Optimal Adaptive Bandwidth Monitoring for QoS Based Retrieval

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1 1 Optimal Adaptive Badwidth Moitorig for QoS Based Retrieval Yizhe Yu, Iree Cheg ad Aup Basu (Seior Member) Departmet of Computig Sciece Uiversity of Alberta Edmoto, AB, T6G E8, CAADA {yizhe, aup, ABSTRACT etwork aware multimedia delivery applicatios are a class of applicatios that provide certai level of QoS guaratees to ed users while ot assumig uderlyig etwork resource reservatios. These applicatios guaratee QoS parameters like media object trasmissio time limit by actively moitorig the available badwidth of the etwork ad adaptig the object to a target size that ca be trasmitted withi a give time limit. A critical problem is how to obtai a accurate eough estimatio of available badwidth while ot wastig too much time i badwidth testig. I this paper, we preset a algorithm to determie optimal amout of badwidth testig give a probabilistic cofidece level for etwork-aware multimedia object retrieval applicatios. The model treats the badwidth testig as samplig from a actual badwidth populatio. It uses statistical estimatio method to quatify the beefit of each ew badwidth-testig sample, which is used to determie the optimal amout of badwidth testig by balacig the beefit with the cost of each sample. Our implemetatio ad experimets shows the algorithm determies the optimal amout of badwidth testig effectively with miimum computatio overhead. EDICS: 5 Multimedia Commuicatio ad etworkig (5-QOSV Quality of Service)

2 I. ITRODUCTIO The Iteret was origially desiged to offer oly oe level of service ("best effort") to all its service users. I such a eviromet, all the data packets put oto the Iteret has the same priority; the etworks do ot provide ay guaratee o whe the packet will be delivered or eve whether it will be delivered at all. However, as ew applicatios cotiue to emerge, more ad more applicatios ow eed to provide some kid of quality guaratees to its users. How to provide this cosistet service quality is ow a active research area termed "Quality of Service" (QoS) [1-5]. Oe solutio to the problem is to allow the applicatio to reserve a certai quality of etwork services i advace [6,7,8]. Oce such a level of service is reserved, the uderlyig etwork will commit the correspodig resources eeded ad guaratee the delivery of such service. Ufortuately, some etwork architecture ad implemetatios may ot support reservatio or oly support reservatio to a limited degree. Oe alterative approach that has geerated a lot of iterest i QoS based multimedia delivery applicatio is the so-called etworkaware applicatio [9,10,11]. These applicatios do ot rely o the uderlyig etwork to provide them a guarateed quality. Istead, they actively moitor the performace variatio of the etwork, ad attempt to adjust its resource demads i respose. For example, if a applicatio eeds to deliver a certai multimedia object to a cliet, the applicatio adapts the volume of data (i tur the quality of media) to be trasferred to the user give differet etwork badwidth availability. If the badwidth is low, the applicatio reduces the size of the object ad thus reduces demads o the etwork; ad if the badwidth is high, the applicatio icreases its demad to fully utilize the additioal resource. I such a method, the applicatio ca guaratee to deliver the object to its cliet withi a pre-defied time limit. Several prototype systems have

3 3 demostrated this idea. I [1], the authors preset a Tele-Learig system that deliver multimedia course materials (icludig JPEG images) to studets over the Iteret. The system allows the users to specify QoS parameters (time limit, resolutio ad quality of image) at the cliet side, ad adapt the resolutio ad quality of the requested image o the server side to provide best quality image to the user while restrictig trasmissio time withi the time limit. Geerally speakig, i order to deliver the user-requested media adaptively, the etwork-aware applicatio eeds to go through three phases. First, the applicatio eeds to moitor the curret etwork coditio ad estimate the available badwidth i a short future period durig which requested media will be trasmitted. Secod, give the specified time limit ad estimated badwidth, the applicatio ca calculate the volume of data the etwork is able to hadle, ad the adapts the media to that size. For example, if the media is a JPEG image, the applicatio ca adapt the image i terms of resolutio, JPEG compressio factor to produce a tailored JPEG image with a certai target size. The i the third phase, the tailored file is trasmitted to a user. Obviously, the badwidth moitorig ad estimatio phase is of critical importace. Oly a accurate estimatio of available badwidth allows the user request to be precisely met. Two approaches exist to address the problem of accurate badwidth estimatio. I the first approach, the media server maitais a coectio for each cliet curretly active. The server seds testig packets periodically to the cliet ad calculates the available badwidth based o the time to trasmit the testig packet. This maitais a history of badwidth samples. Whe media delivery request is made to the server, the server estimates the available badwidth of the future period by usig a type of average of the history badwidth samples. This method is used

