In Section 2, it has been stated that, since all TDs are statistically identical, the global behavior of the network can be modeled by analyzing a single TD. By exploiting the properties of probabilistic broadcast protocols with silencing (described in Section 5), the following assumptions hold: (i) the inter-node distance is characterized by a (memoryless) exponential distribution, so that the topology of every TD is (statistically) identical; (ii) the PAF only depends on the distance and is, therefore, memoryless; (iii) the IEEE 802.11b contention mechanism is memoryless, in the sense that it is restarted at every retransmission. Under these assumptions, every retransmission act can be interpreted as a renewal that resets the statistics of the forwarding process. Moreover, since all TDs are statistically identical, without loss of generality we can focus on the first TD.

Therefore, a complete analytical performance evaluation framework can be derived in the following manner: (i) characterizing the first TD with *local* performance metrics (e.g., the successful transmission probability and the delay); (ii) deriving *global* performance metrics (e.g., D, RE, TE), by means of a recursive approach.

In Section 6.1, the local performance (i.e., single TD) is investigated under the assumption of a given number of equally spaced nodes, by considering, without loss of generality, the first TD. In Section 6.2, we derive the global metrics for an overall deterministic network scenario, where the nodes are equally spaced in the interval (0, *L*). Then, in Section 6.3 the results obtained in the deterministic scenario are extended to the original PPP-based scenario.

### 6.1 Local (single TD) performance analysis with a given number of nodes

Without loss of generality, we focus on the first TD, corresponding to the interval \mathcal{I} introduced in Section 3. We consider a deterministic scenario with a fixed number *n* of nodes equally spaced in the interval \mathcal{I}=\left(0,z\right)\subset \mathbb{R}. Every node in a TD is identified by an index *i* ∈ {1, 2, ..., *n*}. The nodes are thus positioned as in Figure 3 and the positions vector **R**^{(n)}is defined as in (1).

According to the operational principles of the considered protocol, after the reception of a packet in a given TD, each node decides to (or not to) retransmit according to the protocol's PAF. The nodes that lose the contention set their BCs to ∞, while the winners set their BCs according to the policy of the specific broadcast protocol. The protocol execution could lead to three different outcomes: (i) nobody decides to retransmit; (ii) some nodes decide to retransmit, but all their transmitted packets collide; (iii) some nodes decide to retransmit, and a single node transmits successfully (when its BC because zero, no other BC is zero). It is useful to define the following events, associated to the forwarding process in a TD:

\begin{array}{c}{\mathcal{F}}_{1}\triangleq \{\text{nobody}\phantom{\rule{0.5em}{0ex}}\text{decidestoretransmit}}\\ \text{=}{\text{{BC}}_{i}=\infty ,\phantom{\rule{0.5em}{0ex}}\forall i\in \{0,1,\mathrm{...},n\}\}\\ {\mathcal{F}}_{2}\triangleq \{\text{allthetransmittedpacketscollide}}\\ =\{\forall i\in \{0,1,\mathrm{...},n\}:{\text{BC}}_{i}\infty ,\exists j\in \{0,1,\mathrm{...},n\},j\ne i,{\text{BC}}_{j}\infty \phantom{\rule{0.5em}{0ex}}{\text{suchasBC}}_{i}={\text{BC}}_{j}\}\\ \mathcal{F}\triangleq \{\text{nobodywinsthecontention}=}{\mathcal{F}}_{1}\cup {\mathcal{F}}_{2}\\ {\mathcal{S}}_{i}\triangleq \left\{\text{thenode}i\text{successfullyretransmits}\right\}\phantom{\rule{0.5em}{0ex}}i\in \{1,\mathrm{...},n\}\\ =\{{\text{BC}}_{i}\infty ,{\text{BC}}_{i}=\mathrm{min}({\left\{{\text{BC}}_{m}\right\}}_{m=1}^{n})\phantom{\rule{0.5em}{0ex}}\\ \phantom{\rule{0.5em}{0ex}}\cup \{\text{if}\phantom{\rule{0.5em}{0ex}}\exists j\in \{1,\mathrm{...},n\},i\ne j:{\text{BC}}_{j}{\text{BC}}_{i},\text{then}\phantom{\rule{0.5em}{0ex}}\exists m\in \{1,\mathrm{...},n\},m\ne j,m\ne i:\\ \phantom{\rule{0.5em}{0ex}}{\text{BC}}_{j}={\text{BC}}_{m}\}\phantom{\rule{0.5em}{0ex}}i\in \{1,\mathrm{...},n\}\\ \mathcal{S}\triangleq \{\text{anodesuccessfullyretransmits}=}{\displaystyle \underset{i=1}{\overset{n}{\cup}}{\mathcal{S}}_{i}}\end{array}

