Performance Evaluation of Weighted Fair Queuing System Using Matrix Geometric Method

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1 Performace Evaluatio of Weighted Fair Queuig System Usig Matrix Geometric Method mia l-saaai, Irfa a, ad od Fretell Mobile omputig, etorks ad Security esearch Group School of Iformatics, Uiversity of radford, radford, D7 DP, U.K. bstract. This paper aalyses a multiple class sigle server M/M//K queue ith fiite capacity uder eighted fair queuig WFQ disciplie. The Poisso process has bee used to model the multiple classes of arrival streams. The service times have expoetial distributio. We assume each class is assiged a virtual queue ad icomig obs eter the virtual queue related to their class ad served i FIFO order.we model our system as a to dimesioal Markov chai ad use the matrix-geometric method to solve its statioary probabilities. This paper presets a matrix geometric solutio to the M/M//K queue ith fiite buffer uder WFQ service. I additio, the paper shos the state trasitio diagram of the Markov chai ad presets the state balace equatios, from hich the statioary queue legth distributio ad other measures of iterest ca be obtaied. umerical experimets corroboratig the theoretical results are also offered. Keyords: Weighted Fair Queuig WFQ, First Iter First Out FIFO, Markov chai. Itroductio Queue based o eighted fair queuig is a service policy i multiclass system. osider a eighted service lik that provides service for customers belogig to differet classes. I WFQ, traffic classes are served o the fixed eight assiged to the related queue. The eight is determied accordig to the QoS parameters, such as service rate or delay. The WFQ [] is a schedulig disciplie usually applied to QoS eabled routers. I this ork, a to class sigle server M/M//K queue ith a fiite capacity uder WFQ schedulig disciplie is aalyzed. The Poisso process is used to model to classes of arrival streams. The service times have expoetial distributio. We assume each class is assiged a virtual queue ad icomig obs eter the virtual queue related to their class ad served i FIFO order. The queue i is served at rate i for some i > he queue i is ot empty ad at rate uity he that queue is empty. We model our system as a to dimesioal Markov chai ad use matrix geometric method to solve for its statioary probabilities. L. Fratta et al. Eds.: ETWOKIG 9, LS 555, pp , 9. IFIP Iteratioal Federatio for Iformatio Processig 9

2 Performace Evaluatio of Weighted Fair Queuig System 67 There ere may solutios proposed to offer the solutio for the WFQ system ith to classes of customers o ifiite buffer i [4] [5] ad [6]. Hoever, the class based ith fiite buffer is applied i a lot of computer ad commuicatio system ad more realistic. The mai aim of our ork is to provide a solutio to the WFQ system ith a fiite buffer. To the best of our koledge to aalytical ad umerical solutios for such a system have ot bee i the literature. The rest of this paper is orgaized as follos: overvie of related ork is sho i Sectio. The queueig model of M/M//K queue ith WFQ is described i Sectio 3. Sectio 4, offers a matrix geometric solutio for M/M//K queue uder WFQ disciplie, hile sectio 5 presets ad explais the umerical results of the model, folloed by the coclusios ad future ork i Sectio 6. elated Work The WFQ schedulig disciplie is a importat method for providig bouded delay, bouded throughput ad fairess amog traffic flos [], [3]. The subect of WFQ has bee ivestigated by may authors. Models similar to our WFQ system ith to classes of customers have bee aalyzed for Poisso arrivals ad expoetial service times but ith ifiite buffers [4] ad [5]. I [4], a to class system ith to queues is cosidered; hoever the WFQ system is approximated ith a to server, to queue systems ad a umerical solutio is provided. I [5], the same problem i [4] is cosidered; hoever a aalytical solutio for the system is give. I [6], the authors provide a aalytical solutio for a WFQ system ith a log rage depedet traffic iput ith a ifiite buffer ad prove that aalytical results provide a accurate estimatio of queue legth distributio ad ca be helpful i choices of WFQ eights. The ork described i [7], defies a aalytical solutio of a WFQ system ith to classes of customers ith expoetial service times i a usteady state ith a ifiite buffer. The authors itroduce a aalytical model for the system ad derive the exact expressio of the tail of the probability distributio of the umbers of customers. I [8], a aalytical solutio for the WFQ system ith more tha to queues ad time coected variable service rates ith a ifiite buffer is provided. I additio a differet desig proposal for a dyamic WFQ scheduler is aalyzed to provide quality of service guaratees. This paper, e aalyses the WFQ system ith to classes of customers ith a fiite buffer i steady state ad studies the effect of eights o the system mea a queue legth, throughput ad mea respose time. To the best of our koledge, the aalysis of a WFQ system ith fiite buffer has ot bee proposed i the literature. 3 The WFQ System Model Throughout this paper, e cosider the same model i [4]; hoever istead of a ifiite buffer, e use a fiite buffer ith size K. The maximum umber of customers ho ca be i the system at ay time is K ad ay additioal arrivig customers ill be refused etry to the system ad ill depart immediately ithout service.

