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(** * The Sporadic Task Model *) (** In the following, we define the arrival process commonly known as the sporadic task model, where jobs may arrive at any time provided any two jobs of a task are separated by at least the minimum inter-arrival time (or period) of the task. *) (** ** Task Parameter for the Sporadic Task Model *) (** Under the sporadic task model, each task is characterized by its minimum inter-arrival time, which we denote as [task_min_inter_arrival_time]. *) Class SporadicModel (Task : TaskType) := task_min_inter_arrival_time : Task -> duration. (** ** Model Validity *) (** Next, we define the semantics of the sporadic task model. *) Section ValidSporadicTaskModel. (** Consider any type of sporadic tasks. *) Context {Task : TaskType} `{SporadicModel Task}. (** A valid sporadic task should have a non-zero minimum inter-arrival time. *) Definition valid_task_min_inter_arrival_time tsk := task_min_inter_arrival_time tsk > 0. (** Further, in the context of a set of such tasks, ... *) Variable ts : TaskSet Task. (** ... every task in the set should have a valid inter-arrival time. *) Definition valid_taskset_inter_arrival_times := forall tsk : Task, tsk \in ts -> valid_task_min_inter_arrival_time tsk. (** Next, consider any type of jobs stemming from these tasks ... *) Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}. (** ... and an arbitrary arrival sequence of such jobs. *) Variable arr_seq : arrival_sequence Job. (** We say that a task respects the sporadic task model if the arrivals of its jobs in the arrival sequence are appropriately spaced in time. *) Definition respects_sporadic_task_model (tsk : Task) := forall (j j': Job), (** Given two different jobs j and j' ... *) j <> j' -> (** ...that belong to the arrival sequence... *) arrives_in arr_seq j -> arrives_in arr_seq j' -> (** ... and that stem from the given task, ... *) job_task j = tsk -> job_task j' = tsk -> (** ... if the arrival of j precedes the arrival of j' ..., *) job_arrival j <= job_arrival j' -> (** then the arrival of j and the arrival of j' are separated by at least one period. *) job_arrival j' >= job_arrival j + task_min_inter_arrival_time tsk. (** Based on the above definition, we define the sporadic task model as follows. *) Definition taskset_respects_sporadic_task_model := forall tsk, tsk \in ts -> respects_sporadic_task_model tsk. End ValidSporadicTaskModel.