Minimum Congestion Redundant Assignments

Keywords and Synonyms

Minimum fault-tolerant congestion; Maximum fault‐tolerant partition        

Problem Definition

Figure 1

Two redundant assignments of 4 unit size messages to 8 identical links. Both assign every message to 4 links and 2 messages to every link. The corresponding graph is depicted below each assignment. The assignment on the left is the most reliable 2‑partitioning assignment ${ \phi_2 }$. Lemma 3 implies that for every failure probability f, ${ \phi_2 }$ is at least as reliable as any other assignment ϕ with ${ \text{Cong}(\phi) \leq 2 }$. For instance, ${ \phi_2 }$ is at least as reliable as the assignment on the right. Indeed the reliability of the assignment on the right is ${ 1 - 4f^{\,4} + 2f^{\,6} + 4f^{\,7} - 3f^{\,8} }$, which is bounded from above by ${ \text{Rel}(\phi_2) = 1 - 2f^{\,4} + f^{\,8} }$ for all ${ f \in [0, 1] }$