Response Time versus Utilization in Scheduler Overhead Accounting

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1 Response Time versus Utiization in Scheduer Overhead Accounting Siviu S. Craciunas Christoph M. Kirsch Ana Sokoova Department of Computer Sciences University of Sazburg, Austria Emai: Abstract We propose two compementary methods to account for scheduer overhead in the scheduabiity anaysis of Variabe Bandwidth Servers (VBS), which contro process execution speed by aocating variabe CPU bandwidth to processes. Scheduer overhead in VBS may be accounted for either by decreasing process execution speed to maintain CPU utiization (caed response accounting), or by increasing CPU utiization to maintain process execution speed (caed utiization accounting). Both methods can be combined by handing an arbitrary fraction of the tota scheduer overhead with one method and the rest with the other. Distinguishing scheduer overhead due to reeasing and due to suspending processes aows us to further improve our anaysis by accounting for reeasing overhead in a separate, virtua VBS process. Athough our anaysis is based on the VBS mode, the genera idea of response and utiization accounting may aso be appied to other, reated scheduing methods. I. INTRODUCTION We study scheduer overhead accounting in the scheduabiity anaysis of Variabe Bandwidth Servers (VBS) [1], [2]. VBS are a generaization of Constant Bandwidth Servers (CBS) [3]. A CBS aocates a constant fraction of CPU time to a process (the server bandwidth) at a constant granuarity (the server period). Mutipe CBS processes are EDF-schedued using the server periods as deadines. A VBS is simiar to a CBS but aso aows the process to change its execution speed at any time, i.e., change both server bandwidth and server period, as ong as the resuting CPU utiization remains under a given bandwidth cap. The portion of process code from a change in speed to the next is caed an action. A VBS process is therefore a sequence of actions. The tota time to execute an action is expicity modeed and caed the response time of the action. The key resut of VBS is that, for each action of a VBS process, there exist ower and upper bounds on response times and thus aso on jitter that are independent of any other concurrenty executing VBS processes, as ong as system utiization (the sum of a bandwidth caps) is ess than or equa to 100% [1], [2] (Section III). This paper generaizes the VBS scheduing resut by incuding the overhead of scheduer execution. We first determine an upper bound on the number of scheduer invocations that may occur during a given amount of time (Section IV). This is possibe because process reease and suspend times are known in VBS. We then show that there are two compementary methods to account for scheduer overhead, either by Supported by the EU ArtistDesign Network of Exceence on Embedded Systems Design and the Austrian Science Funds P18913-N15 and V decreasing the speed at which processes run to maintain CPU utiization (caed response accounting), or by increasing CPU utiization to maintain the speed at which processes run (caed utiization accounting). Response accounting decreases the net server bandwidth avaiabe to a process by dedicating some of its bandwidth to the scheduer. Utiization accounting increases the server bandwidth to maintain the net bandwidth avaiabe to a process. In other words, with utiization accounting, the bounds on response times are maintained whie CPU utiization is increased whereas, with response accounting, the upper bounds on response times and thus on jitter are increased whie utiization is maintained. Both methods can be combined by handing an arbitrary fraction of the tota scheduer overhead with one method and the rest with the other. We aso show, by exampe, that the fraction may be chosen within server- and process-dependent intervas such that CPU utiization decreases whie the response-time bounds and thus the speed at which processes run remain the same (Section V). Next, we observe that there is a natura division of the VBS scheduer overhead into overhead due to reeasing and due to suspending processes. Moreover, we note that our previousy mentioned upper bound on the number of scheduer invocations can be improved for the scheduer invocations due to reeasing processes by accounting for them in a separate, virtua VBS process instead of the given VBS processes (Section VI). The virtua process is then accounted for in increased CPU utiization. The remaining overhead due to suspending processes may then be accounted for in either more increased CPU utiization, or increased response-time bounds and thus decreased speed at which processes run. The former method is pure utiization accounting, the atter method is combined accounting. Pure response accounting does not appy here. Up unti this point we have assumed that there is a constant upper bound on the execution time of any scheduer invocation. The VBS scheduing agorithm itsef is indeed constanttime. Even queue management in VBS can be done in constant time. However, our VBS impementation features a pugin architecture for queue management so that other pugins with non-constant-time compexity (but ess space compexity) can be used. In addition to the constant-time pugin, there are aso a ist-based, quadratic-time and an array-based, inear-time pugin (in the number of processes) [1], [2]. Our method can readiy be generaized to account for non-constant scheduer overhead.

