arXiv subjects

Jose Blanchet

Publications and source records attributed to Jose Blanchet.

At least 181 records · Page 10Linked to original sources

arXiv2015

Steady-state simulation of reflected Brownian motion and related stochastic networks

This paper develops the first class of algorithms that enable unbiased estimation of steady-state expectations for multidimensional reflected Brownian motion. In order to explain our ideas, we first consider the case of compound Poisson (possibly Markov modulated) input. In this case, we analyze the complexity of our procedure as the dimension of the network increases and show that, under certain assumptions, the algorithm has polynomial-expected termination time. Our methodology includes procedures that are of interest beyond steady-state simulation and reflected processes. For instance, we use wavelets to construct a piecewise linear function that can be guaranteed to be within $\varepsilon$ distance (deterministic) in the uniform norm to Brownian motion in any compact time interval.

math.PR↗

arXiv2015

Perfect Sampling of GI/GI/c Queues

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backwards in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

math.PR↗

arXiv2015

On the Limiting Ratio of Current Age to Total Life for Null Recurrent Renewal Processes

If the inter-arrival time distribution of a renewal process is regularly varying with index $α\in\left( 0,1\right) $ (i.e. the inter-arrival times have infinite mean) and if $A\left( t\right) $ is the associated age process at time $t$. Then we show that if $C\left( t\right) $ is the length of the current cycle at time $t$, \[ A\left( t\right) /C\left( t\right) \Rightarrow U^{1/α}, \] where $U$ is $U\left( 0,1\right) $. This extends a classical result in renewal theory in the finite mean case which indicates that the limit is $U\left( 0,1\right) $.

math.PR↗

arXiv2015

Tail Asymptotics for Delay in a Half-loaded GI/GI/2 Queue with Heavy-tailed Job Sizes

We obtain asymptotic bounds for the tail distribution of steady-state waiting time in a two server queue where each server processes incoming jobs at a rate equal to the rate of their arrivals (that is, the half-loaded regime). The job sizes are taken to be regularly varying. When the incoming jobs have finite variance, there are basically two types of effects that dominate the tail asymptotics. While the quantitative distinction between these two manifests itself only in the slowly varying components, the two effects arise from qualitatively very different phenomena (arrival of one extremely big job (or) two big jobs). Then there is a phase transition that occurs when the incoming jobs have infinite variance. In that case, only one of these effects dominate the tail asymptotics, the one involving arrival of one extremely big job.

math.PR↗

arXiv2015

Exact Sampling of Stationary and Time-Reversed Queues

We provide the first algorithm that under minimal assumptions allows to simulate the stationary waiting-time sequence of a single-server queue backwards in time, jointly with the input processes of the queue (inter-arrival and service times). The single-server queue is useful in applications of DCFTP (Dominated Coupling From The Past), which is a well known protocol for simulation without bias from steady-state distributions. Our algorithm terminates in finite time assuming only finite mean of the inter-arrival and service times. In order to simulate the single-server queue in stationarity until the first idle period in finite expected termination time we require the existence of finite variance. This requirement is also necessary for such idle time (which is a natural coalescence time in DCFTP applications) to have finite mean. Thus, in this sense, our algorithm is applicable under minimal assumptions.

math.PR↗

arXiv2014

State-independent Importance Sampling for Random Walks with Regularly Varying Increments

We develop importance sampling based efficient simulation techniques for three commonly encountered rare event probabilities associated with random walks having i.i.d. regularly varying increments; namely, 1) the large deviation probabilities, 2) the level crossing probabilities, and 3) the level crossing probabilities within a regenerative cycle. Exponential twisting based state-independent methods, which are effective in efficiently estimating these probabilities for light-tailed increments are not applicable when the increments are heavy-tailed. To address the latter case, more complex and elegant state-dependent efficient simulation algorithms have been developed in the literature over the last few years. We propose that by suitably decomposing these rare event probabilities into a dominant and further residual components, simpler state-independent importance sampling algorithms can be devised for each component resulting in composite unbiased estimators with desirable efficiency properties. When the increments have infinite variance, there is an added complexity in estimating the level crossing probabilities as even the well known zero-variance measures have an infinite expected termination time. We adapt our algorithms so that this expectation is finite while the estimators remain strongly efficient. Numerically, the proposed estimators perform at least as well, and sometimes substantially better than the existing state-dependent estimators in the literature.

