Response-Time Control of a Processor-Sharing System Using Virtualized Server Environments Kjaer, Martin Ansbjerg; Kihl, Maria; Robertsson, Anders

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1 Resonse-ime Control of a -Sharing System Using Virtualized Server Environments Kjaer, Martin Ansbjerg; Kihl, Maria; Robertsson, Anders Published in: Proc. of the 7th IFAC World Congress Published: 8-- Link to ublication Citation for ublished version (APA): Kjaer, M. A., Kihl, M., & Robertsson, A. (8). Resonse-ime Control of a -Sharing System Using Virtualized Server Environments. In Proc. of the 7th IFAC World Congress. IFAC. General rights Coyright and moral rights for the ublications made accessible in the ublic ortal are retained by the authors and/or other coyright oners and it is a condition of accessing ublications that users recognise and abide by the legal requirements associated ith these rights. Users may donload and rint one coy of any ublication from the ublic ortal for the urose of rivate study or research. You may not further distribute the material or use it for any rofit-making activity or commercial gain You may freely distribute the URL identifying the ublication in the ublic ortal? ake don olicy If you believe that this document breaches coyright lease contact us roviding details, and e ill remove access to the ork immediately and investigate your claim. Donload date: 9. May. 6 L UDUI VERS I Y PO Box7 L und +66

2 Resonse ime Control of a Sharing System using Virtualized Server Environments Martin A. Kjær Maria Kihl and Anders Robertsson Deartment of Automatic Control, LH, Lund University, Box 8, SE- Lund, Seden, ( {Martin.A.Kjaer,Anders.Robertsson}@control.lth.se) Deartment of Electrical and Information echnology, LH, Lund University, Box 8, SE- Lund, Seden, ( Maria.Kihl@eit.lth.se) Abstract: Feedback in server systems has during last years gained much interest in order to fulfill still increasing demands on erformance and otimization regarding, for examle, quality of service (QoS) requirements. In this aer e exand a reviously ublished feedback based rediction scheme for controlling a single server queue together ith a ne control strategy. hese control structures have the benefit over other reviously suggested control structures that no off line estimation of the required ork is needed. In addition, our solutions maintain or imrove the erformance, regarding average resonse time and loss of comutational resources.. IRODUCIO Server systems are used in most (tele)communication netorks. he server system can rocess jobs from either other netork entities or from clients. In the Internet, eb servers send eb ages to clients. Different control mechanisms may be used to control the server system. Admission control revents system overload by rejecting some jobs hen needed, see for examle Kihl [999]. Load balancing otimizes the load in a distributed system, see Kremien and Kramer [99]. Scheduling can otimize the resonse times hen jobs have different QoS requirements as in Lu et al. []. Virtualized server environments have become oular ithin the server hosting for many reasons (isolation, latform indeendentness, and other). Being able to isolate the resource allocation and consumtion beteen several alications no allo designs for better utilization of the resources than if the alications ere to share the resources or, alternatively, hosted by searated hysical resources. Virtualized server environments are still restricted by limited hysically resources, hich imose interesting control roblems; ho to divide resources otimally beteen the virtualized server environments, and ho to guarantee that the virtualized server environments are ensure, and restricted to, the resources assigned to them. Some virtualization schemes allo a ne form for actuation, alloing the designer to define the amount of comutational resources dedicated to the virtual environment online, and thus, the amount dedicated to the server, see Xu et al. [6], Wang et al. [7]. Since a server system consists of CPUs (or virtual comutational resources), it has a non-linear behavior. A small increase in load can result in raidly groing resonse times. herefore, the design of the control mechanisms is not a simle task if otimized erformance is desired. In recent years this field has gained a large research interest from both academia and I vendors (Hellerstein et al. [, ]). In this aer e focus on resonse-time control (the resonse time is defined as the time beteen the arrival of a job and the time hen is has been fully served, also often denoted the system time, see Kleinrock [97]); an objective directly couled to the end-user, but ith close relations to e.g. the more server oriented queue length control (Kihl et al. [3]). Hoever, the