The probabilities of the above defined events are the following:

\begin{array}{c}{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)\triangleq \mathsf{\text{P}}\left\{{\mathcal{S}}_{i}\right\}\phantom{\rule{1em}{0ex}}i=1,2,...,n\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{p}_{\mathsf{\text{succ}}}^{\left(n\right)}\triangleq \mathsf{\text{P}}\left\{\mathcal{S}\right\}=\sum _{i=1}^{n}{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{p}_{\mathsf{\text{fail}}}^{\left(n\right)}\triangleq 1-P\left\{\mathcal{S}\right\}=1-\sum _{i=1}^{n}{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right).\end{array}

Let us now introduce the random variable *Y* ∈ {0, 1, 2, ... , *n*} with the following PMF:

{P}_{Y}\left(y\right)=P\left\{Y=y\right\}=\left\{\begin{array}{cc}\hfill {p}_{\mathsf{\text{fail}}}^{\left(n\right)}\hfill & \hfill y=0\hfill \\ \hfill {p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(y\right)\hfill & \hfill y\in \left\{1,2,...,n\right\}.\hfill \end{array}\right.

Since the event {*Y* = 0} identifies the failure event, the random variable *Y* indicates either which node has effectively retransmitted or a failure. Moreover, it can be observed that:

\bigcup _{y=1}^{n}\left\{Y=y\right\}=\mathcal{F}\cup \phantom{\rule{0.3em}{0ex}}\mathcal{S}.

Obviously,

{P}_{Y}\left(y|\mathcal{S}\right)={P}_{Y}\left(Y=y|\mathcal{S}\right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill y=0\hfill \\ \hfill \frac{{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(x\right)}{{\sum}_{t=1}^{n}{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)}\hfill & \hfill y\in \left\{1,2,...,n\right\}.\hfill \end{array}\right.

In other words, if there is a retransmission (\mathcal{S}), then {P}_{Y}\left(y|\mathcal{S}\right)\phantom{\rule{2.77695pt}{0ex}}\left(y\in \left\{0,1,2,...,n\right\}\right) is the probability that the *y-th* node has retransmitted.

As shown in Appendix 2, the transmission probabilities \left\{{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)\right\} can be expressed as follows:

{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)={p}_{i}\sum _{m=1}^{n}{q}^{\left(m\right)}{p}_{{V}_{i}^{\left(n\right)}}\left(m-1\right)

(6)

where: *p*_{
i
}denotes the value of the PAF (4) for the *i*-th node and depends on the considered protocol; *q*^{(m)}is the probability that the *i*-th node wins the contention among a set of *m* competing nodes (the same for a given value of *n*); {V}_{i}^{\left(n\right)}\in \left\{0,...,n-1\right\} is the following discrete random variable:

{V}_{i}^{\left(n\right)}\triangleq \left\{\mathsf{\text{numberofnodes}},\mathsf{\text{amongthe}}n\mathsf{\text{nodes,competingwiththe}}i-\mathsf{\text{thnode}}\right\}.

