Adaptve Resource Allocaton Control wth On-Lne Search for Far QoS Level Fumko Harada, Toshmtsu Usho, Graduate School of Engneerng Scence Osaka Unversty {harada@hopf, usho@}sysesosaka-uacjp Yukkazu akamoto EC System Platforms Research Laboratores nakamoto@ctjpneccom Abstract Recently, applcatons of control theory to avod overload condtons have been studed But, n conventonal feedback theory, control nput s determned by an error between a reference value of a controlled sgnal and ts current value In real-tme systems, a set of actve tasks may be tme-varyng so that the reference value may change accordng to the set In ths paper, we proposed a novel control method for a far resource allocaton and QoS levels of tasks are used as controlled sgnals The proposed adaptaton controller allocates a CPU utlzaton factor to each task wth on-lne search for the far QoS level, and the proposed control rule s based on errors between the current QoS levels and ther average Its computaton tme s small so that t does not yeld a heavy overhead Introducton In real-tme systems, overload condtons brng up the sgnfcant decrease of system response predctablty and performance[] To avod the overload condtons n the real-tme systems, admsson controls are utlzed n whch a guarantee test s executed whether a task executon s allowed at ts arrval Recently methodologes are proposed where CPU utlzaton factors are decreased accordng to a QoS(qualty of servce) level allocated to each task to avod the overload condtons Flexble applcatons exst n whch QoS s mproved as long as ts computaton tme and resources ncrease[] In flexble applcatons, the QoS level ndcates a satsfacton level of users for an executon result of released jobs As an example of flexble applcaton, there s an MPEG vdeo playback applcaton In an MPEG applcaton, I- frame playbacks, B-frame and P-frame playbacks are demanded B-frame and P-frame playbacks handle dfferental frames wth I-frame As CPU tme and resources for B-frame and P-frame playbacks ncrease, more precse mages n the playback are produced When each flexble applcaton n a system ncreases the QoS level smultaneously, however, the system becomes overload condtons To avod the condtons, arbtratng the QoS levels of the compettve applcatons are requred There are several researches to arbtrate the compettve applcaton or tasks n real-tme systems Abdelzaher et al have proposed a method where the acceptable QoS levels of tasks n overload condtons are descrbed a pror and the system degrades the QoS levels of the tasks based on the QoS level descrpton at overload condtons[3] However, ths study does not consder dynamc negotaton of QoS levels of tasks Buttazzo et al have ntroduced an elastc task executon model where task executons can be expanded and shrunk lke an elastc sprng and proposed an algorthm to degrade the task executon tme to handle overload condtons[4, 5] A software desgner must model applcaton programs as the elastc model Rajkumar et al have proposed the QoS-based resource allocaton model to solve problems when applcatons n real-tme systems have smultaneous access to multple resource[] In ther works, they obtaned condtons where overall system utlty s maxmzed under the constrant that each applcaton can meet ts mnmum needs Okawa and Rajkumar have proposed software archtecture and module APIs to manage resource reservaton for tmely guaranteed behavor of real-tme programs and developed the portable kernel module based on the APIs[7] The kernel module provdes an admsson control functon to manage the resource reservaton Recently, much attenton has been pad to applcatons of the control theory to real-tme systems The control theory s appled to manage CPU usage n real-tme systems n [8] In ths study, a PID(Proportonal-ntegral-dervatve) controller s used to control CPU utlzaton tme However, condtons of the PID controller, under whch a system status s transted to an expected one, are not obtaned theoretcally To mantan CPU utlzaton factors and deadlne
