2,581 research outputs found
Continuity theorems for the queueing system
In this paper continuity theorems are established for the number of losses
during a busy period of the queue. We consider an queueing
system where the service time probability distribution, slightly different in a
certain sense from the exponential distribution, is approximated by that
exponential distribution. Continuity theorems are obtained in the form of one
or two-sided stochastic inequalities. The paper shows how the bounds of these
inequalities are changed if further assumptions, associated with specific
properties of the service time distribution (precisely described in the paper),
are made. Specifically, some parametric families of service time distributions
are discussed, and the paper establishes uniform estimates (given for all
possible values of the parameter) and local estimates (where the parameter is
fixed and takes only the given value). The analysis of the paper is based on
the level crossing approach and some characterization properties of the
exponential distribution.Comment: Final revision; will be published as i
Join the Shortest Queue with Many Servers. The Heavy-Traffic Asymptotics
We consider queueing systems with n parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy-traffic regime. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length exactly two weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process, while processes counting longer queues converge to a deterministic system decaying to zero in constant time. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, only a vanishing fraction of customers will have to wait, but those who do incur a constant order waiting time. Keywords: queueing theory; parallel queues; diffusion model
Statistical analysis of single-server loss queueing systems
In this article statistical bounds for certain output characteristics of the
and loss queueing systems are derived on the basis of
large samples of an input characteristic of these systems, such as service time
in the queueing system or interarrival time in the
queueing system. The analysis of this article is based on application of
Kolmogorov's statistics for empirical probability distribution functions.Comment: This is the version of the paper that addresses the reviewer's
report. A software for this paper (executable file) can be found in my
homepag
Martingale proofs of many-server heavy-traffic limits for Markovian queues
This is an expository review paper illustrating the ``martingale method'' for
proving many-server heavy-traffic stochastic-process limits for queueing
models, supporting diffusion-process approximations. Careful treatment is given
to an elementary model -- the classical infinite-server model , but
models with finitely many servers and customer abandonment are also treated.
The Markovian stochastic process representing the number of customers in the
system is constructed in terms of rate-1 Poisson processes in two ways: (i)
through random time changes and (ii) through random thinnings. Associated
martingale representations are obtained for these constructions by applying,
respectively: (i) optional stopping theorems where the random time changes are
the stopping times and (ii) the integration theorem associated with random
thinning of a counting process. Convergence to the diffusion process limit for
the appropriate sequence of scaled queueing processes is obtained by applying
the continuous mapping theorem. A key FCLT and a key FWLLN in this framework
are established both with and without applying martingales.Comment: Published in at http://dx.doi.org/10.1214/06-PS091 the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The effective bandwidth problem revisited
The paper studies a single-server queueing system with autonomous service and
priority classes. Arrival and departure processes are governed by marked
point processes. There are buffers corresponding to priority classes,
and upon arrival a unit of the th priority class occupies a place in the
th buffer. Let , denote the quota for the total
th buffer content. The values are assumed to be large, and
queueing systems both with finite and infinite buffers are studied. In the case
of a system with finite buffers, the values characterize buffer
capacities.
The paper discusses a circle of problems related to optimization of
performance measures associated with overflowing the quota of buffer contents
in particular buffers models. Our approach to this problem is new, and the
presentation of our results is simple and clear for real applications.Comment: 29 pages, 11pt, Final version, that will be published as is in
Stochastic Model
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