Timeliness of Slotted Aloha-Based Wireless Broadcasting and Flooding

In this paper, we consider a machine-type communication (MTC) system where the sensor nodes aim to spread their observed updates over the whole network through slotted Aloha based broadcasting and flooding. We investigate the timeliness of the broadcasting process and the rapidity of the flooding process from the perspective of transmitters, which are referred to as the broadcast age of information (bAoI) and the spreading velocity, respectively. Specifically, bAoI quantifies the age of the latest successfully broadcasted packet of each node, and the spreading velocity characterizes the message moving rapidity when the nodes spread their packets over the network. To this end, we develop an analytical transmission model to derive the transmission probability, the collision probability, and the service rate of the nodes. We also propose an analytical traffic model for the network by using the Poisson approximation method. With these two models, we obtain the average bAoI and the spreading velocity over the network in closed-form. Our results show that 1) the total traffic rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$</tex-math> </inline-formula> (including the forwarding and original traffic) of an averaged number of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n_{2r}$</tex-math> </inline-formula> nodes within the interfering range is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n_{2r}}p\lt \exp (-1)$</tex-math> </inline-formula>; 2) the broadcasting capacity (maximum original traffic rate of each node) achieves its maximum when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n_{2r}}\tau =1$</tex-math> </inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\tau $</tex-math> </inline-formula> is the probability of transmission attempt of each node in each slot; 3) as the original traffic rate is increased from zero to the broadcasting capacity, the spatial spreading velocity of message over the network decreases from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2r/3$</tex-math> </inline-formula> to zero, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$</tex-math> </inline-formula> is the transmit range; 4) the average bAoI is explicitly convex in the original traffic rate and is increasing with the transmission range of each node, while the spreading velocity is concave in the transmission range and decreases as the original traffic rate increases. Therefore, the temporal metric of bAoI and the spatial index of spreading velocity exhibit a kind of duality. Moreover, they can be optimized by adjusting the traffic rate and the transmission range, respectively.