Optimal minimal routing and priority assignment for priority-preemptive real-time NoCs

Abstract

The Network-on-Chip (NoC) architecture is an interconnect network with a good performance and scalability potential. Thus, it comes as no surprise that NoCs are among the most popular interconnect mediums in nowadays available many-core platforms. Over the years, the real-time community has been attempting to make NoCs amenable to the real-time analysis. One such approach advocates to employ virtual channels. Virtual channels are hardware resources that can be used as an infrastructure to facilitate flit-level preemptions between communication traffic flows. This gives the possibility to implement priority-preemptive arbitration policies in routers, which is a promising step towards deriving real-time guarantees for NoC traffic. So far, various aspects of priority-preemptive NoCs were studied, such as arbitration, priority assignment, routing, and workload mapping. Due to a potentially large solution space, the majority of available techniques are heuristic-centric, that is, either pure heuristics, or heuristic-based search strategies are used. Such approaches may lead to an inefficient use of hardware resources, and may cause a resource over-provisioning as well as unnecessarily high design-cost expenses. Motivated by this reality, we take a different approach, and propose an integer linear program to solve the problems of priority assignment and routing of NoC traffic. The proposed method finds optimal routes and priorities, but also allows to reduce the search space (and the computation time) by fixing either priorities or routes, and derive optimal values for remaining parameters. This framework is used to experimentally evaluate both the scalability of the proposed method, as well as the efficiency of existing priority assignment and routing techniques.

Appendix

1.1 1. Linearising the implication operator

Let A be a non-negative integer variable, and let B be a binary variable. Moreover, let us linearise the following implication: $(B = 1) \Rightarrow (A = 0)$. That is, if B is equal to 1, then A must be zero. Conversely, if B is equal to 0, then A can have any value. This reasoning can be implemented with the following constraint:

$$\begin{aligned} M \cdot (1 - B) \ge A \end{aligned}$$

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Note, that the symbol M represents a “big enough constant”, which is commonly used in linear programming, and it is called the big M method. In our example, if B is equal to 0, then A can be any positive number and the condition is redundant. Conversely, if B is equal to 1, then A must be 0, which is the desired implication.

1.2 2. Linearising the equivalence operator

Let A and B be binary variables, and let us linearise the following equivalence: $A \Leftrightarrow B$. This can be done with two mutual implications (the linearisation of which was explained in the previous subsection), as demonstrated below:

$$\begin{aligned} A \Rightarrow B \end{aligned}$$

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$$\begin{aligned} B \Rightarrow A \end{aligned}$$

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1.3 3. Linearising the ceiling operator

Let A be an integer variable, and let B be a decimal variable. Moreover, let us linearise the following constraint: $A = \lceil B \rceil$. The following two constraints can be written:

$$\begin{aligned} A \ge B \end{aligned}$$

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$$\begin{aligned} A \le B + 1 - \epsilon \end{aligned}$$

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In the second constraint, $\epsilon$ is an infinitesimally small positive number. The first constraint provides a lower bound on the value of A, while the second constraint provides a corresponding upper bound. There exists only one integer in the range between B and $B + 1 - \epsilon$, and that is the value of A.