Worst-case end-to-end delays evaluation for SpaceWire networks

Abstract

SpaceWire is a standard for on-board satellite networks chosen by the ESA as the basis for multiplexing payload and control traffic on future data-handling architectures. However, network designers need tools to ensure that the network is able to deliver critical messages on time. Current research fails to address this needs for SpaceWire networks. On one hand, many papers only seek to determine probabilistic results for end-to-end delays on Wormhole networks like SpaceWire. This does not provide sufficient guarantee for critical traffic. On the other hand, a few papers give methods to determine maximum latencies on wormhole networks that, unlike SpaceWire, have dedicated real-time mechanisms built-in. Thus, in this paper, we propose an appropriate method to compute an upper-bound on the worst-case end-to-end delay of a packet in a SpaceWire network.

Appendix

1.1 Proof of termination

We first define the link dependency graph D for a network represented by the graph $\mathcal{G}(N,L)$ as the directed graph $D=\mathcal{G}(L,E)$ where the nodes L of D are the links of $\mathcal{G}(N,L)$ and the edges E are the pairs of links used successively by a flow:

$$ E=\{(l_i,l_j) \in L~for~which~\exists f \in F | next(f,l_i)=l_j\} $$

Theorem 1

For any flow f in F and any link l in L, the computation of d(f, l) finishes if and only if D is acyclic.

Proof

• ⇒ We reason by contradiction. Suppose there’s a cycle in D. Since next(f, l) cannot return l for any l (a packet cannot use the same link twice in a row), this cycle’s length is at least 2. Given the definition of d, this means we can find a flow f and a link l such that computing d(f, l) requires calling d(f, l) during the recursion. Thus the computation of d(f, l) does not finish.

• ⇐ Suppose D is acyclic. We can then use topological sorting to assign a total order to the vertices of D such that if (l _i , l _j ) ∈ E then l _i  > l _j (if D is not connected in the graph theory sense we can order each connected component individually then arbitrarily order the component). From the definition of E and next(f, l), it follows that computing d(f, l) only requires calls to d with links inferior to l as parameters. Since F is a finite set and d(f, l) is called only once for some given f and l, it follows that computing d(f, l) always terminates. □

Note that as shown in Dally and Seitz (1987), for an interconnection network using Wormhole Routing under assumptions similar to ours, it is equivalent to say that the link dependency graph is acyclic and that the network is deadlock-free. Since deadlock prone networks would not be valid for satellite use, Theorem 1 shows that the computation finishes in every pertinent case. This is also coherent with the fact that, when the computation does not finish, d(f, l) becomes infinite which is characteristic of a deadlock.

A simple way to check beforehand if the computation will finish is to use the adjacency matrix of the dependency graph and iterate it. If it converges toward the zero matrix then there are no cycles in the graph. However, if a power of the matrix has only 1 on its diagonal, it means that the dependency graph is cyclic and the sum will diverge.

1.2 Complexity

The exact number of operations required to compute d(f, l) on a given network will vary with the topology and the number of conflicting flows on each link. However, we can compute an upper-bound on the number of operations without difficulties.

Let us note n the total number of flows and m the total number of links. If we compute navely the delay for each flow, the size of the computation will increase exponentially with the number of flows in conflicts on each link. However, it is easy notice that the number of possible calls to d(f, l) is finite and bounded by n.m. Thus, if we store the results of all the calls during the computation and given that there can be at most n additions or maximum computations during a call of d(f, l), the complexity of the computation will be O(n ².m).

If Group Adaptative Routing is used, since we have to enumerate all the partitions of the set of conflicting flows, the computation becomes exponential.