558 research outputs found
Simple and explicit bounds for multi-server queues with (and sometimes better) scaling
We consider the FCFS queue, and prove the first simple and explicit
bounds that scale as (and sometimes better). Here
denotes the corresponding traffic intensity. Conceptually, our results can be
viewed as a multi-server analogue of Kingman's bound. Our main results are
bounds for the tail of the steady-state queue length and the steady-state
probability of delay. The strength of our bounds (e.g. in the form of tail
decay rate) is a function of how many moments of the inter-arrival and service
distributions are assumed finite. More formally, suppose that the inter-arrival
and service times (distributed as random variables and respectively)
have finite th moment for some Let (respectively )
denote (respectively ). Then
our bounds (also for higher moments) are simple and explicit functions of
, and
only. Our bounds scale gracefully even when the number of
servers grows large and the traffic intensity converges to unity
simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale
better than in certain asymptotic regimes. More precisely,
they scale as multiplied by an inverse polynomial in These results formalize the intuition that bounds should be tighter
in light traffic as well as certain heavy-traffic regimes (e.g. with
fixed and large). In these same asymptotic regimes we also prove bounds for
the tail of the steady-state number in service.
Our main proofs proceed by explicitly analyzing the bounding process which
arises in the stochastic comparison bounds of amarnik and Goldberg for
multi-server queues. Along the way we derive several novel results for suprema
of random walks and pooled renewal processes which may be of independent
interest. We also prove several additional bounds using drift arguments (which
have much smaller pre-factors), and make several conjectures which would imply
further related bounds and generalizations
RESTLESS BANDIT MARGINAL PRODUCTIVITY INDICES II: MULTIPROJECT CASE AND SCHEDULING A MULTICLASS MAKE-TO-ORDER/-STOCK M/G/1 QUEUE
This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author´s companion paper “Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue”, based on characterization of a project´s marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)´s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter´s near-optimality across the cases investigated.
On Monotonicity and Propagation of Order Properties
In this paper, a link between monotonicity of deterministic dynamical systems
and propagation of order by Markov processes is established. The order
propagation has received considerable attention in the literature, however,
this notion is still not fully understood. The main contribution of this paper
is a study of the order propagation in the deterministic setting, which
potentially can provide new techniques for analysis in the stochastic one. We
take a close look at the propagation of the so-called increasing and increasing
convex orders. Infinitesimal characterisations of these orders are derived,
which resemble the well-known Kamke conditions for monotonicity. It is shown
that increasing order is equivalent to the standard monotonicity, while the
class of systems propagating the increasing convex order is equivalent to the
class of monotone systems with convex vector fields. The paper is concluded by
deriving a novel result on order propagating diffusion processes and an
application of this result to biological processes.Comment: Part of the paper is to appear in American Control Conference 201
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