4 4 i [1]. However, there are several problems with it. Sice the method uses a relatively log history of badwidth to predict future badwidth, the estimatio result is ot very accurate. Ad, the approach assumes the availability of history badwidth iformatio which sometime just ot the case. More seriously, periodically sedig radom bits alog the etwork for badwidth testig purpose whe o real data trasmissio is actually required ca be a sigificat waste of total badwidth of etworks, sice it may iterfere with traffic of other users ruig cogestio cotrol protocol. To address the problem metioed above, we could measure the badwidth "o the fly." Whe the cliet requests a multimedia object ad specifies a time limit, the server first speds a fractio of the time o badwidth testig. The remaiig portio of the time limit is the used to adapt ad trasmit the media. I such a method, the badwidth iformatio obtaied is most up to date; the etire traffic icludig both badwidth testig ad real trasmissio will be bouded by the user specified time limit; ad there will ot be ay uecessary badwidth testig whe trasmissio is ot required. Because of the cogestio avoidace mechaism of various TCP protocol implemetatios, the throughput achieved by a coectio teds to fluctuate aroud a rage close to the maximum badwidth available over the coectio. Therefore, by takig multiple badwidth samples at the startig fractio of a time iterval, we ca make a best estimatio of the badwidth available i the whole time period. A limitatio of this method is that sice we use the badwidth samples at the begiig of a time iterval to represet the etire duratio of the time iterval, the iterval caot be very log; otherwise the overall traffic over the etire path may chage sigificatly ad upredictably ifluece the available badwidth. As we shall see later, i our experimets, secod time-itervals seem to be appropriate for our

5 5 experimetal eviromet. A problem associated with the proposed approach is that the fractio of time limit used for badwidth testig is a pure overhead, reducig the portio of user specified time that really used for actual data trasmissio. The challege is oe of balacig the beefits of more accurate estimatio of badwidth (at the cost of more badwidth testig time) ad more trasmissio time (at the cost of less accurate estimatio). I this research, we propose a mathematical model to quatify the beefit ad loss of badwidth estimatio spedig ad use it to achieve a optimal proportio of badwidth testig. Badwidth testig is treated as samplig from a populatio, ad providig estimates based o probabilistic cofidece levels. Whe each ew sample is take, we calculate how much ew kowledge of the populatio has bee gaied. We the compare this beefit with the cost of oe sample (the time spet). We cotiue the samplig util the margial beefit is less tha the cost. We implemet the method i a Java program ad test it o two typical etwork eviromets, campus etwork ad dial-up etwork. To our kowledge, this is the first research that provides quatitative results as to how effective the badwidth moitorig part is i a etwork-aware applicatio. We preset the mathematical model i Sectio II ad describe our implemetatio ad experimetatio i Sectio III. Sectio IV gives a compariso of our research to related works by other authors. Sectio V is the coclusio of the paper.

6 6 II. MATHEMATICAL MODEL We assume a server ad a cliet eed to make a oe-off trasmissio of a multimedia object (from server to cliet). o previous badwidth iformatio is available for the coectio betwee the server ad cliet. The user o the cliet side has specified a time limit T for the trasmissio. either exceedig the time limit T or uder-utilizig the time T is desirable. The first fractio t of T will be used for badwidth testig to obtai the badwidth estimatio B for the future period ( T t) B T t. The server the adapts the multimedia object to be exactly of the size ad the trasmits it to the cliet. A over-estimatio of badwidth will make the trasmissio exceed the time limit, while uder-estimatio will uder-utilize the time limit, thus actually deliverig a media object that could have had better quality (with a larger size). Our problem is how to determie the appropriate t, ad deliver as much data as possible to the cliet withi the time limit specified. To simplify the model, we igore the time eeded for adaptatio of a media object, sice objects ca be stored with multiple resolutio levels or be dyamically adapted i a short time give a fast processor. To estimate the badwidth, we use slices of equal legth. Suppose each time slice has legth t s, we the have T / ts time slices. Each time slice has a badwidth value C i x i = (i=1, t s ts T / ), where C i is a cout of bytes received durig the i th time slice. These badwidth values ca be viewed as radom variables. We obtai the badwidth value of the first the average of these samples to estimate the average of the T / ts populatio. t / ts slices, ad the use We first defie the followig variables for further discussio.