The derivation of *q*^{(m)}and of the PMF of {V}_{i}^{\left(n\right)} can also be found in Appendix 2.

After deriving {p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right), it is possible to compute the per-hop delay, denoted as *D*_{
i
}, of a retransmission from the *i*-th node. Since the per-hop delay is meaningful only if the *i*-th node decides to retransmit, it is of interest to study the statistical distribution of *D*_{
i
}conditioned on *S*_{
i
}. For this reason, we introduce the random variable *D*_{i|i}, which can be defined as follows:

{D}_{i|i}\triangleq {T}_{\mathsf{\text{slot}}}\left(DIFS+{N}_{i|i}^{\mathsf{\text{bo}}}\right)+{T}^{\mathsf{\text{tx}}}\phantom{\rule{1em}{0ex}}i=1,...,n

where: *T*^{tx} (dimension: [s]) is the transmission time; *T*_{slot} (dimension: [s/slot]) is the deterministic duration of the backoff slot; *DIFS* (dimension: [slot]) is the duration of the DIFS; and {N}_{i|i}^{\mathsf{\text{bo}}} (dimension: [slots]) is the number of slots spent by the *i*-th node during the *backoff* (conditionally on the event *S*_{
i
}). We assume that both the packet size, defined as *P* (dimension: [bits]), and the transmission rate, denoted as *R* (dimension: [bits/s]), are constant, thus leading to a deterministic packet transmission time *T*^{tx} = *P*/*R*. Taking into account that *DIFS, T*_{slot}, and *T*^{tx} are deterministic, the average value of *D*_{i|i}becomes:

{\overline{D}}_{i|i}={T}_{\mathsf{\text{slot}}}\left(DIFS+{\overline{N}}_{i|i}^{\mathsf{\text{bo}}}\right)+{T}^{\mathsf{\text{tx}}}\phantom{\rule{1em}{0ex}}i=1,...,n

(7)

where, according to the derivation in Appendix 3,

{\overline{N}}_{i|i}^{bo}=\frac{{p}_{i}}{cw{p}_{\mathsf{\text{rtx}}}^{\left(N\right)}\left(i\right)}\sum _{v=0}^{N-1}{p}_{{V}_{i}}^{\left(N\right)}\left(v\right)\sum _{k=1}^{cw-1}\left[k\sum _{j=0}^{{J}_{k,v}}{P}_{v}^{\prime}\left(k,j\right)+{T}^{\mathsf{\text{tx}}}\sum _{j=1}^{{J}_{k,v}}j{P}_{v}^{\prime}\left(k,j\right)\right]

(8)

where {J}_{k,v}\triangleq \text{min}\left(k,\u230a\left(v/2\right)\u230b\right) denotes the maximum number of collisions that can happen in slots 0, 1, ..., *k*-1, while the matrix {P}_{v}^{\prime}=\left\{{P}_{v}\left(k,j\right)\right\} is defined in Appendix 3.

Proceeding in a similar manner, it is also possible to obtain the average number of retransmissions per-hop of the node *i*, denoted as {\overline{N}}_{\mathsf{\text{rtx}}}^{\mathsf{\text{hop}}}\left(i\right):

{\overline{N}}_{\mathsf{\text{rtx}}}^{\mathsf{\text{hop}}}\left(i\right)=\frac{{p}_{i}}{cw{p}_{\mathsf{\text{rtx}}}^{\left(N\right)}\left(i\right)}\left(1+\sum _{v=0}^{N-1}{p}_{{V}_{i}^{\left(N\right)}}\left(v\right)\sum _{k=1}^{cw-1}\sum _{h=2}^{v}h{N}_{k,v}\left(0,h\right)\sum _{j=0}^{{J}_{k,v}}{M}_{k,v}\left(j,h\right)\right)

(9)

where the matrices M_{
k,v
}= *M*_{
k,v
}(*j, h*) and N_{
k,v
}= *N*_{
k,v
}(*j, h*) are defined in Appendix 3.

