Analysis of Buffer Delay in Web Server Control

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, WeB8. Analysis of Buffer Delay in Web Server Control Martin Ansbjerg Kjær and Anders Robertsson Abstract This paper addresses a problem related to control of web servers. Due to architectual issues, relevant measurements can only be taken over a limited part of the server system, which leaves the controller unaware of what happens in buffers proceeding the actual web server (for instance, in the TCP/IP layers). In earlier presented work we modeled such a system and proved that the unmeasured buffering turned a measurement of an input disturbance into a state dependent variable. In this paper we investigate how these new properties effect a traditional control design consisting of feedback in combination with a feed forward mechanism. The stability analysis suggests that under certain circumstances, the system enters limit cycles due to the feedback mechanism introduced by the designed feed forward controller. We verify the analysis by both simulations and experiments. I. INTRODUCTION Resource management has a long history in computer science, as poorly managed resources can degrade the performance of a computer system severely. In larger computer systems, work balancing is done in order to distribute the need for resources uniformly over a number of resource units (servers, CPUs, memory units, I/O etc.), thus avoiding that some units are overloaded while others are idle [], []. When more resources are requested than available, e.g., a higher request rate than service rate at a web-server systems, the risk of overload occurs and to maintain some (reduced) service, admission control is needed [3], [4], [5]. This has unwanted side effects, since the price of not satisfying certain request for resources is, in many cases, loss of profit. In order to keep the side effects as small as possible, a lot of focus has been put into developing methods to maintain a well behaving computer system (not overloaded) while keeping the side effects as small as possible. Server systems have traditionally been designed from a queuing theory perspective, but in recent years there has been an increasing interest in both academia and industry to apply control-theoretic design methods for improved performance. Server applications are seldom run directly on the hardware of the computer, but several layers of software are implemented below the web server, often collected in the operating system. The philosophy behind the layering is that as long as the defined interfaces between the layers are respected, the implementation of one layer can be exchanged without affecting the functionality of the system. A whole framework of layering is defined in the OSI model and the TCP/IP stack, which both define a number of layers, where each layer only has to know the interface to the layer above M. A. Kjær and A. Robertsson are with the Department of Automatic Control, LTH, Lund University, Box 8, SE- Lund, Sweden {Martin.A.Kjaer,Anders.Robertsson}@control.lth.se and the layer below [6]. One of the methods to keep the layers independent of the behavior of the other layers is to insert buffers. When a layer has something to sent to the layer above, it might put the job in its own out going buffer, and is then ready to proceed with other tasks. When the layer above receives a job, it might put it in its own in going buffer until it is ready to handle it. If this is done throughout the whole stack of layers, a request sent to a web server might pass through several buffers of the system before reaching the actual web server. All this buffering optimizes throughput, since the individual layers do not have to wait for neither the upper nor the lower layer to respond. The drawback is that it is not possible to determine how long it will take for a request to propagate through the whole stack, which may cause unexpected problems for control design. When trying to control for instance the average response time in a server, the buffering will influence the dynamic properties of the whole queuing system. This paper analyzes the effect of these underlying buffers and their influence on the controlled server dynamics. The investigated system is a web server system with response time control, as investigated in e.g., [4], [7], [8], [9]. The objective of the control is to maintain a pre defined average response time, while minimizing the CPU consumption. The focus of the paper is not the control design itself, but rather the stability properties of the system when including or excluding the dynamical effects of the buffers. In [] we presented a model of a web server together with an analysis of a feed forward control strategy. In this paper a control strategy consisting of both feedback control and feed forward control is analyzed. This control configuration is considered to be far more relevant. II. MODELING Assume that the target problem is a web server running on top of a software stack as described above. The number of buffers is unknown, and naturally, it is not possible to measure any quantities (such as buffer length or buffering times) in these buffers. It is assumed that all the buffers implemented before the web server are ideal buffers that is, they do not impose any delay if the proceeding element in the chain is willing to accept the job. The effect of other jobs passing through the same buffers is neglected, assuming that the requests to the web server are not delayed by jobs of other purpose, such as ftp and ssh. The buffers proceeding the web server (the buffers which are passed when the answer is sent to the client) are also neglected. The simplified view of a web server system is illustrated in Fig //$6. AACC 47