3 68. l-saaai, I. a, ad. Fretell Fig.. The Weighted Fair Queuig System For our WFQ system, e assume to classes of obs; Jobs of class ad class arrive accordig to a Poisso process ith rate i, i, ad require expoetial service times ith mea /µi, i,. Each class is assiged a virtual queue ad arrivig obs eter the virtual queue related to their class ad are served i FIFO order. The queue i is served at rate i for some i > he queue i is ot empty ad at rate uity he that queue is empty. The coefficiets i are such that. The server is ork coservig, i.e., it serves obs oly if at least oe queue is ot empty. Fig. depicts the WFQ system. Fig., illustrates the state-trasitio-rate diagram of the WFQ system here each state deotes the umber of customers i the system. geeralized Markov model ca be described by a to dimesioal Markov chai ith state i,, here i ad are the umber of customers i class ad class at each state, respectively. Whe the process is i state i,, it ca trasfer to the state i,, i,, i, if i >, i,. The trasitio rate from state i, to i, here i K is the arrival rate of class, i.e., of the Poisso process. trasitio out of state i, to i, here K is the arrival rate of class, i.e., of the Poisso process. Whe o customers of class are i the system, the trasitio rate from, to, is the service rate of class, i.e.. The chage from state i, to i, is the service rate of class, i.e.. Hoever, the trasitio rate from state i, to i, is the service rate of class multiplied by the eight of class, i.e.. trasitio from i, to i, is the service rate of class multiplied by the eight of class, i.e.. The ifiitesimal geerator of this process is give by: ad Q Q Q Q i,, i, i,, i, i,, i, i,, i,, i, i,

4 Performace Evaluatio of Weighted Fair Queuig System 69,,,,, i Q i i The geerator matrix Q of the Markov chai is give by Q 3 3 Where,,, I additio 4 ad 5 The block etries,,, are square matrices, hile the block etries,,, ad,,, are rectagular matrices. The Markov chai ith geerator matrix Q is irreducible; the matrices alog the diagoal are o-sigular. elo is the state equilibrium equatios for all of the states of the Markov model of Fig..

5 7. l-saaai, I. a, ad. Fretell Fig.. The state trasitio diagram for a M/M//K queue uder WFQ schedulig disciplie 4 Matrix Geometric Solutio: 4. State Equilibrium Equatios The statioary probability vector for Q is geerally partitioed as,,, ]. Solvig Q alog ith ormalizig equatio e [, yields the folloig set of equatios i matrix form: 6 7, < < 8 9 osiderig the above state equilibrium equatios, it ca be assumed that betee ay to states there is flo i, flo out equilibrium ithout ay effect o the other remaiig eighborig states. 4. Matrix Geometric Method Suppose there exists a matrix as. the, e get by successive substitutios ito the state equilibrium equatios that.