### 6.2 Global performance analysis with fixed number of nodes

Once the per-TD performance has been analyzed (as described in Section 6.1), the global performance metrics introduced in Section 2.2 (namely, RE, TE, and D) can be computed by following a recursive approach, based on the inductive principle. This recursive approach is extensively described, for the evaluation of D, in Appendix 4, but can be directly re-adapted for the evaluation of RE and TE. In the remainder of this subsection, we outline the final results, trying to provide the reader with the intuition behind them.

Recall that we consider a deterministic scenario with a fixed number *N* of nodes equally spaced in the interval (0, *L*) ⊂ ℝ, where *L* = *z* ℓ_{norm}. For simplicity, we assume that a generic TD contains *n* = *N*/ℓ_{norm} nodes. This corresponds to a best-case scenario, where the farthest node of each TD is the domain forwarder (the "silencer," as denoted in Section 5).

*Delay* The computation of the average D is carried out taking into account only the packets successfully arriving at the end of the network (i.e., at the last reachable node) and ignoring the (remaining) packets which stop earlier. On the basis of the approach described in detail in Appendix 4, the average end-to-end delay can be given the following recursive formulation:

D\triangleq {\overline{D}}^{\left(N\right)}={\overline{T}}_{\mathsf{\text{src}}}^{\mathsf{\text{tx}}}+\sum _{i=1}^{n}\left({\overline{D}}^{\left(N-i\right)}+{\overline{D}}_{i|i}\right){p}_{Y}\left(i|\mathcal{S}\right)

(10)

where {\overline{D}}^{\left(N-i\right)} is the average delay in a network with *N* - *i* nodes and {\overline{T}}_{\mathsf{\text{src}}}^{\mathsf{\text{tx}}} is the average transmission time of the source, which differs from those of the following nodes, since the source does not contend with any other node and its transmission is not affected by collisions. Since the average time spent in the backoff is {\overline{T}}_{\mathsf{\text{src}}}^{\mathsf{\text{tx}}} can be expressed as

{\overline{T}}_{\mathsf{\text{src}}}^{\mathsf{\text{tx}}}\triangleq {T}^{\mathsf{\text{tx}}}+{T}_{\mathsf{\text{slot}}}\left(DIFS+\frac{cw-1}{2}\right).

(11)

*RE* The average RE can be defined as follows:

\mathsf{\text{RE}}\triangleq \frac{\overline{{N}_{\mathsf{\text{reach}}}}}{N}

(12)

where *N*_{reach} is a random variable denoting the number of nodes reached by a packet. As a consequence of our assumptions, *N*_{reach} is lower bounded by *n*, since the transmission from the source reaches *n* nodes (those of the first TD) with probability 1. The average value {\overline{N}}_{\mathsf{\text{reach}}} can be obtained by following the approach described in Appendix 4, but for the replacement of {p}_{Y}\left(i|\mathcal{S}\right) with *p*_{Y}(*i*) and of {\overline{D}}_{i|i} with the number of additional nodes covered by a new transmission. For example: a transmission from the 1-st node of the first TD will reach only one additional node (namely, the (*n* + 1)-th); a transmission from the 3-rd node will reach three additional nodes (namely, the (*n* + 1)-th, (*n* + 2)-th, and (*n* + 3)-th); and so on. Please note that, unlike the delay, in the computation of the RE we are not conditioning on the fact of reaching the *N*-th node of the network, i.e., the last reachable node of the network. Therefore, also the packets which stop being retransmitted are taken into account.

