Principles of Distributed Computing - "processors" and "states"
Based on the wording of this, it seems to imply that there are n of each buffer


Assuming that each edge in the network graph $G=(V,E)$ corresponds to a bidirectional channel, there are $4|E|$ many buffers in total: For an (undirected) edge $(p_i,p_j)$, we have one pair of in/out buffers for $p_i$ and one pair for $p_j$. 


  Each state of the processor $p_i$ contains $2r$ special components: $inbuf_i[l]$ and $outbuf_i[l]$.


Let's look at an example to get an intuition for what the above means:
Suppose that we have some network where $p_3$ is connected to $p_1$ and $p_2$.
Let's assume that process $p_3$ sends a message $m$ to process $p_1$ and has received (but not yet processed) a message $m'$ from process $p_2$. Moreover, $p_3$ has a local variable $counter$. Then we could represent the current state $\sigma_3$ of $p_3$ as  $$\sigma_3 = (inbuf_3[1],inbuf_3[2],outbuf_3[1],outbuf_3[2],counter).$$
According to the above example, $m' \in inbuf_3[2]$ and $m \in outbuf_3[1]$. Note that the state of the network is given by $(\sigma_0,\dots,\sigma_n-1)$.