67 research outputs found
On Lerch's transcendent and the Gaussian random walk
Let be independent variables, each having a normal distribution
with negative mean and variance 1. We consider the partial sums
, with , and refer to the process as
the Gaussian random walk. We present explicit expressions for the mean and
variance of the maximum These expressions are in terms
of Taylor series about with coefficients that involve the Riemann
zeta function. Our results extend Kingman's first-order approximation [Proc.
Symp. on Congestion Theory (1965) 137--169] of the mean for .
We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802],
and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin
summation as key ingredients.Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
BRAVO for many-server QED systems with finite buffers
This paper demonstrates the occurrence of the feature called BRAVO (Balancing
Reduces Asymptotic Variance of Output) for the departure process of a
finite-buffer Markovian many-server system in the QED (Quality and
Efficiency-Driven) heavy-traffic regime. The results are based on evaluating
the limit of a formula for the asymptotic variance of death counts in finite
birth--death processes
Factorization identities for reflected processes, with applications
We derive factorization identities for a class of preemptive-resume queueing
systems, with batch arrivals and catastrophes that, whenever they occur,
eliminate multiple customers present in the system. These processes are quite
general, as they can be used to approximate Levy processes, diffusion
processes, and certain types of growth-collapse processes; thus, all of the
processes mentioned above also satisfy similar factorization identities. In the
Levy case, our identities simplify to both the well-known Wiener-Hopf
factorization, and another interesting factorization of reflected Levy
processes starting at an arbitrary initial state. We also show how the ideas
can be used to derive transforms for some well-known
state-dependent/inhomogeneous birth-death processes and diffusion processes
Optimal Tradeoff Between Exposed and Hidden Nodes in Large Wireless Networks
Wireless networks equipped with the CSMA protocol are subject to collisions
due to interference. For a given interference range we investigate the tradeoff
between collisions (hidden nodes) and unused capacity (exposed nodes). We show
that the sensing range that maximizes throughput critically depends on the
activation rate of nodes. For infinite line networks, we prove the existence of
a threshold: When the activation rate is below this threshold the optimal
sensing range is small (to maximize spatial reuse). When the activation rate is
above the threshold the optimal sensing range is just large enough to preclude
all collisions. Simulations suggest that this threshold policy extends to more
complex linear and non-linear topologies
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