Proof of Node Densities

In this paper we present an analytical model accurately describing the forwarding behaviour of a multi-hop broadcast protocol. Our model covers the scenario in which a message is forwarded over a straight road and inter-node distances are distributed exponentially. Intermediate forwarders draw a small random delay before forwarding a message such as is done in flooding protocols to avoid the broadcast storm problem. The analytical model presented in this chapter focuses on having a message forwarded a specific distance. For a given forwarding distance and a given node density our model analysis is able to capture the full distribution of \emph{(i)} the end-to-end delay to have the message forwarded the entire distance, \emph{(ii)} the required number of hops to have the message forwarded the entire distance, \emph{(iii)} the position of each intermediate forwarder, \emph{(iv)} the success probability of each hop, \emph{(v)} the length of each hop, and \emph{(vi)} the delay of each hop. The first three metrics are calculated assuming that the message is successfully forwarded the entire forwarding distance. The model provides the results in terms of insightful, fast-to-evaluate closed-form expressions. The model has been validated by extensive simulations: modelling results stayed within typically 10\%, depending on the source-to-sink distance and the node density