7,917 research outputs found
Validity of heavy traffic steady-state approximations in generalized Jackson Networks
We consider a single class open queueing network, also known as a generalized
Jackson network (GJN). A classical result in heavy-traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a reflected Brownian motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network. In this paper we resolve this open
problem by proving that the re-scaled stationary distribution of the GJN
converges to the stationary distribution of the RBM, thus validating a
so-called ``interchange-of-limits'' for this class of networks. Our method of
proof involves a combination of Lyapunov function techniques, strong
approximations and tail probability bounds that yield tightness of the sequence
of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Redundancy Scheduling with Locally Stable Compatibility Graphs
Redundancy scheduling is a popular concept to improve performance in
parallel-server systems. In the baseline scenario any job can be handled
equally well by any server, and is replicated to a fixed number of servers
selected uniformly at random. Quite often however, there may be heterogeneity
in job characteristics or server capabilities, and jobs can only be replicated
to specific servers because of affinity relations or compatibility constraints.
In order to capture such situations, we consider a scenario where jobs of
various types are replicated to different subsets of servers as prescribed by a
general compatibility graph. We exploit a product-form stationary distribution
and weak local stability conditions to establish a state space collapse in
heavy traffic. In this limiting regime, the parallel-server system with
graph-based redundancy scheduling operates as a multi-class single-server
system, achieving full resource pooling and exhibiting strong insensitivity to
the underlying compatibility constraints.Comment: 28 pages, 4 figure
Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime
In this paper we derive a technique of obtaining limit theorems for suprema
of L\'evy processes from their random walk counterparts. For each , let
be a sequence of independent and identically distributed
random variables and be a L\'evy processes such that
, and as . Let .
Then, under some mild assumptions, , for some random variable and some function
. We utilize this result to present a number of limit theorems
for suprema of L\'evy processes in the heavy-traffic regime
Heavy-traffic analysis of the maximum of an asymptotically stable random walk
For families of random walks with we consider their maxima . We
investigate the asymptotic behaviour of as for
asymptotically stable random walks. This problem appeared first in the 1960's
in the analysis of a single-server queue when the traffic load tends to 1 and
since then is referred to as the heavy-traffic approximation problem. Kingman
and Prokhorov suggested two different approaches which were later followed by
many authors. We give two elementary proofs of our main result, using each of
these approaches. It turns out that the main technical difficulties in both
proofs are rather similar and may be resolved via a generalisation of the
Kolmogorov inequality to the case of an infinite variance. Such a
generalisation is also obtained in this note.Comment: 9 page
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