A Two-Phase Cyclic Nonhomogeneous Markov Chain Performability Evaluation by Explicit Approximate Inverses Applied to a Replicated Database System

The need for a more accurate modeling of the performance of systems whose functioning mainly dependant on external time parameters such as the number of requests during a particular time phase, led us to a novel approach, taking into account the time parameters involved. This is achieved through the evaluation of a performability indicator modeled by means of a two-phase cyclic nonhomogenous Markov chain considering periodical time-dependant arrival request probabilities and applied to a replicated database system. The computation of the performability indicator modeled by cyclic nonhomogeneous Markov chain requires the use of efficient computational methods by using explicit approximate inverse preconditioning methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Performability Analysis of Fork-join Queueing Systems

Fork-join queueing systems offer a natural modelling paradigm for parallel processing systems and for assembly operations in automated manufacturing. The analysis of fork-join queueing systems has been an important subject of research in recent years. Existing analysis methodologies—both exact and approximate—assume that the servers are failure-free. In this study, we consider fork-join queueing systems in the presence of server failures and compute the cumulative distribution of performability with respect to the response time of such systems. For this, we employ a computational methodology that uses a recent technique based on randomization. We compare the performability of three different fork-join queueing models proposed in the literature: the distributed model, the centralized splitting model, and the split-merge model. The numerical results show that the centralized splitting model offers the highest levels of performability, followed by the distributed splitting and split-merge models.     

Probabilistic modeling of computer system availability

System availability is becoming an increasingly important factor in evaluating the behavior of commercial computer systems. This is due to the increased dependence of enterprises on continuously operating computer systems and to the emphasis on fault-tolerant designs. Thus, we expect availability modeling to be of increasing interest to computer system analysts and for performance models and availability models to be used to evaluate combined performance/availability (performability) measures. Since commercial computer systems are repairable, availability measures are of greater interest than reliability measures. Reliability measures are typically used to evaluate nonrepairable systems such as occur in military and aerospace applications. We will discuss system aspects which should be represented in an availability model; however, our main focus is a state of the art summary of analytical and numerical methods used to solve computer system availability models. We will consider both transient and steady-state availability measures and for transient measures, both expected values and distributions. We are developing a program package for system availability modeling and intend to incorporate the best solution methods.     

Approximate solution for two stage open networks with Markov-modulated queues minimizing the state space explosion problem

Analytical solutions for two-dimensional Markov processes suffer from the state space explosion problem. Two stage tandem networks are effectively used for analytical modelling of various communication and computer systems which have tandem system behaviour. Performance evaluation of tandem systems with feedbacks can be handled with these models. However, because of the numerical difficulties caused by large state spaces, considering server failures and repairs at the second stage employing multiple servers has not been possible. The solution proposed in this paper is approximate with a high degree of accuracy. Using this approach, two stage open networks with multiple servers, break downs, and repairs at the second stage as well as feedback can be modelled as three-dimensional Markov processes and solved for performability measures. Results show that, unlike other approaches such as spectral expansion, the steady state solution is possible regardless of the number of servers employed.     

证券价格长期波动控制系统的最优控制策略

探讨证券价格长期波动控制系统的最优控制问题.建立了在有效市场条件下证券价格长期波动的控制系统模型.为了使证券价格和内在价值按照人们预期的目标变化,探讨了对它们服从的系统采用经典信息结构下的随机最优控制策略问题.设计了使系统的输出跟踪证券内在价值的估计值,同时使调节控制的幅度尽可能小的性能指标,给出了最优控制策略的求解公式和计算过程,并给出了考虑系统性能的计算过程,对相应结果进行了分析.主要结论是:当价值对价格的均衡回归调整不足,或投资者对前期价值的增值预期乐观时,最优控制策略所起的作用在加强;而当价值对价格的均衡回归调整过度,或投资者对前期价值的增值预期悲观时,最优控制策略所起的作用在减弱.这些结果可以为完善证券市场和上市公司的监管提供理论依据     

Optimality principle and synthesis for a stochastic control problem in hilbert spaces

We consider a linear system with additive noise in Hilbert space and minimize a convex functional associated with this process. A necessary and sufficient condition for a control to be optimal is derived by evaluating the subdifferential of the cost function. Then the subdifferential of the value function is characterized. Finally using these results and a conditional value function, optimal controls are characterized as a feedback law in terms of the value function.     

