Require:  Chain nodes are numbered $0,1,2,\dots ,N-1$   Require:  $1<N$   Require:  $\alpha ,\beta \in \left\{0,1,2,\dots ,N-1\right\}$   Require:  $P\left[i,j\right]$ is the probability of branching from node $j$ to node $i$    1: if $\alpha <\beta$ then
   2:  $h\leftarrow 1$; $t\leftarrow N-2$; $s\leftarrow 1$
   3: else
   4:  $h\leftarrow N-2$; $t\leftarrow 1$; $s\leftarrow -1$
   5: end if
   6:  $L\leftarrow 1$
   7: for all $i\in \left\{h,h+s,h+2s,\dots ,\alpha \right\}$ do
   8:  $L\leftarrow 1∕\left(1-P\left[i-s,i\right]P\left[i,i-s\right]L\right)$
   9: end for
   10:  $\Pi \leftarrow 1$
   11: for all $i\in \left\{\alpha +s,\alpha +2s,\alpha +3s,\dots ,\beta \right\}$ do
   12:  $\Pi \leftarrow P\left[i-s,i\right]\mathit{L\Pi }$
   13:  $L\leftarrow 1∕\left(1-P\left[i-s,i\right]P\left[i,i-s\right]L\right)$
   14: end for
   15:  $R\leftarrow 1$
   16: for all $i\in \left\{t,t-s,t-2s,\dots ,\beta \right\}$ do
   17:  $R\leftarrow 1∕\left(1-P\left[i+s,i\right]P\left[i,i+s\right]R\right)$
   18: end for
   19: return  $\mathit{LR\Pi }∕\left(L-\mathit{LR}+R\right)$

  Require:  $1<N$   Require:  $a,b\in \left\{0,1,2,\dots ,N-1\right\}$   Require:  $W$ is a boolean array of size $N$ with every element initially set to True   Require:  $N_W$ is the number of True elements in array $W$ (initialised to $N$)   Require:  $P\left[i,j\right]$ is the probability of branching from node $j$ to node $i$   Require:  $\mathit{AdjIn}\left[i\right]$ and $\mathit{AdjOut}\left[i\right]$ are the lists of indices of all nodes connected to node $i$ via incoming and outgoing edges, respectively Recursive function $F\left(\alpha ,\beta ,W,N_W\right)$    1:  $W\left[\beta \right]\leftarrow$ False
   2:  $N_W\leftarrow N_W-1$
   3: if $\alpha =\beta$ and $N_W=0$ then
   4:  $\Sigma \leftarrow 1$
   5: else
   6:  $\Sigma \leftarrow 0.0$
   7: for all $i\in \mathit{AdjOut}\left[\beta \right]$ do
   8: for all $j\in \mathit{AdjIn}\left[\beta \right]$ do
   9: if $W\left[i\right]$ and $W\left[j\right]$ then
   10:  $\Sigma \leftarrow \Sigma +P\left[\beta ,j\right]F\left(j,i,W,N_W\right)P\left[i,\beta \right]$
   11: end if
   12: end for
   13: end for
   14:  $\Sigma \leftarrow 1∕\left(1-\Sigma \right)$
   15: if $\alpha \ne \beta$ then
   16:  $\Lambda \leftarrow 0.0$
   17: for all $i\in \mathit{AdjOut}\left[\beta \right]$ do
   18: if W[i] then
   19:  $\Lambda \leftarrow \Lambda +F\left(\alpha ,i,W,N_W\right)P\left(i,\beta \right)$
   20: end if
   21: end for
   22:  $\Sigma \leftarrow \mathit{\Sigma \Lambda }$
   23: end if
   24: end if
   25:  $W\left[\beta \right]\leftarrow$ True
   26:  $N_W\leftarrow N_W+1$
   27: return  $\Sigma$

  Require:  Nodes are numbered $0,1,2,\dots ,N-1$   Require:  Sink nodes are indexed first, source nodes - last   Require:  $i$ is the index of the first intermediate node   Require:  $s$ is the index of the first source node   Require:  In case there are no intermediate nodes $i=s$, otherwise $i<s$   Require:  $1<N$   Require:  $i,s\in \left\{0,1,2,\dots ,N-1\right\}$   Require:  $\tau \left[\alpha \right]$ is the waiting time for node $\alpha$, $\alpha \in \left\{i,i+1,i+2,\dots ,N-1\right\}$   Require:  $P\left[i,j\right]$ is the probability of branching from node $j$ to node $i$    1: for all $\gamma \in \left\{i,i+1,i+2,\dots ,s-1\right\}$ do