connection beteen the queue length and the resonse time deends on numerous factors such as the nature of the arriving traffic and the service times. Liu et al. [6a] designed a resonse time control system based on admission control, here the admission robability as adjusted to obtain the desired resonse time. he control set u included a queuing theory based feed forard here a measurement of the average required ork as needed. his as measured off line under lo load, and then alied in online control. he off line identification of the average required ork as also the rocedure of the ork by Henriksson et al. [] here feed forard as used as art of control set us ith resource allocation actuators. his method can result in serious degradation of erformance if the average required ork changes, as often haen in real alications. he required ork is in general not easy to define in real server systems that often oerate by rocessor sharing, and it can be hard to give any qualified estimates of this quantity. In this aer, e exand a reviously ublished control scheme (see Kjær et al. [7]) to coe ith eriodical actuation based on comutational resources allocation, as e.g. ith server virtualization schemes. We also assume rocessor sharing serving instead of single server rocessing, and more bursty traffic. he rediction method is utilized for a ne resonse time control scheme. his controller is shon, through simulations, to maintain bet-

3 a λ PS Required ork / Fig.. Left: Inut outut model of the rocessor system. he allocated comutational resources is the inut, he arrival times a and the required ork are disturbances, and the resonse time and the number of jobs in the system are the oututs. Right: Queuing theory model of the rocessor. Jobs arrive at the system ith arrival rate λ, and are served ith service rate /. ter desired robustness and transient roerties than other ublished control strategies. SYSEM DESCRIPIO AD MODELIG We consider a rocessor sharing system. At a given time, the comutational resources available to the rocessor are divided uniformly beteen the jobs resent. Since the rocessor ill accet any incoming job, no queue is resent. In virtualized server environments the CPU resources are often shared beteen several alications. We treat our rocessor system as one alication here the dedicated (reserved) share of the CPU is treated as an abstract comutational resource, hich can be set online at certain time intervals in the interval [., ]. In the case here more comutational resources are allocated (reserved) to the rocessor than the rocessor requires, the over allocation is considered as a loss of comutational resources, hich could else have been utilized by other alications. Also, it is assumed that certain calculations are alloed at each dearture, and that measurements of former resonse times, arrival times, and number of jobs, are available as measurements. For the develoment of controllers, no secific distributions for the arrivals and the required ork are assumed. We do not consider the case here the rocessor can hold only a limited number of jobs. he system is shon as an inut outut system in Fig.. he arrival times and the required ork are treated as disturbances, hereof only the arrival times are measurable.. Purose of Control Loss of comutational resources occurs hen resources are dedicated (reserved) to the rocessor, but no jobs are resent to use the resources. herefore, from a resource otimization oint of vie, it is desirable to ensure that there alays are job to serve. Since jobs can not be invented by the rocessor, the only solution is to reduce the service rate, and thereby holding jobs longer in the rocessor. his has a cost on the alication erformance, since the resonse times ill increase. he trade off is then to ensure that there are jobs to serve, but at the same time, to maintain an accetable resonse time. he goal of control is to ensure that the resonse time is as secified by ref, and minimizing the loss of comutational resources. A third erformance metric is the variation of the resonse times. he smaller it is, the better. Acumulated jobs Queuing time Processing time Knon t no Predicted Fig.. Accumulated jobs in a single server queue.. Modeling ŵ/ time he modeling ork focuses on the rediction of the resonse time of the jobs resent in the rocessor system at a given time. In queuing theory it is ell knon that a M/M/ and a M/G/ PS system have the same average resonse time, see Kleinrock [967] and M. Sakata and Oizurnih [97]. his means that a rocessor sharing system ithout a queue behaves similarly (in terms of average resonse time) as a single server ith an infinite queue. Modeling the system by a single server (ith an infinite queue) means that the jobs can not ass one another in time (the first job that enters is the first job to leave), hich simlifies estimation significantly. Since the system is an event driven system, e chose to model hat haens hen an event occurs. Consider the case here a job leaves the rocessor at time t no, leaving remaining jobs. his is illustrated in Fig.. Assuming that all jobs ill have the same required ork ŵ and that the allocated comutational resources remain constant, a rediction of the average resonse time ˆ is given by ˆ = (t no a i ) + i ( + )ŵ here a i is the arrival time of job i. he first term to the right of the equallity sign reresents the amount of time that the jobs have already send in the rocessor. his quantity is knon by measurement, and is illustrated as the area left of t no in Fig.. he second term of the equation reresents the exected time that the jobs ill have to stay in the rocessor for comleation. his quantaty is based on an estimate, and is illustrated as the area right of t no in Fig.. (see Henriksson et al. [] and Kjær et al. [7] for a more detailed exlanation). 3. ESIMAIO he model described above can be useful for online adjustment of the resonse time, but it relies on several measurements. he number of jobs in the system, and their arrival times a i are quantities often registered by a real system. If the rediction is executed at the dearture, t no is simly found by reading the clock. he average required ork ŵ is a lot harder to find. In a single server ith a queue it could be estimated by measuring former service times, corrected ith the allocated comutational resources, but for rocessor sharing systems, this is not ossible. herefore, relaying on a measurement of the required ork ill reduce the alicability of the estimation. We therefore seek another strategy. () In classical linear estimation roblems models are used to estimate non measured quantities, see for examle

4 a a ref PI controller z PI controller Prediction error Prediction ˆ ref state Inverse redictor Dynamic redictor z Dynamic redictor model, a Fig. 3. Block diagram of feedback based redictor. he redictor can be interreted as an observer ith state z. Franklin et al. [99] or Åström and Wittenmark [997]. Often the measurable variables are comared to the estimated values, and a feedback mechanism tries to minimize the estimation error. We consider as a measurable outut and as a state to be estimated. In Kjær et al. [7] e roosed a redesign of the resonse time rediction resented in Henriksson et al. [], resembling the structure of a classical observer. Because the dynamics of the system are not ell knon, e suggest to use a PI controller due to its robustness. o stress that the estimator does not rely on measurements of, e imose an artificial variable z to act as the inut to the model. he estimator no takes the form ˆ = (t no a i ) + i ( + ) z () e j = j ˆ j (3) I k = I k + h kk e k + h k (v k z k ) i a () v j = K e j + I j () { v for v > z = (6) else here i and K are controller arameters. he variable h k is the time beteen the revious and the current samling. Using a varying samling eriods in the integrator has earlier been shon by Årzén [999] to be suerior to fixed samle eriods for some event based systems. Integrator anti indu is included as the last term in (), here z(k) is the achieved control signal (z is not alloed to be negative). he arameter a determines the convergence rate of the anti ind u, see Åström and Wittenmark [997]. he arameters i =., K =., a =. ere chosen for the investigations to be resented. he estimation scheme is also illustrated in Fig. 3. Equations () and () form a general PI controller, here the term z k is exchanged ith the relevant control signal. Fig.. Block diagram of the suggested state FB controller.. Suggested Controllers. COROLLER DESIG he control roblem has to deal ith a significant time delay; a change in ill only roagate to after a certain time. In classical control theory, the erformance of systems ith delays can be imroved by different rediction techniques, such as the Smith redictor, see for examle Åström and Hägglund []. his idea can also be alied using the estimator in (). o control strategies have been designed based on this rincile. Due to lack of theoretical tools to analyse event driven nonlinear systems no analysis of the stability of the controlled systems have been erformed. Insiration to ho this could be done can be seen in e.g. Kihl et al. [7]. State Feedback Controller (state FB) he redictor in ()-(6) is seen as an observer, and z as a state. A state feedback can no be formed by inverting the rediction model () + state = ( ref i (t no a i )) z. (7) as suggested in Kjær et al. [7]. o handle model errors, a eriodic PI controller as in () and () is added (K =, i =., a =.). he control structure is illustrated