2 Fig.. Requests from clients λ Buffer External buffers Buffer λ i Server Response to clients Requests passes through several buffers before reaching the server. The web server is modeled as a single processor sharing server with a limited number of M jobs allowed simultaneously. The individual external buffers do not hold any independent processors and the external buffers should therefore not be modeled as individual queuing systems, but rather as one long queue. The only processor in the system is that of the actual web server, and the dynamics of the entire queuing system is therefore modeled as one infinite queue and a single processor. The queue length model for the entire queuing system follows Tipper s model ẋ(t) = λ(t) p(t) w x(t) + x(t) where x is the expected queue length when assuming exponentially distributed inter arrival times and service times [], [], [3], [4]. Here, the arriving traffic is that entering the first external buffer, which might not be measurable in a real system. It is assumed that the service rate can be changed online, by altering the reserved CPU bandwidth p. The parameter w is the average time that the jobs would require to be served, if only one job were processed at a time. The response time, from the requests arrive at the first queue to the requests finish at the processor, is modeled according to Little s law. The response time is measured as an average over the sample interval, which imposes low filtering, which is modeled as a continuous first order filter, such that d(t) = d(t) + x(t) T d T d λ(t) where T d is the filter constant, which is chosen equal to the sampling interval. The internal average number of jobs is denoted x i. A relation between the internal queue length and external buffer lengths is derived in [], which is repeated here ( ( ) ) M x(t) ( ) x i(t) = x(t) =: f i x(t) (3) + x(t) The internal arrival rate λ i is the flow of requests entering the web server from the external buffers. It is measured simply by counting the number of requests arriving at the web server over a time interval. A model for the internal arrival rate was presented in [], based on an integrator model: () () λ i(t) = λ(t) d (x(t) xi(t)) (4) dt The internal average response time d i is regarded as the time it takes from a requests enters the web server (from In the sequel, subscript i indicates that the variable appears inside the server. λ p β Entire queue γ s α Internal measurements γ γ f s s T d+ γ i γ i λ d x s T d+ λ i d i x i Fig.. Block diagram of the server with outputs both from the full queue, and from the web server. the external buffers) until it has been fully processed. This variable is often measurable in a real system. Following the derivation of the model for the response time of the entire queue (d), the sampling is modeled as a low pass filter, and using Little s law, the model becomes d i (t) A. Linearization = T d d i (t) + T d x i (t) λ i (t) The result of the linearization of the model is illustrated as a block diagram in Fig. with the parameters α = p w (5) (x +), β = x w(x +) (6) γ = γ i = λ, γ = x (λ ), γ i = x i (λ ) (7) γ f = x + x ( ) x M ( x ) M ( M x ) M x + x + x + x + (8) The figure shows that the full queuing system can be represented by a simple first order system from the control input p to the output d. Also, the disturbance λ affects the system in a fairly simple manner. It is observed that if the external buffers really can be neglected (γ f =), the measured quantities match those of the whole queue (λ i (t) = λ(t) and d i (t) = d(t)). The model clearly reveals that the measurement of the disturbance λ i now becomes state dependent, which later will turn out to have some important effects on the control design. A. Testbed III. TESTBED AND SYSTEM IDENTIFICATION In order to evaluate the model towards a real system, a testbed was developed. The server computer was a Pentium 4, GB memory, 3 GHz PC, with a Linux Fedora 8 operating system and a modified kernel Virtualization was achieved with Control Group functionality [5]. Also, an Apache server, version..8, configured by using theprefork module, was installed on the server computer, [6]. client computers, Athlon,.5 GHz PC with GB memory, Linux Fedora 9 and kernel , generated traffic with the CRIS tool [7], which is a java program developed on data obtained from real life applications (from 48

3 log files of a news site). The client computers were connected to the server by an Mb Ethernet switch. A master computer was connected through a local Ethernet network for administration tasks. The Apache web server was chosen mainly because it is one of the most used web servers on the Internet. A detailed description of the Apache architecture and model structure is found in e.g. [8], [9]. The Linux.6 kernel provides functionality to group different processes and perform scheduler specific operations on group basis. The project is called Control Group, and is accessed through a virtual file system [5], []. According to measurements, not presented here, the access is done on the scale of. ms, and the scheduler can be considered as true processor sharing and without dynamics down to a time resolution of around ms. More details on the testbed are presented in [], []. B. Parameter Estimation and Model Validation The nonlinear model was implemented in Simulink R for Matlab R. The model consists of the full queuing system (() and ()), the internal measurements ((3), (4), and (5)), along with a first order filter to filter the arrival rate: λ f (t) = T λ λ f (t) + T λ λ i(t) (9) where T λ is the filter constant, chosen to T λ = s. The model has only two parameters; the average time for service of a single job, w, and the limit of the internal queue, M. The parameter w is usually not set explicitly, so it has to be estimated. The parameter M is often set explicitly by the system administrator ( e.g., MaxClients in Apache). To excite the system for good identification results, we limit the internal queue (M=8), forcing the external buffers to hold more jobs. In practice, this short internal queue is not realistic, but it is chosen to demonstrate the effect of the external buffers, which could become significant, even under normal configuration, but under high load. The estimation of w was done by steady state measurements, and were presented in []. It revealed a reasonable value of =5. ms. The dynamical model is validated against experiment data. In order to show the dynamic influence of the external buffers, the control signal illustrated in the bottom of Fig. 3 was applied in an experiment. The arrival rate was kept constant at 5.3 requests/second throughout the entire experiment. The results of the experiment are shown in Figs. 3 and 4 along with the corresponding simulation results. The top of Fig. 3 shows the behavior of the internal queue. When the control signal is decreased and the server is overloaded, the queue builds up, but because it is limited to 8, it does not grow that much. The top of Fig. 4 shows the effect of the queue building up; the response times become larger, but still, the full effect is not seen, since the response times over the full queue cannot be measured. The bottom of the figure shows some interesting effects of the external buffers. When the control signal is decreased, the service rate becomes smaller than the arrival rate. As the internal buffer is often full, the rate of the jobs entering the web server is limited by p (%) xi Internal Queue Length experiment simulation CPU Bandwidth Reserved to the Web Server Fig. 3. Validation of the model versus experimental data. The lower part of the figure shows the control signal p, and the top of the figure shows the behavior of the internal queue length. di λf Internal Response Time Filtered Internal Arrival Rate experiment simulation Fig. 4. Validation of the model versus experimental data. The upper part of the figure shows the internal response time, and the lower part of the figure shows the filtered internal arrival rate. the service rate, and therefore, the measured internal arrival rate decreases. When the control signal is increased, the flow into the web server will still be limited by the service rate, until the external buffers have been emptied. At this time, the service rate is around 87 req/s, which is clearly observed as the top peak in the bottom of Fig. 4. When the queue has settled at the new operating point, the arrival rate measured inside the web server matches that of the offered. Note that the arrival rate entering the full queuing system remains constant under the whole experiment. The match between the model and the measurements is acceptable. The problems of matching the right levels of response times and internal queue length observed in the steady state experiments remains, but the dynamics of the system is matched quite well by the model. Especially the filtered internal arrival rate matches well. IV. CONTROL DESIGN (NEGLECT. EXTERNAL BUFFERS) If the system is operated with medium or light load, the external buffers can be neglected, simplifying the control design significantly. The question is then how the control system behaves when operating under higher loads, where the external buffers become significant, but where the control system does not encounter the effect of the extra queuing. That is the scope of the following. 49

4 P β Λ s + α P ff P fb T λ s + -K (T i s+) T i s X i γ i X Λ i D i γ f s γ i T d s + Feed forward controller Feedback controller Fig. 5. Block diagram of the server with feedback and feed forward control. Note that the feed forward controller actually imposes a feedback mechanism. A control strategy based on both feedback from the output d, and feed forward from the disturbance λ is considered, where it is assumed that the external buffers can be neglected. The response time is chosen as main metric, since it is important from a client perspective. An analysis then follows, where the control structure and parameters remains the same, but the system now includes the external buffers. Applying a PI controller as feedback ensures that the reference d r is reached in steady state. The PI controller is defined as P fb (s) = D r K (Tis + ) T is Λ ( ) D r(s) D(s) () according to normal practice []. The first minus sign is included to compensate for the negative gain from the control signal to the output of the server model. Several recent publications propose to measure the disturbance λ and to utilized it for feed forward compensation/control, and thereby reduce the undesired effects of changes in the work load. The term feed forward is used to stress that the only measurement used for this controller is an input i.e., no outputs are used. Later, the measurement used for the feed forward controller will be exchanged with an internal measurement of the arrival rate which is an output of the process. The feed forward mechanism will then actually become a feedback mechanism, but it will still be denoted as feed forward to state that it was originally intended as a feed forward