JCS&T Vol. 9 No. 2 October 2009 Reduced Computational Cost in the Calculation of Worst Case Response Time for Real Time Systems José M. Urriza, Lucas Schorb Javier D. Orozco, Ricardo Cayssials Facultad de Ingeniería- Departamento de Informática Departamento de Ingeniería Eléctrica y Computadoras Universidad Nacional de la Patagonia San Juan Bosco Universidad Nacional del Sur - CONICET Puerto Madryn, Argentina Bahía Blanca, Argentina e-mail: [email protected] e-mail: [email protected] Abstract— Modern Real Time Operating Systems require methods, which have been developed, have higher CC in reducing computational costs even though the microprocessors comparison with the one of Liu & Layland. While this become more powerful each day. It is usual that Real Time statement is true, it also shows a fertile field to explore the Operating Systems for embedded systems have advance discipline in order to obtain the analytical techniques with features to administrate the resources of the applications that low CC, even when they are higher than the bound they support. In order to guarantee either the schedulability of the system or the schedulability of a new task in a dynamic established by Liu & Layland. This increase is rewarded by a Real Time System, it is necessary to know the Worst Case better utilization of resources or by a better treatment of Response Time of the Real Time tasks during runtime. In this systems with more complex requirements paper a reduced computational cost algorithm is proposed to Below is a brief introduction to the RTS. determine the Worst Case Response Time of Real Time tasks. A. Introduction to the Real Time Systems Keywords- Schedulability; Response Time Analysis; RM; DM In the classical definition (Stankovic’s [3]): RTS are those in which results must not be only correct from an I. INTRODUCTION arithmetic-logical point of view but also produced before a certain instant, called deadline. Nowadays there exist several Real Time Operating Depending on the deadlines of the tasks, the RTS can be Systems (RTOS). Among the best known we can name: classified into three types. The first one does not allow any ChibiOS/RT, eCos, FreeRTOS, Fusion RTOS, Nucleus task to lose its deadline, so they’re called hard or critical. RTOS, OSE, OSEK, QNX, RT-11, RTEMS, RTLinux, The second allows to miss some deadlines, so they are called Talon DSP RTOS, Transaction Processing Facility, and soft. Finally, the further dissemination of RTS has required VxWorks, among others. typify those that allow only a certain amount of loss under a The most popular, probably, is that developed by Wind specified statistical criterion. These are called firm. River Systems, VxWorks, for its use in: The Spirit and In the hard RTS, to lose the deadline of a task can have Opportunity Mars Exploration Rovers, The Mars severe consequences for the system integrity and, probably, Reconnaissance Orbiter, The Deep Impact space probe, for the environment, in the case that the system strongly Stardust (spacecraft), The Boeing 787 airliner, The Boeing interacts with it: avionics, robotics, etc. In order to guarantee 747-8 airliner, The BMW iDrive system, Linksys the accomplishment of the tasks before their deadline it is WRT54Gxx wireless routers, to name some of the devices necessary to make an analysis of the schedulability of the that have the RTOS of this company. Its main features are, its system. If the test is successful, it is said that the system is simplicity and that can guarantee that a hard Real Time schedulable and ensures the accomplishment of all its System (RTS) can be scheduled by the discipline of fixed temporal constraints. priorities, Rate Monotonic (RM). In [1], the schedulability of mono-resource and In VxWorks, in order to ensure the schedulability of all multitasking systems is considered. A priority discipline tasks, the RTOS limits the maximum utilization factor establishes a linear order on the set of tasks, allowing the admissible for the system, through an upper bound presented scheduler to define at each instant of activation, which task in 1973 by Liu & Layland in [1]. Its main advantage is to will use the shared resource. have a negligible Computational Cost (CC), in exchange for To formalize the analysis of schedulability, it is providing pessimistic results which imply an over- necessary to model the set of tasks based on their temporal dimensioning of the system. requirements and their interdependencies. The election of this method by Wind River Systems ([2]) Usually it is considered that the tasks are periodic, is probably due to their low CC, or perhaps because the new independent and