80 research outputs found
Some new results on sample path optimality in ergodic control of diffusions
We present some new results on sample path optimality for the ergodic control
problem of a class of non-degenerate diffusions controlled through the drift.
The hypothesis most often used in the literature to ensure the existence of an
a.s. sample path optimal stationary Markov control requires finite second
moments of the first hitting times of bounded domains over all
admissible controls. We show that this can be considerably weakened: may be replaced with , thus reducing
the required rate of convergence of averages from polynomial to logarithmic. A
Foster-Lyapunov condition which guarantees this is also exhibited. Moreover, we
study a large class of models that are neither uniformly stable, nor have a
near-monotone running cost, and we exhibit sufficient conditions for the
existence of a sample path optimal stationary Markov control.Comment: 10 page
A variational formula for risk-sensitive control of diffusions in
We address the variational problem for the generalized principal eigenvalue
on of linear and semilinear elliptic operators associated with
nondegenerate diffusions controlled through the drift. We establish the
Collatz-Wielandt formula for potentials that vanish at infinity under minimal
hypotheses, and also for general potentials under blanket geometric ergodicity
assumptions. We also present associated results having the flavor of a refined
maximum principle.Comment: 19 page
Infinite Horizon Average Optimality of the N-network Queueing Model in the Halfin-Whitt Regime
We study the infinite horizon optimal control problem for N-network queueing
systems, which consist of two customer classes and two server pools, under
average (ergodic) criteria in the Halfin-Whitt regime. We consider three
control objectives: 1) minimizing the queueing (and idleness) cost, 2)
minimizing the queueing cost while imposing a constraint on idleness at each
server pool, and 3) minimizing the queueing cost while requiring fairness on
idleness. The running costs can be any nonnegative convex functions having at
most polynomial growth.
For all three problems we establish asymptotic optimality, namely, the
convergence of the value functions of the diffusion-scaled state process to the
corresponding values of the controlled diffusion limit. We also present a
simple state-dependent priority scheduling policy under which the
diffusion-scaled state process is geometrically ergodic in the Halfin-Whitt
regime, and some results on convergence of mean empirical measures which
facilitate the proofs.Comment: 35 page
Infinite horizon asymptotic average optimality for large-scale parallel server networks
We study infinite-horizon asymptotic average optimality for parallel server
network with multiple classes of jobs and multiple server pools in the
Halfin-Whitt regime. Three control formulations are considered: 1) minimizing
the queueing and idleness cost, 2) minimizing the queueing cost under a
constraints on idleness at each server pool, and 3) fairly allocating the idle
servers among different server pools. For the third problem, we consider a
class of bounded-queue, bounded-state (BQBS) stable networks, in which any
moment of the state is bounded by that of the queue only (for both the limiting
diffusion and diffusion-scaled state processes). We show that the optimal
values for the diffusion-scaled state processes converge to the corresponding
values of the ergodic control problems for the limiting diffusion. We present a
family of state-dependent Markov balanced saturation policies (BSPs) that
stabilize the controlled diffusion-scaled state processes. It is shown that
under these policies, the diffusion-scaled state process is exponentially
ergodic, provided that at least one class of jobs has a positive abandonment
rate. We also establish useful moment bounds, and study the ergodic properties
of the diffusion-scaled state processes, which play a crucial role in proving
the asymptotic optimality.Comment: 35 pages. arXiv admin note: text overlap with arXiv:1602.0327
Ergodic Diffusion Control of Multiclass Multi-Pool Networks in the Halfin-Whitt Regime
We consider Markovian multiclass multi-pool networks with heterogeneous
server pools, each consisting of many statistically identical parallel servers,
where the bipartite graph of customer classes and server pools forms a tree.