mechanism. Because the measurement of the disturbance can be quite noisy, it is filtered with the first order low pass filter from (9). The feed forward controller signal p ff (t) = d r + λ f (t) () is based on queuing theoretical relationships found in [] V. ANALYSIS INCLUDING THE EXTERNAL BUFFERS Consider the case where the control design is conducted as described above (assuming that the external buffers can be neglected), but the external buffers are now significant. This has the consequence that the arrival rate λ, which before was treated as a disturbance, now becomes a dynamical state (λ i ), which introduces an extra feedback loop, as illustrated in Fig. 5. TABLE I OPERATING POINTS Variable p λ µ x x i d d i Unit % req./sec req./sec req. req. sec. sec. Value Imaginary part of s Root locus for the server when w is changed = 4.6 ms = 4.6 ms Real part of s Describing function Ψ Describing Function Amplitude A M = 4 M = 5 M = 56 M = 6 M = Fig. 6. Stability analysis. Left: Root locus for the server when used in the feed forward controller is changed from ms to ms. A fixed PI controller is also imposed. Right: Describing function for x = 7, for different values of M. The expressions for stability analysis becomes too complicated to draw any general conclusions if no parameters are fixed. Therefore, it is chosen to evaluate the stability for testbed system. The parameters and operating point is chosen according to the following arguments. The model parameters are chosen according to the system identified in Section III-B: M=8 and =.54 sec. The operation point is chosen as acceptable response time for the end user. Since the full response time is not measured, it is chosen as reference for the internal response time. That is, d i =. s. Also, a constant arrival rate of λ =5.3 reg/s is chosen, since this was the arrival rate used for the parameter estimation for of the model. The remaining operating point parameters are listed in Table I. The feedback control parameters are chosen to T i =. and K=.. These values are chosen to give large robustness margins (infinite amplitude margin and a phase margin of almost 9 ) when neglecting the external buffers. When taking the zero induced by the external buffers into account, the phase margin is reduced to 7 while the amplitude margin becomes.8 db, which are still on the conservative side. The cross over frequency was.38 rad/s and. rad/s, neglecting and including the external buffers, respectively. The chosen operating point represents a high load situation. Figure 6 shows the root locus when all parameter but in the feed forward controller part are kept constant. The analysis suggests that this system is unstable for > 4. ms, which is about three times higher than the actual value. Several nonlinearities are present in the server system. The nonlinear model prevents the queue from holding negative jobs (x < ) but this is not captured by the linear model. The second nonlinearity is an actuator saturation, since CPU bandwidth is limited from both below and from above. These nonlinearities are not relevant at the investigated operating points, and therefore are not treated any further. The third nonlinearity is the static function of (3), relating the internal queue length to the full queue length, which holds for x. In the case where x <, the function must be limited from below, such that x i. Furthermore, the function is 5

5 U Cut X i γ i s γ i Λ i f i ( ) Static nonlinearity K (T i s+) (T λ s + )T i s T λ s + Linear system G(s) P ff P fb P β s + α X Y Cut Fig. 7. Block diagram of the server with feedback and the feed forward control. The system is divided into a linear part and a nonlinear part, interacting through the signals U(s) and Y (s). Region of interest Im Re text A Region of Interest G(S) -/Ψ(A) /Ψ(A = 7.4) = /Ψ(A = 5.4) = G(.i) = text Fig. 8. Nyquist diagram for the controlled system for =6 ms, along with a plot of the negative inverse describing function ( /Ψ(A)). altered to operate on the linearized variables: { ( ) fi x + x x f i for x x i( x) := x i for x < x () Instability can expose itself in two different ways; either the system goes to an extreme configuration, determined by the limitations of the system (such as saturated actuator, saturated states, etc.), and remains there until external events (such as user interaction) occur, or the system starts to oscillate. Methods to predict the type of instability exist, and one such method is the describing function method. The notation presented here follows that of [3], but also [4] gives a comprehensive treatment. The describing function Ψ is not evaluated analytically due to the complicated integrals, but is computed numerically and presented in Fig. 6 for different values of M. Cuts are made in the block diagram on each side of the nonlinearity, and the two variables U(s) and Y (s) are inserted at the places of cutting. The whole block diagram is rearranged so these variables becomes input and output of a linear system G(s), as shown in Fig. 7. For =5 ms the transfer function becomes G(s) =.88s s +.47s +.38 (3) s s 3.44s +.8s which is an unstable system with two unstable poles, one stable pole, and one pole in the origin. The right hand side of the Laplace plane in encircled by a closed curve S. The origin is not included in the encapsulation, as is the normal procedure when evaluating Nyquist diagrams. The transfer function G(s) is evaluated along the curve S, and mapped into a new complex plane to draw the Nyquist diagram, which is illustrated in Fig. 8. A detailed magnification of the region of interest is shown in the right