartfelt thas to all my frieds ad seiors for their ivaluable cooperatio ad costat ispiratio durig my project wor. I owe a special debt gratitude to my guruji sri sri Aadamurtijii ad my revered parets for their blessigs ad ispiratios. Rourela, 7698 May, alash Sahoo Roll No: 49MA7 Departmet of Mathematics Natioal Istitute of Techology Rourela7698 Orissa, Idia
5 Abstract Networ traffic moitorig is a importat way for etwor performace aalysis ad moitor. The curret project wor explores how to build the basic model of etwor traffic aalysis based o Queuig Theory. I the preset wor, two queuig models M/M/: +/FFS ad M/M/: +/FFS have bee applied to determie the forecast way for the stable cogestio rate of the etwor traffic. Usig this we ca obtai the etwor traffic forecastig ways ad the stable cogestio rate formula. ombiig the geeral etwor traffic moitor parameters, we ca realize the estimatio ad moitor process for the etwor traffic ratioally.
6 otets age o. hapter Itroductio hapter  Model : The queuig model with oe server M/M/: +/FFS 4 hapter  Queueig Theory ad the etwor traffic moitor 9 hapter 4 Model : The queuig model with additioal oe server M/M/: +/FFS hapter 5 oclusio 9 Bibliography
7 hapter  Itroductio Networ traffic moitorig is a importat way for etwor performace aalysis ad moitor. The research wor sees to explore how to build the basic model of etwor traffic aalysis based o Queuig Theory []. Usig this, we ca obtai the etwor traffic forecastig ways ad the stable cogestio rate formula, combiig the geeral etwor traffic moitor parameters. osequetly we ca realize the estimatio ad moitio process for the etwor traffic ratioally. Queuig Theory, also called radom service theory, is a brach of Operatio Research i the field of Applied Mathematics. It is a subject which aalyze the radom regulatio of queuig pheomeo, ad builds up the mathematical model by aalyzig the date of the etwor. Through the predictio of the system, we ca reveal the regulatio about the queuig probability ad choose the optimal method for the system. Adoptig Queuig Theory to estimate the etwor traffic, it becomes the importat ways of etwor performace predictio, aalysis ad estimatio ad, through this way, we ca imitate the true etwor, it is useful ad reliable for orgaizig, moitorig ad defedig the etwor. age
8 The mathematical model of the queuig theory I etwor commuicatio, from sedig, trasferrig to receivig data ad the proceedig of the data codig, decodig ad sedig to the higher layer, i all these process, we ca fid a simple queuig model. Accordig to the Queuig Theory, this correspod procedure ca be abstracted as Queuig theory model [], lie figure. osiderig this id of simple data trasmittig system satisfies the queue model []. Nq T s T N Decodig Dispatchig Hadlig T J T D T From the above figure, Figure :The abstract model of commuicatio process : T N : : N q : : Sedig rate of the seder. Trasportatio delay time. Arrivig speed of the data pacets Quatity of data pacets stored i the buffer temporary storage. acets rate which have mistae i sedig from receiver i.e. lost rate of the receiver. a g e
9 T s : Service time of data pacets i the server where T s =T J +T D +T, T J: T D : T : Decodig time Dispatchig time alculatig time or, evaluatig time or hadlig time. a g e
10 hapter Model: The Queuig model with oe server M/M/:+:FFS I model M/M/, the two M represet the sedig process of the seder ad the receivig process of the receiver separately. They both follow the Marov rocess [4], also eep to oisso Distributio, while the umber stads for the chael. Let Nt= be the legth of the queue at the momet of t. So the probability of the queue whose legth is be I this model, t prob [Nt= ] We have the trasitio rate diagram, = Rate of arrival ito the state =Rate of departure from the state.... Figure: State trasitio diagram 4 a g e
11 The system of differetial differece equatio is. d dt { t t} t t t, for d dt Ad t t ; I model M/M/, we let t Ad Where λ ad µ are costats. for = The ad reduces to d t t t dt t d dt ; for Ad t t ; t for = Here, λ is cosidered as the arrival rate while µ as the service rate. I the steady state coditio Lt t t d Ad Lt { t} t dt Hece from ad whe t we get 4 Ad r, 5 a g e
12 From 4 whe =, we get or I geeral or, where ad is called server utilizatio factor or traffic itesity. We ow, Also This implies that or, or,, where < Hece, =,,, 5 Suppose, L stads for the legth of the queue uder the steady state coditio. It icludes the average volume of all the data pacets which eter the processig module ad store i the buffer. 6 a g e
13 L Hece L 6 Also L sice, 7 If N q shows the average volume of the buffers data pacets. N q L 8 Also N q If the processig module is regarded as a closed regio, the parameter is brought ito the formula 8. Usig the Little s law, we have = average service time of the server = Ts Ts ad here 9 Usig 9, 8 reduces to N q or, Ts Nq Ts 7 a g e ' or, T T N N, sice, s s q q
14 From the above equatio we coclude that, amog three variables viz. T s service time Sedig rate N q Quatity of data pacets stored i the buffer. If we ow ay two variables it is easy to obtai the umerical value of the third oe. So, these three variables are ey parameters for measurig the performace of the trasmissio system. 8 a g e