2 We present the resuts of severa experiments that measure the accuracy of the estimated number of scheduer invocations (Section VII) and concude the paper (Section VIII). Note that, athough our anaysis is based on the VBS mode, the genera idea of response and utiization accounting may aso be appied to CBS as we as to RBED [4], which is another rate-based scheduer cosey reated to VBS. Reated work is discussed next in more detai. II. RELATED WORK We first put Variabe Bandwidth Servers (VBS) [1], [2] in the context of earier work on scheduing. We then identify exampes of response and utiization accounting in reated work on scheduer overhead. We aso acknowedge previous work on a wider range of issues deaing with genera system overhead. The Generaized Processor Sharing (GPS) approach introduced an ideaized mode of resources for fair scheduing based on the assumption that resource capacities are infinitey divisibe [5]. Proportiona-share agorithms (PSA) such as [6] approximate GPS using quantum-based CPU scheduing techniques. Constant Bandwidth Servers (CBS) [3] are reated to PSA [7] but aocate a constant fraction of CPU time to each process (server bandwidth) at a constant granuarity (server period). VBS is a generaization of CBS that aows processes to change their servers bandwidth and period (aso caed virtua periodic resource, or resource reservation [8]) at runtime as ong as the CPU capacity is not exceeded. VBS is cosey reated to RBED, which is a rate-based scheduer extending resource reservations for hard rea-time, soft rea-time, and best effort processes [4]. Like VBS, RBED uses EDF scheduing and aows dynamic bandwidth and rate adjustments. Whie the capabiities are simiar, RBED and VBS differ in the eve of abstraction they provide. In VBS we mode processes as sequences of actions to quantify the response times of portions of process code, where each transition from one action to the next marks an adjustment in bandwidth and rate. Other scheduing techniques reated to VBS incude eastic scheduing [9], handing of quaity of service [10] and overoad scenarios [11], and runtime server reconfiguration in CBS using benefit functions [12]. Next, we identify exampes of response and utiization accounting in reated work deaing with scheduer overhead. In [13], given a set of dynamicay schedued periodic interrupts and tasks, interrupt hander overhead is accounted for in the processor demand of the tasks (reated to utiization accounting). In [14], given a set of periodic tasks, so-caed expicit overhead through system cas invoked by task code is accounted for in the worst-case execution times of tasks (response accounting) and so-caed impicit overhead through scheduer invocations and interrupts is accounted for in CPU utiization (utiization accounting). In [15], given a set of periodic tasks, interrupt and fixed-priority scheduing overhead is accounted for in the response times of the tasks (response accounting). In [16], given a CBS system, scheduer overhead due to suspending processes is accounted for in response-time bounds of so-caed jobs (response accounting). The responsetime bounds are tighter than ours expoiting the fact that server parameters cannot be changed and scheduer invocations due to reeasing processes are not considered. Note that our contribution is not ony scheduer overhead accounting for VBS, but aso recognizing conceptua principes and trade-offs that have ead us to a unifying approach. In a wider context, exampes of previous work deaing with genera system overhead are an anaysis of event- and time-driven impementations of fixed-priority scheduing [17], agorithms to compute the number of preemptions in sets of periodic, DM- and EDF-schedued tasks [18], and a study of RM and EDF scheduing that incudes a comparison of context-switching overhead [19]. There is aso work on reducing context-switching overhead through a modified fixedpriority scheduer [20] and on reducing the number of preemptions in fixed-priority scheduing without modifying the scheduer [21]. The effects of cache-reated preemption deays on the execution time of processes are anayzed in [22] and [23]. III. VBS SCHEDULING We have introduced VBS scheduing in previous work [1], [2]. Here, we briefy reca the necessary definitions and resuts. A variabe bandwidth server (VBS) is an extension of a constant bandwidth server (CBS) where throughput and atency of process execution can vary in time under certain restrictions. Given a virtua periodic resource [24], defined as a pair of a period and a imit (with bandwidth equa to the ratio of imit over period), a CBS executes a singe process no more than the amount of time given by the resource imit in every time interva given by the resource period. A VBS may vary the virtua periodic resource (change the resource periods and imits), as ong as the bandwidth does not exceed a predefined bandwidth cap. A process running on a VBS can initiate a resource switch at any time. The process code from one switch to the next is caed a process action. The execution of a process is thus potentiay an unbounded sequence of actions. In practice each action can be seen as a piece of sequentia code that has a virtua resource associated with it. We impemented a VBS-based scheduing agorithm [1], [2], with four aternative queue management pugins based on ists, arrays, matrices, and trees. The pugins aow trading-off time and space compexity. A. Process Mode A process P (u) corresponding to a VBS with utiization u is a finite or infinite sequence of actions, P (u) α 0 α 1 α 2... for α i Act, where Act N R, with R being the finite set of virtua periodic resources [24], expained beow. An action α Act is a pair α (, R) where (for oad) is a natura number which denotes the exact amount of time the process wi perform the action on the virtua periodic resource R. The oad of an action can be understood as the worst-case execution time of the piece of code that constitutes

