Response-Time Control of a Single Server Queue

Transcription

1 Resonse-Tme Control of a Sngle Server Queue Kjaer, Martn Ansbjerg; Khl, Mara; Robertsson, Anders Publshed n: Proc. of the 46th IEEE Conference on Decson and Control 27 Lnk to ublcaton Ctaton for ublshed verson (APA): Kjaer, M. A., Khl, M., & Robertsson, A. (27). Resonse-Tme Control of a Sngle Server Queue. In Proc. of the 46th IEEE Conference on Decson and Control ( ). IEEE--Insttute of Electrcal and Electroncs Engneers Inc.. General rghts Coyrght and moral rghts for the ublcatons made accessble n the ublc ortal are retaned by the authors and/or other coyrght owners and t s a condton of accessng ublcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and rnt one coy of any ublcaton from the ublc ortal for the urose of rvate study or research. You may not further dstrbute the materal or use t for any roft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the ublcaton n the ublc ortal Take down olcy If you beleve that ths document breaches coyrght lease contact us rovdng detals, and we wll remove access to the work mmedately and nvestgate your clam. L UDUI VERS I TY PO Box7 22L und

2 Proceedngs of the 46th IEEE Conference on Decson and Control ew Orleans, LA, USA, Dec. 2-4, 27 ThB6.4 Resonse Tme Control of a Sngle Server Queue Martn A. Kjær, Mara Khl and Anders Robertsson Abstract Feedback n server systems has durng last years ganed much nterest n order to fulfll stll ncreasng demands on erformance and otmzaton regardng, for examle, qualty of servce (QoS) requrements. In ths aer we nvestgate a feedback based redcton scheme for controllng a sngle server queue. Ths control structure has the beneft over other revously suggested control structures that no measurement of the requred work of each job s needed. However, our soluton mantans the same attractve roertes, regardng average resonse tme and varance. I. ITRODUCTIO Server systems are used n most (tele)communcaton networks. The server system can rocess jobs from ether other network enttes or from clents. In the Internet, web servers send web ages to clents. In cellular networks, Home Locaton Regsters handle subscrber nformaton that may be requested by other nodes n the network. Dfferent control mechansms may be used to control the server system. Admsson control [] revents system overload by rejectng some jobs when needed. Load balancng [2] otmzes the load n a dstrbuted system. Schedulng [3] can otmze the delays when jobs have dfferent QoS requrements. Snce a server system conssts of CPU:s (the servers) and lmted job queues, t has a non-lnear behavor. A small ncrease n load can result n radly growng queues and delays. Therefore, the desgn of the control mechansms s not a smle task f otmzed erformance s desred. In recent years ths feld has ganed a large research nterest from both academa and IT vendors [4], []. In ths aer we focus on resonse-tme control; an objectve drectly couled to the end-user, but wth close relatons to e.g., the more server orented queue length control [6]. However, the connecton between the queue length and the resonse tme deends on numerous factors such as the nature of the arrvng traffc and the servce tmes. Lu et al. [7] desgned a resonse tme control system based on admsson control, where the admsson robablty was adjusted to obtan the desred resonse tme. The control set u ncluded a queung theory based feed forward where a measurement of the average requred work was requred. Ths was measured offlne under low load, and then aled n on lne control. The offlne dentfcaton of the average requred work was also the rocedure of the work by Lu et al. [8] and the work of Henrksson et al. [9] where M. A. Kjær and A. Robertsson are wth the Deartment of Automatc Control, LTH, Lund Unversty, Box 8, SE-22 Lund, Sweden {Martn.A.Kjaer,Anders.Robertsson}@control.lth.se. M. Khl s wth the Deartment of Electrcal and Informaton Technology, LTH, Lund Unversty, Box 8, SE-22 Lund, Sweden Mara.Khl@et.lth.se. Hgh rorty queue Arrvng jobs Low rorty queue Arrvng jobs Common server Dearturng jobs Fg.. Two queues wth common server. The fracton of the server caacty s dedcated to the hgh rorty queue, and the fracton s dedcated to the low rorty queue. feed forward was used as art of a control set us wth resource allocaton actuators. Ths method can result n serous degradaton of erformance f the average requred work