math.PR↗

arXiv2014

A Weak Convergence Criterion Constructing Changes of Measure

Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability conditions (such as the well-known Novikov condition). The weak convergence approach that we propose allows to replace integrability conditions by a suitable tightness condition. We then provide several applications of this approach ranging from simplified proofs of classical results to characterizations of processes conditioned on first passage time events and changes of measures for jump processes.

math.PR↗

arXiv2014

Total variation approximations and conditional limit theorems for multivariate regularly varying random walks conditioned on ruin

We study a new technique for the asymptotic analysis of heavy-tailed systems conditioned on large deviations events. We illustrate our approach in the context of ruin events of multidimensional regularly varying random walks. Our approach is to study the Markov process described by the random walk conditioned on hitting a rare target set. We construct a Markov chain whose transition kernel can be evaluated directly from the increment distribution of the associated random walk. This process is shown to approximate the conditional process of interest in total variation. Then, by analyzing the approximating process, we are able to obtain asymptotic conditional joint distributions and a conditional functional central limit theorem of several objects such as the time until ruin, the whole random walk prior to ruin, and the overshoot on the target set. These types of joint conditional limit theorems have been obtained previously in the literature only in the one dimensional case. In addition to using different techniques, our results include features that are qualitatively different from the one dimensional case. For instance, the asymptotic conditional law of the time to ruin is no longer purely Pareto as in the multidimensional case.

math.ST↗

arXiv2014

Theoretical analysis of a Stochastic Approximation approach for computing Quasi-Stationary distributions

This paper studies a method, which has been proposed in the Physics literature by [8, 7, 10], for estimating the quasi-stationary distribution. In contrast to existing methods in eigenvector estimation, the method eliminates the need for explicit transition matrix manipulation to extract the principal eigenvector. Our paper analyzes the algorithm by casting it as a stochastic approximation algorithm (Robbins-Monro) [23, 16]. In doing so, we prove its convergence and obtain its rate of convergence. Based on this insight, we also give an example where the rate of convergence is very slow. This problem can be alleviated by using an improved version of the algorithm that is given in this paper. Numerical experiments are described that demonstrate the effectiveness of this improved method.

math.PR↗

arXiv2013

Perfect sampling for infinite server and loss systems

We present the first class of perfect sampling (also known as exact simulation) algorithms for the steady-state distribution of non-Markovian loss networks. We use a variation of Dominated Coupling From The Past for which we simulate a stationary infinite server queue backwards in time and analyze the running time in heavy traffic. In particular, we are able to simulate stationary renewal marked point processes in unbounded regions. We use the infinite server queue as an upper bound process to simulate loss systems. The running time analysis of our perfect sampling algorithm for loss systems is performed in the Quality-Driven (QD) and the Quality-and-Efficiency-Driven regimes. In both cases, we show that our algorithm achieves sub-exponential complexity as both the number of servers and the arrival rate increase. Moreover, in the QD regime, our algorithm achieves a nearly optimal rate of convergence.

math.PR↗

arXiv2013

Continuous-time Modeling of Bid-Ask Spread and Price Dynamics in Limit Order Books

We derive a continuous time model for the joint evolution of the mid price and the bid-ask spread from a multiscale analysis of the whole limit order book (LOB) dynamics. We model the LOB as a multiclass queueing system and perform our asymptotic analysis using stylized features observed empirically. We argue that in the asymptotic regime supported by empirical observations the mid price and bid-ask-spread can be described using only certain parameters of the book (not the whole book itself). Our limit process is characterized by reflecting behavior and state-dependent jumps. Our analysis allows to explain certain characteristics observed in practice such as: the connection between power-law decaying tails in the volumes of the order book and the returns, as well as statistical properties of the long-run spread distribution.