3 eary strategy ate strategy a i r i a i r c i i f i c i fi 24 ms ms Fig. 1. Scheduing an action α i with oad i 5 using both reease strategies, where i 2 and π i 4 the action. The virtua periodic resource R is a pair R (, π) of natura numbers with π, where denotes the imit and π the period of the resource. The imit specifies the maximum amount of time the process P can execute on the virtua periodic resource within the period π, whie performing the action α. The utiization of R is u R π u. B. Scheduabiity anaysis without overhead Let P be a finite set of processes. A schedue for P is a partia function σ : N P from the time domain to the set of processes. The function σ assigns to each moment in time a process that is running in the time interva [t, t + 1). If no process runs in the interva [t, t + 1) then σ(t) is undefined. Any scheduer σ of VBS processes determines a unique function σ R : N P R that specifies which virtua periodic resource is used by the running process. We are ony interested in we-behaved schedues with the property that for any process P P, any resource R R with R (, π), and any natura number k N {t [kπ, (k + 1)π) σ R (t) (P, R)}. Hence, in such a we-behaved schedue, the process P uses the resource R at most units of time per period of time π. Our scheduing agorithm produces we-behaved schedues. For each process action α i we denote the foowing absoute moments in time, which are aso depicted in Figure 1: Arriva time a i of the action α i is the time instant at which the previous action of the same process has finished. The first action of a process has zero arriva time. Competion time c i of the action α i is the time at which the action competes its execution. It is cacuated as c i min {c N i {t [a i, c) σ(t) P } }. Termination or finishing time f i of the action α i is the time at which the action terminates or finishes its execution, f i c i. We adopt the foowing termination strategy: The termination time is at the end of the period within which the action has competed. Reease time r i is the eariest time when α i can be schedued, r i a i. As shown in Figure 1, we consider two reease strategies. In the eary reease strategy r i a i, i.e., the process is reeased immediatey upon arriva with a fraction of its imit computed for the remaining interva unti the period ends. The ate reease strategy deays the reease of an action unti the beginning of the next period. The eary reease strategy may improve average response times. The schedued response time s i of the action α i under the scheduer σ is the difference between the finishing time and the arriva time, i.e., s i f i a i. We digress at this point for a brief eaboration on the type of tasks that are generated by VBS processes, which are used to prove the VBS scheduabiity resut. Let τ (r, e, d) be an aperiodic task with reease time r, execution duration e, and deadine d. We say that τ has type (, π) where and π are natura numbers, π, if the foowing conditions hod: d (n + 1)π for a natura number n such that r [nπ, (n + 1)π), and e (d r) π. The task type (, π) represents a virtua periodic task which we use in order to impose a bound on the aperiodic task τ. If the reease of τ is at time nπ, the execution duration e is imited by. Otherwise, if the reease is not at an instance of the period π, the execution duration is adjusted so that the task τ has utiization factor at most π over the interva [r, d]. Let S be a finite set of task types. Let I be a finite index set, and consider a set of tasks with the properties: {τ (r, e, d ) i I, j 0} Each τ has a type in S. We wi write (, ) for the type of τ. The tasks with the same first index are reeased in a sequence, i.e., r +1 d and r i,0 0. We refer to such a set as a typed set of tasks. Lemma 1 ([2]): Let {τ i I, j 0} be a typed set of tasks. If i I max j 0 1, then this set of tasks is scheduabe using the EDF strategy at any point of time, so that each task meets its deadine. The proof of Lemma 1 foows the standard periodic EDF proof of sufficiency with the addition that the periods of the tasks may change in time. The fu proof can be found in our previous work on VBS [2]. The same resut in a different formuation has aso been shown in [4]. Using the definition of VBS processes and actions we can give both response-time bounds for an action and a constanttime scheduabiity test for an idea system, i.e., a system without overhead. Let P {P i (u i ) 1 i n} be a finite set of n processes with corresponding actions α (, R ) for j 0. Each P i (u i ) α i,0 α i,1... corresponds to a VBS with utiization u i. Let R (, ) be the virtua periodic resource associated with the action α with,, and being the oad, the imit, and the period for the action α. The upper bound for the response time of action α is b u 1 +.