appropriable. A periodic task is one that 72 JCS&T Vol. 9 No. 2 October 2009 after a certain time requests execution. The task is said to be sufficient condition that guarantee the feasibility of the independent if it doesn’t need the result of the execution of system. some other task for its own execution. By definition, a FP of a function f is a number t, such Finally, it is said that the task is appropriable when the that tft= (). As in this case the FP equation is a function of scheduler can suspend its execution and withdraw it from the time. A t point and a t instant, are equivalent expressions. resource at any time. The method of FP for an RTS, was first developed by Generally, the parameters of each task, under this Joseph and Pandya ([12]). In this method, it is proved that framework, are: its execution time, which is noted as Ci, its there isn’t an analytical construction to resolve such period; noted Ti, and its deadline Di. So a set of n tasks S(n) problems and it is only possible to calculate it by iterative is specified by S(n)={(C1, T1, D1), (C2, T2, D2),..., (Cn, Tn, computations. Dn)}. Joseph’s method is initialized in the critical instant In [1] it was proved that the worst generation scheme, for where all the tasks are simultaneously invoked. As shown in a mono-resource system, is one in which all the tasks request [12], the result is the WCRT of task i, of a subset of tasks to be executed in the same instant and it is called the critical S(i). Below is the same equation with only a change in instant. It was showed also that if this state is schedulable, nomenclature and the addition of a superscript (q) to indicate the RTS is schedulable for any other state, under the priority in which iteration is: discipline used. The utilization factor (U) of a set of hard tasks S(n) i−1 q q+1 ⎡⎤t determines the level of utilization of the resource. tC=+ij∑ ⎢⎥ C (1) In [1], the schedulability of the RTS is guaranteed by j=1 ⎢⎥Tj calculating an upper bound, whose CC is of the order of the number of tasks, because it is needed to calculate the U. If The solution, if it exists, is only valid when there is a FP qq+1 the U is lower than or equal to the level above, it ensures the ( ttii= ) and if the response time to attend the task i, is q schedulability of the RTS. lower than or equal to its deadline ( tDi ≤ i ). Numerous schedulability tests have been developed since 1973 with the same technique ([4, 5, 6, 7, 8, 9].). In 1982, A. Iterative algorithm Leung ([10]), defined Deadline Monotonic (DM), but a Following is the analysis of Joseph’s iterative method, in schedulability test wasn’t defined. In 1991 Audsley ([11]) order to propose from this analysis, an algorithm to obtain presented a solution for this problem. In 1986, Joseph and the FP with a lower CC. Pandya ([12]) presented an iterative method of Fixed Point Suppose a schedulable RTS, S(n) = {(C1, T1, D1), (C2, T2, (FP) to evaluate a necessary and sufficient condition to D2),..., (Cn, Tn, Dn)}. validate the feasibility of an RTS, using a RM scheduler. When applying the method of [12] to a task i, with Several works have been published with equivalent solutions iin ∈ 1<≤ , m +1 iterations will occur before reaching a FP in [13, 14, 15, 16]. In 1998, Sjödin ([17]) incorporated an with m ≥ 0 . Below is a generic trace: improvement to Joseph’s test. Which is to begin the iteration of the task i+1 at the moment where the method of Joseph Iteration 0: found the worst case response time (WCRT) of task i plus 0 m the execution time of task i +1 ( ttCiii+1 =+). 00 0 ⎡⎤tt ⎡⎤ ⎡⎤ t 1 In 2004, Bini ([18]) proposed a new method called ⎢⎥CC12+ ⎢⎥ ++... ⎢⎥ CCtii− 1 + = ⎢⎥TT12 ⎢⎥ ⎢⎥ Ti− 1 Hyperplanes Exact Test (HET) to determine the schedulability of an RTS. Iteration 1: The paper is organized as follows: Section 2 describes the FP iterative methods used to determine the schedulability ⎡⎤tt11 ⎡⎤ ⎡⎤ t 1 of an RTS. Furthermore it proves theorems to improve the 2 ⎢⎥CC12+ ⎢⎥ ++... ⎢⎥ CCtii− 1 + = search of the mentioned FPs. In section 3, an example is ⎢⎥TT12 ⎢⎥ ⎢⎥ Ti− 1 presented. In section 4, the CC of the proposed method with the CC of the algorithms presented in Real Time literature are compared. The analysis results are presented in Section .......................................... 5. In Section 6, the conclusion and further works are presented Iteration m-1: II. FIXED POINT ITERATIVE METHODS mm−−11 m − 1 ⎡⎤tt ⎡⎤ ⎡⎤ t m ⎢⎥CC12+ ⎢⎥++... ⎢⎥ CCtii− 1 + = Since 1986, most of the schedulability tests developed for ⎢⎥TT12 ⎢⎥ ⎢⎥ Ti− 1 the disciplines of fixed priorities RM or DM, are based on applying a FP method for ensuring the necessary and 73 JCS&T Vol.