Schedulability tests for tasks with Variable Rate-dependent Behaviour under fixed priority scheduling

Abstract

Abstract-Automotive embedded real-time systems such as Engine Management utilise cyclic tasks that are activated periodically based on angular rotation rather than time. As well as having variable inter-arrival times, these tasks also have deadlines and worst-case execution times that are dependent on angular velocity i.e. engine speed or rpm. Such tasks exhibit Variable Rate-dependent Behaviour (VRB). In this paper, we introduce response time analysis for systems comprising VRB and sporadic tasks under fixed priority scheduling. Sufficient schedulability tests are introduced; from simple linear upper bounds on interference, to a more complex analysis using information about the physical limitations of the system to provide constraints for an ILP formulation of the problem.

Parametric utilization bounds for implicit-deadline periodic tasks in automotive systems

“…In the literature, such tasks are also called variable-rate-behaviour tasks [11] or multi-mode tasks [16]. Schedulability tests of such angle-synchronous tasks under FP scheduling have been proposed in [11,16].…”

“…There are two existing methods to calculate the interference due to an angle-synchronous task in an interval length ∆. One is to nd the worst-case workload by investigating the worst-case release patterns using integer linear programming (ILP) [11] or dynamic programming [16]. Another, as shown in the following lemma, is to safely approximate the interference due to an anglesynchronous task τ i by using…”

“…This is based on Theorem 1 from Davis et al [11] and Lemma 2 from Huang and Chen [16]. Without loss of generality, let the interval start from 0.…”

“…Without loss of generality, let the interval start from 0. Theorem 1 in [11] shows that the maximum interference by an angle-synchronous task τ i to a lower-priority job arriving at time 0 happens in the following worst-case pattern: a) release the rst job at time 0, b) follow the minimum period needed in the particular execution mode, and c) release the last job with execution time C max i before ∆. We consider the two cases individually.…”

Performance-Driven Design of Engine Control Tasks

“…Pollex et al [14] derived a sufficient feasibility test for fixed priority scheduling, but assuming a constant engine speed. Davis et al [10] used an Integer Linear Programming (ILP) formulation to derive a sufficient schedulability test of AVR tasks under fixed-priorities, also taking acceleration into account, but considering a finite set of (discretized) initial engine speeds.…”