mss ratos wth specfed values, the feedback control theores are utlzed For example, n [9], gven relatve reference values of task s deadlne, deadlne mss ratos are controlled wthn specfed certan ranges by reservng CPU tmes wth a PID controller based on the dfference between the reference values and actual deadlnes Lu et al have proposed a gan feedback control method wth gven reference values for CPU utlzaton factors and deadlne mss ratos[0, ] Abdelzaher et al use PI controllers for control of the CPU utlzaton factors n Web server end-systems Moreover, recently, hybrd controllers have been proposed n [] All of them requre reference values of controlled varables a pror In real-tme systems where controlled tasks changes as the tme elapses, however, the reference values depend on controlled tasks and must be recalculated In the recalculaton, solvng nonlnear equatons requres O(n ) computaton tme, where n s the number of the tasks, and characterstcs of QoS levels of all tasks must be known When CPU utlzaton factors are decreased wth the same rato for all tasks to avod overload condtons or to arbtrate the QoS levels of compettve tasks, the qualty of servces of tasks vares and the devaton of the QoS levels could occur To prevent unfarness of QoS levels, ths paper proposes a QoS adaptve control algorthm to equalze the QoS levels of all actve tasks under a constrant where the total CPU utlzaton factor s constant The proposed controller obtans a QoS level of each task as feedback data through a montor and allocates a CPU utlzaton factor to each task adaptvely The tasks adapt ther jobs parameters to complete the jobs usng the allocated CPU utlzaton factors and release ther jobs A novelty of the proposed method s that t searches a desrable value on-lne wthout ncrease of computaton tme n the controller The paper organzes as follows: Secton ntroduces a normalzed QoS level and defne a farness of QoS levels Secton 3 proposes a resource resource allocaton archtecture and QoS adaptaton controller to acheve farness of QoS In Secton 4, we descrbe about analyss and desgn of the QoS adaptaton controller Secton 5 shows the results of smulaton experment of our method Fnally, Secton concludes ths paper Task model and QoS farness In ths paper, we deal wth a real-tme system wth a set of perodc ndependent tasks {τ, τ,, τ } and one CPU resource Each task τ releases the l-th job J l wth a CPU utlzaton factor r l at tme tl, perodcally, and t s assumed that every job J l s executed and completed such that the allocated CPU utlzaton factor r l s satsfed We ntroduce a QoS adaptaton controller whch allocates a CPU utlzaton factor r l to task τ Shown n Fg s an llustra- ~ 4 5 & & '()+*-,/0 +3 W $ # XZY\[! " # @BAC > C DFE/AC 7+G 7+8 9;:=<:+>? ]F^/_aÙb=c/d/ded"fPgehjakUl mnoqpertsvuvuvu waxzy;{q e} H;IKJLFM POQPR S IUTPV Fgure Real-tme systems wth QoS adaptaton control ton of the control archtecture whose detals wll be shown n the next secton An executon result of every job released by the tasks s evaluated as a QoS level Denoted by QoS l s a QoS level of job J l Obvously, QoSl depends on both rl and a QoS characterstcs of task τ Each task has a mnmal and a maxmal QoS requrement QoS mn and QoS max, respectvely, where QoS mn represents the worst QoS level acceptable to execute τ and the QoS max the best QoS level τ requres In order to acheve QoS mn and QoS max, task τ needs specfed CPU utlzaton factors r mn and r max, respectvely In ths paper, we assume the QoS level Q (r) of task τ s modeled by a real-valued functon of the CPU utlzaton factor r whch s contnuous and monotoncally ncreasng In mult-task real-tme systems, resource allocaton may cause unfarness n the followng sense: we consder two tasks τ and τ, and both tasks release jobs wth the same CPU utlzaton factor r If r = r max = r mn holds, one s executed wth ts maxmal QoS level and the other wth ts mnmal QoS level Ths stuaton s not preferable In order to evaluate such unfarness, we ntroduce a normalzed QoS level φ for task τ as follows: φ (r) := 0 f r < r mn Q (r) QoS mn, f r mn QoS max QoS mn f r > r max r r max, ote that φ (r) s a monotoncally ncreasng functon from 0 to wth respect to r, and represents the rate of achevement of the QoS level n the sense that φ (r mn ) = 0 means that task τ releases a job wth ts mnmal QoS level ()