Customers form their own queue and are served in the first-come first-served
discipline, and can abandon while waiting in queue. Service rates are both
class and pool dependent. The objective is to study the limiting diffusion
control problems under the long run average (ergodic) cost criteria in the
Halfin--Whitt regime. Two formulations of ergodic diffusion control problems
are considered: (i) both queueing and idleness costs are minimized, and (ii)
only the queueing cost is minimized while a constraint is imposed upon the
idleness of all server pools. We develop a recursive leaf elimination algorithm
that enables us to obtain an explicit representation of the drift for the
controlled diffusions. Consequently, we show that for the limiting controlled
diffusions, there always exists a stationary Markov control under which the
diffusion process is geometrically ergodic. The framework developed in our
earlier work is extended to address a broad class of ergodic diffusion control
problems with constraints. We show that that the unconstrained and constrained
problems are well posed, and we characterize the optimal stationary Markov
controls via HJB equations.Comment: 32 page
A counterexample to a nonlinear version of the Krein-Rutman theorem by R. Mahadevan
In this short note we present a simple counterexample to a nonlinear version
of the Krein-Rutman theorem reported in [Nonlinear Anal. 11 (2007), 3084-3090].
Correct versions of this theorem, and related results for superadditive maps
are also presented.Comment: To appear in Nonlinear Analysis, 6 page
On a Class of Stochastic Differential Equations With Jumps and Its Properties
We study stochastic differential equations with jumps with no diffusion part.
We provide some basic stochastic characterizations of solutions of the
corresponding non-local partial differential equations and prove the Harnack
inequality for a class of these operators. We also establish key connections
between the recurrence properties of these jump processes and the non-local
partial differential operator. One of the key results is the regularity of
solutions of the Dirichlet problem for a class of operators with locally weakly
H\"older continuous kernels.Comment: 40 page
On the Capacity of Multiple Access Channels with State Information and Feedback
In this paper, the multiple access channel (MAC) with channel state is
analyzed in a scenario where a) the channel state is known non-causally to the
transmitters and b) there is perfect causal feedback from the receiver to the
transmitters. An achievable region and an outer bound are found for a discrete
memoryless MAC that extend existing results, bringing together ideas from the
two separate domains of MAC with state and MAC with feedback. Although this
achievable region does not match the outer bound in general, special cases
where they meet are identified.
In the case of a Gaussian MAC, a specialized achievable region is found by
using a combination of dirty paper coding and a generalization of the
Schalkwijk-Kailath, Ozarow and Merhav-Weissman schemes, and this region is
found to be capacity achieving. Specifically, it is shown that additive
Gaussian interference that is known non-causally to the transmitter causes no
loss in capacity for the Gaussian MAC with feedback.Comment: Prelimenary result appears in ISIT 200
Strict monotonicity of principal eigenvalues of elliptic operators in and risk-sensitive control
This paper studies the eigenvalue problem on for a class of
second order, elliptic operators of the form , associated with
non-degenerate diffusions. We show that strict monotonicity of the principal
eigenvalue of the operator with respect to the potential function fully
characterizes the ergodic properties of the associated ground state diffusion,
and the unicity of the ground state, and we present a comprehensive study of
the eigenvalue problem from this point of view. This allows us to extend or
strengthen various results in the literature for a class of viscous
Hamilton-Jacobi equations of ergodic type with smooth coefficients to equations
with measurable drift and potential. In addition, we establish the strong
duality for the equivalent infinite dimensional linear programming formulation
of these ergodic control problems. We also apply these results to the study of
the infinite horizon risk-sensitive control problem for diffusions, and
establish existence of optimal Markov controls, verification of optimality
results, and the continuity of the controlled principal eigenvalue with respect
to stationary Markov controls.Comment: 45 page
Ergodicity of L\'evy-driven SDEs arising from multiclass many-server queues
We study the ergodic properties of a class of multidimensional piecewise
Ornstein-Uhlenbeck processes with jumps, which contains the limit of the
queueing processes arising in multiclass many-server queues with heavy-tailed
arrivals and/or asymptotically negligible service interruptions in the
Halfin-Whitt regime as special cases. In these queueing models, the It\^o
equations have a piecewise linear drift, and are driven by either (1) a
Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy
process with independent one-dimensional symmetric -stable components,
or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process.
We also study the class of models driven by a subordinate Brownian motion,
which contains an isotropic (or rotationally invariant) -stable L\'evy
process as a special case. We identify conditions on the parameters in the
drift, the L\'evy measure and/or covariance function which result in
subexponential and/or exponential ergodicity. We show that these assumptions
are sharp, and we identify some key necessary conditions for the process to be
ergodic. In addition, we show that for the queueing models described above with
no abandonment, the rate of convergence is polynomial, and we provide a sharp
quantitative characterization of the rate via matching upper and lower bounds.Comment: 42 page
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