Reduction of the search space for Abstract RTA

In this file, we prove that in order to calculate the worst-case response time it is sufficient to consider only values of A that lie in the search space defined below.

The response-time analysis we are presenting in this series of documents is based on searching over all possible values of A, the relative arrival time of the job respective to the beginning of the busy interval. However, to obtain a practically useful response-time bound, we need to constrain the search space of values of A. In this section, we define an approach to reduce the search space.

First, we provide a constructive notion of equivalent functions.

Consider an arbitrary type T...

...and two function from nat to T.

Let B be an arbitrary constant.

Then we say that f1 and f2 are equivalent at values less than B iff for any natural number x less than B f1 x is equal to f2 x.

And vice versa, we say that f1 and f2 are not equivalent at values less than B iff there exists a natural number x less than B such that f1 x is not equal to f2 x.

Let tsk be any task that is to be analyzed

To ensure that the analysis procedure terminates, we assume an upper bound B on the values of A that must be checked. The existence of B follows from the assumption that the system is not overloaded (i.e., it has bounded utilization).

Instead of searching for the maximum interference of each individual job, we assume a per-task interference bound function IBF(A, x) that is parameterized by the relative arrival time A of a potential job (see abstract_RTA.definitions.v file).

Recall the definition of ε, which defines the neighborhood of a point in the timeline. Note that ε = 1 under discrete time. To ensure that the search converges more quickly, we only check values of A in the interval [0, B) for which the interference bound function changes, i.e., every point x in which interference_bound_function (A - ε, x) is not equal to interference_bound_function (A, x).

In this section we prove that for every A there exists a smaller A_sp in the search space such that interference_bound_function(A_sp,x) is equal to interference_bound_function(A, x).

Let A be any constant less than B.

We prove that there exists a constant A_sp such that: (a) A_sp is no greater than A, (b) interference_bound_function(A_sp, x) is equal to interference_bound_function(A, x) and (c) A_sp is in the search space. In other words, either A is already inside the search space, or we can go to the "left" until we reach A_sp, which will be inside the search space.

In this section we prove that any solution of the response-time recurrence for a given point A_sp in the search space also gives a solution for any point A that shares the same interference bound.

Suppose A_sp + F_sp is a "small" solution (i.e. less than B) of the response-time recurrence.

Next, let A be any point such that: (a) A_sp ≤ A ≤ A_sp + F_sp and (b) interference_bound_function(A, x) is equal to interference_bound_function(A_sp, x) for all x less than B.

We prove that there exists a constant F such that A + F is equal to A_sp + F_sp and A + F is a solution for the response-time recurrence for A.

In this section, we prove a simple lemma that allows one to switch IBFs inside of the is_in_search_space predicate.

Consider any type of tasks.

Let tsk be any task that is to be analyzed.

Similarly to the previous section, to ensure that the analysis procedure terminates, we assume an upper bound B on the values of A that must be checked.

Given two IBFs IBF1 and IBF2 such that they are equal for all inputs, if an offset A is in the search space of IBF1, then A is in the search space of IBF2.