QoS mn! " # $& Fgure Illustraton of functon φ whle φ (r max ) = means that the job s executed wth ts maxmal QoS level QoS max Defnton (QoS Farness) It s called that a far resource allocaton s acheved by a CPU utlzaton factors r ( =,,, ) f the normalzed QoS levels φ satsfy the followng equaton: φ (r ) = φ (r ) = = φ (r ) () When every job of task τ s released wth the CPU utlzaton factor r that satsfes Eq (), all tasks can be executed and completed wth the same degree of performance Snce QoS s monotoncally ncreasng, the shape of φ (r) can be llustrated as Fg In the followng, a normalzed QoS level wll be called a QoS level for short We assume φ s unknown except the followng condtons: φ (r) s dfferentable n r (r mn, r max ) For any r (r mn, r max ), 0 < dφ dr D In order to guarantee that every job s completed by ts deadlne, = rmn R = rmax, where R s the desred total CPU utlzaton factor for whch the task set s schedulable 3 QoS adaptaton control 3 Archtecture In order to avod an overload condton and to acheve the far allocaton of CPU utlzaton factors, we propose an adaptaton resource allocaton control archtecture wth on-lne search for the far QoS level shown n Fg, whch conssts of a basc scheduler, a QoS adaptaton controller, and a montor The montor evaluates a normalzed QoS level of each completed job and feeds t back to the QoS Adaptaton controller The QoS adaptaton controller actvates perodcally or aperodcally, and allocates the CPU utlzaton factors to each task wth searchng a far QoS level on-lne Ths allocaton s based on the normalzed QoS levels fed back from the montor A control rule used n the QoS adaptaton controller wll be dealt wth n the next secton The basc scheduler works wth a specfed schedulng algorthm (eg, EDF, RM, or DM algorthm) Released jobs are scheduled based on the CPU utlzaton factor allocated to these correspondng tasks by the QoS adaptaton controller We assume that the least upper bound of the total CPU utlzaton factor U lub for the basc scheduler s known a pror The allocaton by the QoS adaptaton controller s done such as the total CPU utlzaton factor s less than or equal to U lub The desred total CPU utlzaton factor R s less than or equal to U lub The followng notatons wll be used n ths paper: t k : the tme of k-th actvaton of the QoS adaptaton controller r (k) = r (t k ): the CPU utlzaton factor allocated to task τ by the QoS adaptaton controller actvated at tme t k Q (k) = φ (r (t k )): the normalzed QoS level of the completed job of task τ whose CPU utlzaton factor s equal to r (t k ) To perform the far QoS adaptaton control, ths controller dynamcally allocates r (k) for achevng under Q (k) = Q (k) = = Q (k) (3) r (k) = R (4) = Snce φ s contnuous and monotoncally ncreasng, a set of the far utlzaton factors {r f,, rf }, whch satsfes Eqs (3) and (4), s unquely determned under the total CPU utlzaton factor R and denoted by Q f s a QoS level where the far QoS allocaton s acheved If we know Q f, t s easy to acheve a far resource allocaton by usng the conventonal control theory In order to obtan Q f, however, we have to dentfy the characterstcs of φ exactly, and t may be a heavy overload to calculate Q f on-lne snce φ s nonlnear and we have to solve a set of nonlnear algebrac equatons So resource allocaton method usng Q f s unrealstc and we propose a novel archtecture where the QoS adaptaton controller allocates a CPU utlzaton to each task wth avodng the overload condton, searches