6 Performace Evaluatio of Weighted Fair Queuig System 7 solutio of the is called a matrix geometric solutio [9]. The explaatio for solvig a matrix geometric system is to state the matrix, the rate matrix, hich is discussed belo. 4.3 omputatio of the ate Matrices y a simple algebraic stage maagemet of the state equilibrium equatios ' s is formed as follos: From Equatio 6 ad e ko that is o-sigular ad e obtai, 3 Equatio leads to the folloig expressio for ad here is required to be o-sigular, 4 Equatio 9 leads to the folloig expressio of ad Fially from equatio 8 e get a geeral relatio betee, ad, < ca be calculated from lgorithm. lgorithm. alculate : : if the 3: for do 4: 5: ed for 6: retur 7: ed if 8: if the 9: retur : ed if 5 6

7 7. l-saaai, I. a, ad. Fretell 4.4 Statioary Probabilities Theorem. For ay QD process ith a fiite state space, havig a ifiitesimal geerator matrix give by Equatio Q, the statioary probabilities are give i matrix-geometric form by 7 Where ad is computed usig lgorithm. Proof. The system of liear equatios is solved for, ad from equatio 4, is obtaied as: 8 Solvig Equatio 6 ad 7 for ad use leads to, 9 Thus, after a exchage ad mathematical maipulatio, Equatio 8 follos from ormalizig coditio e ad equatio 6. e e It is orth otig that Theorem givig the structure of the vector ad that of the vector still eeds to be determied. The vector could be computed from either oe of the equatios, < < ad e e

8 Performace Evaluatio of Weighted Fair Queuig System 73 lgorithm. alculate : for do : 3: ed for lgorithm is used to compute the statioary probabilities. 5 Performace alyses I this sectio, e preset the results of the umerical calculatios for a M/M//K queue uder WFQ usig 5 ith differet values for,, ad. The maximum buffer size for the system is 5; hece this model has 36 states. The folloig is the umerical evaluatio of the aalytical model results for some performace measuremets based o ad derived from the Markov chai described i the previous sectio. 5. Mea Queue Legth The mea queue legth L is calculated from the model as follos: K L i i i Where i,,, K. The folloig equatios are derived from to fid the mea queue legth for class, L ad class, L, respectively: K K i L i i, i K K L i, i here i, is the steady state probability at state i,. Fig. 3 depicts the mea queue legth for class ad class for differet arrival rates ad ith. 5. We choose to oly examie the effect of the queue eights. Usig these parameters, the WFQ system reduces to to M/M//K systems. It ca be see from Fig. 3 that the mea queue legth for both classes icreases ith the icrease of the arrivig traffic util the L L The total of L ad L equals the buffer size: K5. L ad L have idetical values for all arrival rates. 3 4

9 74. l-saaai, I. a, ad. Fretell I Fig. 4, e choose. 6 ad. 4 for differet values of he,. It idicates that the mea queue legth for class is loer tha the mea queue legth for class, as expected. lass is served faster tha class at rate.. We fid the same results for L, he. 4,. 6 ad,. Fig. 3. Mea queue legth,. 5 ad, Fig. 4. Mea queue legth,.6,. 4 ad,, >, > Fig. 5 depicts the mea queue legth for differet values of class ith a arrival rate > ad fixed arrival rate. ith.6,. 4. We make a ote of that as icreases, ad the L icreases util the mea queue legth approaches to 5 ad the it is stable. Hoever, the mea queue legth of class icreases ad the decreases to zero. The L is loer i this case compared to L.The reaso hy L is higher tha L is the fact that class is gettig a higher service rate,. ad its

10 Performace Evaluatio of Weighted Fair Queuig System 75 icreasig arrivig traffic causes class to be refused etry to the system. This ca be observed i Fig System Throughput The folloig equatio is used to fid the system throughput T for the M/M//K queue from the aalytical model: is the output rate, he the server is busy. T 5 server is busy - server is idle. he the server is idle, the output rate equals zero. Equatio 5 is used to derive the throughput T for class, T, ad class, T, respectively: T 6 T 7 Fig. 5. Mea queue legth,.6,. 4 ad,, >,. Fig. 6 shos T ith. 6 ad T ith. 4 for differet arrival rates of class ad class, >, >. It idicates that T depeds o the eight of class ad that it meas T ad T. The system throughput ill ot exceed the services rate i this example:. 6. I Fig. 7, e obtai somethig like the results for T ith. 6 ad T ith. 4 for differet arrival rates of class ad class, >, >. Hoever, the graph ill be stable ith arrival rate. 3for both classes.