4 The ower response-time bound varies depending on the strategy used, namey, for ate reease b, for eary reease. Note that the ower bound in the eary reease strategy is achieved ony if divides, in which case. From these bounds we can derive that the responsetime jitter, i.e., the difference between the upper and ower bound on the response time, is at most 1 for the ate reease strategy and at most 2 1 for the eary reease strategy. It is possibe to give more precise bounds for the eary strategy (using (2) and (3) beow) and show that aso in that case the jitter is at most 1. We say that the set of VBS processes is scheduabe with respect to the response-time bounds b u and b if there exists a we-behaved schedue σ such that for every action α the schedued response time s satisfies b s b u. It is important to note that each action α of a process produces a sequence of tasks with type (, ) that are reeased at period instances of the virtua periodic resource used by the action. Scheduing VBS processes therefore amounts to scheduing typed sets of tasks. We ca the reease of such a task an activation of the action. Proposition 1 ([1], [2]): Given a set of VBS processes P {P i (u i ) 1 i n}, if i I u i 1, then the set of processes P is scheduabe with respect to the responsetime bounds b u and b. Proof: As mentioned above, each process P i provides a typed set of tasks since each action α of the process is spit into severa tasks that a have the type (, ). The tasks are reeased in a sequence, i.e. the reease time of the next task is aways greater than or equa to the deadine of the current task (aso due to the termination strategy). We can thus appy Lemma 1 and concude that the set of tasks is scheduabe. To check the upper bound, we distinguish a case for the eary reease strategy and one for the ate reease strategy in terms of the first task of an action. The detais can be found in [2]. We present the proof for the ower bounds which is not discussed in [2]. For each action α, according to the termination strategy and the ate reease strategy, we have f r + (1) where the reease times are given by r n j for some natura number n j such that the arriva times are a f 1 ((n j 1), n j ]. Therefore, for the ate strategy we have (1) s f a + r a b, for ate reease. For the eary reease strategy we distinguish two cases depending on whether the foowing inequaity hods m (2) where m n j r and r a f 1 ((n j 1), n j ]. The finishing time for the action α is f In both cases f a + + m, if (2) hods a + + m, otherwise + a, so s f a b, for eary reease. In the foowing sections we anayze VBS scheduabiity in the presence of scheduer overhead. In particuar, we define an upper bound on the number of scheduer invocations that occur within a period of an action, and perform an anaysis of the changes in the response-time bounds and utiization due to the scheduer overhead. IV. VBS SCHEDULING WITH OVERHEAD Our first aim is to bound the number of scheduer invocations over a time interva. In particuar, we need the worstcase number of preemptions, and hence scheduer invocations, that an action of a VBS process experiences during a period. Hence, by overhead we mean the overhead due to scheduer invocations (and not interrupts or other types of system overhead). Typicay the duration of a scheduer invocation is severa orders of magnitude ower than a unit execution of an action. Therefore, we assume that a periods beong to the set of discrete time instants M {c n n 0} N, for a constant vaue c N, c > 1. Hence, for any action α with its associated virtua periodic resource R (, ) we have that c with π N. We ca c the scae of the system. Intuitivey we can say that there are two different timeines, the fine-grained timeine given by the set of natura numbers and the coarse-grained timeine given by the set M. Resource periods are defined on the coarse-grained timeine, whie the execution time of the scheduer is defined on the fine-grained timeine. In VBS scheduing, a process P i is preempted at a time instant t if and ony if one of the foowing situations occurs: 1) Competion. P i has competed the entire work reated to its current action α (, R ). 2) Limit. P i uses up a resource imit of the current resource R. 3) Reease. A task of an action is reeased at time t, i.e., an action of another process is activated. Note that a preemptions due to reease occur at time instants on the coarse-grained timeine, the set M. (3)

5 P1 P2 P3 R L L R L R R,L C R,C Fig. 2. Exampe schedue for processes P 1, P 2, and P 3 Exampe 1: Consider the first action of three processes P 1, P 2, and P 3, with corresponding virtua periodic resources R 1,0 ( 1,0, π 1,0 ) (10, 40), R 2,0 ( 2,0, π 2,0 ) (10, 60), and R 3,0 ( 3,0, π 3,0 ) (50, 100), and oads 30, 20, and 100, respectivey. Figure 2 shows an exampe schedue of the three actions up to time 120 and the time instants at which preemptions occur. Preemptions that occur because of the reease of an action instance (action activation) are abeed with R, preemptions that occur because an action has finished its imit are abeed with L, and preemptions that are due to an action finishing its entire oad have the abe C. At time 0, 40, 60, 80, and 100 preemptions occur due to an activation of one of the three actions, whie at time 10, 20, 50, and 