In this section, we will introduce the periodic and sporadic task parameters defini- tions in detail, as well as the definitions of concrete task set and abstract task set.
2.1 Traditional definition
Since a lot of deterministic and successful results have been obtained for a task set anyone of which has the deadline equalling to the period size, in what follows, the pa- rameter of deadline will be ignored since it equals to period. Therefore, a task is τi for- mally modelled by a pair (ci, pi) where:
• ci is the computational cost: the maximum amount of processor time required to exe- cute (the sequential program of) task τi on a dedicated unit-processor.
• pi is the period: the minimal interval between instances of task τi.
Throughout this paper, we assume time is discrete and clock ticks are indexed by the natural numbers. Task instances occur and task executions begin at the clock ticks;
each of the parameters ci and pi is expressed as a multiple of (the interval between) clock ticks. If a task τi with cost ci begins execution at time t and is executed without interrup- tion on a uniprocessor, then the execution is completed at time t+ci. We consider two
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paradigms of task instance: periodic and sporadic. If τi is periodic, the period pi specifies a constant interval between instances. If τi is sporadic, pi specifies a minimum interval between instances. The definition of the behaviour of a task depends on whether it is pe- riodic or sporadic. The behaviour of a periodic task τi =(ci, pi) is given by the following rules for the instance and execution of τi. If ti,k is the time of the kth instance of task τi, then:
• The (k+1)th instance of taskτi will occur at time ti,k+1 = ti,k+pi..
• The kth execution of taskτi must begin no earlier than ti,k and be completed no later than the deadlines of ti,k+pi. This requires that ci units of processor time be allo- cated to executions of τi in the interval [ti,k, ti,k+ pi]
The behaviour of a sporadic task is slightly less constrained than that of a periodic task. The behaviour of a sporadic taskτi = (ci, pi) is given by the following rules for the instance and execution of τi. If ti,k is the time of the kth instance of taskτi, then:
• The (k+1)th instance of τi will occur no earlier than time ti,k+pi; thus ti,k+1 ≥ ti,k+pi.
• The kth execution of taskτi must begin no earlier than ti,k and be completed no later than the deadline of ti,k+pi.
Thus the behaviours of periodic and sporadic tasks differ only in the first rule. We assume instances of sporadic tasks are independent in the sense that the time a sporadic task is invoked depends only upon the time of its last instance and not upon the instance times of any other task. Note that the worst-case behaviour of a sporadic taskτi=(ci, pi) (“worst” in the sense of requiring the most processor time), occurs when τi behaves like a periodic task, that is, τi is invoked every pi time steps.
2.2 Concrete and Abstract Task set
The difficulty of scheduling periodic tasks can be affected by the times that tasks are first invoked [Jeffay91]. In addition, we are interested in the predictability of sched- ulability which can service the network resource dimensioning or evaluate the QoS be- fore the transmission. Therefore, the study concerns the task set with any release time, and concrete and abstract task set definitions should be introduced [Jeffay91].
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A concrete task is a pair (τi, ri), where τi is a task, and ri is a non-negative integer that is the time of the first instance, or the release time, of τi. The behaviour of (τi ,ri) is the behavior of τi constrained further by the rule that the first instance of τi occurs at time ri. Once released, tasks are invoked repeatedly forever.
A set of periodic (sporadic) tasks Γ = {τ1,τ2, …,τn } is a set of tasks indexed from 1 to n, where for each i, 1≤ i ≤ n, τi = (ci, pi). A concrete set of periodic (sporadic) tasks ω = {(τ1, r1), (τ2, r2) ..., (τn, rn)} is a set of concrete tasks indexed from 1 to n, where ri is the release time of taskτi. There is a natural many-to-one relation between concrete tasks and (non-concrete) tasks. We say the taskτi generates a concrete task (τi, ri) and a con- crete task (τi, ri) is generated from the taskτi. This relation extends naturally to a relation between concrete task sets and abstract (non concrete) task sets. Let Γ= {τ1,τ2, …,τn } be a task set and let ω={(τ1, r1), (τ2, r2) ..., (τn, rn)} be a concrete task set, then the task set Γ generates the concrete task set ω and ω is generated from Γ.
Note that a task set is schedulable if and only if the tasks can be scheduled for any set of release times. In contrast, each member of a concrete task set has a specified re- lease time, and showing that a concrete task set is schedulable only establishes that its specified release times can be accommodated. Earliest Deadline (EDF) First scheduling algorithm is considered as a successful algorithm since it is optimal for either preemptive or non-preemptive task set. A schedulability of a non-preemptive general task set can be decided by the following theorem given in [Jeffay91].
Theorem 1: [Jeffay91] Let Γ = {τ1,τ2, …,τn } be a set of sporadic or periodic takss sorted in non-decreasing order by periods (i.e., for any pair of tasks τi and τj, if i > j, then pi≤
pj). If Γ is schedulable then
(1) 1
1
∑
≤= n
i i
i
p c
(2) ∀i, 1< i≤ n; ∀L, p1 < L < pi:
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1
1 i 1
i j
j j
c L c L
p
−
=
−
+ ≤
∑
And if Γ satisfies conditions (1) and (2) then the non-preemptive EDF scheduling algo- rithm will schedule any concrete set of periodic or sporadic tasks generated from Γ.
In the article [George00], a necessary and sufficient condition is introduced for a concrete periodic task set for non-idling, non-preemptive scheduling policy, which is given in the form of the following theorem:
Theorem 2 [George00]: Let Γ be a concrete periodic task set (defined as τi =(ri , ci ,di , pi) for i=1...n with 0< ci≤ di ≤ pi ) Γ is feasible on one processor if and only if
(i)
1
1
n i
i i
c
= p
∑
≤(ii) a schedule exists where all deadlines in the interval [0, r+2P] are met for all the tasks in the periodic task set.
where P = least common multiple of {p1, ..., pn } the periods of a task set Γ, r = max{r1 ,...,rn }
In what follows, we will introduce the fault-tolerant real-time system under (m,k)- firm constraint. The research interest is to guarantee the level of fault-tolerance in condi- tion of overload, and the performance evaluation or resource allocation strategies will be focused on.