in Fig.. PI ith Predictive Feedback Controller () A eriodic PI controller as in () and () is used for feedback control of the resonse time (K =, i =., a =.). his suffers from the significant time delay discussed earlier. Feedback from the redicted resonse time can allo much faster reaction. We chose to use a roortional controller given by = K ( ref ˆ). (8) ith K =.. he structure is illustrated in Fig... Samle Periods It is an assumtion that the comutational resources dedicated to the rocessor only can be changed at some fixed time eriod, s. Hoever, the estimation is not restricted by the samling eriod. he estimation is udated for each dearture, ensuring that the estimate ˆ and the state z alays incororate the neest measurements. When a samle incident occurs, the control signal can be based

5 ref PI controller P controller ref ˆ a Dynamic redictor average resonse time of a M/G/-PS in stationarity is given by (Kleinrock [967], M. Sakata and Oizurnih [97]) x = λ x. () Equation (), or varieties hereof, have formed the base for other feed forard designs in the literature, e.g. Liu et al. [6b], Lu et al. [3]. In our case, x = / if is constant. Assuming that is knon (and exact), and λ is estimated by some indoing mechanism (ˆλ), a feed forard signal can be formed as Fig.. Block diagram of the suggested controller. ref Feedback Controller Feed-forard Controller ff a Fig. 6. Block diagram of a combined feedback, feed forard setu for the rocessor system. on data accumulated not just at the former samling incidences, but also on data obtained in beteen..3 Controllers for Comarison Earlier ork resents solutions here feed forard and feedback are combined as illustrated Fig. 6, see e.g Liu et al. [6b], Lu et al. [3], and Henriksson et al. []. hese all assume that an estimate of the average required ork is available. An often used rocedure to obtain the estimate is to measure the resonse times at very lo arrival rates. Assuming that only one job is resent, the resonse times can be used to estimate the required ork. he estimate is then used online at higher arrival rates, assuming that it ill remain constant. his method is not robust toards changes in the required ork, as, for examle, hen the relative oularity ithin a certain ebsite suddenly changes. We resent to controllers based on this rincile for comarison. PI ith Inverse Prediction Feed Forard (PI IPFF) A eriodic PI controller as in () and () is used for feedback (K =., i =., a =.). Equation () is rearranged, and the ˆ is exchanged ith the desired value ref + ff = ( ref i (t no a i ))ŵ. (9) his is a slightly modified version of the feed forard resented in Henriksson et al. []. PI ith Queuing heoretic Feed Forard (PI QFF) A eriodic PI controller as in () and () is used for feedback (K =., i =., a =.). he feed forard is based on queuing theoretic models. he. Filtering Issues ff = + ˆλ ref ref () he involved signals can be quite irregular, hich can lead to irregular estimation and oor control erformance. All the tested eriodic PI controllers use the comarison of the reference and a filtered resonse time ; k = ( k + k s)/ here k s is the average resonse time of the jobs that dearted under the interval beteen samling k and k. o udate the estimator, the estimated resonse time is comared to a filtered measured resonse time; f i =.999 f i +. i, here i indicates the dearting job number, and i is the resonse time of job i. he resonse time estimate can also be quite irregular. An obvious idea is to aly a filter directly to the estimate ˆ. his has an undesirable effect due to the nonlinear structure. Linear filtering of the term / ould eight small values of and could lead to rong estimates. Also, a linear filtering of the term i (t no a i ) ould eight the jobs that have a long service time over those having a short resonse time, thus increasing the average estimate. he filtering must therefore be laced ith care. he best results have been obtained by simly filtering ; f i =.999 f i +. i, hich is an event based filter. o controllers are based on inverse rediction (the state FB controller and the controller). hat is, any resonse time error is comensated in one udate. A similar idea is used in classical minimum variance control, hich is knon to have oor robustness roerties, see Åström [6]. In our case, the result is an undesirable irregular control signal, and some filtering is imosed. Again, alying a filter to the control signal ould drive the average control signal off due to the nonlinearity of the fraction in (7) and (9). herefore, the numerator and denominator are filtered searately; P i =.999 P i +. ( + )z () Q i =.99 Q i +. (t no a i ) (3) state = P i ( ref Q i ) i () for the state FB controller. For the controller, state is substituted for ff, and z for.