changes, as mght haen n real alcatons. The requred work s not usually easy to defne n real server systems that often oerate by rocessor sharng, and t can be hard to gve any qualfed estmates of the ths quantty. In ths aer, we exand exstng redcton schemes by addng a feedback mechansm. The new redcton method s utlzed for a new resonse tme control scheme. Ths controller s shown, through smulatons, to mantan the same desred robustness and transent roertes as the revous ublshed controllers, but here obtaned wthout the measurement of the requred work. The controller s comared to control strateges usng the same or more measured varables. II. QUEUIG MODEL The target system n ths aer s a sngle-server system that rocesses jobs comng from clents, as llustrated n Fg.. Clents are consdered satsfed f the resonse tme T (the tme between the arrval of a job and the tme where s has been fully served, also often denoted the system tme []) for a job s less than a certan tme lmt, T ref, n average. Also, there are low-rorty jobs n the server systems, for examle, mantenance and montorng jobs. The low-rorty jobs have no strct real-tme requrements, but they should be rocessed. We assume that jobs comng from clents have hgh rorty, but snce we need to assure that also low-rorty jobs can be served, the hgh-rorty jobs only have a share,, of the server caacty (where ). should be set so that the average resonse tme, T, for a hgh-rorty job s ket at the reference value, T ref wth low varance. We consder a system wth two classes of requests, hghrorty jobs and low-rorty jobs, each wth ts own job queue. It s assumed that there are always low-rorty jobs watng for servce, whch means that the low-rorty job /7/$2. 27 IEEE. 382 Authorzed lcensed use lmted to: IEEE Xlore. Downloaded on March,2 at :46:33 EDT from IEEE Xlore. Restrctons aly.

3 46th IEEE CDC, ew Orleans, USA, Dec. 2-4, 27 ThB6.4 class uses all ts avalable server caacty. Jobs have a requred work w that descrbes whch rocessng tme the job would have f t was gven the full server caacty. Ths gves the relatonsh x = w/, where x s the rocessng tme. a w T III. COTROL SYSTEM A block dagram of the server system s llustrated n Fg. 2. The nuts are those varables that are drectly accessble by the controller, and n the target system, the only nut s. The dsturbances are the nuts that are not affectable by the controller. They mght be measurable, but n many cases they are not. In the target system, the nter arrval tmes of new jobs, denoted a, and requred work, w, can be regarded as dsturbances. Oututs are measurable quanttes that deend on the system under nvestgaton, and the revous behavor of the nuts. In the target system, the resonse tme for a job, denoted T, and the job queue length, denoted, are the oututs. In ths aer, the objectves of the control system are the followng. Frst, to mantan an average resonse tme E{T } = T ref for hgh-rorty jobs wth as control varable. Second, to reduce the average allocated server caacty E{} whle mnmzng the resonse tme varance E{T 2 } E{T } 2. Only the queue length () and the resonse tme (T,) the tme of arrvals (a), and, f necessary, the requred work (w) should be used by the control system. Also, effects of varyng arrval rates should be ket mnmal. The dynamcs of a queung system can change dramatcally deendng of the load and the length of the queue. If the queue holds a lot of jobs, the nfluence of the statstcs wll be reduced because of the averagng effect. Snce the resonse tme for a job can be measured when the job leaves the system, t s natural to adjust at dearture nstants. However, ths control acton wll not only affect the resonse tme of the job now beng served, but t wll also affect the resonse tme of all the other jobs watng n the queue. Lkewse, the resonse tme of the job beng served wll have been affected by the control sgnals set whle the job was watng n the queue. Ths means that f the queue on average holds many jobs, a sgnfcant, and non constant, tme delay wll arse, whch can reduce the control erformance. If, on the other hand, the queue s almost emty at all tme, the stochastc nature of both the nter arrval tmes and the rocessng tmes wll have large affect on the obtaned resonse tmes, thus comlcatng the control actons. Furthermore, a short queue s often obtaned on the cost of a hgher control sgnal on average. IV. COTROLLERS Due to the robustness roertes nherted n every