80 preemptions occur because an action has used a its imit for the current period. At times 90 and 100 preemptions occur due to competion of an action. At certain time instants such as 80 and 100 there is more than one reason for preemption. In this exampe the scae c of the system is 10. Let us first consider preemptions due to reease, i.e., activation of an action. Each activation happens when a new period of the used virtua periodic resource starts. Hence, an action α with a virtua periodic resource (, ) wi ony be activated at time instants k with k N. Therefore, in the worst-case scenario, the preemptions caused by action activations of a processes in the system occur at a time instants in the set {k gcd({ i I, j 0}) k N}. Note that, because the periods of a actions in the system are on the coarse-grained timeine, gcd({ i I, j 0}) c. We compute the scheduer overhead for each action using an upper bound on the number of preemptions during one period of the respective action. Lemma 2: Let P {P i (u i ) 1 i n} be a set of VBS processes with actions α and corresponding virtua periodic resources (, ). There are at most N N R + N L scheduer invocations every period for the action α, where N R (4) gcd({π m,n m I, n 0, m i}) and N L 1. Proof: There are scheduer invocations due to reease at most at every k gcd({π m,n m I, n 0}), for k N. An action α beonging to a process P i (u) can ony be preempted by task reeases of actions not beonging to P i (u). Hence, there can be at most N R preemptions due to reease over a period of the considered action. By design of the scheduing agorithm, there is exacty one more scheduer invocation due to the action using a its imit or competing its entire oad in each period, i.e., N L 1. This is a pessimistic approximation which coud be improved by running a runtime anaysis that considers ony the periods of the active actions at a specific time. Nevertheess, we take the pessimistic approach in order to be abe to anayze the system off-ine. Another bound on the number of preemptions due to reease for sets of periodic tasks has been given in [14]. For an action α, it cacuates the number of period instances from other actions that occur during the period. The worst-case number of scheduer invocations for a period of an action due to reease is thus π. (5) k I k i >0 π k, Depending on the periods in the system one of the two given bounds approximates the worst-case number better than the other. Consider for exampe a case where the periods in the system are 3, 5, and 7. In this case (5) approximates the number of preemptions better than (4). In another exampe, if the periods in the system are 2, 4, and 6, then (4) provides a more accurate bound than (5). We choose to consider (4) for a bound on the number of preemptions due to reease, for reasons that wi be expained in Section VI. V. SCHEDULABILITY ANALYSIS WITH OVERHEAD In a rea system the scheduer overhead manifests itsef as additiona oad which must be accounted for within the execution of a process in order not to invaidate the schedue of the system. Let ξ denote the duration of a singe scheduer invocation. The tota scheduer overhead for one period of action α is therefore δ N ξ. Hence, the tota overhead is made up of N pieces of ξ workoad. An important aspect in the anaysis is that a scheduer invocation with overhead ξ is nonpreemptabe. Note that N, and therefore δ, depends on the finitey many periods in the system and not on the oad of the action. As a resut, there are ony finitey many possibe vaues for δ even if there are infinitey many actions. Accounting for the overhead can be done in two ways. One way is to aow an action to execute for ess time than its actua imit within one period and use the remaining time to account for the scheduer overhead. The other way is to increase the imit such that the action wi be abe to execute its origina imit and the time spent on scheduer invocations within one period. Intuitivey, the first method increases the response-time bounds, and the second increases the utiization of an action. We recognize a fundamenta trade-off between an increase in response time versus utiization by distributing the amount of scheduer overhead. Namey, we write that the overhead is δ δ b + δ u, where δ b is the overhead that extends the response-time bounds of the respective action and δ u increases the utiization. Note that no scheduer invocation is divisibe, i.e., both δ b and δu are mutipes of ξ.

6 Case Overhead distribution Load Utiization Scheduabiity test m RA δ b δ, δ u 0 + δ δ u P i I π max j 0 1 m UA δ b 0, δu δ + δ ı ı RUA δ b, δu > 0 + δ b, + δ u δ b There are three cases: Response accounting (RA), δ δ b, when the entire overhead is executing within the imit of the action, keeping both the imit and period (and thus the utiization) of the actions constant but increasing the response-time bounds. Utiization accounting (UA), δ δ u, when the entire overhead increases the imit of the action, and thus the utiization, but the response-time bounds remain the same. Combined accounting (RUA), with δ δ b + δu, δ b > 0, and δu > 0, which offers the possibiity to trade-off utiization for response time, for each action, in the presence of scheduer overhead. For an action α, in the presence of overhead, we