Response time analysis in distributed real-time systems: An
improvement based on best-case finalization time analysis
Citation for published version (APA):Bril, R. J., Cucu-Grosjean, L., & Goossens, J. (2009). Response time analysis in distributed real-time systems: An improvement based on best-case finalization time analysis. In J. Blazewicz, M. Drozdowski, G. Kendall, & B. McCollum (Eds.), Proceedings 4th Multidisciplinary International Scheduling Conference (MISTA 2009, Dublin, Ireland, August 10-12, 2009) (pp. 1-4)
Document status and date: Published: 01/01/2009
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
providing details and we will investigate your claim.
MISTA 2009
Response time analysis in distributed real-time systems
An improvement based on best-case finalization time analysis
R.J. Bril · L. Cucu-Grosjean · J. Goossens
1 Introduction
Existing end-to-end response time analysis in distributed real-time systems [2], where the finalization of one task on a processor activates another task on another processor, is pes-simistic. By “pessimistic” we mean that not all systems deemed to be unschedulable by the analysis are in fact unschedulable. This pessimism has two causes: (i) the existing analysis is based on best-case response times rather than best-case finalization times and (ii) those best-case response times are based on analysis for (worst-case) deadlines at most equal to periods minus (absolute) activation jitter [1]. In this paper, we present analytical means to determine best-case finalization times of independent real-time tasks with deadlines larger than periods minus activation jitter under uniprocessor fixed-priority preemptive scheduling (FPPS) and arbitrary phasing, allowing an improvement of the existing analysis. We will illustrate the improvement by means of an example.
We assume a single processor and a setT of n periodically released, independent tasks
τ1, τ2, . . . , τnwith unique, fixed priorities. At any moment in time, the processor executes the
highest priority task that has work pending, i.e. tasks are scheduled using FPPS.
Each task τigenerates an infinite sequence of jobs ιik with k ∈Z. The inter-activation
times of τiare characterized by a (fixed) period Ti∈R+and an (absolute) activation jitter
AJi∈R+∪ {0}, where AJi< Ti. Moreover, τiis characterized by a (fixed) computation time
Ci∈R+, a phasing ϕi∈R, a (relative) worst-case deadline WDi∈R+, and a (relative)
best-case deadline BDi∈R+∪ {0}, where BDi≤ WDi. The set of phasings ϕiis termed the
phasing ϕ of the task setT. We assume that we have no control over the phasing ϕ, so any
arbitrary phasing may occur. The deadlines BDiand WDiare relative to the activations.
Reinder J. Bril
Technische Universiteit Eindhoven (TU/e), Mathematics and Computer Science Department, Den Dolech 2, 5600 AZ Eindhoven, The Netherlands.
E-mail: R.J.Bril@tue.nl Liliana Cucu-Grosjean
INRIA Nancy-Grand Est, TRIO team, 615 rue du Jardin Botanique, Villers les Nancy, 54600, France. E-mail: Liliana.Cucu@loria.fr
Jo¨el Goossens
Universit´e Libre de Bruxelles (ULB), Computer Science Department, Boulevard du Triomphe - C.P.212, 1050 Brussels, Belgium.
Note that the activations of τi do not necessarily take place strictly periodically with
period Ti, but somewhere in an interval of length AJithat is repeated with period Ti. The
activation times aikof τisatisfy supk,`(aik(ϕi) − ai`(ϕi) − (k − `)Ti) ≤ AJi, where ϕidenotes
the start of the interval in which job zero is activated, i.e. ϕi+ kTi≤ aik≤ ϕi+ kTi+ AJi. A
task with activation jitter equal to zero is termed a strictly periodic task.
task τi time Ti WDi aik fik Rik wdik ai,k+1 preemptions by higher priority tasks execution release
(absolute) worst-case deadline Legend:
W
(absolute) best-case deadline
B W B BDi bdik AJi
(absolute) activation jitter
ϕi+kTi ϕi+(k+1)Ti
Fik
Fig. 1 Basic model for a periodic task τiwith (absolute) activation jitter AJi.
The (relative) finalization time Fikof job ιikis defined relative to the start of the interval in
which ιikis activated, i.e. Fik= fik− (ϕi+ kTi). The active interval of job ιikis defined as the
time span between the activation time of that job and its finalization time, i.e. [aik, fik). The
response time Rikof job ιikis defined as the length of its active interval, i.e. Rik= fik− aik.
Figure 1 illustrates the above basic notions for an example job of a periodic task τi. Whereas
Fik= Rikholds for a strictly periodic task τi, the following relation holds in general
Fik≥ Rik. (1)
For notational convenience, we assume that the tasks are given in order of decreasing
prior-ity, i.e. task τ1has highest priority and task τnhas lowest priority.
2 Existing results
The worst-case response time WRi and the best-case response time BRi of a task τi are
the largest and the smallest (relative) response time of any of its jobs, respectively, i.e.
