Dynamic Connectivity and Packet Propagation Delay in ALOHA Wireless Networks

We consider an ad hoc network which uses multihop and slotted ALOHA for its MAC contention. We then formulate the minimum time required for a packet to reach the destination from the origin. We define this delay as the minimum time required for a causal path to form between the source and destination. We derive the distributional properties of the connection time using ideas from first passage percolation and epidemic processes. We show that the delay scales linearly with the distance and also provide asymptotic results (w.r.t time) for the positions of the nodes which are able to connect to a transmitter located at the origin. We also provide simulation results to support the theoretical results.