   2: for all $\beta \in \left\{\gamma +1,\gamma +2,\dots ,N-1\right\}$ do
   3: if $P\left[\gamma ,\beta \right]>0$ then
   4:  $\tau \left[\beta \right]\leftarrow \left(\tau \left[\beta \right]+\tau \left[\gamma \right]P\left[\gamma ,\beta \right]\right)∕\left(1-P\left[\beta ,\gamma \right]P\left[\gamma ,\beta \right]\right)$
   5: for all $\alpha \in \left\{0,1,2,\dots ,N-1\right\}$ do
   6: if $\alpha \ne \beta$ and $\alpha \ne \gamma$ then
   7:  $P\left[\alpha ,\beta \right]\leftarrow \left(P\left[\alpha ,\beta \right]+P\left[\alpha ,\gamma \right]P\left[\gamma ,\beta \right]\right)∕\left(1-P\left[\beta ,\gamma \right]P\left[\gamma ,\beta \right]\right)$
   8: end if
   9: end for
   10:  $P\left[\gamma ,\beta \right]\leftarrow 0.0$
   11: end if
   12: end for
   13: for all $\alpha \in \left\{0,1,2,\dots ,N-1\right\}$ do
   14:  $P\left[\alpha ,\gamma \right]\leftarrow 0.0$
   15: end for
   16: end for
   17: for all $\alpha \in \left\{s,s+1,s+2,\dots ,N-1\right\}$ do
   18: for all $\beta \in \left\{s,s+1,s+2,\dots ,N-1\right\}$ do
   19: if $\alpha \ne \beta$ and $P\left[\alpha ,\beta \right]>0$ then
   20:  $P_{\alpha ,\beta }\leftarrow P\left[\alpha ,\beta \right]$
   21:  $P_{\beta ,\alpha }\leftarrow P\left[\beta ,\alpha \right]$
   22:  $T\leftarrow \tau \left[\alpha \right]$
   23:  $\tau \left[\alpha \right]\leftarrow \left(\tau \left[\alpha \right]+\tau \left[\beta \right]P_{\beta ,\alpha }\right)∕\left(1-P_{\alpha ,\beta }{P}_{\beta ,\alpha }\right)$
   24:  $\tau \left[\beta \right]\leftarrow \left(\tau \left[\beta \right]+T{P}_{\alpha ,\beta }\right)∕\left(1-P_{\alpha ,\beta }{P}_{\beta ,\alpha }\right)$
   25: for all $\gamma \in \left\{0,1,2,\dots ,i-1\right\}\cup \left\{s,s+1,s+2,\dots ,N-1\right\}$ do
   26:  $T\leftarrow P\left[\gamma ,\alpha \right]$
   27:  $P\left[\gamma ,\alpha \right]\leftarrow \left(P\left[\gamma ,\alpha \right]+P\left[\gamma ,\beta \right]P_{\beta ,\alpha }\right)∕\left(1-P_{\alpha ,\beta }{P}_{\beta ,\alpha }\right)$
   28:  $P\left[\gamma ,\beta \right]\leftarrow \left(P\left[\gamma ,\beta \right]+T{P}_{\alpha ,\beta }\right)∕\left(1-P_{\alpha ,\beta }{P}_{\beta ,\alpha }\right)$
   29: end for
   30:  $P\left[\alpha ,\beta \right]\leftarrow 0.0$
   31:  $P\left[\beta ,\alpha \right]\leftarrow 0.0$
   32: end if
   33: end for
   34: end for

  Require:  $1<N$   Require:  $\gamma \in \left\{0,1,2,\dots ,N-1\right\}$   Require:  $\tau \left[i\right]$ is the waiting time for node $i$   Require:  $\mathit{Adj}\left[i\right]$ is the ordered list of indices of all nodes connected to node $i$ via outgoing edges   Require:  $\left|\mathit{Adj}\left[i\right]\right|$ is the cardinality of $\mathit{Adj}\left[i\right]$   Require:  $\mathit{Adj}\left[i\right]\left[j\right]$ is the index of the $j$th neighbour of node $i$   Require:  $P\left[i\right]$ is the ordered list of probabilities of leaving node $i$ via outgoing edges, $\left|P\left[i\right]\right|=\left|\mathit{Adj}\left[i\right]\right|$   Require:  $P\left[i\right]\left[j\right]$ is the probability of branching from node $i$ to node $\mathit{Adj}\left[i\right]\left[j\right]$    1: for all ${\beta }_{\gamma }\in \left\{0,1,2,\dots ,\left|\mathit{Adj}\left[\gamma \right]\right|-1\right\}$ do
