Laws of Little in an open queueing network

The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented

Performance analysis of queueing networks via robust optimization

Performance analysis of queueing networks is one of the most challenging areas of queueing theory. Barring very specialized models such as product-form type queueing networks, there exist very few results that provide provable nonasymptotic upper and lower bounds on key performance measures. In this paper we propose a new performance analysis method, which is based on the robust optimization. The basic premise of our approach is as follows: rather than assuming that the stochastic primitives of a queueing model satisfy certain probability laws—such as i.i.d. interarrival and service times distributions—we assume that the underlying primitives are deterministic and satisfy the implications of such probability laws. These implications take the form of simple linear constraints, namely, those motivated by the law of the iterated logarithm (LIL). Using this approach we are able to obtain performance bounds on some key performance measures. Furthermore, these performance bounds imply similar bounds in the underlying stochastic queueing models. We demonstrate our approach on two types of queueing networks: (a) tandem single-class queueing network and (b) multiclass single-server queueing network. In both cases, using the proposed robust optimization approach, we are able to obtain explicit upper bounds on some steady-state performance measures. For example, for the case of TSC system we obtain a bound of the form C(1 – {rho})–1 ln ln((1 – {rho})–1) [C(1-p) superscript -1 ln ln ((1 - p) superscript -1)]on the expected steady-state sojourn time, where C is an explicit constant and {rho} is the bottleneck traffic intensity. This qualitatively agrees with the correct heavy traffic scaling of this performance measure up to the ln ln((1 – {rho})–1) [ln ln((1 - p) superscript -1)] correction factor.National Science Foundation (U.S.) (Grant DMI-0556106)National Science Foundation (U.S.) (Grant CMMI-0726733

Central Limit Theorems and Large Deviations for Additive Functionals of Reflecting Diffusion Processes

This paper develops central limit theorems (CLT's) and large deviations results for additive functionals associated with reflecting diffusions in which the functional may include a term associated with the cumulative amount of boundary reflection that has occurred. Extending the known central limit and large deviations theory for Markov processes to include additive functionals that incorporate boundary reflection is important in many applications settings in which reflecting diffusions arise, including queueing theory and economics. In particular, the paper establishes the partial differential equations that must be solved in order to explicitly compute the mean and variance for the CLT, as well as the associated rate function for the large deviations principle

Diffusion Asymptotics for Sequential Experiments

We propose a new diffusion-asymptotic analysis for sequentially randomized experiments, including those that arise in solving multi-armed bandit problems. In an experiment with $n$ time steps, we let the mean reward gaps between actions scale to the order $1/\sqrt{n}$ so as to preserve the difficulty of the learning task as $n$ grows. In this regime, we show that the behavior of a class of sequentially randomized Markov experiments converges to a diffusion limit, given as the solution of a stochastic differential equation. The diffusion limit thus enables us to derive refined, instance-specific characterization of the stochastic dynamics of adaptive experiments. As an application of this framework, we use the diffusion limit to obtain several new insights on the regret and belief evolution of Thompson sampling. We show that a version of Thompson sampling with an asymptotically uninformative prior variance achieves nearly-optimal instance-specific regret scaling when the reward gaps are relatively large. We also demonstrate that, in this regime, the posterior beliefs underlying Thompson sampling are highly unstable over time

Asymptotic behavior of local times of compound Poisson processes with drift in the infinite variance case

Consider compound Poisson processes with negative drift and no negative jumps, which converge to some spectrally positive L\'evy process with non-zero L\'evy measure. In this paper we study the asymptotic behavior of the local time process, in the spatial variable, of these processes killed at two different random times: either at the time of the first visit of the L\'evy process to 0, in which case we prove results at the excursion level under suitable conditionings; or at the time when the local time at 0 exceeds some fixed level. We prove that finite-dimensional distributions converge under general assumptions, even if the limiting process is not c\`adl\`ag. Making an assumption on the distribution of the jumps of the compound Poisson processes, we strengthen this to get weak convergence. Our assumption allows for the limiting process to be a stable L\'evy process with drift. These results have implications on branching processes and in queueing theory, namely, on the scaling limit of binary, homogeneous Crump-Mode-Jagers processes and on the scaling limit of the Processor-Sharing queue length process.Comment: Final version accepted for publication in Journal of Theoretical Probabilit

Sample Path Properties of Bifractional Brownian Motion

Let $B^{H, K}= \big\{B^{H, K}(t), t \in \R_+ \big\}$ be a bifractional Brownian motion in $\R^d$. We prove that $B^{H, K}$ is strongly locally nondeterministic. Applying this property and a stochastic integral representation of $B^{H, K}$, we establish Chung's law of the iterated logarithm for $B^{H, K}$, as well as sharp H\"older conditions and tail probability estimates for the local times of $B^{H, K}$. We also consider the existence and the regularity of the local times of multiparameter bifractional Brownian motion $B^{\bar{H}, \bar{K}}= \big\{B^{\bar{H}, \bar{K}}(t), t \in \R^N_+ \big\}$ in $\R^d$ using Wiener-It\^o chaos expansion

Applications of robust optimization to queueing and inventory systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 105-111).This thesis investigates the application of robust optimization in the performance analysis of queueing and inventory systems. In the first part of the thesis, we propose a new approach for performance analysis of queueing systems based on robust optimization. We first derive explicit upper bounds on performance for tandem single class, multiclass single server, and single class multi-server queueing systems by solving appropriate robust optimization problems. We then show that these bounds derived by solving deterministic optimization problems translate to upper bounds on the expected steady-state performance for a variety of widely used performance measures such as waiting times and queue lengths. Additionally, these explicit bounds agree qualitatively with known results. In the second part of the thesis, we propose methods to compute (s,S) policies in supply chain networks using robust and stochastic optimization and compare their performance. Our algorithms handle general uncertainty sets, arbitrary network topologies, and flexible cost functions including the presence of fixed costs. The algorithms exhibit empirically practical running times. We contrast the performance of robust and stochastic (s,S) policies in a numerical study, and we find that the robust policy is comparable to the average performance of the stochastic policy, but has a considerably lower standard deviation across a variety of networks and realized demand distributions. Additionally, we identify regimes when the robust policy exhibits particular strengths even in average performance and tail behavior as compared with the stochastic policy.by Alexander Anatolyevich Rikun.Ph.D