Real Time Systems Design And Analysis

Chapter 7.2.5 - Response-Time Analysis for Fixed-Period Systems

7.2.5   Response-Time Analysis for Fixed-Period Systems

In general, utilization-based tests are not exact and provide good estimates for a
very simplified task model. In this section, a necessary and sufficient condition
for schedulability based on worst-case response time calculation is presented.

For the highest-priority task, its worst-case response time evidently will be
equal to its own execution time. Other tasks running on the system are subjected
to interference caused by execution of higher-priority tasks. For a general task
τi, response time, Ri, is given as

 

where Iiis the maximum amount of delay in execution, caused by higher priority
tasks, that task τi is going to experience in any time interval [t, t + Ri). At a
critical instant Ii will be maximum, that is, the time at which all higher-priority
tasks are released along with task τi .

Consider a task τj of higher priority than τi.Within the interval [0, Ri), the time
of release of τj will be . Each release of task τj is going to contribute
to the amount of interference τi is going to face, and is expressed as:

 Maximum interference = ej(7.7)

Each task of higher priority is interfering with task τi. So,

 

where hp(i) is the set of higher-priority tasks with respect to τi. Substituting this
value in Ri = ei + Ii yields

 

Due to the ceiling functions, it is difficult to solve for Ri. Without getting into
details, a solution is provided where the function R is evaluated by rewriting it
as a recurrence relation

 

where Rin is the response in the nth iteration.

To use the recurrence relation to find response times, it is necessary to compute
Rin+1 iteratively until the first value m is found such that Rim+1 = Rim+1×Rim is then
the response time Ri. It is important to note that if the equation does not have
a solution, then the value of Ri will continue to rise, as in the case when a task
set has a utilization greater than 100%.

 

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