6 . Simulations We investigated the controllers ith simulations. We used a discrete-event simulation rogram ritten in Java. he Java-rogram included classes for the traffic generator, the rocessor, the observer, and the controller. Steady state and transient simulations ere erformed. All steady state results ere evaluated after all transients had been removed. ransient behavior as investigated after convergence to steady state, and as alloed to run for sufficiently long time for the transient to settle. o achieve sufficient burstiness, e used second order hyer exonential (H ) distributions to generate inter arrival times and required ork. A H distributed variable x is ith robability α a realization of logarithmic distributed variable ith exected value a, and ith robability ( α) a realization of logarithmic distributed variable ith exected value a. We used the folloing arameters: α = (C )/(C + 6) () a =. x (6) a = α x; (7) α here C and x ere the variance coefficient and average value of the H distributed sequence, resectively. he value of C ere chosen equal for the inter arrival times and the required ork distributions, and had the value C = unless stated differently. Average values and confidence intervals ere evaluated on don samled data to remove correlation beteen the measurement oints. he size of the 9% confidence intervals for the resonse time did not exceed % of the mean value. he size of the confidence interval for the loss of comutational resources did not exceed % of the maximum available comutational resources. o evaluate the caability to maintain lo variation of the resonse times the variation cost function as defined as J s = = ( k ) (8) k= k (9) k= In the transient investigation e could not rely on long simulation runs to obtain sufficient accurate results hen evaluating erformance. herefore, the transient eriods ere averaged over a large number M of exeriments. o comare the erformance of the simulation runs, e used the cost functions J (M) = M ( ref k ) () M i= t k t J (M) = M (q k ) () M i= t k t here t as the transient eriod, and q k as the loss of comutational resources. Average res. (s ) Average arrival rate λ (/s) Average arrival rate λ (/s) Average arrival rate λ (/s) Fig. 7. Averaged steady state simulations for different arrival rates λ. Variance coefficient C =, average required ork =. s, off line estimated required ork ŵ =. s, samling eriod s = s. resonse time reference ref =.. RESULS AD DISCUSSIO In this section e ill resent and discuss the results from the investigations.. Steady State Simulations raffic Load he traffic is often described by to traffic quantities; the arrival rate (λ) and the nominal service rate (/ ). Often, the traffic is quantified by the traffic factor ρ = λ. A lo value of ρ means a lightly loaded system, here as values close to, but belo, one means a heavy loaded system. If ρ exceeds one, the system lack resources to serve incoming requests, and the system is overloaded. In the folloing, the effect of the arrival rate and the service rate is investigated. Fig. 7 shos the erformance metrices hen the arrival rate as varied in a range corresonding to ρ =..9. It indicates that all the controllers managed to kee the average resonse time near the reference, hoever, small off sets ere observed. All controllers erformed best hen the arrival rate as relative high, since the loss of comutational resources as small. Esecially the controller seemed to erform oorly at very lo arrival rates. In the simulations, the estimate of the required ork ŵ corresonded exactly to the average required ork for the controllers for comarison. he suggested controllers estimated this arameter online, but shoed no significant degradation in erformance because of this. he shoed a higher variation cost J s, hich indicates a less smooth resonse time than the other controllers. Fig. 8 shos the erformance metrices hen the required ork as varied in a range corresonding to ρ =..87. It indicates that all the controllers managed to kee the average resonse time near the reference, hoever, small off sets ere observed. Also here, the queuing based rediction shoed to erform rather oorly over the full range, since it yield a higher loss of comutational resources, but also a large variation cost J s. In the simulations, the estimate of the required ork ŵ did not