denote the new oad by, the new imit by, and the new utiization by u. Using these new parameters for an action we determine the new upper and ower response-time bounds which we denote with b u and b, respectivey. The upper response-time bound b u for action α is b u + 1. (6) TABLE I SCHEDULER OVERHEAD ACCOUNTING u + δ u + δ u P i I max j 0 P i I max j 0 + δ 1 + δ u 1 provide information on the actua number of scheduer invocations during one period. Hence, when determining the upper bound on response-time jitter, the assumption that an action aways experiences the worst-case number of preemptions is invaid. Therefore, the jitter is at most the difference between the new upper response-time bound b u and the idea ower response-time bound b, as shown in Figure 3. Tabe I summarizes the three cases which we discuss in detai in the remainder of this section. We eaborate on the change in response-time bounds, utiization, and jitter for each case. A. Response Accounting In the response accounting case each action executes for ess time than its actua imit so that enough time remains for the scheduer execution. As a resut, the action wi not execute more than its imit even when scheduer overhead is considered. In this way the utiization of the action remains the same but its response-time bounds increase. We have that δ δ b. R L R,L R L R R,L L R,L The ower response-time bound b for α using the ate reease strategy is b, (7) P1 P2 P whereas using the eary reease strategy is b. (8) In the previous section we have given an upper bound on the number of preemptions that an action wi experience during one period. This number is used to compute the scheduer overhead and hence the new response-time bounds for the respective action. As shown in Figure 3 and discussed in the remainder of this section, both the upper and the ower bounds increase if we assume that the worst-case number of preemptions actuay occurs. However, our anaysis does not Fig. 3. b b jitter b u b u Bounds and jitter with and without overhead response time Fig. 4. Exampe of response accounting of scheduer overhead Exampe 2: Consider the same processes as in Exampe 1. The schedue in Figure 4 shows how the scheduer overhead can be accounted for. At each scheduer invocation due to reease (abeed with R), the process that is schedued to run accounts for the overhead. Note that in our anaysis, unike in this exampe schedue, we account for an over-approximation of the number of scheduer invocations due to reease. In the case of preemptions due to imit or competion (abeed with L or C) the preempted process accounts for the overhead. This impies that in case of preemptions due to imit, the action has to be preempted before the imit is exhausted so that there is time for the scheduer to execute within the imit. The response-time bounds for each action differ from the ones given in Section III-B due to the fact that each action can execute ess of its own oad in the presence of scheduer overhead. We compute the new oad of the action α as + δ. δ

7 An obvious condition for the system to be feasibe is δ <. Intuitivey, if δ, the action α woud make no progress within one period as it wi just be executing the scheduer, and hence it wi never compete its oad. If δ >, given that the execution of the scheduer is nonpreemptabe, α wi exceed its imit, which may resut in process actions missing their deadines or processes not being scheduabe anymore. The new imit and utiization of the action are the same as without overhead, i.e. and u u. Since > and, we get that both the upper and the ower response-time bound increases in case of response accounting. Proposition 2: Let P {P i (u i ) 1 i n} be a set of VBS processes each with bandwidth cap u i. If i I u i 1 and δ <, with δ, as defined above, then the set of processes are scheduabe with respect to the new response-time bounds b u and b, in the presence of worstcase scheduer overhead. Proof: The proof foows from Proposition 1, and the discussion above, when substituting the oad with the new oad. The jitter for any action α in the response accounting case is at most b u b. For further reference, we write the new oad in the response accounting case as a function B. Utiization Accounting RA(,, δ) + δ δ. In the utiization accounting case an action is aowed to execute for more time than its origina imit within a period in order to account for scheduer overhead. Thus, we have that δ δ u. The new oad of action α becomes + δ. The new imit is + δ, and the new utiization is u + δ. Proposition 3: Given a set of processes P {P i (u i ) 1 i n}, et u + δ i max. j 0 If i I u i 1, then the set of processes P is scheduabe with respect to the origina response-time bounds b u and b defined in Section III-B, in the presence of worst-case scheduer overhead. Proof: We consider a modified set of processes P {Pi (u i ) 1 i n}. By Proposition 1, this set of processes is scheduabe with respect to the response-time bounds b u and b cacuated using the new oad and the new imit. We wi prove that the upper response-time bounds with the new oad and imit for action α origina upper bounds without overhead (b u bu are the same as the ), and the new ower bound is not ower than the od ower bound (b b ). We start by showing that. Let d R be the difference d. It suffices to estabish that d [0, 1) in order to prove our caim. We have, + δ d. + δ + δ Both If 0 and + δ > 0, so d 0., then d 0 and we are done. If, we can write + for 0 < <. Therefore, + δ < ( ), d < 1. + δ Hence we have estabished. We then show that. There are three cases to consider: 1) If does not divide, and does not divide, then. 2) If does not divide, but divides, then >. 