716 research outputs found
Many-Sources Large Deviations for Max-Weight Scheduling
In this paper, a many-sources large deviations principle (LDP) for the
transient workload of a multi-queue single-server system is established where
the service rates are chosen from a compact, convex and coordinate-convex rate
region and where the service discipline is the max-weight policy. Under the
assumption that the arrival processes satisfy a many-sources LDP, this is
accomplished by employing Garcia's extended contraction principle that is
applicable to quasi-continuous mappings.
For the simplex rate-region, an LDP for the stationary workload is also
established under the additional requirements that the scheduling policy be
work-conserving and that the arrival processes satisfy certain mixing
conditions.
The LDP results can be used to calculate asymptotic buffer overflow
probabilities accounting for the multiplexing gain, when the arrival process is
an average of \emph{i.i.d.} processes. The rate function for the stationary
workload is expressed in term of the rate functions of the finite-horizon
workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page
Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic
A result of Ward and Glynn (2005) asserts that the sequence of scaled offered
waiting time processes of the queue converges weakly to a
reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the
traffic intensity approaches one. As a consequence, the stationary distribution
of a ROU process, which is a truncated normal, should approximate the scaled
stationary distribution of the offered waiting time in a queue;
however, no such result has been proved. We prove the aforementioned
convergence, and the convergence of the moments, in heavy traffic, thus
resolving a question left open in Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (1961) showing that an exponential
distribution approximates the scaled stationary offered waiting time
distribution in a queue in heavy traffic, our result confirms that
the addition of customer abandonment has a non-trivial effect on the queue
stationary behavior.Comment: 29 page
Diffusion models and steady-state approximations for exponentially ergodic Markovian queues
Motivated by queues with many servers, we study Brownian steady-state
approximations for continuous time Markov chains (CTMCs). Our approximations
are based on diffusion models (rather than a diffusion limit) whose
steady-state, we prove, approximates that of the Markov chain with notable
precision. Strong approximations provide such "limitless" approximations for
process dynamics. Our focus here is on steady-state distributions, and the
diffusion model that we propose is tractable relative to strong approximations.
Within an asymptotic framework, in which a scale parameter is taken large,
a uniform (in the scale parameter) Lyapunov condition imposed on the sequence
of diffusion models guarantees that the gap between the steady-state moments of
the diffusion and those of the properly centered and scaled CTMCs shrinks at a
rate of . Our proofs build on gradient estimates for solutions of the
Poisson equations associated with the (sequence of) diffusion models and on
elementary martingale arguments. As a by-product of our analysis, we explore
connections between Lyapunov functions for the fluid model, the diffusion model
and the CTMC.Comment: Published in at http://dx.doi.org/10.1214/13-AAP984 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Gaussian queues in light and heavy traffic
In this paper we investigate Gaussian queues in the light-traffic and in the
heavy-traffic regime. The setting considered is that of a centered Gaussian
process with stationary increments and variance
function , equipped with a deterministic drift ,
reflected at 0: We
study the resulting stationary workload process
in the limiting regimes (heavy
traffic) and (light traffic). The primary contribution is that we
show for both limiting regimes that, under mild regularity conditions on the
variance function, there exists a normalizing function such that
converges to a non-trivial
limit in
Load Balancing in the Non-Degenerate Slowdown Regime
We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as
the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a
classical load balancing policy for queueing systems with multiple parallel
servers. Parallel server queueing systems are regularly analysed and
dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling
regime. However, when jobs must be dispatched to a server upon arrival, we
advocate the Non-Degenerate Slowdown regime (NDS) to compare different
load-balancing rules.
In this paper we identify novel diffusion approximation and timescale
separation that provides insights into the performance of JSQ. We calculate the
price of irrevocably dispatching jobs to servers and prove this to within 15%
(in the NDS regime) of the rules that may manoeuvre jobs between servers. We
also compare ours results for the JSQ policy with the NDS approximations of
many modern load balancing policies such as Idle-Queue-First and
Power-of--choices policies which act as low information proxies for the JSQ
policy. Our analysis leads us to construct new rules that have identical
performance to JSQ but require less communication overhead than
power-of-2-choices.Comment: Revised journal submission versio
- …