This document presents the modeling and stability proof of a scheduling algorithm applied to Eventing (Concours Complet d'Équitation). (https://github.com/jefeste/Competition-Planner/)

The problem studied is that of a No-Wait Flow-Shop with shared resources. Concretely, a sequence of tasks (riders) must sequentially pass through machines (tests) without possible intermediate storage (no waiting once the process has started). The difficulty lies in the resource looping: the first and last stages share the same physical infrastructure, introducing non-trivial blocking constraints.

Empirically, we observe that applying an "earliest start" launch policy leads the system toward regular behavior, forming a "staircase block" pattern.
The objective of this proof is to formally demonstrate, via Max-Plus Algebra, that:

We will first define the algebraic structure , then translate the physical constraints of the competition into linear state equations and finally explain how to optimize resources.

We work within the idempotent semiring , equipped with the following operations:

For matrices , the matrix product is defined as:

Let us consider a discrete event system composed of  resources processing a sequence of items indexed by . The state of the system at step , denoted by , is governed by a set of linear constraints in .

Item  is subject to a set of constraints. Let  be the set of constraints. For each constraint , there exists a capacity  and a weight .

This yields a Recurrence of Order  (where  is the maximum capacity):

To apply the spectral theorem, we introduce the augmented state vector representing the system's recent history:

The system can then be written in the form , with the state transition matrix :

(Note: Row 1 corresponds to equation (1), while the subsequent rows simply satisfy the identity )

Since the graph associated with  is strongly connected (irreducible matrix), the Baccelli-Cohen-Olsder-Quadrat Theorem states that there exist:

Such that :

Thus, we obtain the scalar relation:

We apply this theoretical framework to the scheduling of an equestrian Eventing competition (Concours Complet).

For each stage, we have the following constant parameters:

Rider  cannot begin stage  until they have completed stage . Furthermore, the process allows for no intermediate waiting times once initiated.
By recurrence, letting the cumulative lag be , we have:
Note: This relation reduces the dimensionality of the problem, as the state of the entire system is coupled to the start time .

If a stage  has a capacity , rider  can only enter it once rider  has vacated it.
Consider two stages  and  sharing a critical resource or dependency. The constraint is written as:

By combining A and B, the inequality becomes a recurrence on the decision variable  (here simply denoted as ):

The graph associated with the system is connected because there is a trivial sequential dependency between  and  (job  is not launched before job ), which links all states; the matrix is therefore irreducible.

By applying the preceding theorem, the optimal scheduling of the competition necessarily converges to a stable periodic regime:

In the steady state regime, the completion time of the -th participant is:
Minimizing  is the key to minimizing the total duration.
The bottleneck is the stage that dictates the eigenvalue .
In the context of the Concours Complet :

Reducing the transition is twice as effective as reducing the duration of a test.