84 research outputs found

### Spectral gap of the Erlang A model in the Halfin-Whitt regime

We consider a hybrid diffusion process that is a combination of two
Ornstein-Uhlenbeck processes with different restraining forces. This process
serves as the heavy-traffic approximation to the Markovian many-server queue
with abandonments in the critical Halfin-Whitt regime. We obtain an expression
for the Laplace transform of the time-dependent probability distribution, from
which the spectral gap is explicitly characterized. The spectral gap gives the
exponential rate of convergence to equilibrium. We further give various
asymptotic results for the spectral gap, in the limits of small and large
abandonment effects. It turns out that convergence to equilibrium becomes
extremely slow for overloaded systems with small abandonment effects.Comment: 48 page

### 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

### Optimal subgraph structures in scale-free configuration models

Subgraphs reveal information about the geometry and functionalities of
complex networks. For scale-free networks with unbounded degree fluctuations,
we obtain the asymptotics of the number of times a small connected graph
occurs as a subgraph or as an induced subgraph. We obtain these results by
analyzing the configuration model with degree exponent $\tau\in(2,3)$ and
introducing a novel class of optimization problems. For any given subgraph, the
unique optimizer describes the degrees of the vertices that together span the
subgraph.
We find that subgraphs typically occur between vertices with specific degree
ranges. In this way, we can count and characterize {\it all} subgraphs. We
refrain from double counting in the case of multi-edges, essentially counting
the subgraphs in the {\it erased} configuration model.Comment: 50 pages, 2 figure

### On the Sojourn Time Distribution in a Finite Population Markovian Processor Sharing Queue

We consider a finite population processor-sharing (PS) queue, with Markovian
arrivals and an exponential server. Such a queue can model an interactive
computer system consisting of a bank of terminals in series with a central
processing unit (CPU). For systems with a large population $N$ and a
commensurately rapid service rate, or infrequent arrivals, we obtain various
asymptotic results. We analyze the conditional sojourn time distribution of a
tagged customer, conditioned on the number $n$ of others in the system at the
tagged customer's arrival instant, and also the unconditional distribution. The
asymptotics are obtained by a combination of singular perturbation methods and
spectral methods. We consider several space/time scales and parameter ranges,
which lead to different asymptotic behaviors. We also identify precisely when
the finite population model can be approximated by the standard infinite
population $M/M/1$-PS queue.Comment: 60 pages and 3 figure

### Variational principle for scale-free network motifs

For scale-free networks with degrees following a power law with an exponent
$\tau\in(2,3)$, the structures of motifs (small subgraphs) are not yet well
understood. We introduce a method designed to identify the dominant structure
of any given motif as the solution of an optimization problem. The unique
optimizer describes the degrees of the vertices that together span the most
likely motif, resulting in explicit asymptotic formulas for the motif count and
its fluctuations. We then classify all motifs into two categories: motifs with
small and large fluctuations

### Cluster tails for critical power-law inhomogeneous random graphs

Recently, the scaling limit of cluster sizes for critical inhomogeneous
random graphs of rank-1 type having finite variance but infinite third moment
degrees was obtained (see previous work by Bhamidi, van der Hofstad and van
Leeuwaarden). It was proved that when the degrees obey a power law with
exponent in the interval (3,4), the sequence of clusters ordered in decreasing
size and scaled appropriately converges as n goes to infinity to a sequence of
decreasing non-degenerate random variables.
Here, we study the tails of the limit of the rescaled largest cluster, i.e.,
the probability that the scaling limit of the largest cluster takes a large
value u, as a function of u. This extends a related result of Pittel for the
Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random
graphs with infinite third moment degrees. We make use of delicate large
deviations and weak convergence arguments.Comment: corrected and updated first referenc

### Novel scaling limits for critical inhomogeneous random graphs

We find scaling limits for the sizes of the largest components at criticality
for rank-1 inhomogeneous random graphs with power-law degrees with power-law
exponent \tau. We investigate the case where $\tau\in(3,4)$, so that the
degrees have finite variance but infinite third moment. The sizes of the
largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting
times of a "thinned" L\'{e}vy process, a special case of the general
multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854]
and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should
be contrasted to the case \tau>4, so that the third moment is finite. There,
instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the
excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous
[Ann. Probab. 25 (1997) 812-854] for the Erd\H{o}s-R\'{e}nyi random graph and
extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden
[Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/11-AOP680 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Triadic closure in configuration models with unbounded degree fluctuations

The configuration model generates random graphs with any given degree
distribution, and thus serves as a null model for scale-free networks with
power-law degrees and unbounded degree fluctuations. For this setting, we study
the local clustering $c(k)$, i.e., the probability that two neighbors of a
degree-$k$ node are neighbors themselves. We show that $c(k)$ progressively
falls off with $k$ and eventually for $k=\Omega(\sqrt{n})$ settles on a power
law $c(k)\sim k^{-2(3-\tau)}$ with $\tau\in(2,3)$ the power-law exponent of the
degree distribution. This fall-off has been observed in the majority of
real-world networks and signals the presence of modular or hierarchical
structure. Our results agree with recent results for the hidden-variable model
and also give the expected number of triangles in the configuration model when
counting triangles only once despite the presence of multi-edges. We show that
only triangles consisting of triplets with uniquely specified degrees
contribute to the triangle counting

### Asymptotically Optimal Load Balancing Topologies

We consider a system of $N$ servers inter-connected by some underlying graph
topology $G_N$. Tasks arrive at the various servers as independent Poisson
processes of rate $\lambda$. Each incoming task is irrevocably assigned to
whichever server has the smallest number of tasks among the one where it
appears and its neighbors in $G_N$. Tasks have unit-mean exponential service
times and leave the system upon service completion.
The above model has been extensively investigated in the case $G_N$ is a
clique. Since the servers are exchangeable in that case, the queue length
process is quite tractable, and it has been proved that for any $\lambda < 1$,
the fraction of servers with two or more tasks vanishes in the limit as $N \to
\infty$. For an arbitrary graph $G_N$, the lack of exchangeability severely
complicates the analysis, and the queue length process tends to be worse than
for a clique. Accordingly, a graph $G_N$ is said to be $N$-optimal or
$\sqrt{N}$-optimal when the occupancy process on $G_N$ is equivalent to that on
a clique on an $N$-scale or $\sqrt{N}$-scale, respectively.
We prove that if $G_N$ is an Erd\H{o}s-R\'enyi random graph with average
degree $d(N)$, then it is with high probability $N$-optimal and
$\sqrt{N}$-optimal if $d(N) \to \infty$ and $d(N) / (\sqrt{N} \log(N)) \to
\infty$ as $N \to \infty$, respectively. This demonstrates that optimality can
be maintained at $N$-scale and $\sqrt{N}$-scale while reducing the number of
connections by nearly a factor $N$ and $\sqrt{N} / \log(N)$ compared to a
clique, provided the topology is suitably random. It is further shown that if
$G_N$ contains $\Theta(N)$ bounded-degree nodes, then it cannot be $N$-optimal.
In addition, we establish that an arbitrary graph $G_N$ is $N$-optimal when its
minimum degree is $N - o(N)$, and may not be $N$-optimal even when its minimum
degree is $c N + o(N)$ for any $0 < c < 1/2$.Comment: A few relevant results from arXiv:1612.00723 are included for
convenienc

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