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

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

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

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

Achievable Performance in Product-Form Networks

We characterize the achievable range of performance measures in product-form networks where one or more system parameters can be freely set by a network operator. Given a product-form network and a set of configurable parameters, we identify which performance measures can be controlled and which target values can be attained. We also discuss an online optimization algorithm, which allows a network operator to set the system parameters so as to achieve target performance metrics. In some cases, the algorithm can be implemented in a distributed fashion, of which we give several examples. Finally, we give conditions that guarantee convergence of the algorithm, under the assumption that the target performance metrics are within the achievable range.Comment: 50th Annual Allerton Conference on Communication, Control and Computing - 201