Task execution model

3.3 Preemptive Scheduler Synthesis Framework

3.3.1 Task execution model

3.3 Preemptive Scheduler Synthesis Framework

Table 3.3: Description of events (for preemptive execution)

Event Description

a_i Arrival of a task τ_i

p_i Acceptance of a task τ_i r_i Rejection of a taskτ_i

ei Execution of a segment of τi on the processor ci Execution of the last segment ofτi on the processor

event (t). All events in Σ are considered to be observable. SupposeEi is equal to 1, i.e., each instance of τ_i requires only a single segment of execution; then Σ_i ={a_i, p_i, r_i, c_i}.

In this work, the notion of acceptance captures the following fact: Once a real-time task is accepted by the scheduler, then it will not be rejected in the middle of its execution.

Similarly, rejection captures the scenario in which a real-time task is not accepted by the scheduler in order to guarantee timely execution of the already accepted tasks in the system. We have modeled the notion of acceptance and rejection using (mutually exclusive) transitions on events p_i and r_i, respectively. The transition structure of T_i (shown in Figure 3.12) is explained as follows:

T\Ti

ai p_i r_i

T\Ti T\Ti

ei t ci

t T\Ti

# of ei = Ei - 1 e_i

#1 #2 #3

#4 #6

#7

#8

#5

Figure 3.12: Task execution model T_i for task τ_i hE_i, D_i, P_ii.

• #1 −−−→^Σ\Σi #1: Ti stays at State #1 until the occurrence of arrival event ai by executing the events in Σ\Σ_i. Here, Σ\Σ_i contains the events such as arrival, rejection, acceptance, execution and completion of a task, that may occur with respect to other tasks in the system. In addition, Σ\Σ_i contains t which is used to model the arbitrary arrival of τ_i, i.e., a_i (no delay), ta_i (after a tick from the system start), tta_i (after two ticks) etc.

• #1 −^a→ⁱ #2: On the occurrence of a_i, T_i moves to State #2. Here, the scheduler takes the decision whether to accept (p_i) or reject (r_i).

• #2−→^ri #1: Suppose, the scheduler rejects τ_i, then T_i transits back to State #1.

• #2 −^p→ⁱ #3: Suppose, the scheduler accepts τ_i, then T_i transits to State #3. The self-loop transition Σ\Σ_i is similar to the one present inState #1.

• #3−→^ei #4: T_i moves from State #3 toState #4 through τ_i’s execution evente_i.

• #4−→^t #5: After the occurrence of tick (t),T reaches State#5 which is similar to State #3. It may be noted that eit represents execution of τi for one time unit.

• #5−→^ei #6. . .#7−→^ci #8: By continuing in this way,Ti reachesState#7 and moves to State #8 through the completion event c_i. Note that the number of e_i events executed between the acceptance (p_i) event and the completion (c_i) event isE_i−1.

• #8 −→^t #1: After the occurrence of a tick event, T_i moves back to the initial (as well as marked) state #1 to represent taskτ_i’s completion.

Remark 3.3.1. Generated and Marked Behavior of Ti: The generated behavior L(Ti) is the set of all strings that Ti (shown in Figure 3.12) can generate. The marked be- havior L_m(T_i) is the subset of all strings in L(T_i) for which the terminal state belongs to Q_m(= {#1}). The marked language may be described as: L_m(T_i) = (s_i0(a_i(r_i + p_is_i1e_its_i2e_it...s_iEic_it))^∗)^∗, where s_ij ∈(Σ\Σ_i)^∗ and j ={0,1, . . . , E_i}.

Definition: Deadline-meeting sequence (with respect to preemptive execution): A se- quencex=x₁a_ix₂c_ix₃ ∈L_m(T_i), wherex₁, x₃ ∈Σ^∗,x₂ ∈(Σ\ {c_i})^∗ isdeadline-meeting, if tickcount(x₂)≤ D_i−1. Otherwise,x is a deadline-missing sequence.

Theorem 3.3.1. L_m(T_i) contains both deadline-meeting and deadline-missing sequences of task τ_i.