q-fin.TR↗

arXiv2012

Sampling Point Processes on Stable Unbounded Regions and Exam Simulation of Queues

Given a marked renewal point process (assuming that the marks are i.i.d.) we say that an unbounded region is stable if it contains finitely many points of the point process with probability one. In this paper we provide algorithms that allow to sample these finitely many points efficiently. We explain how exact simulation of the steady-state measure valued state descriptor of the infinite server queue follows as a simple corollary of our algorithms. We provide numerical evidence supporting that our algorithms are not only theoretically sound but also practical. Finally, having simulation optimization in mind, we also apply our results to gradient estimation of steady-state performance measures.

math.PR↗

arXiv2012

Efficient Rare-event Simulation for Perpetuities

We consider perpetuities of the form D = B_1 exp(Y_1) + B_2 exp(Y_1+Y_2) + ... where the Y_j's and B_j's might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the Y_j's satisfy the so-called Cramer condition with associated root theta_{ast} in (0,infty) and that the tails of the B_j's are appropriately behaved so that D is regularly varying with index theta_{ast}. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the Y_j's according to theta_{ast} fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.

math.PR↗

arXiv2012

Rare-Event Simulation for Many-Server Queues

We develop rare-event simulation methodology for the analysis of loss events in a many-server loss system under quality-driven regime, focusing on the steady-state loss probability (i.e. fraction of lost customers over arrivals) and the behavior of the whole system leading to loss events. The analysis of these events requires working with the full measure-valued process describing the system. This is the first algorithm that is shown to be asymptotically optimal, in the rare-event simulation context, under the setting of many-server queues involving a full measure-valued descriptor.

math.PR↗

arXiv2011

Characterizing Optimal Sampling of Binary Contingency Tables via the Configuration Model

A binary contingency table is an m x n array of binary entries with prescribed row sums r=(r_1,...,r_m) and column sums c=(c_1,...,c_n). The configuration model for uniformly sampling binary contingency tables proceeds as follows. First, label N=\sum_{i=1}^{m} r_i tokens of type 1, arrange them in m cells, and let the i-th cell contain r_i tokens. Next, label another set of tokens of type 2 containing N=\sum_{j=1}^{n}c_j elements arranged in n cells, and let the j-th cell contain c_j tokens. Finally, pair the type-1 tokens with the type-2 tokens by generating a random permutation until the total pairing corresponds to a binary contingency table. Generating one random permutation takes O(N) time, which is optimal up to constant factors. A fundamental question is whether a constant number of permutations is sufficient to obtain a binary contingency table. In the current paper, we solve this problem by showing a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as N goes to \infty. Our finding shows surprising differences from recent results for binary symmetric contingency tables.

math.PR↗

arXiv2011

On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case

Two of the most popular approximations for the distribution of the steady-state waiting time, $W_{\infty}$, of the M/G/1 queue are the so-called heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the traffic intensity, $ρ$, is close to 1 and the processing times have finite variance, the heavy-traffic approximation states that the distribution of $W_{\infty}$ is roughly exponential at scale $O((1-ρ)^{-1})$, while the heavy tailed asymptotic describes power law decay in the tail of the distribution of $W_{\infty}$ for a fixed traffic intensity. In this paper, we assume a regularly varying processing time distribution and obtain a sharp threshold in terms of the tail value, or equivalently in terms of $(1-ρ)$, that describes the point at which the tail behavior transitions from the heavy-traffic regime to the heavy-tailed asymptotic. We also provide new approximations that are either uniform in the traffic intensity, or uniform on the positive axis, that avoid the need to use different expressions on the two regions defined by the threshold.

math.PR↗

arXiv2011

Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.

math.PR↗

arXiv2010

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2β+1}) function evaluations suffice to achieve a given relative precision, where β is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.

math.PR↗