15 hapter Queuig theory ad the etwor traffic moitor Forecastig the etwor traffic usig Queuig Theory The etwor traffic is very commo [5], The system will be i worse coditio, whe the traffic becomes uder extreme situatio, i which leads to the etwor cogestio [6]. There are a great deal of research about moitorig the cogestio at preset,besides, the documets which mae use of Queuig Theory to research the traffic rate appear more ad more. For forecastig the traffic rate, we ofte test the data disposal fuctio of the router used i the etwor. osiderig a router s arrival rate of data flow i groups is, ad the average time which the routers use to dispose each group is, the buffer of the routers is, if a certai group arrives, the waitig legth of the queue i groups has already reached, so the group has to be lost. Whe the arrivig time of group timeouts, the group has to resed. Suppose, the group s average waitig time is. We idetify i t to be the arrival probability of the queue legth for the routers group at the momet of t, supposig the queue legth is i: t = t, t,..., i t, i =,,...,+. The the queuig system of the router s date groups satisfies simple Marov rocess [7], accordig to Marov rocess, we ca fid the diversio stregth of matrix of model as follow: 9 a g e
16 Networ ogestio Rate Networ cogestio rate is chagig all the time [8]. The istataeous cogestio rate ad the stable cogestio rate are ofte used to aalysis the etwor traffic i etwor moitor. The istataeous rate A c t is the cogestio rate at the momet of t. The A c t ca be obtaied by solvig the system legth of the queue s probability distributig, which is called c+ t. Let tk=,,...,+ to be the arrival probability of the queue legth for the routers group at the momet of t by cosiderig the queue legth is. The the queuig system of the router s date groups satisfies simple Marov rocess. Accordig to Marov rocess, t satisfies the followig system of differetial differece equatios. Let, a g e t = prob { o. of data pacets preset i the system i time t }
17 ad t+ t = prob { o. of data pacets preset i the system i time t + t } ase : For t+ t = rob { o. of data pacets preset i the system at time t } prob { o data pacets arrival i time t } prob {o data pacet departure i time t } + rob {  o. of data pacets preset i the system at time t } prob { data pacet arrival i time t } prob { o data pacet departure i time t } + prob { + o. of data pacets preset i the system at time t } prob { o data pacets arrival i time t } prob { data pacet departure i time t }+... t t t { t o t} { t o t} t{ t o t} { t o } t + t { t o t} { t o t} o t t t t t t t t t t o t Dividig both sides by t ad taig limit as t d dt { t a g e t} t t
18 o t Sice, lim t t Here i state data pacet arrival is i.e., Also i state data pacet departure is i.e., Hece reduces to d dt { t t} t { } t, where =,,, ase : For =, we have t+ t = prob {o data pacet preset i the system i time t+ t } = prob {o data pacet preset i time t } prob { o data pacet arrival i time t } + prob {oe data pacet preset i time t} prob {o data pacet arrival I time t } prob { oe data pacet departure i time t }. = t t { t o t} t { t o t} { t o } t t t t t t o a g e
19 Dividig both sides by t ad taig limit as t, we get d { t} t t, for = dt sice, Ad ase : For =+, we have + t+ t = prob { + o. of data pacet preset i the system i time t+ t } = prob { o. of data pacet preset i time t } prob { data pacet arrival i time t } prob { o data pacet departure i time t } + rob { + o of data pacets preset i time t } prob { o data pacet departure i time t } = t { t o t} { t o t} t { t o t} t t t t o t t Dividig both sides by t ad taig limit as t we get d dt d { t} t t dt { t t} t Sice, 4 By solvig this differetial equatio system, we ca get the istataeous cogestio rate A as t a g e A t t e t
20 The istataeous cogestio rate ca ot be used to measure the stable operatig coditio of the system, so we must obtai the stable cogestio rate of the system. The socalled stable cogestio rate meas it will ot chage with the time chagig, whe the system wors i a stable operatig coditio. The defiitio of the stable cogestio rate is A lim A t t osiderig, lim t as the distributig of the stable legth of the queue ad as t the buffer of the router, the stable cogestio rate ca be obtaied i two ways: firstly, we obtai the istataeous cogestio rate,the mae its limit out. Accordig to its defiitio, it ca be obtaied with the distributig of the legth of the queue. Secodly, accordig to the Marov rocess, we ow that the distributig of the stable legth of queue ca be get through system of steady state equatios. From,,4, we have the system of differetial differece equatios as follows d dt d dt d dt { t t} t { } t for =,,,.., 5 { t t} t for = 6 { t t} t for =+ 7 4 a g e
21 Accordig to some properties of Marov process, we ow that i t i=,,,,+ satisfies the above differetial equatio. Here t [ t, t,..., ] t [,,..., ],,,..., d For steady state coditio lim t t dt ad lim t t Uder steady state coditio,5,6,7 trasform to followig balace equatios. = { } for =,,,, 8, for = 9 for =+ The above system of steady state equatios ca be writte i matrix from as Q i i Where,,..., ad 5 a g e