C A 45/7 8:9,'-/ 0')/3!"$# '&)(+* ;=<?>$@BA D D EF G HJI K LJM OQP R SUT T &(')+*-, 0/34 B+CD EGFH3IJ! " " #! $ kj! " " #! $ lnm K0LM3O 5(87+9: ; <>=@? UVXW>Y[Z \^]`_ba@c VXWZY\[ ] ^ _ `Bacb P(Q8R+ST! " " #! $ oqp d efbgh Fgure 3 Varaton of QoS levels(j k : the k-th job of task τ, =, ) the far QoS level on-lne, and acheves a far resource allocaton fnally 3 Actvaton of controller As descrbed n the prevous secton, the controller actvates at dscrete tmes t k (k =,, ) When the QoS adaptaton controller actvates at the tme t = t k, t updates r (k),, r (k) based on QoS levels Q (k ),, Q (k ), whch are fed back The CPU utlzaton factors of all jobs released by task τ n every tme nterval [t k, t k+ ) are equal to r (k) After these jobs are completed, the QoS level Q (k) s fed back to the QoS adaptaton controller through the montor Snce the allocated CPU utlzaton factors to all jobs released by τ n the nterval are same, we can assume that ther QoS levels are the same n the nterval Varaton of Q (k) s llustrated as Fg 3 In order to update the QoS levels of all tasks after the actvaton of the QoS adaptve controller, we assume that the nterval of actvaton of the QoS adaptaton controller s large enough for at least one job of each task to be released and completed n [t k, t k+ ) 33 Control rule Conventonal feedback control rules such as PID control are based on an error between a reference value and ts current value n general, and the prevous studes on applcatons of control theory to real-tme systems assume that the reference values (dependng on control specfcatons) are gven a pror [8, 0] Fgure 4 QoS adaptaton controller In ths paper, control specfcatons are that all QoS levels of tasks are equal to the far QoS level Q f But, snce t depends on a set of actve tasks whch wll be tme varyng, t s dffcult to use the conventonal feedback control rules for the control specfcatons So ths paper proposes a novel control rule whch uses only the current QoS levels Q (k), Q (k),, Q (k) The proposed control rule s based on an error between Q (k) and the average QoS level Q(k) of all QoS levels Q (k) defned by Q(k) = Q (k) (5) = If the error s zero, all QoS levels take the same value and the far resource allocaton s acheved So, n order for the error to converge to zero, we propose the followng control rule for the allocaton of the CPU utlzaton factor r (k): r (k + ) = r (k) + α(q(k) Q (k)), () where the real number α s a control parameter Shown n Fg 4 s a block dagram of the adaptaton controller based on the above control rule The man objectve of ths rule s to acheve the far resource allocaton, and ths rule s based on dscrete-tme I(Integral)-control ote that ths control rule s smple and ts calculaton requres O(n) computaton tme Eq () ndcates that f the error between the average QoS level Q(k) and Q (k) becomes zero, r (k) takes a constant value, whch means a far resource allocaton s acheved If Q(k) Q (k) > 0, then the allocated CPU utlzaton factor r (k + ) at the next actvaton of the QoS adaptaton controller s ncreased so that released jobs of τ wll be completed at a larger QoS level On the other hand, f
Q(k) Q (k) < 0, then r (k +) s decreased so that ts released job wll be completed at a smaller QoS level Thus, all QoS levels are expected to converge to the same value Moreover, Eq () guarantees that, at each actvaton tme of the QoS adaptaton controller, the total CPU utlzaton factor = r (k) s always constant Ths s derved from the followng equaton: ( r (k + ) = r (k) + α Q(k) Q (k) ) = = = r (k) = = = = r (k ) = = r (0) (7) = Thus, f the ntal resource allocaton {r (0),, r (0)} satsfes = r (0) = R, then = r (k) = R for any k In Eq (), the selecton of the gan parameter α s mportant It must be selected such that the followng two condtons are satsfed: [Feasblty condton] Each r (k)(r =,,, ) s postve for every k [Stablty condton] Every CPU utlzaton factor r converges to the far resource allocaton In the next secton, we wll deal wth the above two condtons Remark: Many studes use PI- or PID-control[8, 9] From the control theoretcal pont of vew, our control specfcaton s a knd of servo wth a pecewse-constant reference, whch s unknown a pror, and, by the nternal model prncple, we