We consider a system consisting of N parallel servers, where jobs with different resource requirements arrive and are assigned to the servers for processing. Each server has a finite resource capacity and therefore can serve only a finite number of jobs at a time. We assume that different servers have different resource capacities. A job is accepted for processing only if the resource requested by the job is available at the server to which it is assigned. Otherwise, the job is discarded or blocked. We consider randomized schemes to assign jobs to servers with the aim of reducing the average blocking probability of jobs in the system. In particular, we consider a scheme that assigns an incoming job to the server having maximum available vacancy or unused resource among d randomly sampled servers. We consider the system in the limit where both the number of servers and the arrival rates of jobs are scaled by a large factor. This gives rise to a mean field analysis. We show that in the limiting system the servers behave independentlya property termed as propagation of chaos. Stationary tail probabilities of server occupancies are obtained from the stationary solution of the mean field which is shown to be unique and globally attractive. We further characterize the rate of decay of the stationary tail probabilities. Numerical results suggest that the proposed scheme significantly reduces the average blocking probability of jobs as compared to static schemes that probabilistically route jobs to servers independently of their states.

We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. We focus on the scenario where the agents are biased towards one of the opinions called the preferred opinion. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is $$\varTheta (\log N)$$Θ ( log N ) , where N is the number of agents in the network. Furthermore, under the majority rule model, we show that consensus can be achieved on the preferred opinion with high probability even if it is initially the opinion of the minority. We also study the majority rule model when stubborn agents with fixed opinions are present. We find that the stationary distribution of opinions in the network in the large system limit using mean field techniques.

For broadcast channels, a power allocation scheme is proposed to maximize the number of active receivers, for each of which, a minimum rate R_min > 0 can be achieved. Under the assumption of independent Rayleigh-fading channels for different receivers, as the total number of receivers n goes to infinity, the maximum number of active receivers is shown to be arbitrarily close to ln(Pln n)/R_min with probability approaching 1, where P is the total transmit power.

We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in ${\mathbb{R}}^n$ or is restricted to a given compact subset of ${\mathbb{R}}^n$. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of $n-1$ or less. When the input is restricted to a compact subset of ${\mathbb{R}}^n$, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in ${\mathbb{R}}^n$.