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AbstradIt is shown that the position of any fixed percentile of the maximal displacement of standard branching Brownian motion in one dimension is 2112 t -3 T 3 1 2 log t + O( 1) at time t, the secondorder term having been previously unknown. This determines (to within O(1)) the position of the travelling wave of the semilinear heat equation, u, =lux, +f(u), in the classic paper by Kolmogorov-Petrovsky-Piscounov, "Etude de l'iquations de la diffusion avec croissance de la quantiti de la matitre et son application a un probltme biologique", 1937.

We consider the discrete two-dimensional Gaussian free field on a box of side length N , with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.

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We obtain rigorous bounds on the long-time behavior of the densities PAU) and peit) of species A and B, which diffuse and annihilate upon meeting, i.e., A+B-•inert.For equal initial densities p^(0) = p#(0), the density goes to zero asymptotically as t ~d lA for dimensions d<4 and as t~x for 4. When PA(0) : 3.

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Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-αN Poisson process, with α < 1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. [19], it was shown that when each arriving job is assigned to the shortest of D, D ≥ 2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where queue sizes decay exponentially.The reasoning in [19] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. [5]. The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N → ∞. This allows computation of queue size distributions and other performance measures of interest.In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult prob- lem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

Randomized load balancing greatly improves the sharing of resources in a number of applications while being simple to implement. One model that has been extensively used to study randomized load balancing schemes is the supermarket model. In this model, jobs arrive according to a rate-nλ Poisson process at a bank of n rate-1 exponential server queues. A notable result, due to Vvedenskaya et.al. (1996), showed that when each arriving job is assigned to the shortest of d ≥ 2 randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as n → ∞. This is a substantial improvement over the case d = 1, where queue sizes decay exponentially.The method of analysis used in the above paper and in the subsequent literature applies to jobs with exponential service time distributions and does not easily generalize. It is desirable to study load balancing models with more general, especially heavy-tailed, service time distributions since such service times occur widely in practice.This paper describes a modularized program for treating randomized load balancing problems with general service time distributions and service disciplines. The program relies on an ansatz which asserts that any finite set of queues in a randomized load balancing scheme becomes independent as n → ∞. This allows one to derive queue size distributions and other performance measures of interest. We establish the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate (this includes heavy-tailed service times). Assuming the ansatz, we also obtain the following results: (i) as n → ∞, the process of job arrivals at any fixed queue tends to a Poisson process whose rate depends on the size of the queue, (ii) when the service discipline at each server is processor sharing or LIFO with preemptive resume, the distribution of the number of jobs is insensitive to the service distribution, and (iii) the tail behavior of the queue-size distribution in terms of the service distribution for the FIFO service discipline.