Algorithmic Approach for Taylor Series Expansions with applications to Queueing Networks
Karim Abbas (University of Bejaia, Algeria)
In this talk we present a new formula based on higher-order derivatives of transition matrices of ergodic finite-state Markov chains for obtaining performance measures in a functional form. By using this formula, we propose a numerical approach based on a Taylor series expansion of the performance measures of the underlying Markov chain. The Taylor series coefficients are expressed in closed form as functions of the fundamental matrix associated to the Markov chain. A typical advantage of our approach is that the fundamental matrix, which is required as an input, is numerically easy to compute. We apply this numerical approach to queueing systems. Finally, we illustrate the performance of the proposed approach by some numerical examples.
Ghost Simulation Model for Discrete Event Systems, an Application to a local Bus Service
Felisa Vazquez-Abad, (Hunter College of the City University of New York, United States)
In this talk we present a simulation model for large networks that increases the efficiency compared to a discrete event simulation model. These networks have two di erent time scales: a fast one and a slow one. The main idea is to replace some of the faster point processes by a “fluid” (called the ghost processes) thus accelerating the execution of the simulation. Using local modularity for the code, there is no need to keep a list of events. Clocks are not necessarily synchronized. When a local clock advances due to a slower event, retrospective calculations recover the ne detail lost in the fluid model. Mathematically, the model is a special case of the Filtered Monte Carlo method. Efficiency improvement results not only from the speed of execution, but also from variance reduction. We provide proofs of unbiasedness. Throughout the paper we use a case scenario of an airport car park.
Rehabilitation of the Finite Di erence Estimator
Haralambie Leahu (VU University Amsterdam)
We note that under mild assumptions the di erential operator on the class of analytical functions can be expressed as a randomized nite-di erence quotient. We discuss possible applications in stochastic
gradient estimation for parameter-dependent random systems and establish connections with the existing methods. The key observation is that unbiased gradient estimators can be obtained whenever the
performance of the underlying system can be estimated without bias, for any instance of the parameter. In other words, (properly constructed) FD estimators are not necessarily biased!
Workforce Planning: Comparing three Algorithms for nding an optimal Recruitment Strategy
Charlotte Bech (VU University Amsterdam)
This talk provides insights into strategic workforce planning. For many organizations it would be very useful to know how their staff composition, also called workforce, will change in the future. “How many employees will leave the organization and from which function level?” “How many employees will make a transition internally and towards which function levels?” These questions lead to one fundamental question which is the motivation for this thesis: How many employees, and from which function level, have to be hired in the future? In other words, the challenge is to find an optimal recruitment strategy such that a desired workforce is reached as close as possible over a fixed period of time. Therefore, in this talk, three different algorithms are introduced and investigated. First, a Markov Decision Process is proposed to find an optimal recruitment strategy based on the probability distribution of the evolution of the workforce over time. Second, to make it more practical, a deductive algorithm is proposed that finds an optimal recruitment strategy based on the expected evolution of the workforce over time. Then, the idea arises that it is not always necessary to find an optimal recruitment strategy, but that sometimes a good recruitment strategy is already good enough. Therefore, the third algorithm proposed uses Ordinal Optimization. The latter is based on choosing a subset of strategies out of all possible strategies and investigate them intensively, resulting in a ranking of these strategies and the strategy that is ranked first is said to be a good recruitment strategy. Which of the three algorithms performs best depends on the organization and its preferences, and we discuss five criteria for characterizing the performances of the algorithms. In a case study we compare the algorithms based on their scores on these criteria.
A Statistical Perspective on Sensitivity Analysis for European Call Options
Alexia Bolotaki (VU University Amsterdam)
The European call option in the Black-Scholes world is a stochastic model based on the values of the parameters which are the strike price, the expiration day, the risk-free interest rate and the volatility. The first two parameters are controllable and, thus, known. But what happens when the risk-free interest rate and the volatility are inuenced by exogenous noises? This talk will investigate if the sensitivity analysis and the statistical estimation can be adjusted into one framework. Using the Taylor series of the expected value of the nancial option up to the third derivative (i.e. including skewness), an attempt for acknowledging the distributions of the exogenous noises will be made. In this talk, we will outline future research on including error estimation and exploration of its behavior w.r.t the noises. Hereby, studying the impact of the error characteristics such as variance and skewness will be a crucial issue. Finally, simulating the model and its error, numerical results and a comparison between the Black-Scholes model and the new one will be inspected.