   2:  $\beta \leftarrow \mathit{Adj}\left[\gamma \right]\left[{\beta }_{\gamma }\right]$
   3:  ${\gamma }_{\beta }\leftarrow -1$
   4: for all $i\in \left\{0,1,2,\dots ,\left|\mathit{Adj}\left[\beta \right]\right|-1\right\}$ do
   5: if $\mathit{Adj}\left[\beta \right]\left[i\right]=\gamma$ then
   6:  ${\gamma }_{\beta }\leftarrow i$
   7: break
   8: end if
   9: end for
   10: if not ${\gamma }_{\beta }=-1$ then
   11:  $P_{\beta ,\beta }\leftarrow 1∕\left(1-P\left[\beta \right]\left[{\gamma }_{\beta }\right]P\left[\gamma \right]\left[{\beta }_{\gamma }\right]\right)$
   12:  $P_{\beta ,\gamma }\leftarrow P\left[\beta \right]\left[{\gamma }_{\beta }\right]$
   13:  $\mathit{Adj}\left[\beta \right]\leftarrow \left\{\mathit{Adj}\left[\beta \right]\left[0\right],\mathit{Adj}\left[\beta \right]\left[1\right],\dots ,\mathit{Adj}\left[\beta \right]\left[{\gamma }_{\beta }-1\right],\mathit{Adj}\left[\beta \right]\left[{\gamma }_{\beta }+1\right],\dots \right\}$
   14:  $P\left[\beta \right]\leftarrow \left\{P\left[\beta \right]\left[0\right],P\left[\beta \right]\left[1\right],\dots ,P\left[\beta \right]\left[{\gamma }_{\beta }-1\right],P\left[\beta \right]\left[{\gamma }_{\beta }+1\right],\dots \right\}$
   15: for all ${\alpha }_{\gamma }\in \left\{0,1,2,\dots ,\left|\mathit{Adj}\left[\gamma \right]\right|-1\right\}$ do
   16:  $\alpha \leftarrow \mathit{Adj}\left[\gamma \right]\left[{\alpha }_{\gamma }\right]$
   17: if not $\alpha =\beta$ then
   18: if exists edge $\beta \to \alpha$ then
   19: for all $i\in \left\{0,1,2,\dots ,\left|\mathit{Adj}\left[\beta \right]\right|-1\right\}$ do
   20: if $\mathit{Adj}\left[\beta \right]\left[i\right]=\alpha$ then
   21:  $P\left[\beta \right]\left[i\right]\leftarrow P\left[\beta \right]\left[i\right]+P_{\beta ,\gamma }P\left[\gamma \right]\left[{\alpha }_{\gamma }\right]$
   22: break
   23: end if
   24: end for
   25: else
   26:  $\mathit{Adj}\left[\beta \right]\leftarrow \left\{\alpha ,\mathit{Adj}\left[\beta \right]\left[0\right],\mathit{Adj}\left[\beta \right]\left[1\right],\mathit{Adj}\left[\beta \right]\left[2\right],\dots \right\}$
   27:  $P\left[\beta \right]\leftarrow \left\{P_{\beta ,\gamma }P\left[\gamma \right]\left[{\alpha }_{\gamma }\right],P\left[\beta \right]\left[0\right],P\left[\beta \right]\left[1\right],P\left[\beta \right]\left[2\right],\dots \right\}$
   28: end if
   29: end if
   30: end for
   31: for all $i\in \left\{0,1,2,\dots ,\left|P\left[\beta \right]\right|-1\right\}$ do
   32:  $P\left[\beta \right]\left[i\right]\leftarrow P\left[\beta \right]\left[i\right]P_{\beta ,\beta }$
   33: end for
   34:  ${\tau }_{\beta }\leftarrow \left.{\tau }_{\beta }+P_{\beta ,\gamma }{\tau }_{\gamma }\right)P_{\beta ,\beta }$
   35: end if
   36: end for