Persistent-Idle Load-Distribution

A parallel server system is considered in which a dispatcher routes incoming jobs to a fixed number of heterogeneous servers, each with its own queue. Much effort has been previously made to design policies that use limited state information (e.g., the queue lengths in a small subset of the set of servers, or the identity of the idle servers). However, existing policies either do not achieve the stability region or perform poorly in terms of job completion time. We introduce Persistent-Idle (PI), a new, perhaps counterintuitive, load-distribution policy that is designed to work with limited state information. Roughly speaking, PI always routes to the server that has last been idle. Our main result is that this policy achieves the stability region. Because it operates quite differently from existing policies, our proof method differs from standard arguments in the literature. Specifically, large time properties of reflected random walk, along with a careful choice of a Lyapunov function, are combined to obtain a Lyapunov condition over sufficiently long-time intervals. We also provide simulation results that indicate that job completion times under PI are low for different choices of system parameters, compared with several state-of-the-art load-distribution schemes.

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDEs, where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

A Continuous Semigroup Approach to the Distributional Stability of Nonlinear Models

We prove the existence of an invariant measure for the continuous semigroup associate with a nonlinear model under the compact set Lyapunov condition. Further,adding the ergodicity of the semigroup operator, we prove the asymptotic stability in distribution for the semigroup. We give a criteria of the asymptotic stability in distribution for the type of evolution equation having a linear generator. Our method is based on continuous semigroup and its generator.We illustrate the result by the Lorenz chaotic model and prove the existence of the natural invariant measure for Lorenz chaotic model.