A Correction Term for the Covariance of Renewal-Reward Processes with Multivariate Rewards

We consider a renewal-reward process with multivariate rewards. Such a process is constructed from an i.i.d.\ sequence of time periods, to each of which there is associated a multivariate reward vector. The rewards in each time period may depend on each other and on the period length, but not on the other time periods. Rewards are accumulated to form a vector valued process that exhibits jumps in all coordinates simultaneously, only at renewal epochs. We derive an asymptotically exact expression for the covariance function (over time) of the rewards, which is used to refine a central limit theorem for the vector of rewards. As illustrated by a numerical example, this refinement can yield improved accuracy, especially for moderate time-horizons

ESTIMATING TAIL PROBABILITIES OF RANDOM SUMS OF INFINITE MIXTURES OF PHASE-TYPE DISTRIBUTIONS

We consider the problem of estimating tail probabilities of random sums of infinite mixtures of phase-Type (lMPH) distributions -A class of distributions corresponding to random variables which can be represented as a product of an arbitrary random variable with a classical phase-Type distribution. Our motivation arises from applications in risk and queueing problems. Classical rare-event simulation algorithms cannot be implemented in this setting because these typically rely on the availability of the CDF or the MGF, but these are difficult to compute or not even available for the class of IMPH distributions. In this paper, we address these issues and propose alternative simulation methods for estimating tail probabilities of random sums of IMPH distributions; our algorithms combine importance sampling and conditional Monte Carlo methods. The empirical performance of each method suggested is explored via numerical experimentation

Frequency combs and optical feedback in Quantum Cascade Lasers: a unifying theoretical framework

We propose a unified theoretical framework for the frequency comb formation and optical feedback effects in semiconductor lasers. We use the quantum cascade laser as a suitable device to develop the framework unifying these two research areas, so far treated distinctly. We generate a novel feedback regime diagram and we provide a proof of principle that feedback can be used to induce and manipulate frequency combs, selecting their harmonic order. These results open a pathway towards new methodologies for hyperspectral imaging, multimode coherent sensing, and multi-channel communication

Transient provisioning and performance evaluation for cloud computing platforms: A capacity value approach

User demand on the computational resources of cloud computing platforms varies over time. These variations in demand can be predictable or unpredictable, resulting in ‘bursty’ fluctuations in demand. Furthermore, demand can arrive in batches, and users whose demands are not met can be impatient. We demonstrate how to compute the expected revenue loss over a finite time horizon in the presence of all these model characteristics through the use of matrix analytic methods. We then illustrate how to use this knowledge to make frequent short term provisioning decisions — transient provisioning. It is seen that taking each of the characteristics of fluctuating user demand (predictable, unpredictable, batchy) into account can result in a substantial reduction of losses. Moreover, our transient provisioning framework allows for a wide variety of system behaviors to be modeled and gives simple expressions for expected revenue loss which are straightforward to evaluate numerically

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance

Combining Optimisation and Simulation Using Logic-Based Benders Decomposition

Operations research practitioners frequently want to model complicated functions that are are difficult to encode in their underlying optimisation framework. A common approach is to solve an approximate model, and to use a simulation to evaluate the true objective value of one or more solutions. We propose a new approach to integrating simulation into the optimisation model itself. The idea is to run the simulation at each incumbent solution to the master problem. The simulation data is then used to guide the trajectory of the optimisation model itself using logic-based Benders cuts. We test the approach on a class of stochastic resource allocation problems with monotonic performance measures. We derive strong, novel Benders cuts that are provably valid for all problems of the given form. We consider two concrete examples: a nursing home shift scheduling problem, and an airport check in counter allocation problem. While previous papers on these applications could only approximately solve realistic instances, we are able to solve them exactly within a reasonable amount of time. Moreover, while those papers account for the inherent variance of the problem by including estimates of the underlying random variables as model parameters, we are able to compute sample average approximations to optimality with up to 100 scenarios.Comment: 30 page