nonpreemptive.v 6.14 KB
Newer Older
1
Require Export rt.restructuring.analysis.fixed_priority.rta.nonpr_reg.response_time_bound.
2 3
Require Export rt.restructuring.analysis.basic_facts.preemption.task.nonpreemptive.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.nonpreemptive.
Sergey Bozhko's avatar
Sergey Bozhko committed
4
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
Sergey Bozhko's avatar
Sergey Bozhko committed
5

6 7
(** Throughout this file, we assume ideal uniprocessor schedules. *)
Require Import rt.restructuring.model.processor.ideal.
8

9 10 11
(** Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model. *)
Require Import rt.restructuring.model.readiness.basic.

Sergey Bozhko's avatar
Sergey Bozhko committed
12
(** Throughout this file, we assume the fully non-preemptive task model. *)
13
Require Import rt.restructuring.model.task.preemption.fully_nonpreemptive.
Sergey Bozhko's avatar
Sergey Bozhko committed
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137

(** * RTA for Fully Non-Preemptive FP Model *)
(** In this module we prove the RTA theorem for the fully non-preemptive FP model. *)
Section RTAforFullyNonPreemptiveFPModelwithArrivalCurves.

  (** Consider any type of tasks ... *)
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  
  (**  ... and any type of jobs associated with these tasks. *)
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

  (** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
  Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

  (** Consider an arbitrary task set ts, ... *)
  Variable ts : list Task.

  (** ... assume that all jobs come from the task set, ... *)
  Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
  
  (** ... and the cost of a job cannot be larger than the task cost. *)
  Hypothesis H_job_cost_le_task_cost:
    cost_of_jobs_from_arrival_sequence_le_task_cost arr_seq.

  (** Let max_arrivals be a family of valid arrival curves, i.e., for
      any task tsk in ts [max_arrival tsk] is (1) an arrival bound of
      tsk, and (2) it is a monotonic function that equals 0 for the
      empty interval delta = 0. *)
  Context `{MaxArrivals Task}.
  Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
  Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

  (** Let tsk be any task in ts that is to be analyzed. *)
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

  (** Next, consider any ideal non-preemptive uniprocessor schedule of
      this arrival sequence ... *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.
  Hypothesis H_nonpreemptive_sched : is_nonpreemptive_schedule sched.

  (** ... where jobs do not execute before their arrival or after completion. *)
  Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
  Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

  (** Consider an FP policy that indicates a higher-or-equal priority relation,
     and assume that the relation is reflexive and transitive. *)
  Variable higher_eq_priority : FP_policy Task.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.
  Hypothesis H_priority_is_transitive : transitive_priorities.

  (** Let's define some local names for clarity. *)
  Let task_rbf := task_request_bound_function tsk.
  Let total_hep_rbf := total_hep_request_bound_function_FP _ ts tsk.
  Let total_ohep_rbf := total_ohep_request_bound_function_FP _ ts tsk.
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.  

  (** Assume we have sequential tasks, i.e, tasks from the same task
      execute in the order of their arrival. *)
  Hypothesis H_sequential_tasks : sequential_tasks sched.

  (** Next, we assume that the schedule is a work-conserving schedule ... *)
  Hypothesis H_work_conserving : work_conserving arr_seq sched.
  
  (** ... and the schedule respects the policy defined by the
     [job_preemptable] function (i.e., jobs have bounded nonpreemptive
     segments). *)
  Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.    

  (** Next, we define a bound for the priority inversion caused by tasks of lower priority. *)
  Let blocking_bound :=
    \max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk) (task_cost tsk_other - ε).
  
  (** Let L be any positive fixed point of the busy interval recurrence, determined by 
      the sum of blocking and higher-or-equal-priority workload. *)
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

  (** To reduce the time complexity of the analysis, recall the notion of search space. *)
  Let is_in_search_space (A : duration) := (A < L) && (task_rbf A != task_rbf (A + ε)).
  
  (** Next, consider any value R, and assume that for any given arrival A from search space
      there is a solution of the response-time bound recurrence which is bounded by R. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum:
    forall (A : duration),
      is_in_search_space A -> 
      exists (F : duration),
        A + F = blocking_bound
                + (task_rbf (A + ε) - (task_cost tsk - ε))
                + total_ohep_rbf (A + F) /\
        F + (task_cost tsk - ε) <= R.
  
  (** Now, we can leverage the results for the abstract model with
      bounded nonpreemptive segments to establish a response-time
      bound for the more concrete model of fully nonpreemptive
      scheduling. *)
  Theorem uniprocessor_response_time_bound_fully_nonpreemptive_fp:
    response_time_bounded_by tsk R.
  Proof.
    move: (posnP (@task_cost _ H tsk)) => [ZERO|POS].
    { intros j ARR TSK.
      have ZEROj: job_cost j = 0.
      { move: (H_job_cost_le_task_cost j ARR) => NEQ.
        rewrite /job_cost_le_task_cost TSK ZERO in NEQ.
          by apply/eqP; rewrite -leqn0.
      }
        by rewrite /job_response_time_bound /completed_by ZEROj.
    }
    eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with
        (L0 := L).
    all: eauto 2 with basic_facts. 
  Qed.

End RTAforFullyNonPreemptiveFPModelwithArrivalCurves.