Decentralized Task Reallocation on Parallel Computing Architectures Targeting an Avionics Application

Abstract

This work presents an online decentralized allocation algorithm of a safety-critical application on parallel computing architectures, where individual Computational Units can be affected by faults. The described method includes representing the architecture by an abstract graph where each node represents a Computational Unit. Applications are also represented by the graph of Computational Units they require for execution. The problem is then to decide how to allocate Computational Units to applications to guarantee execution of a safety-critical application. The problem is formulated as an optimization problem with the form of an Integer Linear Program. A state-of-the-art solver is then used to solve the problem. Decentralizing the allocation process is achieved through redundancy of the allocator executed on the architecture. No centralized element decides on the allocation of the entire architecture, thus improving the reliability of the system. Inspired by multi-core architectures in avionics systems, an experimental illustration of the work is also presented. It is used to demonstrate the capabilities of the proposed allocation process to maintain the operation of a physical system in a decentralized way while individual components fail.

Appendix A

This appendix provides the proof that the coefficients in the objective function from Eq. (3) allow to meet all three requirements stated in Sect. 3.3.

Theorem A.1

$$\begin{aligned} \forall \ {\tilde{k}} \in \llbracket 1,\ N_{\mathrm{apps}} \rrbracket :\alpha _{{\tilde{k}}} > (\beta + 1)\sum _{j = 1}^{N_{\mathrm{nodes}}} 1 + \sum _{k = 1}^{N_{\mathrm{realloc}}}\sum _{j = 1}^{N_{\mathrm{CUs}}}\sum _{i = 1}^{N_{\mathrm{paths}}} 1, \end{aligned}$$

that is, the contribution to the value of the objective function for executing application $\mathrm {app}_{{\tilde{k}}}$ is greater than the maximum contribution for reducing the number of reallocations and the length of the communication paths.

Proof

$\forall \ {\tilde{k}} \in \llbracket 1,\ N_{\mathrm{apps}} \rrbracket : \alpha _{{\tilde{k}}} \ge (\beta +1) \times N_{\mathrm{nodes}} + \beta ,$ by definition of $\alpha _{{\tilde{k}}}$.

Now,

$$\begin{aligned} (\beta + 1)\sum _{j = 1}^{N_{\mathrm{nodes}}} 1 + \sum _{k = 1}^{N_{\mathrm{realloc}}}\sum _{j = 1}^{N_{\mathrm{CUs}}}\sum _{i = 1}^{N_{\mathrm{paths}}} 1 = (\beta +1) \times N_{\mathrm{nodes}} + \beta . \end{aligned}$$

So

$$\begin{aligned} \alpha _{{\tilde{k}}} > (\beta + 1)\sum _{j = 1}^{N_{\mathrm{nodes}}} 1 + \sum _{k = 1}^{N_{\mathrm{realloc}}}\sum _{j = 1}^{N_{\mathrm{CUs}}}\sum _{i = 1}^{N_{\mathrm{paths}}} 1. \end{aligned}$$

$\square$

Theorem A.1 proves that requirement 1 is met.

Theorem A.2

$$\begin{aligned} (\beta + 1) \times 1 > \sum _{k = 1}^{N_{\mathrm{realloc}}}\sum _{j = 1}^{N_{\mathrm{CUs}}}\sum _{i = 1}^{N_{\mathrm{paths}}} 1, \end{aligned}$$

that is, the contribution to the value of the objective function for not reallocating one Application node is greater then the maximum contribution for reducing the length of communication paths.

Proof

$\beta + 1 > \beta = N_{\mathrm{realloc}} \times N_{\mathrm{CUs}} \times N_{\mathrm{paths}} = \sum _{k = 1}^{N_{\mathrm{realloc}}}\sum _{j = 1}^{N_{\mathrm{CUs}}}\sum _{i = 1}^{N_{\mathrm{paths}}} 1$. $\square$

Theorem A.2 proves that requirement 2 is met.

Theorem A.3

$$\begin{aligned} \forall \ {\tilde{k}} \in \llbracket 1,\ N_{\mathrm{apps}} - 1 \rrbracket :\alpha _{{\tilde{k}}} > \sum _{l = {\tilde{k}}+1}^{N_{\mathrm{apps}}} \alpha _l \end{aligned}$$

that is, the contribution to the value of the objective function for executing application $\mathrm {app}_{{\tilde{k}}}$ is greater than the contribution for executing every applications with lower priority than $\mathrm {app}_{{\tilde{k}}}$, which are $\mathrm {app}_{{\tilde{k}}+1}$ to $\mathrm {app}_{N_{\mathrm{apps}}}$.

Proof

$\forall \ {\tilde{k}} \in \llbracket 1,\ N_{\mathrm{apps}} \rrbracket :\alpha _{{\tilde{k}}} = \sum _{l = {\tilde{k}}+1}^{N_{\mathrm{apps}}} \alpha _l + (\beta +1) \times N_{\mathrm{nodes}} + \beta + 1 > \sum _{l = {\tilde{k}}+1}^{N_{\mathrm{apps}}} \alpha _l$

since $(\beta +1) \times N_{\mathrm{nodes}} + \beta + 1 > 0$. $\square$

Theorem A.3 proves that requirement 3 is met.