147 research outputs found
Necessary condition for null controllability in many-server heavy traffic
Throughput sub-optimality (TSO), introduced in Atar and Shaikhet [Ann. Appl.
Probab. 19 (2009) 521-555] for static fluid models of parallel queueing
networks, corresponds to the existence of a resource allocation, under which
the total service rate becomes greater than the total arrival rate. As shown in
Atar, Mandelbaum and Shaikhet [Ann. Appl. Probab. 16 (2006) 1764-1804] and Atar
and Shaikhet (2009), in the many server Halfin-Whitt regime, TSO implies null
controllability (NC), the existence of a routing policy under which, for every
finite , the measure of the set of times prior to , at which at least one
customer is in the buffer, converges to zero in probability at the scaling
limit. The present paper investigates the question whether the converse
relation is also true and TSO is both sufficient and necessary for the NC
behavior. In what follows we do get the affirmation for systems with either two
customer classes (and possibly more service pools) or vice-versa and state a
condition on the underlying static fluid model that allows the extension of the
result to general structures.Comment: Published in at http://dx.doi.org/10.1214/13-AAP1001 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation
A nonlinear stochastic differential equation with the order of nonlinearity
higher than one, with several discrete and distributed delays and time varying
coefficients is considered. It is shown that the sufficient conditions for
exponential mean square stability of the linear part of the considered
nonlinear equation also are sufficient conditions for stability in probability
of the initial nonlinear equation. Some new sufficient condition of stability
in probability for the zero solution of the considered nonlinear non-autonomous
stochastic differential equation is obtained which can be considered as a
multi-condition of stability because it allows to get for one considered
equation at once several different complementary of each other sufficient
stability conditions. The obtained results are illustrated with numerical
simulations and figures.Comment: Published at https://doi.org/10.15559/18-VMSTA110 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Queueing systems with many servers: Null controllability in heavy traffic
A queueing model has heterogeneous service stations, each consisting
of many independent servers with identical capabilities. Customers of
classes can be served at these stations at different rates, that depend on both
the class and the station. A system administrator dynamically controls
scheduling and routing. We study this model in the central limit theorem (or
heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model
on with a singular control term that describes the scaling
limit of the queueing model. The singular term may be used to constrain the
diffusion to lie in certain subsets of at all times . We
say that the diffusion is null-controllable if it can be constrained to
, the minimal closed subset of containing all
states of the prelimit queueing model for which all queues are empty. We give
sufficient conditions for null controllability of the diffusion. Under these
conditions we also show that an analogous, asymptotic result holds for the
queueing model, by constructing control policies under which, for any given
, all queues in the system are kept empty on the time
interval , with probability approaching one. This introduces a
new, unusual heavy traffic ``behavior'': On one hand, the system is critically
loaded, in the sense that an increase in any of the external arrival rates at
the ``fluid level'' results with an overloaded system. On the other hand, as
far as queue lengths are concerned, the system behaves as if it is underloaded.Comment: Published at http://dx.doi.org/10.1214/105051606000000358 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Stability of a stochastically perturbed model of intracellular single-stranded RNA virus replication
Replication of single-stranded RNA virus can be complicated, compared to that
of double-stranded virus, as it require production of intermediate antigenomic
strands that then serve as template for the genomic-sense strands. Moreover,
for ssRNA viruses, there is a variability of the molecular mechanism by which
genomic strands can be replicated. A combination of such mechanisms can also
occur: a fraction of the produced progeny may result from a stamping-machine
type of replication that uses the parental genome as template, whereas others
may result from the replication of progeny genomes. F. Mart\'{\i}nez et al. and
J. Sardany\'{e}s at al. suggested a deterministic ssRNA virus intracellular
replication model that allows for the variability in the replication
mechanisms.
To explore how stochasticity can affect this model principal properties, in
this paper we consider the stability of a stochastically perturbed model of
ssRNA virus replication within a cell. Using the direct Lyapunov method, we
found sufficient conditions for the stability in probability of equilibrium
states for this model. This result confirms that this heterogeneous model of
single-stranded RNA virus replication is stable with respect to stochastic
perturbations of the environment
Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation
A nonlinear stochastic differential equation with the order of nonlinearity
higher than one, with several discrete and distributed delays and time varying
coefficients is considered. It is shown that the sufficient conditions for
exponential mean square stability of the linear part of the considered
nonlinear equation also are sufficient conditions for stability in probability
of the initial nonlinear equation. Some new sufficient condition of stability
in probability for the zero solution of the considered nonlinear non-autonomous
stochastic differential equation is obtained which can be considered as a
multi-condition of stability because it allows to get for one considered
equation at once several different complementary of each other sufficient
stability conditions. The obtained results are illustrated with numerical
simulations and figures.Comment: Published at https://doi.org/10.15559/18-VMSTA110 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Optimal control of Volterra type stochastic difference equations
AbstractMany processes in automatic regulation, physics, etc. can be modelled by stochastic difference equations. One of the main problems of the theory of difference equations and their applications is connected with stability and optimal control [1]. In this paper we discuss the optimal control of second-kind Volterra type stochastic difference equations. In [2–9] for Volterra type stochastic integral equations, analogous results were obtained
Power Strip Packing of Malleable Demands in Smart Grid
We consider a problem of supplying electricity to a set of
customers in a smart-grid framework. Each customer requires a certain amount of
electrical energy which has to be supplied during the time interval . We
assume that each demand has to be supplied without interruption, with possible
duration between and , which are given system parameters (). At each moment of time, the power of the grid is the sum of all the
consumption rates for the demands being supplied at that moment. Our goal is to
find an assignment that minimizes the {\it power peak} - maximal power over
- while satisfying all the demands. To do this first we find the lower
bound of optimal power peak. We show that the problem depends on whether or not
the pair belongs to a "good" region . If it does - then
an optimal assignment almost perfectly "fills" the rectangle with being the sum of all the energy demands - thus
achieving an optimal power peak . Conversely, if do not belong to
, we identify the lower bound on the optimal value of
power peak and introduce a simple linear time algorithm that almost perfectly
arranges all the demands in a rectangle
and show that it is asymptotically optimal
Critically loaded queueing models that are throughput suboptimal
This paper introduces and analyzes the notion of throughput suboptimality for
many-server queueing systems in heavy traffic. The queueing model under
consideration has multiple customer classes, indexed by a finite set
, and heterogenous, exponential servers. Servers are dynamically
chosen to serve customers, and buffers are available for customers waiting to
be served. The arrival rates and the number of servers are scaled up in such a
way that the processes representing the number of class- customers in the
system, , fluctuate about a static fluid model, that is
assumed to be critically loaded in a standard sense. At the same time, the
fluid model is assumed to be throughput suboptimal. Roughly, this means that
the servers can be allocated so as to achieve a total processing rate that is
greater than the total arrival rate. We show that there exists a dynamic
control policy for the queueing model that is efficient in the following strong
sense: Under this policy, for every finite , the measure of the set of times
prior to , at which at least one customer is in the buffer, converges to
zero in probability as the arrival rates and number of servers go to infinity.
On the way to prove our main result, we provide a characterization of
throughput suboptimality in terms of properties of the buffer-station graph.Comment: Published in at http://dx.doi.org/10.1214/08-AAP551 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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