3) The case when divides, but does not divide is not possibe, since if divides, then d 0 (see above) impying that divides. Hence, the set of processes is scheduabe with respect to the od upper bound and the new ower bound (which is greater than or equa to the od ower bound), which makes it scheduabe with respect to the od bounds. In the utiization accounting case, the jitter for any action is the same as in the anaysis without overhead, because the response-time bounds are the same. We write the new oad again as a function C. Combined Accounting UA(,, δ) + δ. In the combined accounting case, both the response-time bounds and the utiization of an action increase. We have that δ δ b + δu, δb > 0, and δu > 0. Given an action α with its associated virtua periodic resource R (, ), and oad, the new oad is computed in two steps. First we account for the overhead that increases the response time + δ b δ b

8 R L R,L R L R R L L R L 260 b b * b u b u* u u * ate strategy 50 response time [ms] jitter utiization (%) P P P PS δ b [µs] Fig. 5. Response time and utiization with δ b varying in [0µs, 100µs] for α (7300µs, (400µs, 1000µs)) using the ate strategy and then we add the overhead that increases the utiization + δ. u The oad function for the combined case is therefore RUA(,, δ b, δ u ) UA(RA(,, δ b ),, δ u ). The new imit for action α utiization becomes u + δ u. is + δ u, and the The upper response-time bound b u for action α is now RUA( b u,, δ b, δu ) + δ u + 1. The ower response-time bound b for the same action using the ate reease strategy is RUA( b,, δ b, δu ) + δ u, and using the eary reease strategy is RUA( b,, δ b, δu ) + δ u. Proposition 4: Given a set of processes P {P i (u i ) 1 i n}, et u + δ u i max. j 0 If i I u i 1, then the set of processes P is scheduabe with respect to the response-time bounds b u and b, in the presence of worst-case scheduer overhead. Proof: This scheduabiity resut is derived by combining Proposition 2 and Proposition 3 in the response accounting and utiization accounting case, respectivey. Figure 5 shows the effect of the scheduer overhead distribution on the response time and utiization for an exampe action. We consider the action α with imit 400µs, period π 1000µs, and oad 7300µs. The tota scheduer overhead of δ 100µs corresponds to 100 scheduer invocations, each with overhead ξ 1µs. If δ b 0 (utiization accounting), the ower and upper response-time bounds b and b u remain the same as the respective bounds b and b u without scheduer Fig. 6. Exampe of scheduer overhead as a separate, virtua VBS process P S for the combined accounting case overhead accounting but the utiization u is 10% higher than the utiization u without scheduer overhead accounting. If δ b δ (response accounting), we have that u u but b and b u are severa periods arger than b and b u, respectivey, aso resuting in a arger bound b u b on the response-time jitter. As δ b increases, u decreases whie b u increases aong with b and the jitter bound. Note that, depending on the invoved parameters, we can change the overhead distribution within some intervas such that the utiization decreases but the response-time bounds remain the same, for exampe, when δ b varies between 0µs and 16µs. VI. OPTIMIZATION There is a natura division of the VBS scheduer overhead into overhead due to reease δ R and overhead due to imit/- competion δ L. In Section IV we bounded the overhead due to reease for each period of an action as δ R N R ξ. In this estimate, each action accounts for the reease of each of the other actions in the system, or in other words, when an action is activated a other processes account for the overhead, athough at most one process is truy preempted. This is ceary an over-approximation. As aready discussed, a preemptions due to reease occur at time instants given by the set {k gcd({ i I, j 0}) k N}. Hence, instead of etting other processes account for the reease overhead, the overhead can be modeed as a separate, virtua VBS process with the same action repeated infinitey many times. We ca this process the scheduer process. Introducing a virtua process aows accounting for the overhead due to reease ony once, and not (as before) in every process. As a resut, the estimate on scheduer invocations may improve. A sufficient condition for improvement of the estimate is that there exists a process P m that accounts, in the origina estimate, for as many scheduer invocations as the scheduer process, i.e., gcd({ i I, j 0}) equas gcd({ i I, j 0, i m}). In addition to a set of VBS processes (as before) there is the scheduer process P S with a actions equa α S,j (ξ, R S ) where R S ( S, π S ) (ξ, gcd({ i I, j 0}). Thus, the utiization of the scheduer process is u S ξ gcd({ i I, j 0}). Note that the scheduer process accounts ony for the preemptions due to the reease of an action but not for the actions using a their imit or competing their oad.