11 76. l-saaai, I. a, ad. Fretell Fig. 6. Throughput,.6,. 4 ad,, >, > Fig. 7. Throughput,.6,. 4 ad,, >,..3 The Mea espose Time Usig Little s la []: L T S 8 We ca derive the mea respose time for class, S, ad class, S, from 8 respectively as: S L /T 9 S L /T 3 here T ad T are measured system throughput he packet loss is cosidered T Effective. Fig. 7 shos the mea respose time of class ad class he, ad.6,. 4. We ca observe that S is higher tha S ad class stays loger i the system. The reaso hy the mea respose time has tured out to be higher for class is the fact that it has a lo eight.

12 Performace Evaluatio of Weighted Fair Queuig System 77 Fig. 8. Mea respose time,.6,. 4 ad,, >, > Fig. 8 depicts the mea respose time for differet values of class ith the arrival rate > ad the fixed arrival rate. ith.6,. 4. We ca calculate that as icreases, S icreases util the mea respose time approximately equals 4 ad the it is stable. Hoever, the mea respose time of class icreases util.4 ad the decreases to zero. The reaso hy S is a higher tha S is the fact class is receivig differet arrival rates ad a higher service rate,.. Fig. 9. Mea respose time,.6,. 4 ad,, >,. For all obtaied performace measuremets, the WFQ system behaved as expected. These umerical results cofirm the validity of the aalytical model. 6 oclusios ad Future Work I this paper, e provide a aalysis of the to class sigle server M/M//K queue ith a fiite capacity uder a eighted fair queuig schedulig disciplie. The

13 78. l-saaai, I. a, ad. Fretell Poisso process has bee used to model the multiple classes of arrival streams. The service times have expoetial distributio. aalytical expressio for the flo balaced equatios has bee derived usig a Markov chai. Queue legth distributio has bee derived by solvig these expressios. We derived a geeral expressio for the steady state probabilities for ay fiite buffer ith size K. I additio, e foud the steady state probabilities i, for M/M//5 queue ith WFQ as a example ad e preseted the umerical results. Future ork ill focus o derivig geeral equatios for the performace measure for a M/M//K queue ith a fiite capacity uder a eighted fair queuig WFQ ith more tha to classes. We ill also exted it ith more realistic traffic models. efereces. Li,., Tsao, S., he, M., Su, Y., Huag, Y.: Proportioal Delay Differetiatio Service ased o Weighted Fair Queuig. I: Proceedigs ith Iteratioal oferece o omputer ommuicatios ad etorks,, pp Parkekh,.K., Gallager,.G.: geeralized processor sharig approach to flo cotrol i itegrated services etorks: The sigle-ode case. IEEE Tras. O etorkig 3, Parekh,.K., Gallager,.G.: geeralized processor sharig approach to flo cotrol i itegrated services etorks: The multiple ode case. IEEE Tras. O etorkig, Horváth, G., Telek, M.: pproximate aalysis of to class WFQ systems. I: Workshop o Preformability Modelig of omputer ad ommuicatio Systems - PMS 3, rligto, IL, US, September 3, pp Guillemi, F., Picho, D.: alysis of the Weighted Fair Queuig System ith To lasses of ustomers ith Expoetial Service Times. Joural of pplied Probability 4 6. shour, M., Le-goc, T.: Performace of Weighted Fair Queuig System ith Log age Depedet Traffic Iputs. I: Electrical ad omputer Egieerig, pp Davis, P.F., abioitz, P.: Methods of umerical itergratio, d ed. cademic Press, Lodo shour, M., Le-goc, T.: Performace of Weighted Fair Queuig System ith Variable Service ates. I: Iteratioal oferece, pp Daigle, J.: Queueig Theory for Telecommuicatios, st ed., pp ddiso- Wesley Logma 99. da, I., esig, J.: Queueig Theory, Eidhove Uiversity of Techology, Departmet of Mathematics ad omputig Sciece, February 4

B. Maddah ENMG 622 ENMG /27/07

B. Maddah ENMG 622 ENMG /27/07 B. Maddah ENMG 622 ENMG 5 3/27/7 Queueig Theory () What is a queueig system? A queueig system cosists of servers (resources) that provide service to customers (etities). A Customer requestig service will