7 7 DEFIITIO 1: Basic otatio T is the time limit for the multimedia object trasmissio specified by a user. t is the time used for badwidth testig. t s is the legth of each time slice used for obtaiig badwidth samples. T =, is the umber of time slices i period T, which geerates our badwidth t s populatio, X 1, X,..., X. = t t s, is the umber of time slices i period t, which geerates our badwidth samples, x 1, x,..., x. X i µ =, µ is the average of badwidth populatio, the actual average badwidth we are tryig to calculate. ( ) xi µ σ =, σ is the variace of badwidth populatio. xi x =, x is the average of the badwidth samples x 1, x,..., x. s = ( x x) i 1, s is the variace of the badwidth samples. With the otatio defied above, our problem ca be stated as give,, x, s, estimate µ. I statistical practice, x is usually used as a estimate of µ. Actually, give,, x ad s, the

8 8 radom variable µ has a probability distributio cetered at x. If we assume the radom variable X (the badwidth i oe slice) is a ormal variable (i.e., follows the ormal distributio), the accordig to statistical theory, d = µ x s 1 (1) is a cotiuous radom variable followig the Studet s t-distributio with degree of freedom (umber of samples) [13]. 1 We use character d istead of t to specify the t-distributio radom variable to avoid ame coflictio with the time t we have already defied. We believe that the assumptio of ormality of variable x is reasoable, sice withi a short period the available badwidth teds to approach a costat. Moreover, the t-distributio is said to be quite robust, which meas that the assumptio of ormality of X ca be relaxed without sigificatly chagig the samplig distributio of the t-distributio (Equatio (1)). Thus give,, x, s µ = x + d () 1 s, ad t-distributio, we ow have the probability distributio of µ. This meas that whe we take badwidth samples, we ca expect the average badwidth withi period T to be a radom variable with probability distributio followig Equatio (). With the distributio of µ, we ca ifer the 100( 1 α)% cofidece iterval for µ as follows, 1 Mathematically, the radom variable t is defied as a stadardized ormal variable z divided by the square root of a idepedetly distributed chi-square variable, which has bee divided by its degree of freedom; that is, t z / χ / =.

9 9 s x d ( α, ) µ (3) 1 s (3) tells us that the value of µ is greater or equal to x d ( α, ) with a probability of 1 100( 1 α)%. With this observatio we defie badwidth estimatio as follows. DEFIITIO : Safe Badwidth Estimatio s We defie µ est = x d( α, ) to be a safe estimatio of the future badwidth. 1 Accordig to the theory described above, we have a cofidece of 100( 1 α)% that the actual badwidth will ot be lower. We metioed that µ is a radom variable cetered at x. From Defiitio, we ca see that i order to provide a time limit guaratee we actually uder-estimate the badwidth. This uderestimatio has its cost; if we use µ est as future badwidth estimatio to determie the maximum size of the media object, the actual trasmissio will most likely uder-utilize the remaiig time T-t. It ca be show that the actual time uder-utilized is o the average ( t) µ est ~ T 1. So we defie the uder-utilized time T as follows: µ ( t ) ( T ) µ est t T, or µ

10 10 DEFIITIO 3: Uder-utilized Time ~ T s d( α, ) µ 1 = = x x est ( ) 1 ( T t). It determies the average uderutilizatio time with the safe estimatio. ( T t ) s Give all the defiitio ad aalysis above, we are ready to determie how much badwidth testig to perform i order to maximize the data delivered to a user while coformig to the time limit requiremet set by the user. From Defiitio 3, we assert T ~ is decreasig as grows ad the rate of decrease dimiishes (we provide a iformal proof i Theorem 1 below) whe is relatively small compared to. This meas as we take more badwidth samples, we ca estimate the future badwidth more accurately with the same cofidece α, thus reducig the cost of time uder-utilizatio. We ca ~ ~ ~ deote the beefit of each ew sample as T = T T 1. As we said, T ~ decreases as grows. O the other had, the cost of each ew sample is a costat, t s, which is a time slice. So we ca compare T ~ with t s each time a ew sample is obtaied, the process of badwidth testig cotiues util T ~ method. is less tha t s. The pseudocode i ALGORITHM 1 below illustrates the Theorem 1: T ~ Decrease as grows, ad the rate of decrease dimiishes whe is relatively small compared to.