closed loo control strateges, all the followng control desgn methods wll be based on feedback. A. Classcal feedback controllers The classcal closed loo setu contans a measurement of the objectve varable, comared to the desred reference. Fg. 2. The hgh rorty queue seen from a control ersectve. The arrval tmes (a) and requred work (w) are unaffectable quanttes and are treated as dsturbances. Dynamc varables of the queung system are the queue length (), the resonse tme (T ), whch are treated as oututs. The fracton of the server caacty denoted to the hgh rorty queue () s a user defned arameter, and s therefore treated as nut. T ref Feedback Controller Feed-forward Controller fb ff Fg. 3. The hgh rorty queue wth a classcal feedback controller. The setu can also hold a feed forward mechansm n order to acheve better erformance. Based on the error (and ossble ast errors), the control sgnal s adjusted n a manner ntended to reduce the error. A smle method s to adjust the control sgnal wth a fxed amount as long as the error s above some threshold. Ths strategy wll n the future be denoted ste control. Another method s to use varants of the PID-controller [], [2]. B. Feed forward based redcton The control strateges above are based solely on feedback. Ths means that nformaton s taken from the oututs alone, and sent back to the controller, thus for the controller to react, an error must be resent. It s often ossble to redct, based on measurements of e.g. the dsturbances, that an error s about to arse f no mmedate acton s taken. If such nformaton s avalable, t can be utlzed n the control mechansm n what s often called feed forward. The combnaton of feedback and feed forward for the sngle server queue can be seen n Fg. 3. Here the arrval tmes and the requred work are used for feed forward. If the feed forward block s removed, the control s based exclusvely on feedback. Removng the feedback and relay only on the feed forward results n an oen loo scheme, whch means that the robustness roertes of the closed loo are lost. Measurements of dsturbances can be ncluded n many ways. In control of queung systems, queue theoretc aroaches are often taken, such as n [9], [3]. In [9] a redcton for the resonse tme s derved where the redcton s calculated at the dearture of a job from the server (at tme t now ). Assumng that there are jobs n the queue, the redcted resonse tme of the remanng jobs s gven by the average tme they have sent n the queue lus a redcton of how long tme t wll take to serve them. A a w T 383 Authorzed lcensed use lmted to: IEEE Xlore. Downloaded on March,2 at :46:33 EDT from IEEE Xlore. Restrctons aly.

4 46th IEEE CDC, ew Orleans, USA, Dec. 2-4, 27 ThB6.4 Acumulated jobs Known Predcted w/ a w Queung tme t now tme Processng tme Fg. 4. Server and queung dagram. The average tme of the known art s gven by P (tnow a ). The average tme of the redcted area s (+) w gven by. 2 slghtly modfed verson of the redctor s gven by ˆT = (t now a ) + ( + ) w 2 where w s the exected requred work for all the remanng jobs. A grahcal nterretaton of the redctor s gven n Fg. 4. See also [9]. Usng ths redctor, a feed forward sgnal can be calculated by ( + ) ff = 2(T ref (t now a )) () w (2) otce that an estmate of the requred work s needed n order to use the redctor. Also note, that the feed forward n (2) does not calculate the redcted resonse tme exlctly, and therefore, there s no mechansm to ensure that the redcton s correct. C. Proosed feedback-based redcton and control In ths secton we roose a redesgn of the resonse tme redcton n order to mrove the redcton based on knowledge of earler redctons. Ths s erformed by mosng feedback nto the redcton scheme. Ths s a method often used n lnear state estmaton [2]. The control scheme resented here contans the robustness roertes of feedback as well as the redctve actons from the feed forward. Instead of usng the nformaton from the measurements of the requred work n (), a vrtual arameter z s used nstead. Ths arameter can be nterreted as the comensated averaged requred work. The redctor s now gven by ˆT = (t now a ) + ( + ) 2 z (3) The estmated servce tme s now seen as a new outut and z as an nut. The urose of ths control roblem s to make the resonse tme estmate ˆT follow the real resonse tme T. The control error s therefore gven by e T = T ˆT. Usng a PI controller wth e T as nut and z as outut wll, at least n steady