# Worst-case response time analysis of real-time tasks under fixed-priority scheduling with deferred preemption

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DOI: 10.1007/s11241-009-9071-z

- Cite this article as:
- Bril, R.J., Lukkien, J.J. & Verhaegh, W.F.J. Real-Time Syst (2009) 42: 63. doi:10.1007/s11241-009-9071-z

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## Abstract

Fixed-priority scheduling with deferred preemption (FPDS) has been proposed in the literature as a viable alternative to fixed-priority pre-emptive scheduling (FPPS), that obviates the need for non-trivial resource access protocols and reduces the cost of arbitrary preemptions.

This paper shows that existing worst-case response time analysis of hard real-time tasks under FPDS, arbitrary phasing and relative deadlines at most equal to periods is pessimistic and/or optimistic. The same problem also arises for fixed-priority non-pre-emptive scheduling (FPNS), being a special case of FPDS. This paper provides a revised analysis, resolving the problems with the existing approaches. The analysis is based on known concepts of critical instant and busy period for FPPS. To accommodate for our scheduling model for FPDS, we need to slightly modify existing definitions of these concepts. The analysis assumes a continuous scheduling model, which is based on a partitioning of the timeline in a set of non-empty, right semi-open intervals. It is shown that the critical instant, longest busy period, and worst-case response time for a task are suprema rather than maxima for all tasks, except for the lowest priority task. Hence, that instant, period, and response time cannot be assumed for any task, except for the lowest priority task. Moreover, it is shown that the analysis is not uniform for all tasks, i.e. the analysis for the lowest priority task differs from the analysis of the other tasks. These anomalies for the lowest priority task are an immediate consequence of the fact that only the lowest priority task cannot be blocked. To build on earlier work, the worst-case response time analysis for FPDS is expressed in terms of known worst-case analysis results for FPPS. The paper includes pessimistic variants of the analysis, which are uniform for all tasks, illustrates the revised analysis for an advanced model for FPDS, where tasks are structured as flow graphs of subjobs rather than sequences, and shows that our analysis is sustainable.

### Keywords

Level-*i*active period Level-

*i*busy period Worst-case response time Worst-case occupied time Periodic tasks Fixed-priority scheduling Deferred preemption Real-time systems