Generalized Offset based Response Time Analysis Deepak Vedha Raj Sudhakar INCHRON GmbH Karl-Liebknecht-Strasse, 138 Potsdam, Germany deepak.sudhakar@inchron.de Karsten Albers INCHRON GmbH Karl-Liebknecht-Strasse, 138 Potsdam, Germany karsten.albers@inchron.de Frank Slomka Institute of Embedded Systems /Real-Time Systems Ulm University, Ulm, Germany frank.slomka@uni- ulm.de ABSTRACT In real-time theory, there exist two approaches for comput- ing the response time of tasks: the classical response time analysis (RTA) approach and the modular performance anal- ysis with the Real-Time Calculus (MPA-RTC). The classical RTA techniques are based on the busy window methods and the response time equation needs to be adapted every time a new scheduling policy or a new event activation pattern is considered in the system model. MPA-RTC has its roots in Network Calculus (NC). MPA-RTC offers more powerful abstraction and flexibility than the RTA based techniques, because of its ability to model resource availability and arbi- trary incoming activations with the help of expressive service curves and arrival curves respectively. MPA-RTC models scheduling policy with the help of service curves and can handle hierarchical scheduling better than RTA based tech- niques [1]. In this paper, we propose a generalized approach to model offset dependencies using MPA-RTC based tech- nique to compute tight worst-case response times of tasks. 1. INTRODUCTION Offset dependencies between tasks/messages ensure that there exists a fixed time interval between the activation of a set of tasks/ messages, thereby avoiding simultaneous ac- tivations. Assuming tasks to be independent in the analysis would lead to pessimistic worst-case response times calcula- tion. Therefore, it is necessary to include time offsets into the computational model and to extend the analysis to take account of the time offsets. The classical RTA techniques use the notion of the busy window period introduced by Lehoczky [2]. The level-i busy period is defined as the maximum time interval for which a processor is busy executing tasks of priority higher than or equal to priority of task i. The longest response time of the job of task i occurs during a level-i busy period initi- ated at a critical instant when the task is requested simul- taneously with requests for all higher priority tasks. For systems with task offsets where tasks are not released si- multaneously, identifying the critical instant is not straight- forward. It leads to a combinatorial problem where several busy window periods have to be explored to identify the critical instant of the task. There exist several works [3], [4], [5], [6] that use the busy window technique to perform offset based response time analysis for a specific scheduling policy. All the above RTA based techniques use iterative fixed-point recurrence equation to compute response times and are limited to standard activation patterns. They do not support analysis of complex activation patterns such as sporadically periodic event streams with bursts and arbi- trary event arrival patterns from an event trace. It can be avoided by performing event model conversion to sup- ported event models. However, lossy conversion can lead to pessimistic response time estimates. Modular performance analysis with the Real-Time Calculus (MPA-RTC) [1] per- forms response time analysis with the help of arrival and service curves and is based on the Network Calculus (NC) [7]. Arrival curves model the resource demand and service curves model the computational resource available in any time interval. The worst-case response time of a task is ob- tained by calculating the maximum delay (maximum hor- izontal distance) between the upper arrival curve and the lower service curve. There exist works [8], [9], [10], [11] that model offset dependencies between packet flows in a switched Ethernet, Time-Triggered Ethernet and CAN net- work using Network calculus. All the above works compute aggregate cumulative flows by considering offset dependen- cies between flows and propagate them in a network to com- pute end to end delays of packets. However, in this paper, we extend the work to real-time systems by modelling sys- tems with tasks and offset dependencies and compute the worst case response times of tasks by combining the busy window methods and the MPA-RTC. Our focus in this pa- per is to compute the worst-case response time (WCRT) of tasks with offset dependencies modeled as a greedy process- ing component (GPC) in an MPA-RTC framework and not outgoing curves and end to end delays. The paper is orga- nized as follows: we highlight the limitations of the existing system model of an MPA-RTC framework in handling off- set dependencies in Section 2. In Section 3, we extend the system model of an MPA-RTC framework to consider offset dependencies. In Section 4, we present the response time analysis of the extended system model. The paper is con- cluded in Section 5. 2. MOTIVATION MPA-RTC models event streams that trigger the system using arrival curves. It models the communication and com- putation resources that are available to the system using service curves and model tasks in the system as a GPC. A system model is then obtained by interconnecting the arrival and service inputs and outputs of all the models based on the resource sharing schemes (scheduling policies) and data flow among the GPCs. The arrival and service curves are defined in [1]. The WCRT (r + i ) of a task τi is computed as the maximum horizontal distance between the incoming