state, assure that E{ ˆT } = E{T }. The varable z can now be used to calculate the feed forward control sgnal for the sngle server queue: ( + ) ff = 2(T ref (t now a )) z (4) Predcton error PI controller z Inverse redcton model Predcton model T ref T ˆT Feed-forward nformatons Fg.. The resonse tme redcton s udated by a feedback mechansm. Usng the nternal state z, the nverse queue model gves the control sgnal requred to obtan the desred average resonse tme. The control setu usng ths feedback based redcton scheme s llustrated n Fg.. It s noted that the structure has sgnfcant smlartes to the state feedback control structure, whch s often used n lnear control desgn [2]. Because of ths smlarty, the urosed control scheme wll be denoted state feedback control n the remander of ths aer. Alternatvely, the PI controller could be exanded wth a dfferental art (PID) whch has been successfully used n many other alcatons to acheve faster resonses. The dfferental art s senstve to nose, and snce the system under nvestgaton s extremely nose, the dfferental art has been dscarded at an early stage. V. IVESTIGATIOS We nvestgated the controllers wth smulatons based on a dscrete-event smulaton rogram wrtten n Java. The Javarogram ncluded classes for the traffc generator, the queue, the observer and the controller. In ths secton all detals regardng our nvestgatons wll be descrbed. A. Controller Desgn Fve controllers were evaluated: A ste controller, two ordnary PI-controllers, one PI controller combned wth feed forward, and one state feedback controller. One PI-controller was desgned to obtan fast transents, n the followng called the fast PI controller. The other PI-controller, called slow PI, was desgned to avod oscllatons. All controllers were event based, and was trggered by when a job dearts from the server. To reduce the varance of the control sgnal, a low ass fltered resonse tme was used. The flter, whch also was event based, was gven by T f (k) =.9T f (k ) +.T(k). () The control sgnal was truncated to be n the nterval mn =. to max = before aled to the server. 384 Authorzed lcensed use lmted to: IEEE Xlore. Downloaded on March,2 at :46:33 EDT from IEEE Xlore. Restrctons aly.

5 46th IEEE CDC, ew Orleans, USA, Dec. 2-4, 27 ThB6.4 The ste controller was gven by (k ) + d for e(k) > h, (k ) < max (k) = (k ) d for e(k) < h, (k ) > mn (k ) else (6) where h was the threshold and chosen to h =.. The ste sze was d =.. The fltered resonse tme was used to calculate the control error e(k) = T ref T f (k). The slow PI controller was desgned wth an ant wndu scheme as n [4] I(k) = I(k ) + g( r) e(k) (7) g( r) ( + fb (k ) (k ) ) (8) fb = g r e(k) + I(k) (9) The varable I reresent the ntegral art of the PI controller. The controller constants g and r were chosen to. and., resectvely. The controller was based on the fltered resonse tme e(k) = T f (k) T ref. Alternatvely, the ntegral art of the PI controller could be mlemented wth varyng amlfcaton to take the aerodcally nature of the event based control udate nto account, as n []. The fast PI controller was equvalent to the slow PI controller, excet for the control arameter g =.. The slow PI FF controller used the same feedback loo as the slow PI controller, but a feed forward mechansm was also ncluded. A nverse resonse tme redctor was gven by Q(k) = (t now a (k)) () ff (k) = (k)+ 2 (T ref Q(k)) w f(k) for T ref > Q(k) () else max where a (k) are the arrval tmes of the jobs n the server system at dearture k and t now s the current tme. The varable w f s a fltered verson of the requred work w, gven by w f (k) =.w f (k )+.w(k). It s assumed that w can be measured. The control sgnal was gven by a summaton of ff the outut of the slow PI controller n (9). ote that ths feed forward strategy requres a measurement of the requred work w. The state feedback controller was mlemented wth a feedback corrected redcton scheme as Q(k) = (t now a (k)) (2) ˆT(k) = Q + + z(k ) (3) 2(k ) ˆT f (k) =.9 ˆT f (k ) +. ˆT(k) (4) e(k) = T f (k) ˆT f (k) () z(k) = z(k ) + g ( e(k) r e(k ) ) (6) { (k)+ (k) = 2 (T ref Q(k) z(k) for T ref > Q(k) (7) else max The control arameters were gven by g =. and r =.. ote that ths strategy does not requre a measurement of the requred work w. B. Smulatons Both steady-state and transent smulatons were erformed. All steady state results were evaluated after all transents had been removed. Transent