7 Average res. (s ) Required ork (s) Required ork (s) Required ork (s) Average res. (s ) Resonse ime Reference Resonse ime Reference Resonse ime Reference Fig. 8. Averaged steady state simulations for different average required ork. Variance coefficient C =, arrival rate λ = 7 s, off line estimated required ork ŵ =. s, samling eriod s = s. resonse time reference ref =. corresonds exactly to the average required ork for the controllers for comarison. Desite this, the inverse rediction feed forard controller erformed ell in steady state because of the robustness of the PI controller. Resonse ime Reference By ensuring that there alays are jobs to rocess, the loss of comutational resources ill be minimal according to the rincile behind the designs. his is obtained by alloing a higher average resonse time than hat might be ossible ith more allocated resources. Fig. 9 shos that there as a limit here it as not orth to increase the resonse time reference any further. In the given case, resonse time references above aroximately one did not result in any significant reduction in the loss of comutational resources. It does not matter, for the loss of comutational resources, hether there ere fe (but alays some) or many jobs in the system. hus, a higher resonse time reference ould only generate higher average resonse time, but no imrovements. When the resonse times became too short, the risk of an emty system becomes significant, hich introduces loss of comutational resources. his is clearly indicated for the lo resonse time reference of Fig. 9. raffic variance raffic might be more bursty than in the examles resented above. Fig. shos ho the suggested controllers erformed under other tyes of traffic, at different arrival rates. he figure indicates that generally, the controllers did not deend much on the burstiness of the traffic. Only the state FB controller seemed to have a higher loss of comutational resources at lo burstiness and lo arrival rate. Generally, the variation cost J s increased ith the burstiness, but this is a natural trend, originating from the traffic itself. Samle interval Fig. shos results that are interesting from an imlementation oint of vie. In some cases, the samling eriod for the udate of might not be a free choice, but can be restricted by softare and hardare limitations. Generally, the results resented in Fig. 9. Averaged steady state simulations for different resonse time references. Variance coefficient C =. Arrival rate λ = 7 s. Average required ork =. s, off line estimated required ork ŵ =. s, samling eriod s = s. Average res. (s ) raffic variance coefficient, C PI PFB, λ=, λ= PI PFB, λ=, λ= PI PFB, λ=7, λ=7 6 8 raffic variance coefficient, C 6 8 raffic variance coefficient, C Fig.. Averaged steady state simulations for different arrival rates and different variance coefficient C. Average required ork =. s, off line estimated required ork ŵ =. s, samling eriod s = s. resonse time reference ref =. Fig. sho that the loss of comutational resources increased ith the samling eriod. Esecially for lo traffic, the to suggested controllers shoed degraded erformance as the samling eriod as increased. he variation cost J s increased ith the samling eriod, and as a result, higher loss of comutational resources as observed. For higher arrival rates the controllers shoed accetable erformance ith samling eriods several times higher than the resonse time reference, but as the samling eriod became higher than about seven times the resonse time reference, the erformance started to degrade. his can cause a serious constraint on the resonse time references.. ransient Simulations One strong argument to use feedback in the control is the robustness toards changes in the environment. he above simulations ere all erformed ith steady state environments, hich means that both the required ork

8 Average res. (s ) Samle Samle PI PFB, λ= 3, λ= PI PFB, λ=, λ= PI PFB, λ=7, λ= Samle Fig.. Averaged steady state simulations for different samling eriod s. Arrival rate λ = s. Variance coefficient C =. Average required ork =. s, off line estimated required ork ŵ =. s. resonse time reference ref =. E{q} (%) E{q} (%) E{q} (%) E{q} (%) PI PFB PI IPFF PI QFF E{} (s) E{} (s) E{} (s) E{} (s) PI PFB PI IPFF PI QFF Fig.. ime domain results: ransient simulations case ith changing arrival rate and high traffic burstiness. (C =). Each lot reresents an average of simulation runs, initiated ith different random generators. and the arrival rate had constant distributions. It is of high relevance to investigate the behavior of the controlled system under changes in the environment. he to diagram of Fig. 3 illustrates cost functions for the transient behavior of the system hen exosed to change in the required ork. In the beginning of the exeriment the load as relatively lo ith ρ =. (λ = s, =.8 s). At time t = the average required ork as doubled, such that the