2,825 research outputs found
A Tandem Fluid Network with L\'evy Input in Heavy Traffic
In this paper we study the stationary workload distribution of a fluid tandem
queue in heavy traffic. We consider different types of L\'evy input, covering
compound Poisson, -stable L\'evy motion (with ), and
Brownian motion. In our analysis we separately deal with L\'evy input processes
with increments that have finite and infinite variance. A distinguishing
feature of this paper is that we do not only consider the usual heavy-traffic
regime, in which the load at one of the nodes goes to unity, but also a regime
in which we simultaneously let the load of both servers tend to one, which, as
it turns out, leads to entirely different heavy-traffic asymptotics. Numerical
experiments indicate that under specific conditions the resulting simultaneous
heavy-traffic approximation significantly outperforms the usual heavy-traffic
approximation
Random Fluid Limit of an Overloaded Polling Model
In the present paper, we study the evolution of an overloaded cyclic polling
model that starts empty. Exploiting a connection with multitype branching
processes, we derive fluid asymptotics for the joint queue length process.
Under passage to the fluid dynamics, the server switches between the queues
infinitely many times in any finite time interval causing frequent oscillatory
behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid
limit is random. Additionally, we suggest a method that establishes finiteness
of moments of the busy period in an M/G/1 queue.Comment: 36 pages, 2 picture
Scaling limits for infinite-server systems in a random environment
This paper studies the effect of an overdispersed arrival process on the
performance of an infinite-server system. In our setup, a random environment is
modeled by drawing an arrival rate from a given distribution every
time units, yielding an i.i.d. sequence of arrival rates
. Applying a martingale central limit theorem, we
obtain a functional central limit theorem for the scaled queue length process.
We proceed to large deviations and derive the logarithmic asymptotics of the
queue length's tail probabilities. As it turns out, in a rapidly changing
environment (i.e., is small relative to ) the overdispersion
of the arrival process hardly affects system behavior, whereas in a slowly
changing random environment it is fundamentally different; this general finding
applies to both the central limit and the large deviations regime. We extend
our results to the setting where each arrival creates a job in multiple
infinite-server queues
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
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