behavor was nvestgated after the queung system had converged to steady state, and was allowed to run for suffcently long tme to make sure that all transent behavor was observed. In all smulatons, we used a Posson rocess for modelng the arrval rocess of new jobs. Also, the requred work of a job was exonentally dstrbuted. In the steady state smulatons, the mean arrval rate of new jobs, λ, was frst vared between and 9 jobs er second. The mean requred work for a job was. seconds. If = (.e., an uncontrolled system) and λ = /s, the average resonse tme would become.2 seconds wth a mean queue length of. jobs. The reference value for the resonse tme, T ref, was set to.6 seconds, whch corresond to a queue length of about 3 jobs for λ = /s (Lttle s law [6]). Then, the resonse tme reference T ref was vared from.s to.7s whle the arrval rate was ket constant at λ =. All other arameters was as wth the revous steady state exerment. In the transent smulatons, the requred work dstrbuton was ket constant as an exonental dstrbuton wth mean. seconds. However, the average arrval rate of new jobs changes from 2 to 8 jobs er second at tme t = 8 seconds. Ths s a consderable change n arrval rate, whch brngs the queung system from oeratng wth low load to oeratng n the hgh end of the medum load range. All ntal transent behavor has ded out and the system has reached steady state before the change of arrval rate. VI. RESULTS AD DISCUSSIO In ths secton we wll resent and dscuss the results from the nvestgatons. A. Steady state smulatons Fg. 6 shows some erformance metrcs from the smulatons. The to of the fgure shows how the controllers track the reference as the traffc changes. It s notced from the fgure that both the fast PI controller and the ste controller are not caable of mantanng the desred resonse tme n steady state. Also the roosed state feedback controller suffers from oor steady state reference followng at both hgh and low arrval rates, whereas t erforms satsfactory at medum arrval rates n the range of 2 8/s. The bottom of Fg. 6 shows the varance of the resonse tme. Here the beneft of usng feed forward becomes obvous. The combned PI and feed forward controller and the state feedback controller dam the stochastc varatons of the queung system far better than any of the controllers solely based on feedback. 38 Authorzed lcensed use lmted to: IEEE Xlore. Downloaded on March,2 at :46:33 EDT from IEEE Xlore. Restrctons aly.

6 46th IEEE CDC, ew Orleans, USA, Dec. 2-4, 27 ThB6.4 Average resonse Steady statepi behavour of the resonse tme T Ste control +FF StatePI Feedback Reference.8.6 Ste control +FF State feedback Ideal bla Average arrval rate λ (/s) P(T<x).4 Varance (s 2 ) Varance of the resonse tme Ste control +FF StatePI feedback x Average arrval rate λ (/s) Fg. 6. Performance metrcs for the controllers. To: steady state reference followng of the resonse tme. Bottom: Steady state varance of the resonse tme. otce that the slow PI controller, the slow PI controller wth feed forward, and the state feedback controller concde n some arts n the to fgure. So does the slow PI controller wth feed forward and the state feedback controller n the lower lot. Fg. 7. Average CPU reserved to the hgh rorty queue Ste control +FF StatePI feedback Average arrval rate λ (/s) Server caacty allocaton as functon of arrval rate n steady state. Fg. 7 shows the averaged allocated CPU as functon of the arrval rate. The fgure shows a general trend; the need for server resources ncreases wth the traffc. The fgure even ndcates that ths relatonsh s lnear. By close nsecton t can be seen that the slow PI controller demands an nsgnfcantly smaller amount of server resources than both the combned PI and feed forward controller and the state feedback controller for arrval rates above 2/s. Below ths, the ste controller and the slow PI controller seem to erform slghtly better. In ths regon the stochastcs become extremely domnatng, and redctve control, whch s based on averagng, becomes useless. The slow reactng feedback based mechansms therefore erform better. However, n the domnatng arrval rate san, the feed forward based controllers show sueror erformance. A common way to reresent smulaton and exerment data n the telecommuncaton communty s the dstrbuton as llustrated n Fg. 8. The dashed lne reresents the result Fg. 8. Resonse tme dstrbutons. otce that the the slow PI controller