After the execution of the recursive approach outlined in Appendix 4, it is sufficient to add a constant equal to *n*, corresponding to the number of nodes directly reached by the source at the first hop. The final expression of {\overline{N}}_{\mathsf{\text{reach}}} becomes (using the notation of Appendix 4):

\begin{array}{ll}\hfill {\overline{N}}_{\mathsf{\text{reach}}}& ={\overline{N}}_{\mathsf{\text{reach}}}^{\left(N\right)}=n+\sum _{i=1}^{n}\left({\overline{N}}_{\mathsf{\text{reach}}}^{\left(N-i\right)}+i\right){p}_{Y}\left(i\right)\phantom{\rule{2em}{0ex}}\\ =n+\sum _{i=1}^{n}\left({\overline{N}}_{\mathsf{\text{reach}}}^{\left(N-i\right)}+i\right){p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right)\phantom{\rule{2em}{0ex}}\end{array}

(13)

where {\overline{N}}_{\mathsf{\text{reach}}}^{\left(N-i\right)} corresponds to the average number of nodes reached in a network with *N* - *i* nodes and can be recursively computed in the same way.

*TE* In order to reduce the computational burden, we adopt the following approximated formulation of TE:

\mathsf{\text{TE}}\triangleq \frac{\mathsf{\text{RE}}}{{N}_{\mathsf{\text{rtx}}}}

(14)

where {\overline{N}}_{\mathsf{\text{rtx}}} denotes the average overall number of retransmissions over all hops. From a computation viewpoint {\overline{N}}_{\mathsf{\text{rtx}}} is approximated by {\overline{N}}_{\mathsf{\text{rtx}}}^{{m}^{\left(*\right)}}, where *m** corresponds to the average number of reached nodes-it is a sort of approximated indicator of the "depth" of the propagation process. Since the RE can be interpreted as the ratio between the average number of reached nodes and the total number (*N*) of nodes, *m** can be approximated as follows:

m*\simeq N\cdot \mathsf{\text{RE}}.

At this point, {\overline{N}}_{\mathsf{\text{rtx}}}^{{m}^{\left(*\right)}} can be computed by applying the recursive approach presented in Appendix 4, by replacing (i) {p}_{Y}\left(i|\mathcal{S}\right) with *p*_{
Y
}(*i*) and (ii) {\overline{D}}_{i|i} with the average number of transmissions per hop, denoted by {\overline{N}}_{\mathsf{\text{rtx}}}^{\mathsf{\text{hop}}} and given in (9).

### 6.3 Generalization to a PPP-based scenario

According to the original PPP-based model, described in Section 2, the number of nodes within \mathcal{I}, denoted as *N*_{
z
}, has the following Poisson distribution:

{p}_{{N}_{z}}(n,{\rho}_{\text{s}}z)=\frac{{e}^{-{\rho}_{\text{s}}z}{({\rho}_{\text{s}}z)}^{n}}{n!}n\in \{0,1,2,\mathrm{...}\}.

However, since a real vehicle has a finite length, it is not possible to have an infinite number of vehicles within \mathcal{I}. Therefore, it makes sense to impose an arbitrary limit to the maximum number of nodes within \mathcal{I}, denoted as *N*_{
c
}. The new truncated Poisson random variable, denoted as {N}_{z}^{\prime}, has the following distribution:

{p}_{{{N}^{\prime}}_{z}}\left(n,{\rho}_{\mathsf{\text{s}}}z\right)=\frac{\frac{{e}^{-{\rho}_{\mathsf{\text{s}}}z}{\left({\rho}_{\mathsf{\text{s}}}z\right)}^{n}}{n!}}{{\sum}_{i=1}^{{N}_{c}}\frac{{e}^{-{\rho}_{\mathsf{\text{s}}}z}{\left({\rho}_{\mathsf{\text{s}}}z\right)}^{i}}{i!}}\phantom{\rule{1em}{0ex}}n\in \left\{1,2,...,{N}_{c}\right\}

where we have also removed the event *n* = 0--this would correspond to an empty TD.