11 11 Proof: Sice T ~ > 0, the statemet is equivalet to ( ) 0 ~ < T ad ( ) 0 ~ > T. We write ( ) T ~ as follows, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) h g f c t T d x s t T x s d T s s = = = ~,, α α (4) Sice s ad x are radom variables determied by the characteristics of the badwidth populatio, we view them as part of the costat c. We deote ( ) ( ) d f, α =, ( ) 1 = g, ( ) ( ) t s T h =. Thus, we have ( ) 1 = g, ad ( ) ( ) = g (5). We have ( ) 0 < g, ad ( ) 0 > g for 4 3 <. We also have ( ) 0 < = t s h, ad ( ) 0 = h (6). Thus, if we deote ( ) ( ) ( ) h g l =, the ( ) ( ) ( ) ( ) ( ) + = h g h g l, ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = h g h g h g l (7) Therefore, with (5)(6)(7), we have ( ) 0 < l, ad ( ) 0 > l for 4 3 <.

12 1 Fially, whe eed to show that for f ( ) = d ( α, ), we have similar results that f ( ) < 0 f ( ) > 0, ad. As f() is a very complex fuctio, a strict mathematical proof of the property is beyod the scope of this paper. To prove the property umerically, we calculated the values of α for α { 0.75,0.90,0.95} d (, ), ad = 1 500, ad verified that f( ) - f(-1) < 0 ad f()- f(-1)>f(-1)-f(-). We calculated the values of d ( α, ) usig the algorithm described i [15] ad compared these values with values provided i the t-distributio table i [16] for verificatio purpose. ~ T = + <, Therefore, ( ) f ( ) l( ) f ( ) l( ) 0 ~ T + ( ) = f ( ) l( ) + f ( ) l( ) f ( ) l( ) > 0 for 3 <. 4 ALGORITHM 1: Optimal Badwidth Testig Obtai samples x 1, x, x 3 ; ~ ~ ~ ~ ~ Calculate T, T 3, ad T3 = T3 T ; = 3; while ( T ~ > t ) { s = + 1; Obtai sample x ; Calculate T ~ ~ ~ ~ ad T = T T 1 ;

13 13 } retur µ est ; III. IMPLEMETATIO AD EXPERIMETS We aswer the followig questios through experimets: (a) How well does the model work i achievig our goal of performig optimal amout of badwidth testig i badwidth estimatio? (b) How well does the model deliver the fial QoS result to ed-users? For example, if a user specifies its target trasmissio time limit associated with a cofidece goal (say meet the time limit by 95% time ), ca the model really achieve this level of guaratee? Fially, we are iterested i the complexity of the model. Ca we implemet the model efficietly so that we ca make decisios based o them for retrieval applicatios while estimatig badwidth? I this sectio, we describe our implemetatio of the model i Java ad the preset the experimetatio results that aswer these questios. Our implemetatio cosists of two programs, a server ad a cliet. The server is a backgroud process o the server host. It listes to a kow port o the server machie ad waits for a coectio request from a cliet. The cliet is a Java Applet embedded i a HTML page. It icludes a GUI that allows a user to specify parameters for a object trasmissio. There are three parameters; Time Limit gives the time a user wats to use for object trasmissio, Cofidece Goal is the system s guaratee that with this probability the trasmissio will be fiished withi the time limit, Time Slice is the time used for each badwidth sample. Figure 1

14 14 below is a scree capture of the Cliet program showig the GUI for user parameter settig. Figure 1: Scree capture of the Cliet program Oce the user sets the parameters ad presses a Go! butto, the followig steps are executed: 1. The Cliet first creates a socket to the server program by coectig to the kow port o the server host. It the seds a TEST commad to the server. The server respods to the TEST commad by sedig a stream of radom bytes to the cliet via the coectio.. The Cliet receives the testig stream o the other side of the coectio ad at the same time couts the bytes received. For every time slice, the cliet divides the umber of bytes received by the time slice to determie the badwidth of this slice. This is a ew badwidth sample x. 3. The Cliet calculates x, s, µ est, ad T ~ as defied i Sectio II ad compares T ~ to t s.

