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1991 | Buch

The Square Wave Spectrum of a Markov Renewal Process Kapitel lesen Erstes Kapitel lesen

Stochastic Processes and their Applications

Proceedings of the Symposium held in honour of Professor S.K. Srinivasan at the Indian Institute of Technology Bombay, India, December 27–30, 1990

herausgegeben von: M. J. Beckmann, M. N. Gopalan, R. Subramanian

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Economics and Mathematical Systems

Enthalten in: Professional Book Archive

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Über dieses Buch

A volume of this nature containing a collection of papers has been brought out to honour a gentleman - a friend and a colleague - whose work has, to a large extent, advanced and popularized the use of stochastic point processes. Professor Srinivasan celebrated his sixt~ first 1:!irth d~ on December 16,1990 and will be retiring as Professor of Applied Mathematics from the Indian Institute of Technolo~, Madras on June 30,1991. In view of his outstanding contributions to the theor~ and applications of stochastic processes over a time span of thirt~ ~ears, it seemed appropriate not to let his birth d~ and retirement pass unnoticed. A s~posium in his honour and the publication of the proceedings appeared to us to be the most natural and sui table ~ to mark the occasion. The Indian Societ~ for ProbabU it~ and Statistics volunteered to organize the S~posium as part of their XII Annual conference in Bomba~. We requested a number of long-time friends, colleagues and former students of Professor Srinivasan to contribute a paper preferabl~ in the area of stochastic processes and their applications. The positive response and the enthusiastic cooperation of these distinguished scientists have resulted in the present collection. The contributions to this volume are divided into four parts: Stochastic Theor~ (2 articles), P~sics (6 articles), Biolo~ (4 articles) and Operations Research (12 articles). In addition the ke~note address delivered b~ Professor Srinivasan in the S~posium is also included.

Inhaltsverzeichnis

Frontmatter

Stochastic Theory

The Square Wave Spectrum of a Markov Renewal Process
Abstract
The power spectral density of a stationary square wave process with sign changes at the transitions of a finite state Markov renewal process is derived. Particularly explicit formulas are obtained for the Markovian Arrival Process, a generalization of the Poisson process with a natural matrix formalism which commonly leads to useful explicit results. An application to a procedure for the quantification of burstiness is discussed.
Marcel F. Neuts
Simulation and Estimation Procedures for Stress Release Model
Abstract
The stress release model is a piecewise linear Markov Process similar to the collective risk model,in which the observations consist of the times and sizes of the Jumps, which are taken to represent the times and sizes of large earthquakes. Earlier studies have shown that the asymptotic behaviour of the likelyhood-based tests for this process against a Poisson null hypothesis show anomalous behaviour. The present paper develops simulation methods for the process and uses them to investigate quantitatively some of the qualitative predictions of theoretical studies. In particular it is confirmed that the distribution of the likelyhood ratio statistic is non-standard.
Ann-Lee Wang, David Vere-Jones, Xiao-gu Zheng

Physics

Positive Definite Functions in Quantum Mechanics and in Turbulence
Abstract
Positive-definite functions have various representations which can be interpreted in probability theory, in quantam mechanics,and in turbulence theory. The equivalence of these representations is studied. The nature of the characteristic functions in quantum mechanics is discussed. In turbulence, time-correlations are positive-definite functions, but space-correlations have an ambiguous structure. In the annex, the properties of pseudo-random functions, which are convenient representations of turbulence, are summarized.
J. Bass
Population Monitoring and the Quantum Input-Output Formalism
Abstract
The quantum Input-Output formalism of Gardiner and Collett is used to derive the rate equation describing the photoelectron counting statistics of an electromagnetic field output from a high-Q optical cavity. The evolution of the cavity field is assumed to be governed by an arbitrary Hamiltonian for a single mode of the field. The resulting equation is shown to conform with one previously derived on intuitive grounds, using purely population statistical arguments. As an explicit example, the solution for a freely-evolving cavity field is computed.
J. Jeffers, T. J. Shepherd
An Application of the Kalman Filter in Geoastronomy
Abstract
This paper is presented as a contribution to the Symposium in honour of my distinguished colleague and long-time friend, S.K. Srinivasan. My intention in this paper is to provide a brief overview of an application of stochastic theory to geosastronomy, an area which Professor Srinivasan has not touched, but one in which there have been some remarkable advances in the past decade. Stochastic processes are inextricably mixed up in the recordings of signals from celestial bodies which constitute the basic raw material of geoastronomy. The Kalman filter (Kalman [1960], Kalman and Bucy [1960]), developed as an efficient method of including in an estimation process parameters whose values change during a period over which data are collected, uses stochastic-process models to predict their changes between epochs of observation. The use of the Kalman filter in extracting results of astonishingly high accuracy from the data contaminated by the stochastic “noise” is detailed in a recent paper by Herring et al. [1990], on which the present outline is largely based. The kinds of results that can be obtained relating to the surface and the interior of the Earth are illustrated herein by a few examples.
P. M. Mathews
Conformal Martingales in Stochastic Mechanics
Abstract
We study the probabilistic description of quantum systems in terms of stochastic differential equations, subject to multiplicative noise. The martingale properties of the complex Brownian motion are exploited. The conformal stochastic calculus is the key concept in arriving at the new result in terms of diffusions.
K. V. Parthasarathy
Probability Distributions over Noncommuting Variables
Abstract
The notion of probability distributions and associated characteristic functions over a set of noncommuting variables are studied. With the provision that the marginal distributions restricted to a commuting set should give the nonnegative probability distributions over those variables we find that such generalized master distributions exist which, while incorporating implicit positivity requirements, are not themselves pointwise necessarily nonnegative. The cumulant generating functions are studied and the applicability of the Marcinkiewicz theorem explored.
E. C. G. Sudarshan
Stochastic Quantum Mechanics
Abstract
The need for answering the question whether quantum mechanics is complete by itself and whether quantum fluctuations and thermal fluctuations are of the same genus, is discussed in this report. The hydrodynamic Hamilton Jacobi equations for the quantum Madelung fluid reveal the existence of a quantum potential which corresponds to the mysterious dependence of the individual on the statistical ensemble of which it is a member. Nelson’s analysis, starting from the Brownian motion of a particle moving in the field of a background white noise, leads to coupled equations for the velocity fields under certain conditions; these are described in the first three sections.
Section 4 deals with indeterminacy relation in a novel way based on the two velocities envisaged in Nelson’s work and some examples are given. The concluding section deals with the criticism of these two approaches and promises the unification of the three approaches including the derivation of the Feynman’s path formalism obtained by complexifying the phase accumulation and assigning a proper measure for it. The extension of this approach to relativistic and spinning particles will form the content of part II of this article to be published elsewhere.
R. Vasudevan