wth feed forward and the state feedback controller concde n most arts of the fgure. Average CPU reserved to the low rorty queue Average resonse tme Fg. 9. Steady state smulaton results for the state feedback controller. The resources avalable for the low rorty queue, as functon of resonse tme T, lotted for dfferent values of arrval rate λ. of an deal resonse. The fgure ndcates that none of the controllers are deal, but the two controllers wth feed forward (the slow PI+FF controller and the state feedback controller) erforms sgnfcantly better. The Ste controller and the two ure PI controllers are skewed and show to result n hgh ossblty for long resonse tmes. Fg. 9 shows what s earned by usng control, snce the caacty not reserved for the hgh-rorty queue s avalable for the low-rorty queue. The fgure shows that a lot s ganed when accetng slghtly longer resonse tmes. However, as longer resonse tmes are acceted, the ncrease n caacty for the low-rorty queue dmnshes. B. Transent smulatons One strong argument to use feedback n the control s the robustness towards changes n the envronment. The above smulatons were all erformed wth steady state envronments, whch means that the both the requred work and the arrval rate had constant dstrbutons. It s of hgh relevance to nvestgate the behavor of the controlled queung system under changes n the envronment. Fg. llustrates the λ=3 λ= λ=7 386 Authorzed lcensed use lmted to: IEEE Xlore. Downloaded on March,2 at :46:33 EDT from IEEE Xlore. Restrctons aly.

7 46th IEEE CDC, ew Orleans, USA, Dec. 2-4, 27 ThB6.4. Ste FF State PI Ste FastPI FF State PI VII. COCLUSIOS Performance related control of server systems s a wde research area wth numerous alcatons. In ths aer, we have consdered a sngle-server queue n whch hghrorty jobs should receve as much server caacty as needed to exerence a certan resonse tme. However, the server caacty should be otmzed n order to allow for the executon of low-rorty jobs. We have roosed a state feedback controller that uses a redcton of the resonse tme. We comare the roosed controller wth other controllers found n the lterature. The roosed controller s shown to have a smlar desrable behavor for the target system than the best of other controllers deste that no measurement of the requred work s requred n our soluton. VIII. ACKOWLEDGMETS Ths work has been founded by the Swedsh Research Councl, roject Fg.. Transent behavor of the sngle server queung system. An ncrease n the arrval rate from 2 /s to 8 /s occurs at tme t = 8 s. The left column shows the control sgnal and the rght column shows the resonse tmes. Rows one to fve show results for the ste controller, the slow PI controller, the fast PI controller, the combned PI and feed forward controller, and the state controller, n the mentoned order. transent behavor of the sngle serve queue wth the dfferent controllers. The transent behavors show that the slow desgned controllers based only on feedback result n a large transent at the arrval rate change. Frst the change must be detected n the resonse tme measurements, and then the controllers must buld u the control sgnal to comensate. Durng ths tme, the queue has bult u, resultng n large resonse tmes. The fast PI controller manages to react faster to the dsturbance and thereby mantanng a shorter queue. The transent behavor can be seen n the fgure, whch yelds a oscllatng behavor that s undesrable. It should be mentoned that the fast PI controlled system becomes unstable (ever ncreasng queue length) f the reference s ncreased too much. The two controllers based on both feedback and feed forward seems to manage the change well, whch was also the reason to utlze the feed forward. The feed forward measures the change of arrval rate and starts to comensate for t even before t has affected the resonse tme. There s no sgnfcant dfference between the transent erformance of the two controllers. C. Dscusson Both the steady state smulatons and the transent smulatons show that the caablty to mantan the reference and to reduce the resonse tme reference s mroved sgnfcantly by mosng feed forward. The two feed forward methods show smlar roertes n most of the tested cases. However, our roosed state feedback controller does not requre a measurement of the requred work, whch can be consdered as a consderable advantage. REFERECES [] M. 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