15 15 If T ~ > t s, it repeats Step ; otherwise it proceeds to Step The Cliet stops the testig coectio, makes a ew coectio to server ad seds a TRASMIT commad to server, alog with a target size of the object which equals to ( T t) µ est. 5. The Server seds a byte stream of legth ( T t) µ est to the Cliet. 6. The Cliet receives the byte stream represetig the object, records the fiish time, calculates the statistics about this trasmissio ad updates them o the right side regio of the applet, as show i Figure 1. To miimize the computatio of T ~ whe each ew sample is obtaied, parts of it ca be precomputed ad stored i a static array of costats. We orgaize the compoet d ( α, ) 1 ito a two-dimesioal array of costats, with ( α,) represetig the two dimesios, so that the computatio of T ~ takes oly two floatig poit multiplicatio ad oe divisio give x ad s. Moreover, the computatio s ca be performed usig the fast computatio form s 1 = x 1 i x. So, if we maitai the itermediate values x ad i x i, the computatio costs of x ad s are miimized. We performed experimets i two typical etwork eviromets. The first oe is the Uiversity of Alberta campus LA. Both the server ad the cliet rus Red Hat Liux 6 ad the cliet uses etscape Commuicator 4.0 as browser. We preset the statistics of our experimets i Table 1 below.

16 16 Cofidece Goal 95% 75% Average o. Of Sample Sample Average (kbps) Sample Variace (kbps) Estimatio (kbps) Actual Average (kbps) Actual Trasfer Time (s) Actual Total Time (s) Total o. Of Exceptio i 100 rus 4 0 Table 1: Experimet results o campus etwork, with Time Limit of 10s, Time Slice of 0.0s. Results are for two differet cofidece goals. The results i Table 1 are based o experimets with parameters Time Limit = 10 secods, Time Slice = 0.0 secods. We give the results for Cofidece Goal = 95% ad 75% ( α = 5 adα = 5 ). For each parameter set, we perform the trasmissio 100 times ad report the averages. From the results i Table 1, we ca see that whe α = 5, it takes oly 14.6 samples out of 500 (=10/0.0=500) to achieve the optimal amout of badwidth testig. Oly 4% of the trasmissios exceed the time limit ad we still utilize 94.8% of the time limit for actual applicatio data trasfer. The results for α = 5 are similar, except that it takes fewer testig samples ad thus leads to a less accurate estimate of badwidth. This i tur produces more

17 17 trasmissio exceedig the time limit (0%) but utilizes a larger portio of time limit for actual applicatio data trasmissio (96.6%). I the secod experimet eviromet settig, we still use the server i Uiversity of Alberta, but use a host coected to a local broadbad ISP with cable modem as cliet. A traceroute result shows there are 11 hops betwee the cliet ad the host. The cliet rus Microsoft Widows 000 Professioal ad Iteret Explorer versio 5.0. We preset the same set of results i Table below. Cofidece Goal 95% 75% Average o. Of Sample Sample Average (kbps) Sample Variace (kbps) Estimatio (kbps) Actual Average (kbps) Actual Trasfer Time (s) Actual Total Time (s) Total o. Of Exceptio i 100 rus 7 Table : Experimet results o home etwork, with time Limit of 10s, Time Slice of 0.0. Results are for two differet cofidece goals. Comparig Table with Table 1, we ca see that the variace of the badwidth is sigificatly larger i the home etwork eviromet. Therefore, the badwidth testig part of the program is

18 18 forced to take more samples ad make more coservative estimatio o future badwidth, which i tur reduces the actual data trasfer time. evertheless, eve i this etwork eviromet, 87% of time limit is used i applicatio data trasmissio whe α = 5. We ow show the effectiveess of the method by comparig the results of our method with a o-adaptive badwidth testig method, which meas the cliet always takes the same umber of samples ad use the average of these as the estimate of future badwidth. Figure shows the results for the campus etwork eviromet described above. Figure : Actual trasmissio time with differet sample umber usig o-adaptive badwidth testig.

19 19 I experimets associated with Figure, we use the same parameters as those i Table 1. Time Limit is set to 10 secods; Time Slice is 0.0 secod; Cofidece Goals are 95% ad 75%. The differece is that this time we do ot use the algorithm described i Sectio II to decide how may testig sample to take, but istead use a costat sample umber. We ru the trasmissio for differet pre-set sample umbers ad plot the actual trasmissio time (out of time limit 10 secods) i Figure. For easy compariso, we also plot the lie y=9.48 ad y=9.66 which is the actual trasmissio time usig our algorithm with Cofidece Goal of 95% ad 75% respectively. From the plot we see that our algorithm accurately arrives at the optimal umber of testig samples to maximize the actual trasmissio time (i tur trasmittig the largest possible size of a object) while guarateeig the time limit requiremet. IV. RELATED WORKS [9] presets a framework for costructio of etwork-aware applicatio ad gives a real example of such applicatio, the Chariot image search egie. The authors cocur that good badwidth estimatio is hard to achieve ad eed further study. [10] presets some badwidth estimatio techiques usig trasport ad etwork layer metrics. Our research differs from these papers i several aspects. First, the method described i this paper uses oly iformatio obtaied at the applicatio layer to provide badwidth estimatio, so it does ot assume special support from uderlyig layers i the etwork stack ad is applicable to a wider group of programmers