22 For =, From 9 we get Also, Solvig ad we get Hece A For =, Also, From we get, 6 a g e
23 From 5 we get sice, 7 Usig equatio 6 we get ] [ From 7 we get Hece, A For =, 8 9 Also 7 a g e..
24 From 8 we get ad From 9 we get From we get From we get Hece, A For =, we have a g e
25 6 4 7 From 6 we get Ad From we get From 4 we get From 7 we get 4 Also, 4 9 a g e
26 Hece, 4 A O the aalogy of this, we coclude that,the stable cogestio rate is } { A A A A, for age
27 hapter4 Model : The Queuig Model with additioal oe server M/M/ : +/FFS I this model, umber of servers or chaels are two ad these are arrage i parallel. Here, arrival distributio is poissio distributio with mea rate per uit time. The service time is expoetioal with mea rate per uit time. Each server are idetical i.e. each server gives idetically service with mea rate per uit time.the overall service rate ca be obtaied i two situatios. If there are umbers of data pacets are preset i the system. ase For < There will be o queue. Therefore  server will remai idle ad the combied service rate will be ase For, < The all the servers will busy. So, maximum  + umber of data pacets preset i the queue. The combied service rate will be a g e,
28 Hece combiig case ad case we get., for all,,,, Figure : State trasitio rate diagram The steady state equatios are,, for = 8, for = 9 { }, for 4, for =+ 4 The above system of steady state balace equatios ca be writte i matrix form as Q a g e ad i i
29 Where,,..., ad For = we have 4 Also, 4 From 4 we get The 4 becomes Hece A For = 44 a g e
30 45 46 Also, 47 From 44 we get say ad From 46 we get Sice ] [ Therefore, 4 a g e
31 Ad Hece A For = Also, 5 From 48, we get ad From 49 we get [ sice, ], 5 a g e
32 From 5 we get 4 [Usig the value of ] Sice or, ] 4 [ 4 4 Hece } 4 A For = a g e
33 4 57 From 5 we get Ad From 54 we get From 55 we get ] [ 4 From 57 we get 4 7 a g e
34 8 Also, 4 ] 8 4 [ Hece A O the aalogy of this, we coclude that, the stable cogestio rate is } { A A A A for 8 age
35 hapter 5 oclusio This research program cites the aalysis of the etwor traffic model through Queuig Theory. I the preset aalysis, we describe that how we ca mae a queuig model o the basis of queuig theory ad subsequetly we derive the estimatio after aalyzig the etwor traffic through queuig theory models. I the preset wor two queuig models M/M/: +/FFS ad M/M/:+/FFS have bee applied. These two models are used to determie the forecast way for the stable cogestio rate of the etwor traffic. Usig the Queuig Theory models, it is coveiet ad simple way for calculatig ad moitorig the etwor traffic properly i the etwor commuicatio system. We ca moitor the etwor efficietly, i the view of the ormal, optimal ad or eve for the high overhead etwor maagemet, by moitorig ad aalyzig the etwor traffic rate. Fially, we ca say that etwor traffic rate ca have a importat role i the etwor commuicatio system. 9 a g e
36 Bibliography [] Joh N. Daigle, Queueig Theory with Applicatios to acet Telecommuicatio, ISBN: , Spriger, Bosto, USA, 5. [] Ver axso, Sally Floyd. Why we do t ow how to simulate the iteret. I: proceedig of the 997 witer simulatio coferece.usa: AM,997. [] Re Xiagai, Ziog QiBag. A Applicatio Of Mobile Aget For I Networ Traffic Maagemet, omputer Egierig,,. [4] Li DaQi, She JuYi. Queuig Theory Supervisig KMeas lusterig Algorithm ad ITS applicatio i optimized desig of TT etwor. Joural Of Astroautics,7, [5] Wag eifa, Zhag Shiwei, Li Ju. The Applicatio ad Achievemet of SVG i Networ Netflow Moitor Field. Microelectroics & omputer, 5, 4 [6] Wag Tig, Wag Yu. Survey o a Queue Theory Based Hadover Scheme for UAVS ommuicatio Networ. hiese Joural of Sesors ad Actuators, 7, 4 [7] Guther N. The ractical erformace Aalyst. New Yor: McGrawHill, 998 [8] Ha Jig, Guo Fag, Shi JiHua. Research o the traffic moitorig of the distributed etwor based o huma immue algorithm. Microcomputer Iformatio, 7, 8 age
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