WRidef= supϕ,kRik(ϕ) and BRidef= infϕ,kRik(ϕ). For worst-case deadlines at most equal to
pe-riods minus activation jitter, i.e. WDi≤ Ti− AJi, BRiis given by the largest x ∈R+that
satisfies x = Ci+
∑
j<i µ» x − AJj Tj ¼ − 1 ¶+ Cj. (2)Here, the notation w+ stands for max(w, 0), which is used to indicate that the number of
preemptions of tasks with a higher priority than τican not become negative. To calculate
BRi, we can use an iterative procedure based on recurrence relationships, starting with an
upper bound, e.g. WRi.
The worst-case finalization time WFiand the best-case finalization time BFiof a task τi
are the largest and the smallest (relative) finalization time of any of its jobs, respectively, i.e.
WFidef= supϕ,kFik(ϕ) and BFidef= infϕ,kFik(ϕ). The worst-case (absolute) finalization jitter FJi
of task τiis the largest difference between the finalization times of any two of its jobs, i.e.
FJi
def
= sup
ϕ,k,`
Finalization jitter analysis presented in [2] is based on FJi≤ WFi− BRi, where BRiis
deter-mined using Equation (2). 3 Contributions
From Equation (3) we derive FJi≤ WFi− BFi. The finalization jitter analysis presented
in [2] is therefore pessimistic for two reasons; firstly BFi≥ BRifor AJi> 0; see Equation
(1) and secondly BRias determined by Equation (2) is pessimistic for worst-case deadlines
larger than periods minus activation jitter [1]. We will illustrate this by means of an example and subsequently present a conjecture for best-case finalization time analysis.
Table 1 presents the characteristics of our example task setT1consisting of three tasks,
and Figure 2 shows a time-line forT1with BF3= 3 and BR3= 2.4 of task τ3, hence BF3>
BR3. Using Equation (2) yields a value BR3= 2, which is pessimistic, i.e. too small. A
conjecture for exact best-case response time analysis of tasks with arbitrary deadlines that are scheduled using FPPS has been presented in [4].
task T C AJ BF BR
τ1 4 2 0 2 2
τ2 5 1 0 1 1
τ3 7 2 0.6 3 2.4
Table 1 Task characteristics ofT1and values for best-case finalization times and response times.
-10 0 -15 -5 task τ1 task τ2 time task τ3 F3,-2 = 5 F3,-1 = 5 F3,0 = 3
Fig. 2 Time-line forT1with a best-case finalization time BF3= 3 and a best-case response time BR3= 2.4 for job ι3,0of task τ3.
Conjecture 1 The best-case finalization time BFiof task τiwith Ti− AJi< WDiis given by
BFi= max
0≤k<w`i (BR0
i((k + 1)Ci) − kTi) . (4)
where w`iis the worst-case number of jobs of τiin a level-i active period1, and BR0i((k +
1)Ci) is the best-case response time of a task τ0iwith a computation time Ci0= (k + 1)Ci, a
period equal to its worst-case deadline, i.e. T0
i = WD0i, a worst-case deadline WD0igiven by
WD0i= WDi+
½
0 k = 0
kTi− AJiotherwise, (5)
1 A level-i active period is the longest interval in which the sum of pending loads is higher than 0 for tasks with a priority equal to or higher than the priority of task τi; see [3]. The length of the longest level-i active
period is finite for all 1 ≤ i ≤ n when either (i) the utilization factor UT = ∑
1≤ j≤n
Cj
Tj is smaller than 1 or when (ii) UT is equal to 1, the activation jitter of all tasks ofT are equal to zero, and the least common
and a best-case deadline BDiequal to its computation time, i.e. BDi= (k + 1)Ci. ¤
Based on Conjecture 1, we find w`3= 3 and BF3= max(2, 9 − 7, 17 − 14) = 3.
Note that for w`i= 1, (4) becomes equal to the solution of (2). Hence, the conjecture
therefore applies for tasks with arbitrary deadlines.
References
1. O. Redell and M. Sandfridson, Exact best-case response time analysis of fixed priority scheduled tasks, Proc. 14thEuromicro Conference on Real-Time Systems, 165-172 (2002)
2. R. Henia, R. Racu and R. Ernst, Improved Output Jitter Calculation for Compositional Performance Anal-ysis of Distributed Systems, Proc. IEEE Int. Parallel and Distributed Processing Symp., 1-8 (2007) 3. R.J. Bril, J.J. Lukkien and W.F.J. Verhaegh, Worst-case response time analysis of real-time tasks
un-der fixed-priority scheduling with deferred preemption, Accepted for Real-Time Systems Journal (2009), http://www.springerlink.com/content/f05r404j63424h27/fulltext.pdf (online).
4. R.J. Bril, L. Cucu-Grosjean and J. Goossens, Exact best-case response time analysis of real-time tasks under fixed-priority pre-emptive scheduling for arbitrary deadlines Accepted for Work-in-Progress session of the 21stEuromicro Conference on Real-Time Systems, (2009).
