Two-Hop Connectivity to the Roadside in a VANET Under the Random Connection Model

We compute the expected number of cars that have at least one two-hop path to a fixed roadside unit in a one-dimensional vehicular ad hoc network in which other cars can be used as relays to reach a roadside unit when they do not have a reliable direct link. The pairwise channels between cars experience Rayleigh fading in the random connection model, and so exist, with probability function of the mutual distance between the cars, or between the cars and the roadside unit. We derive exact equivalents for this expected number of cars when the car density $\rho$ tends to zero and to infinity, and determine its behaviour using an infinite oscillating power series in $\rho$, which is accurate for all regimes. We also corroborate those findings to a realistic situation, using snapshots of actual traffic data. Finally, a normal approximation is discussed for the probability mass function of the number of cars with a two-hop connection to the origin. The probability mass function appears to be well fitted by a Gaussian approximation with mean equal to the expected number of cars with two hops to the origin.Comment: 21 pages, 7 figure

Large-scale Join-Idle-Queue system with general service times

A parallel server system with $n$ identical servers is considered. The service time distribution has a finite mean $1/\mu$, but otherwise is arbitrary. Arriving customers are be routed to one of the servers immediately upon arrival. Join-Idle-Queue routing algorithm is studied, under which an arriving customer is sent to an idle server, if such is available, and to a randomly uniformly chosen server, otherwise. We consider the asymptotic regime where $n\to\infty$ and the customer input flow rate is $\lambda n$. Under the condition $\lambda/\mu<1/2$, we prove that, as $n\to\infty$, the sequence of (appropriately scaled) stationary distributions concentrates at the natural equilibrium point, with the fraction of occupied servers being constant equal $\lambda/\mu$. In particular, this implies that the steady-state probability of an arriving customer waiting for service vanishes.Comment: Revision. 11 page