In order to exploit the results of Section 6.1, the stochastic network topology of the PPP needs to be mapped into a deterministic one with equally spaced nodes. In order to do this, the interval \mathcal{I} is partitioned in *N*^{int} sub-intervals of length *z*/*N*^{int}, where *N*^{int} ∈ {*N*_{
c
}, *N*_{
c
}+ 1, *N*_{
c
}+ 2,...} is a design parameter. The computational burden and the accuracy are directly related to the value of *N*^{int}. After some numerical tests, we observed that the value *N*^{int} = 100 is a good tradeoff between precision and computational time. The *i*-th sub-interval thus is:

{\mathcal{I}}_{i}=\left[\frac{\left(i-i\right)z}{{N}^{\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}}},\frac{iz}{{N}^{\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}}}\right]\phantom{\rule{1em}{0ex}}i=1,2,...,{N}^{\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}}.

Every sub-interval can contain at most one node: in general, we assume that in each sub-interval there is a "virtual" node. Consequently, it is possible to associate a transmission probability {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right) to the generic sub-interval {\mathcal{I}}_{i}, defined as {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right), and a corresponding per-node delay, denoted as *D*(*i*)^{eq} (*i* = 1,..., *N*^{int}).

We define as {p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(j\right) the probability of retransmission of the *j*-th node, given that there are exactly *n* nodes in the interval \mathcal{I}. Using the total probability theorem, {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right) can be expressed as follows:

\begin{array}{ll}\hfill {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right)& =\sum _{n=1}^{{N}_{c}}\left({p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right)|{N}_{z}^{\prime}=n\right)P\left({N}_{z}^{\prime}=n\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{n=1}^{{N}_{c}}\sum _{j=1}^{n}{p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(j\right)f\left(i,j,n\right){p}_{{N}_{z}^{\prime}}\left(n,{\rho}_{\mathsf{\text{s}}}z\right)i\in \left\{1,...,{N}^{\mathsf{\text{int}}\phantom{\rule{1em}{0ex}}}\right\}\phantom{\rule{2em}{0ex}}\end{array}

(15)

where *f*(*i,j,n*) is an indicator function defined as follows:

f\left(i,j,n\right)\triangleq \left\{\begin{array}{cc}\hfill 1\hfill & \hfill {\overline{\mathsf{\text{R}}}}_{j}^{\left(n\right)}\in {\mathcal{I}}_{i}\hfill \\ \hfill 0\hfill & \hfill {\overline{\mathsf{\text{R}}}}_{j}^{\left(n\right)}\notin {\mathcal{I}}_{i}.\hfill \end{array}\right.

(16)

The probability {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right) is now a function of {p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right) (*n* ∈ {1, 2,..., *N*_{
c
}}, *i* ∈ {1, 2,..., n}), which can be computed with combinatorics, since it is associated with a deterministic scenario with *n* static nodes equally spaced in [0, *z*].

At this point, by using (6) in Equation (15), it is possible to obtain a closed-form expression for {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right). Leveraging on the knowledge of {p}_{\mathsf{\text{rtx}}}^{\mathsf{\text{eq}}}\left(i\right), by using Equations (15) into (7) and (9), it is possible to obtain, respectively, *D*(*i*)^{eq} (*i* = 1,...,*N*^{int}) and {n}_{\mathsf{\text{rtx}}}^{\mathsf{\text{ho}}{\mathsf{\text{p}}}^{\mathsf{\text{eq}}}}. Then, it is possible to use the framework presented in Section 6.2 to derive RE, TE, and D for a deterministic network composed by *N*_{c}ℓ_{norm} nodes, since *N*_{
c
}is the (imposed) number of nodes in the interval \mathcal{I} (and, thus, in each TD).

As anticipated at the end of Section 1, we remark that the presented analytical framework can be employed to study other types of broadcast protocols, not necessarily probabilistic, by simply re-adapting the definition of {p}_{\mathsf{\text{rtx}}}^{\left(n\right)}\left(i\right) and *D*_{i|i}. This is the subject of our current research activities.