need I-control for elmnatng offset[3] Fortunately, we can show n the next secton that the above two condtons hold smultaneously wthout PD-control f we select the control parameter α approprately 4 Analyss of the control rule 4 Feasblty condton If the gan parameter α of the control rule () s too large, r (k) wll cause a large varaton even f the error Q(k) Q (k) s small So r (k) may take a negatve value, whch s nfeasble for schedulng The followng lemma guarantees the feasblty of the control rule Lemma (Feasblty condton) Assume that r (0) 0 and = r (0) = R If 0 < α then r (k) 0 for every k max D, (8) Proof: Snce dφ (r )/dr D and φ (0) = 0, we have Q (k) D r (k) We wll use the nducton method Suppose that r (k) 0 and = r (k) = R for k Then r (k + ) = r ( (k) + α(q(k) ) Q (k)) Q (k) + αq(k) D max j D j 0 4 Stablty condton Assume that = r (0) = R, whch keeps the total CPU utlzaton factor to R, from Eq (7) A far resource allocaton r (k) wth Q (k) = Q (k) = = Q (k) = Q f s a fxed pont of Eq () We wll derve a stablty condton for the far resource allocaton Let r (k) Q (k) r (k) Q (k) r(k) :=, Q(k) := (9) r (k) Q (k) ote that r (k) s excluded n the vector r(k) snce t s determned by r (k) = R = r (k) By Eq () and Q (k) = φ (r (k)), the real-tme system s modeled by the followng dfference equaton: Q(k) r(k + ) = r(k) + Q(k) Q(k) = r(k) + φ (r (k)) φ (r (k)) φ (R (0) = r (k)) ow let r f r(k) := r(k) r f, () φ (r ) := φ(r r f ) Qf ()
ote that r (k) r f = = r (k) and 0 < d φ d r D Then, Eq (0) can be rewrtten as follows: r(k + ) = r(k) + α φ ( r (k)) φ ( r (k)) φ ( = r (k)) (3) The lnearzed equaton of Eq (3) around the orgn s gven by r(k + ) = A r(k) := (a j ) r(k), (4) { α a j = {( )d + d } f = j, α (d (5) j d ) otherwse, where d = d φ (0) d r = dφ (r f ) dr, =,,, () A stablty condton of the orgn of Eq (4) s gven by the followng lemma Lemma (Stablty condton) If 0 < α ( ) max d + d, (7) then the orgn of Eq (4) s asymptotcally stable Proof: Denoted by s the L matrx norm[4] Snce r(k + ) A r(k), t s suffcent to prove that A <, whch leads to lm k r(k) = 0 Wthout loss of generalty, assume that d d, =,,, (8) Then, usng Eqs (7) and (8), we have A = max a j j [ = = max α j {( )d j + d } α + (d j d ) j,= [ = max α j {( )d j + d } + ] α(d j d ) [ = max α ] j {d j + ( )d } < task τ τ τ 3 τ 4 τ 5 τ perod 500 400 700 00 300 00 phase 0 40 30 50 0 0 D 50 8 90 00 34 3 r mn 0 0 009 0 0 0 r max 040 050 0857 050 050 0333 φ (I) (IV) (II) (I) (III) (III) *: every perod s equal to ts correspondng relatve deadlne Table Task parameters for smulaton 43 Selecton of α Accordng to Lemmas and, Eqs (8) and (7) gve suffcent condtons to acheve a far resource allocaton control Snce max D d j, j =,,,, (9) Eq (8) mples Eq (7) Thus, even f d s unknown, we can use a suffcent condton for both the feasblty and the stablty to hold smultaneously shown n the followng theorem Theorem (Far resource allocaton condton) Assume that r (0) 0 and = r (0) = R If Eq (8) holds, each Q (k) converges to Q f wth satsfyng that 0 r (k) R and = r (k) = R for every k 5 Smulaton We show smulaton experments to evaluate the performance of the proposed QoS adaptaton control We consder a perodc task set {τ, τ,, τ } shown n Table Each task has a relatve deadlne equal to ts perod Each φ s one of the followng functons: (I)Q (k) = (r (k) r mn )/(r max r mn ) (0) (II)Q (k) = sn π(r (k) r mn ) (r max r mn π (III)Q (k) = 05 + 05 sn (r (k) rmax ) r max (IV)Q (k) = + sn π(r (k) r max (r max ) r mn ) r mn r mn () ) () (3) The QoS adaptaton controller actvates every 000 unt tmes, namely t k = 000 k Ths guarantees that at least
