510 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
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
On the Three Methods for Bounding the Rate of Convergence for some Continuous-time Markov Chains
Consideration is given to the three different analytical methods for the
computation of upper bounds for the rate of convergence to the limiting regime
of one specific class of (in)homogeneous continuous-time Markov chains. This
class is particularly suited to describe evolutions of the total number of
customers in (in)homogeneous queueing systems with possibly
state-dependent arrival and service intensities, batch arrivals and services.
One of the methods is based on the logarithmic norm of a linear operator
function; the other two rely on Lyapunov functions and differential
inequalities, respectively. Less restrictive conditions (compared to those
known from the literature) under which the methods are applicable, are being
formulated. Two numerical examples are given. It is also shown that for
homogeneous birth-death Markov processes defined on a finite state space with
all transition rates being positive, all methods yield the same sharp upper
bound
Stein's Method for the Single Server Queue in Heavy Traffic
Following recent developments in the application of Stein's method in
queueing theory, this paper is intended to be a short treatment showing how
Stein's method can be developed and applied to the single server queue in heavy
traffic. Here we provide two approaches to this approximation: one based on
equilibrium couplings and another involving comparison of generators.Comment: 11 pages. To appear in Statistics and Probability Letters, 2019
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