状态和控制均含时滞的受外界扰动的线性系统的最优控制

主要研究一类受外界持续扰动且状态和控制含不同时滞的线性系统的最优控制,首先通过变量代换,将系统化为控制不含时滞的滞后型微分系统.接着使用最优控制的极大值原理的必要条件,得到含超前和滞后项的两点边值问题.为了得到最优控制律的解析解,引进一个灵敏参数ε,得到两点边值问题序列,通过迭代法,得到最优控制律的解析解.并对外界扰动状态构造降维观测器,来实现最优控制律的物理可实现性.最后实例验证了上述方法的有效性.     

On the asymptotics of the optimal value of the performance functional in a linear optimal control problem

We consider the asymptotics of the optimal value of the performance functional in an optimal control problem for a linear system with rapid and slow variables, with convex terminal performance functional depending on the slow variables, and with smooth geometric constraints on the control. We find sufficient conditions for the regularity of asymptotics and construct an algorithm for finding the complete asymptotics of the optimal value of the performance functional.     

带有止步和状态相依的M/Ej/1/N排队系统的矩阵解法

研究了带有止步和服务率依赖于状态的M/Ej/1/N排队系统.顾客到达系统时,以一定的概率选择进入系统或止步(不进入系统).顾客接受服务的服务率依赖于系统中的顾客数,当系统中的顾客数不超过临界值k时,服务员慢速服务;否则,服务员快速服务.利用分块矩阵的方法,推出了稳态概率向量所满足的矩阵形式的迭代公式,给出了稳态概率的表达式和计算过程.作为特例,考虑了N=4时系统稳态概率的计算.在此基础上,还求出了系统的一些性能指标,并建立了以临界值k为控制变量的费用模型.通过数值分析,求出了使费用函数最小的最优临界值k*,并进一步研究了模型参数对最优临界值和最优费用的影响.     

Interval optimal control problem in a Hilbert space

An optimal control problem for a system involving an interval parameter is considered. The concepts of a universal optimal state and a universal optimal control are introduced. The existence and uniqueness of a universal solution to the interval optimal control problem is proved, and an algorithm for its determination is presented. The interval optimal control problem for a system described by the boundary value problem for a second-order ordinary differential equation is solved as an example.     

Constructing Robust Feedback Laws by Set Oriented Numerical Methods

In [8, 6] a numerical method for the construction of optimally stabilizing feedback laws was proposed. The method is based on a set oriented discretization of phase space in combination with graph theoretic algorithms for the computation of shortest paths in directed weighted graphs. The resulting approximate optimal value function is piecewise constant, yielding an approximate optimal feedback which might not be robust with respect to perturbations of the system. In this contribution we extend the approach to the case of perturbed control systems. Based on the concept of a multivalued game we show how to derive a directed weighted hypergraph from the original system and generalize the corresponding shortest path algorithm. The resulting optimal value function yields a robustly stabilizing approximate optimal feedback law. This note is an abbreviated version of [5]. For the proofs of the statements here we refer to the full paper. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

OPTIMAL CONTROL OF FINANCIAL MANAGEMENT SYSTEMS WIITH DELAY

Recentlymoderncontroltheoryhasbeenwidelyusedineconomymanagementsystems.Particularlythetheoryofoptimalcontrolisusedinfinancialmanagementsystemswithgreatsuccess.Butwenoticethatfinancialmanagementsystemsaregenerallycontrolsystemswithdelay.Theoptimizationofth…     

Dynamic Network Inventory Control Model with Interval Nonstationary Demand Uncertainty

We consider a dynamic inventory control system described by a network model with an interval assigned nonstationary demand. We assume that unknown demand may take any value within the interval, which bounds depend on time. In terms of Kaucher interval arithmetic, we derive necessary and sufficient conditions for the existence of a feasible feedback control and sufficient conditions for the existence of an optimal feedback control strategy. We obtain an optimal feasible storage level and estimate the rate of the system convergence to this level. Then we develop the algorithm of finding the optimal control strategy. These results are applied to an example.     

Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls

This work focuses on numerical methods for finding optimal dividend payment and investment policies to maximize the present value of the cumulative dividend payment until ruin; the surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the optimal value function obeys a coupled system of nonlinear integro-differential quasi-variational inequalities. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to approximate the value function and optimal controls. Convergence of the approximation algorithms are proved. Examples are presented to illustrate the applicability of the numerical methods.     

Optimal internal pricing and capacity planning for service facility with finite buffer

An economic model is developed for exploring optimal internal pricing and capacity planning for service facility with finite buffer capacity. Because of the limited buffer capacity, jobs finding the system full upon their arrival would be rejected. Such rejections create a gap between the value collectively perceived by users and the actual achievement of the organizational value. This gap, called a loss externality, has never been studied before and plays an important role for designing optimal pricing scheme. In general, the underlying economic structure may involve multiple equilibria and it is unclear whether or not the system can be controlled through internal pricing. In this regard, a sufficient condition is given under which the system administrator can find two separate prices for accepted and rejected users at any demand level to be desired so that the desired demand level becomes the unique equilibrium of the system. For a short-run problem, it is shown that the optimal pricing scheme can be expressed as the sum of the congestion and the loss externalities. For a long-run problem, the optimal pricing scheme is expressed in a unified manner so that a structural relationship between the short-run problem and the long-run problem at optimality can be readily observed. A necessary and sufficient condition is also given for the marginal capacity pricing to be optimal, i.e., the optimal long-run pricing consists of the marginal cost for processing capacity and the marginal cost for buffer capacity without involving any externality at all.     

Optimal damping ratio for linear second-order systems

In this paper, it is shown that the optimal damping ratio for linear second-order systems that results in minimum-time no-overshoot response to step inputs is of bang-bang type. The optimal damping ratio is zero at the outset and is switched to some maximum value at an appropriate instant of time. The switching time is shown to be a function of the maximum damping ratio and the system natural frequency. Furthermore, it is shown that the larger the maximum damping ratio is, the shorter it takes for the system to reach the desired set point. Finally, it is shown that, if the optimal damping ratio is switched as a function of the system state, then the minimum-time no-overshoot criterion is satisfied, irrespective of the magnitude of the uncertainty in the value of the system natural frequency.     

A Procedure for Constructing Optimum Functional Filters for Linear Stationary Stochastic Systems

Three problems closely related to the classical unbiased optimal filtration problem: an unbiased optimal filtration problem without a control in the system,a biased optimal filtration problem where the bias does not exceed a given value, and the joint problem of stabilization and optimal filtration. It is proposed these problems be reduced to ones of nonlinear optimization. For unbiased filtration with no control, conditions are provided that allow the one for classical unbiasedness to be weakened or excluded for the filter. A new estimate of the bias of the mean filtration error is proposed.     

On the zero dynamics of one class of distributed discrete systems

The problem of optimal control of a linear dynamical system under set-membership uncertainty is studied: it is required to steer the system to the terminal set with a guarantee and to maximize the guaranteed value of the quality criterion. The sets of the initial and current a preposteriori distributions of the states of the dynamical system are introduced; they are used to determine a positional solution of the problem of optimal a preposteriori³ observation with the help of inaccurate measurements of input and output signals of the observation object by two measuring devices. The obtained solution is used for determining a positional solution of the optimal control problem under uncertainty. Depending on the amount of the information used, optimal closable and closed output feedbacks are determined. The method of quasiimplementation of optimal feedbacks by means of optimal estimators and a regulator producing real-time control actions is described. The results are illustrated by examples.     

上升线性需求条件下考虑资金时间价值的变质性物品生产库存模型

变质性物品生产库存系统的研究具有重要实际意义.本文研究了变质性物品生产库存系统在上升趋势线性需求条件下,考虑资金的时间价值,在有限计划时间水平内,如何确定最优生产周期,各周期最优生产率,以及最优库存安排策略.通过本文的研究,得到了一些有用的结论.     

A note on multi-item inventory systems with limited capacity

Functional relationships between the Lagrange multiplier and system parameters for multi-item inventory systems with one restriction are identified and used to establish tight bounds on the optimal multiplier value. A recursive process which rapidly converges to the optimal multiplier is also discussed. Finally, a comparative analysis of the new bounds in relation to existing bounds is presented.