9 Case Overhead distribution Load Utiization Scheduabiity test m RA δ b δl + δr, δu 0 + δ δ u P i I π max j 0 1 π m UA δ b 0, δu δl + δr + ξ u + ξ P i I π max + ξ j 0 RUA δ b δl, δu δr + ξ Exampe 3: Consider the same processes as in Exampe 1. The schedue in Figure 6 shows how the scheduer overhead resuting from reeases is integrated into the scheduer process. In cases where preemption can occur due to mutipe reasons, such as at time 20, we prioritize the reease and et the scheduer process run. Note that the non-optimized estimate on scheduer invocations for this exampe is arger than the optimized estimate, which happens to coincide with the actua number of scheduer invocations in this case. Tabe II summarizes the effect of the optimization to a cases. Note that the response accounting case cannot be optimized, since δ u 0 in this case. A. Combined accounting In the combined accounting case an optimization is possibe ony in the case δ u δr and δb δl ξ. The overhead due to imit/competion continues to add to the response time of each process and thus the new oad for action α is + ξ ξ and the utiization remains the same. The bounds b u m ξ TABLE II CASES WITH OPTIMIZATION and b are given by (6), (7) and (8), as before. The next resut presents an optimization to Proposition 4. Proposition 5: Given a set of processes P {P i (u i ) 1 i n}, if i I u i 1 u S, then the set of processes P is scheduabe with respect to the response-time bounds b u b and, in the presence of worst-case scheduer overhead. Proof: Let P {P i (u i ) 1 i n} be a set of VBS processes each with bandwidth cap u i. Let P S be the scheduer process as defined above. It is important to note that the scheduer process generates tasks that aways have the highest priority when reeased. This is because the action α S,k has the period π S gcd({ i I, j 0})), i I, j 0. Hence, at every invocation time of the action its deadine is earier or at the same time as any other deadine in the system and thus we can be sure it is never preempted by any other task using EDF. By Proposition 4 we get that the extended set of processes (together with the scheduer process) is scheduabe with respect to the new bounds if B. Utiization accounting u S + i I max j 0 1. In the case when the entire overhead increases the utiization of an action, optimization is possibe. For the preemptions due u P i I max j 0 1 u S 1 u S to imit or competion, the imit of the action α becomes + ξ and therefore the utiization is u + ξ. Note that it is not possibe to account for such preemptions in the scheduer process as that woud mean that the scheduer process coud execute at every time instant of the fine-grained timeine. The next resut presents the optimized version of Proposition 3. Proposition 6: Given a set of processes P {P i (u i ) 1 i n}, et u i max j 0 + ξ. If i I u i 1 u S, then the set of processes P is scheduabe with respect to the response-time bounds b u and b defined in Section III-B, in the presence of worst-case scheduer overhead. Proof: The proof is simiar to Proposition 3 and Proposition 5 with the difference that at any time the system consists of tasks with the type ( +ξ, ) generated by the processes in P and tasks generated by the scheduer process. The response-time bounds are proven to remain unchanged by substituting δ with ξ in Proposition 3. The set of processes is not scheduabe if ξ gcd({ i I, j 0}), since then the scheduer process has aready utiization at east 1. The utiization that can be used by processes may drop to 0 if ξ c. The system is thus better suited for vaues of c that respect the condition ξ c gcd({ i I, j 0}). VII. EXPERIMENTS We have conducted a series of experiments, using different simuated processes and actions, that show the accuracy of the estimate on scheduer invocations. First, we measure the number of scheduer invocations during a period of an action with a virtua periodic resource of (256, 512), averaged over the whoe execution of the action. We compare this to the estimate on scheduer invocations computed in Section IV. If the concurrenty running actions have periods that are harmonic to the period of the measured action, the estimate equas the actua number of scheduer invocations. We show the accuracy of the estimate for non-harmonic periods. The first experiment (Figure 7(a)) shows the estimate and the actua number of scheduer invocations (y-axis) when two other actions run concurrenty with the measured action. We vary the concurrent actions such that their periods resut in an increasing gcd (ogarithmic x-axis). The periods are chosen as mutipes of the gcd. As can be seen from the figure,

10 scheduer invocations estimate average for π 1 2 gcd, π 2 3 gcd average for π 1 3 gcd, π 2 5 gcd average for π 1 5 gcd, π 2 7 gcd average for π 1 7 gcd, π 2 11 gcd gcd (a) estimate vs. actua scheduer invocations π π (b) difference of estimate and actua scheduer invocations scheduer invocations non-optimized estimate optimized estimate actua scheduer invocations gcd (c) optimized vs. non-optimized estimate Fig. 7. Accuracy of the estimate on scheduer invocations the difference between the estimate and the actua number of scheduer invocations depends not ony on the gcd of the periods invoved but aso on their vaues, i.e., if smaer periods resut in the same gcd, then the approximation is better. The second experiment (Figure 7(b)) shows the difference between the estimate and the actua number of scheduer invocations (z-axis) when there are two actions running concurrenty with the measured action and the gcd of their periods is 1. We measure the difference for every combination of periods (ogarithmic x- and y-axis) that are chosen from a set of pairwise coprime numbers. This situation corresponds to the worst possibe scenario for the accuracy of the estimate. Again, it can be seen that smaer periods resut in a better approximation. In the third experiment (Figure 7(c)) we compare the goba optimized and non-optimized estimates, and the tota number of actua scheduer invocations measured for three concurrenty running actions over 100, 000 time units. The periods of the actions are chosen so that they resut in an increasing gcd (ogarithmic x-axis). The periods are actuay mutipes of the gcd by 2, 3, and 5. For sma vaues of the gcd the accuracy of the optimized estimate is consideraby better than the accuracy of the non-optimized estimate. For arge vaues of the gcd, the estimates converge. VIII. CONCLUSIONS We have introduced response and utiization accounting of scheduer overhead in the scheduabiity anaysis of VBS. 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