PERIODIC TASK SCHEDULING - Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications - page 80

Φ i denotes the phase of task τ i ; that is, the release time of its first instance

(Φ i = r i, 1 );

D i

denotes the relative deadline of task τ i ;

d i,j denotes the absolute deadline of the j th instance of task τ i ( d i,j =Φ i +

( j

− 1) T i + D i ).

s i,j denotes the start time of the j th instance of task τ i ; that is, the time at

which it starts executing.

f i,j denotes the finishing time of the j th instance of task τ i ; that is, the time

at which it completes the execution.

Moreover, in order to simplify the schedulability analysis, the following hypotheses
are assumed on the tasks:

A1. The instances of a periodic task τ i are regularly activated at a constant

rate. The interval T i between two consecutive activations is the period

of the task.

A2. All instances of a periodic task τ i have the same worst-case execution

time C i .

A3. All instances of a periodic task τ i have the same relative deadline D i ,

which is equal to the period T i .

A4. All tasks in Γ are independent; that is, there are no precedence relations

and no resource constraints.

In addition, the following assumptions are implicitly made:

A5. No task can suspend itself, for example on I/O operations.

A6. All tasks are released as soon as they arrive.

A7. All overheads in the kernel are assumed to be zero.

Notice that assumptions A1 and A2 are not restrictive because in many control ap-
plications each periodic activity requires the execution of the same routine at regular
intervals; therefore, both T i and C i are constant for every instance. On the other hand,
assumptions A3 and A4 could be too tight for practical applications.