Proof. Let us consider the execution of task τi on the processor. According to Re- mark 3.3.1, such execution may be represented by a sequence, x =a_ip_ix_ijc_i, where x_ij

= x_i1e_itx_i2x_it...x_iEi and x_ij ∈ (Σ\Σ_i)^∗ and j = {1,2, . . . , E_i}. It may be noted that x_ij represents the self-loop Σ\Σ_i in T_i and contains t in it. This models the waiting time of task τ_i in the ready queue before it is being assigned onto the processor for execution. Suppose, the task τ_i is not preempted at anytime after its acceptance to completion, then tickcount(x_ij) = E_i −1 (≤ D_i, Assumption 3). However, τ_i may be preempted by the scheduler during its execution. Let y be the total amount of time that task τ_i is kept in the ready queue between its acceptance to completion. If this

3.3 Preemptive Scheduler Synthesis Framework

preemption time (captured by y) is greater than (D_i−E_i) ticks from the arrival of τ_i, then τ_i is guaranteed to miss its deadline. Hence, the sequence x is deadline-meeting if (y+E_i −1) ≤ D_i. Otherwise, x is deadline-missing. Thus, L_m(T_i) contains both deadline-meeting and deadline-missing sequences of τ_i.

As discussed earlier, deadline-misses cannot be tolerated in hard real-time systems.

In later section, we develop a model that can be used to eliminate deadline-missing sequences from L_m(T_i).

From the perspective of real-time task execution, a sporadic sequence may be con- sidered as the most generic one and subsumes within it the definitions periodic and aperiodic sequences. Therefore, models specifying the behavior of periodic and aperi- odic sequences may be easily derived from the model representing a sporadic sequence.

The synthesis approach presented here attempts to derive scheduling sequences that satisfy sporadic task execution behavior.

Definition: Work-conserving execution sequence [23]: An execution sequence is work- conserving if and only if it never idles the processor when there exists at least one accepted task awaiting execution in the system. Otherwise, it is non-work-conserving.

Work-conserving execution has certain inherent advantages. Specifically, task re- sponse times under work-conserving execution may be lower than non-work-conserving execution since no processor time is wasted. Additionally, the processor time saved due to such early response may be utilized to execute other useful applications. Hence, our finally synthesized scheduler is intended to be work-conserving.

By extending the arguments presented in Theorem 3.3.1, it may be proved that L_m(T_i) represented by the task execution model T_i includes (i) deadline-meeting, (ii) deadline-missing, (iii) sporadic, (iv) non-sporadic (v) work-conserving and (vi) non- work-conserving execution sequences for task τ_i.

Example: Let us consider a simple two task system τ₁h3,6,6i and τ₂h2,4,4i. The task execution models T₁ for τ₁ and T₂ for τ₂ are shown in Figures 3.13(a) and 3.13(b), respectively. Let us considerT₁ corresponding to taskτ₁ shown in Figure 3.13(a). Using T₁, we illustrate the different types of execution sequences present inL_m(T₁).

• The sequenceseq₁ =a₁p₁e₁te₁tc₁t∈L_m(T₁) isdeadline-meetingsincetickcount(a₁

0 T\T1

a1 p₁ r₁

1 2

T\T1

3 6

T\T₁

e1 t c1 7

t

4 T\T₁

# of e1 = E1t 1 = 2 e₁ 5 t

0 T\T2

a₂ p2

r₂

1 2

T\T2

3 4

T\T2

e₂ t c₂ 5

# of e2 = E2t 1 = 1 t (a)

(b)

Figure 3.13: TDES Models: (a) T₁ for τ₁h3,6,6i, (b)T₂ for τ₂h2,4,4i.

p1e1te1tc1) = 2≤ D1−1(= 5). In addition, this sequence is also work-conserving since the processor is never kept idle when the taskτ₁ is waiting for execution.

• The sequenceseq₂ =a₁p₁tttte₁te₁tc₁t ∈L_m(T₁) isdeadline-missing sincetickcount(a₁ p₁tttte₁te₁tc₁) = 6 D₁ −1(= 5). In addition, this sequence is also non-work- conserving since the processor is kept idle even when the task τ₁ is waiting for execution.

• The sequenceseq₃ =a₁p₁e₁te₁tc₁tttta₁p₁e₁te₁tc₁t ∈L_m(T₁) issporadicsincetickcount (a₁p₁e₁te₁tc₁tttta₁) = 6 ≥P₁(= 6).

• The sequence seq₄ = a₁p₁e₁te₁tc₁ta₁p₁e₁te₁tc₁t ∈ L_m(T₁) is non-sporadic since tickcount (a₁p₁e₁te₁tc₁ta₁) = 3 P₁(= 6).

A similar discussion holds for T₂.