20 0 cosiderig providig adaptive media delivery features. Secod, we preset the quatitative results showig how effective the badwidth moitorig work. The previous works let the user select a time limit for object trasmissio, ad makes the best effort to delivery such time limit guaratee. Our work itroduces the additioal parameter Cofidece goal. We let the user select this cofidece goal ad guaratee the time limit requiremets are met by a probability set by the Cofidece goal. V. COCLUSIO I this paper, we proposed a model to determie the optimal amout of badwidth testig for etwork-aware multimedia object retrieval applicatios. The model treats the badwidth testig as samplig from a actual badwidth populatio. It uses a statistical estimatio method to determie the estimatio cofidece iterval whe each ew badwidth sample is obtaied. This cofidece iterval is the used to determie whether badwidth testig should cotiue. We implemet the model i a pair of Java Cliet/Server programs ad demostrate the effectiveess ad efficiecy of the model. The strategy effectively determies the optimal amout of badwidth testig to be performed i differet etwork coditios, itroduces egligible computatio overhead ad meets the user specified QoS requiremet (time limit) with meaigful guaratee level (the cofidece goal). We focus o commuicatio betwee a pair of cliet ad server i this research; i future work, we pla to exted this model to the applicatio of moitorig multiple servers o oe cliet.

21 1 REFERECES [1] A. Vogel, B. Kerherv'e, G. vo Bochma, J. Gecsei, Distributed Multimedia ad QoS: a Survey, IEEE Multimedia, Vol. o., Summer [] P. Ferguso, G. Husto Quality of Service: Deliverig QoS o the Iteret ad i Corporate etworks, Wiley Computer Publishig, [3] I. Miloucheva, Quality of Service Research for Distributed Multimedia Applicatios ACM Pacific Workshop o Distributed Multimedia Systems, Hawaii, Hoululu, 1995 [4] G. Pacifici, R. Stadler, A Architecture for Performace Maagemet of Multimedia etworks. I Proceedig of IFIP/IEEE Iteratioal Symposium o Itegrated etwork Maagemet, Sata Barbara, Jue 1995 [5] G. Gopalakrisha, G.Parulkar, Efficiet Quality of Serverice i Multimedia Computer Operatig Systems, Departmet of Computer Sciece, Washigto Uiversity, Techical report, August [6] L. Zhag et al. RSVP: A ew resource reservatio protocol. IEEE etwork, 7(5):8-18, September [7] R. Brade, L. Zhag et al. Resource ReSerVatio Protocol - Versio 1 Fuctioal Specificatio (RFC 05), September [8] D. Ferrari, A. Gupta, ad G. Vetr, Distributed advace reservatio of real-time coectios. ACM/Spriger Verlag Joural o Multimedia Systems, 5(3), [9] J. Bolliger ad T. Gross. A Framework-based Approach to the Developmet of etworkaware Applicatios. IEEE Trasactios o Software Egieerig, 5(5): , May 1998.

22 [10] J. Bolliger, T. Gross et al. Badwidth modelig for etwork-aware applicatios. I Proceedigs of Ifocomm'99, March [11] X. Wag ad H. Schulzrie, Compariso of Adaptive Iteret multimedia applicatios, IEICE Tras. o Commuicatios, vol. E8-B, o. 6, pp , Jue [1] I. Cheg, A. Basu, Y. Zhag ad S. Tripathi, QoS Specificatio ad Adaptive Badwidth Moitorig for Multimedia Delivery. I Proc. Of EUROCO 001, page , Bratislava, Slovak Republic, July 001. [13] D. Harett. Statistical Methods, Third Editio. Addiso-Wesley Publishig Compay, Ic. 198 [14] R. Jai, The Air of Computer Systems Performace Aalysis. Joh Wiley & Sos, Ic., [15] William H. Press et al, umerical Recipes i C, secod Editio. Cambridge Uiversity Press, Jauary [16] Daiel Zwilliger, CRC Stadard Mathematical Tables, 30th Editio. CRC Press, Jauary 1996.

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