1,899 research outputs found
Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution
We consider a heterogeneous queueing system consisting of one large pool of
identical servers, where is the scaling parameter. The
arriving customers belong to one of several classes which determines the
service times in the distributional sense. The system is heavily loaded in the
Halfin-Whitt sense, namely the nominal utilization is where
is the spare capacity parameter. Our goal is to obtain bounds on the
steady state performance metrics such as the number of customers waiting in the
queue . While there is a rich literature on deriving process level
(transient) scaling limits for such systems, the results for steady state are
primarily limited to the single class case.
This paper is the first one to address the case of heterogeneity in the
steady state regime. Moreover, our results hold for any service policy which
does not admit server idling when there are customers waiting in the queue. We
assume that the interarrival and service times have exponential distribution,
and that customers of each class may abandon while waiting in the queue at a
certain rate (which may be zero). We obtain upper bounds of the form
on both and the number of idle servers. The bounds
are uniform w.r.t. parameter and the service policy. In particular, we show
that . Therefore, the
sequence is tight and has a uniform exponential tail
bound. We further consider the system with strictly positive abandonment rates,
and show that in this case every weak limit of
has a sub-Gaussian tail. Namely .Comment: 21 page
SELFISHMIGRATE: A Scalable Algorithm for Non-clairvoyantly Scheduling Heterogeneous Processors
We consider the classical problem of minimizing the total weighted flow-time
for unrelated machines in the online \emph{non-clairvoyant} setting. In this
problem, a set of jobs arrive over time to be scheduled on a set of
machines. Each job has processing length , weight , and is
processed at a rate of when scheduled on machine . The online
scheduler knows the values of and upon arrival of the job,
but is not aware of the quantity . We present the {\em first} online
algorithm that is {\em scalable} ((1+\eps)-speed
-competitive for any constant \eps > 0) for the
total weighted flow-time objective. No non-trivial results were known for this
setting, except for the most basic case of identical machines. Our result
resolves a major open problem in online scheduling theory. Moreover, we also
show that no job needs more than a logarithmic number of migrations. We further
extend our result and give a scalable algorithm for the objective of minimizing
total weighted flow-time plus energy cost for the case of unrelated machines
and obtain a scalable algorithm. The key algorithmic idea is to let jobs
migrate selfishly until they converge to an equilibrium. Towards this end, we
define a game where each job's utility which is closely tied to the
instantaneous increase in the objective the job is responsible for, and each
machine declares a policy that assigns priorities to jobs based on when they
migrate to it, and the execution speeds. This has a spirit similar to
coordination mechanisms that attempt to achieve near optimum welfare in the
presence of selfish agents (jobs). To the best our knowledge, this is the first
work that demonstrates the usefulness of ideas from coordination mechanisms and
Nash equilibria for designing and analyzing online algorithms
Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control
We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA)
problem to design optimal traffic flow controls for freeway networks as modeled
by the Cell Transmission Model, using variable speed limit, ramp metering, and
routing. We consider two optimal control problems: the DTA problem, where
turning ratios are part of the control inputs, and the Freeway Network Control
(FNC), where turning ratios are instead assigned exogenous parameters. It is
known that relaxation of the supply and demand constraints in the cell-based
formulations of the DTA problem results in a linear program. However, solutions
to the relaxed problem can be infeasible with respect to traffic dynamics.
Previous work has shown that such solutions can be made feasible by proper
choice of ramp metering and variable speed limit control for specific traffic
networks. We extend this procedure to arbitrary networks and provide insight
into the structure and robustness of the proposed optimal controllers. For a
network consisting only of ordinary, merge, and diverge junctions, where the
cells have linear demand functions and affine supply functions with identical
slopes, and the cost is the total traffic volume, we show, using the maximum
principle, that variable speed limits are not needed in order to achieve
optimality in the FNC problem, and ramp metering is sufficient. We also prove
bounds on perturbation of the controlled system trajectory in terms of
perturbations in initial traffic volume and exogenous inflows. These bounds,
which leverage monotonicity properties of the controlled trajectory, are shown
to be in close agreement with numerical simulation results
Convergence of the D-iteration algorithm: convergence rate and asynchronous distributed scheme
In this paper, we define the general framework to describe the diffusion
operators associated to a positive matrix. We define the equations associated
to diffusion operators and present some general properties of their state
vectors. We show how this can be applied to prove and improve the convergence
of a fixed point problem associated to the matrix iteration scheme, including
for distributed computation framework. The approach can be understood as a
decomposition of the matrix-vector product operation in elementary operations
at the vector entry level.Comment: 9 page
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