$ ) *, 0 + - / * +, - / 0 3 one job s released and completed for each task n the nterval of actvatons [t k, t k+ ] The gan parameter α s set to be /8( 059), whch satsfes Eq (8) Two typcal schedulng algorthms for basc scheduler are used n smulatons A: We use the Earlest Deadlne Frst(EDF) algorthm We set R = 08 so that t s schedulable[, ] In ths case, the theoretcal value Q f of a far QoS level s equal to 0 by solvng Eqs (3) and (4), numercally B: We use Rate Monotonc(RM) algorthm We set R = 0 so that schedulablty for any perodc task set s guaranteed[, ] In ths case, we have Q f = 07 The results of smulatons A and B are as shown n Fgs 5 and, respectvely These fgures show that far resource allocatons are acheved after some tmes of actvaton of the controller Moreover, n both smulatons, every Q (k) converges to the theoretcal value Q f of the far QoS level and no deadlne mss occurs Thus, these smulatons show that the proposed QoS adaptaton control method acheves a far resource allocaton wthout an overload condton Conclusons We proposed a novel control method for a far resource allocaton based on QoS levels The proposed adaptaton controller allocates a CPU utlzaton factor to each task wth on-lne search for the far QoS level, and the proposed control rule s smple snce allocatons s based on errors between the current QoS levels and ther average So, ts computaton tme s small so that t does not yeld a heavy overhead As future work, we are relaxng the far resource allocaton condton gven by Theorem, whch s a suffcent condton, and t s shown n smulaton that the far resource allocaton can be acheved even f the gan parameter s larger than Eq (8) Moreover, we focused on the stablty and the feasblty, but transent behavor of controlled QoS levels s also an mportant problem It s also future work to generalze the proposed method to a multple resource case and a multple QoS dmensons case References [] G Buttazzo, Hard Real-Tme Computng Systems: Predctable Schedulng Algorthms and Applcatons, Kluwer Academc Publshers, Boston, 997 [] J Lu, Real-Tme Systems, Prentce Hall, Upper Saddle Rver, J, 000 [3] T Abdelzaher, E Atkns, and K Shn, QoS egotaton n Real-Tme Systems and ts Applcaton to Automated Flght 45 3 45 # "! 3 4 5!"$#$# &'( 3 4 5 '& '( ') Fgure 5 Behavor of QoS level and CPU utlzaton wth EDF schedulng and R = 08 Control, Proceedngs of the 3rd IEEE Real-Tme Technology and Applcatons Symposum, Montreal, June 997, pages 8 38 [4] G Buttazzo and L Aben, Adaptve Workload Management through Elastc Schedulng, Real-Tme Systems, July 00, pages 3( ):7 4 [5] G Buttazzo, G Lpar, M Caccamo, and L Aben, Elastc Schedulng for Flexble Workload Management, IEEE Transactons on Computers, March 00, pages 5(3):89 30 [] R Rajkumar, C Lee, J Lehoczky, and D Seworek, A Resource Allocaton Model for QoS Management, Proceedngs of the 8th IEEE Real-Tme Systems Symposum, San Francsco, December 997, pages 98-307 [7] S Okawa and R Rajkumar, Portable RK: a Portable Resource Kernel for Guaranteed and Enforced Tmng Behavor, Proceedngs of the 5th IEEE Real-Tme Technology and Applcatons Symposum, Vancouver, June 999, pages 0
*, +, 0 # $! " # $ & 7 8 5 4 3 4 5 putaton and Control, LCS3, Prague, Aprl 003, pages 389 404 [3] W Wonham, Lnear Multvarable Control: A Geometrc Approach, Thrd edton, Sprnger-Verlag, Y, 985 [4] C Desoer and M Vdyasagar, Feedback Systems: Input- Output Propertes, Academc Press, Y, 975 )*+ ' ( &'+* ('& ' ) -/ / /0 3 4 5 3 "! - / Fgure Behavor of QoS level and CPU utlzaton wth RM schedulng and R = 0 [8] D Steere, A Goel, J Gruenberg, D Mcamee, C Pu, and J Walpole, A Feedback-Drven Proportonal Allocator for Real-Rate Schedulng, Proceedngs of the 3rd Symposum on Operatng Systems Desgn and Implementaton, January 999 [9] L Aben, L Palopol, G Lpar, and J Walpole, Analyss of a Reservaton-based Feedback Scheduler, Proceedngs of the 3rd Real-Tme Systems Symposum, Austn, December 00, pages 7 80 [0] C Lu, J Stankovc, S Son, and G Tao, Feedback Control Real-Tme Schedulng: Framework, Modelng, and Algorthms, Real-Tme Systems, July/September 00, pages 3():85 [] T Abdelzaher, K Shn, and Bhatt, Performance Guarantees for Web Server End-Systems: A Control-Theoretcal Approach, IEEE Transactons on Parallel and Dstrbuted Systems, January 00, pages 3():80 9 [] L Palopol, L Aben, and G Lpar, On the Applcaton of Hybrd Control to CPU Reservaton, Hybrd Systems: Com-