system as exosed to high load traffic, ρ =.8. Fig. and the middle and bottom of Fig. 3 illustrates the results for the transient behavior of the rocessor hen the arrival rate as changed. Initially, the system as exosed to a lo load traffic ith ρ =.3 (λ = s, =.7 s). At time t = s the arrival rate doubled, such that the system as exosed to high load traffic, ρ =.7. An incorrect estimate of the required ork for PI-IPFF and controllers as used for all the transient simulations (ŵ =.). he exeriment..8.6 Jq Jq.. Jq. J J I9(J) I9(Jq) J Fig. 3. Cost function results. o: ransient simulations case ith changing required ork (M=, =7 s). Middle: ransient simulations case ith changing arrival rate and high traffic burstiness. (C =, M=, =3 s). Bottom: ransient simulations case ith changing arrival rate and lo traffic burstiness. (C =., M=, =3 s). resented in the middle and bottom of Fig. 3 had traffic variance coefficients C = and C =., resectively A first observation is that the queuing based feed forard control resonded oorly to changes; in the case of increasing required ork the feed forard did not do any difference, since it only considered the arrival rate (and a rong, static, estimate of ). herefore, the eriodic PI controller had to handle the change, hich as rather slo, resulting in a large deviation of both the resonse time and a large loss of comutational resources. In the cases here the arrival rate changed, the queuing theory based feed forard over comensated, resulting in a large loss of comutational resources. For change in required ork, the and the state FB controller shoed no significant difference in erformance. Immediately after the change, none of the controllers had a good estimate of the required ork. he state FB controller adated to the change, but because the rediction controller as rather slo, this did not have any significant effect during the investigated transient. A difference beteen the to controllers as observed hen the arrival rate as changed and the burstiness as lo (the bottom of Fig. 3). Here, the state FB controller had a valid estimate of the required ork under the transient, and as thus caable of handling the transient better than the controller, hich had a false estimate. When more bursty traffic as imosed, the difference beteen the state FB controller and the controller vanished (the middle of Fig. 3). For all high burstiness simulations our roosed controller shoed the best transient resonse of all the controllers. 6. COCLUSIOS In this aer e have resented design of to controllers for a rocessor sharing system here the dedicated com-

9 utational resources could be set at fixed samle times. he erformance of the to suggested controllers as comared to to other controllers from the literature. A general trend in all the investigation as the oor erformance of the controller. It as based on a fixed, off line estimated, required ork, and only considered long term averages in the feed forard art. Only ith lo arrival rate, here the stochastics of the traffic became dominating, this controller erformed similarly, or a bit better, than the other controllers. he and the state FB controller differ in the estimate of the required ork; off-line estimation and online estimation, resectively. he investigations shoed that the dynamic estimation had most imact on the transient behavior. In the steady state, the feed forard based on a oor estimate of the required ork as comensated by the PI controller. In the transient simulations, the dynamic estimator shoed better erformance if the traffic as not too bursty. In bursty conditions, the stochastics became so dominating that the rediction inverse model as too inaccurate to ensure roer control. In many of the exeriments, our suggested controller shoed the best erformance. he transient behavior as suerior to all the other controllers. Only hen the traffic as lightly bursty (C ) the state FB controller and the shoed a little better erformance, since the inverse model as relative correct. In the steady state analysis the PI PFB controller shoed to be sensitive to small required ork and lo arrival rate (that is, sensitive to lo load). A restriction on the resonse time reference as ut on both of the suggested controllers, hich might cause restriction to their use in ractical imlementations. he suggested controllers did not rely on any off line estimation of the required ork and shoed similar or better erformance than the controllers for comarison. 7. ACKOWLEDGMES his ork has been founded by the Sedish Research Council, roject REFERECES K-E. Årzén. 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