## 3 Analysis
all possible subsets of active transmitters, while \(b(l,n)\) becomes one if the \(nth\) node in \(\mathcal{I}_{i,j}\) is assumed active in the \(lth\) subset examined. For each such subset, indexed by \(l\) in (10), the corresponding success probability of link \((i,j)\), given that this subset of nodes is active, is expressed through \(\mathcal{P}_{i,j,l}\) presented in (13). Note that, when interference cancelation is employed at node \(j\), then the expression for the success probability \(P_{i,j,l}\) is based on (9), presented in Section 2.2. \[\mathcal{T}_{i,j}=\sum_{l=0}^{2^{\mathcal{C}_{i,j}-1}}P_{i/M_{l}}^{j}q_{i,j} \prod_{n=1}^{\mathcal{L}_{i,j}}q_{\mathcal{I}_{i,j}[n]}^{b(l,n)}(1-q_{\mathcal{ I}_{i,j}[n]})^{1-b(l,n)}, \tag{10}\]  
where  
\[q_{i,j}=\left\{ \begin{array}{cc}q_{i}^{\prime\prime}&j=D\\ q_{i}^{\prime\prime}(1-q_{j}^{\prime\prime})&j\
eq D^{\prime}\end{array} \right. \tag{11}\]  
\[q_{i}^{\prime\prime}=\left\{ \begin{array}{cc}q_{i}&i\
eq relay\\ q_{i}\mathbb{1}[q_{Src(r(i))}>0]&i=relay\end{array} \right. \tag{12}\]  
\[P_{i/M_{l}}^{j}=p_{i/I\cup\{\mathcal{I}_{i,j}[n],\;\forall\;m\;b(l,n)\
eq 0\} ^{j})}^{j}, \tag{13}\]  
where \(b(l,n)=l\;\&\;2^{n-1}\), & is the logical bitwise AND operator. In (12), \(\mathbb{1}[q_{Src(r(i))}>0]\) denotes an indicator function whose value becomes one if \(q_{Src(r(i))}>0\) and zero otherwise. The reason for employing this indicator function is discussed at the end of this section. As also described in Section 4, transmission probability and position for every node can be propagated periodically to all other nodes through the routing protocol's topology control messages. Position information is used to infer each link's success probability based on either (5), or (9), depending on whether interference is treated as noise (IAN), or successive interference cancelation (SIC) is employed at the receiver. The average aggregate throughput achieved by all flows is expressed through \(\mathcal{T}_{aggr}=\sum_{k=1}^{m}\mathcal{T}_{r_{k}}\), where \(\mathcal{T}_{r_{k}}=\underset{(i,j)\in\mathcal{T}_{k}}{min}\mathcal{T}_{i,j}\). Aggregate throughput optimal flow rates, that also provide bounded packet delay, for a set of flows and a specific wireless topology can be estimated by solving the following optimization problem:  
\[\underset{S^{\prime}}{\text{Maximize}}\sum_{k=1}^{m}\left\{ \begin{array}{cc}\mathcal{T}_{Src(r_{k}),D},&|r_{k}|=1\\ q_{Src(r_{k})}^{\prime},&|r_{k}|>1\end{array} \right.\] \[s.t.:\] \[\qquad\qquad(S1):\;0\leq q_{Src(r_{k})}\leq 1,\;k=1,...,m\] \[\qquad\qquad(S2):\;\mathcal{\bar{T}}_{Src(r_{k}),i}\leq\mathcal{ \bar{T}}_{j,l},\] \[\qquad\qquad\qquad\{\forall i,j,k,l:(Src(r_{k}),i),\;(j,l)\in r_{ k},|r_{k}|>1\] \[\qquad\qquad\qquad k=1,...,m\}\] \[\qquad\qquad(S3):\;0\leq q_{Src(r_{k})}^{\prime}\leq 1,\;\{\forall k :|r_{k}|>1\}\] \[\qquad\qquad(S4):\;q_{Src(r_{k})}^{\prime}\leq\mathcal{\bar{T}}_ {i,j},\;\{\forall i,j,k:|r_{k}|>1,\;(i,j)\in r_{k}\}\]  
where, \(\mathcal{S^{\prime}}=\{q_{Src(r_{k})},\;k=1,...,m\}\cup\{q_{src(r_{k})}^{ \prime}:\;|r_{k}|>1\}\). In the above optimization problem, constraint set \(S1\) ensures that the maximum data rate for any flow does not exceed one packet per slot, while also allowing paths to remain unutilized. Constraint \((S2)\) ensures that the flow injected on each path, that is the throughput of that path's first link, is limited by the flow that can be serviced by any subsequent link of that path. In this way, data packets are prevented from accumulating at the relay nodes, providing bounded packet delay. For the rest of the paper, this constraint will be referred to as, _bounded delay constraint_ and the corresponding optimization problem as the _flow allocation optimization problem_. Moreover, the scheme that determines the flow to be assigned on each path based on the above optimization problem will be referred to as the _Throughput Optimal Flow Rate Allocation (TOFRA)_ scheme for the rest of the study.