hand side along with the negative inverse describing function. It shows that the two curves intersect twice at A = 7.4 and A = TABLE II DEVIATION FROM THE OPERATING POINT. Parameter λ f d i x x i Operating point 5.3 req/s. s req 5.3 req Upper deviation. req/s.45 s 38 req 7. req Lower deviation.59 req/s.8 s 34 req.7 req 5.4 (G(i) = /Ψ(A) = 3.686). The frequency of the transfer function at the intersection is =. rad/s. Each intersection is a possible limit cycle, which are now investigated separately: A=7.4 : Assume that the system is oscillating with an amplitude slightly above 7.4. The negative inverse describing function will then have moved slightly towards the origin, and this point is encircled clockwise twice by the Nyquist curve. Following the procedure of [3], the limit cycle is stable because the open loop system G(s) has two poles in the right hand side of the Laplace plane. A=5.4: Assume that the system is oscillating with an amplitude slightly above 5.4. The negative inverse describing function will then have moved slightly towards, and this point is not encircled by the Nyquist curve. Following the procedure of [3], the limit cycle is thus unstable because the open loop system G(s) has two poles in the right hand side of the Laplace plane. From the arguments above, the system is expected to have a stable limit cycle with oscillations in x of 7.4 req in amplitude and with period time of approximately π/.=6. s A. Validation by Simulation VI. VALIDATION The simulation model developed for validation was expanded to include both the feed forward controller and the feedback controller. The focus here is the stability, so all simulations were initiated at the operating point, but a small deviation in the control signal in the beginning of the simulation ensured that instability revealed itself. The operating point was as described in Table I. The system remained stable and behaved well for low values of, but entered stable limit cycles for higher values. The limit cycles were characterized by period times of approximately 65 s. All variables oscillated, but not symmetrically around the operating point value. Table II lists the deviations from the operating points for the most important variables. The stability analysis suggested that the limit for stability is 4.6 ms, which corresponds well with the simulation results. The predicted limit cycle period was 6. s, which corresponds well with the period time obtained by the simulations. B. Experimental Validation The controllers were implemented on the testbed described in Section III-A. The condition of all the experiments were as described in Table I. A saturation was imposed to limit the control signal p between. and.89. (% of the CPU bandwidth was reserved to the operating system and other 5

6 Int. response time di (s) Int. arrival rate λi (req/s) = 5. ms = 6. ms = 5. ms = 6. ms = 8. ms = 8. ms Fig. 9. Experimental results showing the internal measured response time d i and arrival rate λ i when utilizing feedback and feed forward control. The estimate used in the feed forward controller is varied from 5 to 8 ms. tasks, and the web server was guaranteed at least % of the CPU bandwidth to ensure resources for measurements). Figure 9 illustrates the results of three experiments. Note that the initial conditions for the experiments were not the same, so only the steady state conditions can be compared. The figures indicate that the system remained stable and well behaved for low values of, but certain signs of instability were observed already at =6 ms. Stable oscillations were observed at =8 ms. Here, the experimental results deviate from the analysis, which predicted instability for > 4.6 ms. VII. CONCLUSIONS This paper has considered the problem where requests from a client pass through an unknown amount of buffers before reaching a web server, and the variables of these buffers were unmeasurable. Based on modeling presented in [] the analysis has been expanded to include both a feed forward mechanism and a feedback mechanism. The investigations have reveled an interesting property. The (internal) measurement of the arrival rate becomes state dependent. This alters the properties of the controlled system significantly, as the feed forward controller turns into a feedback mechanism. The paper has demonstrated the danger of neglecting the buffering during the control design process. The investigations has showed that the system becomes unstable if the gain of the feed forward controller was chosen too high, and that the instability manifested as oscillations. The stability conclusions were supported by theoretical analysis, simulations, and experiments, which all showed correspondence in the qualitative conclusions. The experimentally found stability limit deviated from the theoretically predicted stability limit, but this comes not as a surprise, as the model showed less accurate prediction of the response time (which the feedback was based upon) than for the arrival rate (which the feed forward controller was based upon). VIII. ACKNOWLEDGMENTS This work has been founded by the Swedish Research Council, under project and the Lund Center for Control of Complex Engineering Systems (LCCC). REFERENCES [] Y. Diao, C. Wu, J. Hellerstein, A. Storm, M. Surendra, S. Lightstone, S. Parekh, C. Garcia-Arellano, M. Carroll, L. Chu, and J. Colaco, Comparative studies of load balancing with control and optimization techniques, in American Control Conference, 5. Proc. of the 5, Portland, Oregon, June 5, pp [] Y. Fu, H. Wang, C. Lu, and R. 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