Biology

A New Approach to the Solution of Neurological Models: Application to the Hodgkin-Huxley and the Fitzhugh-Nagumo Equations
Abstract
The solution of the Hodgkin-Huxley and the Fitzhugh-Nagumo equations are demonstrated as applications of the decomposition method [1–3] which can be used as a new and useful approach obtaining analytical and physically realistic solutions to neurological models and other biological problems without perturbation, linearization, discretization, or massive computation.
George Adomian, Matthew Witten, Gerald E. Adomian
Neuronal Variability — Stochasticity or Chaos?
Abstract
Irregularity in neuronal activity can be characterised in terms of either stochastic theory and dynamical system theory. The choice of approach is arbitrary but it may be possible to fuse these approaches in the near future.
A. V. Holden, M. A. Muhamad
A Limit Theorem and Asymptotical Statistical Characteristics for a Selective Interaction Model of a Single Neuron
Abstract
We consider in this article, interaction of two types of inputs, one excitatory and the other inhibitory arriving according to a stationary renewal point process, resulting in a renewal point process of events consisting of neural discharge (response yielding events). Apart from obtaining the explicit expressions for the asymptotical statistical characteristics of time for the first response yielding event, a limit theorem is proved.
A. Rangan
Phase Dependent Population Growth Models
Abstract
Growth of biological populations is phase dependent; individuals can give birth to offspring only after reaching a ‘maturity age’. Assuming that the life time of an individual consists of two phases, we study the implication of phase dependence in deterministic and stochastic population growth models. By taking the first phase to be of constant duration in the deterministic model an explicit expression for the size of the population is derived. Developing a birth and death process in which the birth and death rates are age and phase dependent, an explicit expression for the mean number of individuals has been obtained in the case when the death rate is constant. Two particular cases of the model are also discussed.
C. R. Ranganathan

Operations Research

The Optimal Investment Process in German Industry
Abstract
This paper provides an explanation of investment behaviour of German industrial firms. Regarding the actual development of business investment, there are two main problems of interest:
(1)
its decline between 1970 – 1976,
 
(2)
its recovery, first slow between 1976 – 1984, then sudden after 1984.
 
The theoretical framework of our analysis is provided by the Bonn Model of Firm Development, which represents an econometric model of firm growth consisting of 27 equations dependent on each other. Firms are divided into functional sections which act independent from each other and therefore come to globally suboptimal decisions. Data material of annual statements of accounts of 134 German industrial firms yield empirical support of our theory.
Two models of business investment are motivated, discussed and tested: In the first model, the cost of the production process are minimized subject to a Cobb-Douglas production function with Hicks-neutral technical progress. Factor inputs react highly elastic to changes in relative factor prices. Factor stocks adjust to their optimal level with high speed. As a consequence, investment behaviour is explained by the investment rate of the previous period, weighted averages of growth rate of sales and weighted averages of growth rate of relative prices for capital goods.
Whereas user cost of capital remained relatively stable overtime, wage rates and raw material prices rose significantly. For the reason of revaluation of the deutschmark, relative prices remained rather stable, so that no major investments were undertaken. After 1984, the price indices of factor inputs developed in opposite directions in favour of capital goods prices. As a consequence, investment increased substantially.
In the second model, the intertemporal allocation of resources is optimized simultaneously. Capital and labour are treated as quasi-fix factors. The speed of adaptation is considered rather low since reaching the optimal level of factor input stocks is costly. Therefore our cost function is extended to an adaptation cost term. First, this model reveals a complementary relationship between labour and capital. Secondly, the amount of adaptation cost is crucial for the level of employment. In order to achieve full employment, conditions for equity financing have to be improved. To stress the importance of this ”delicious circle” is one fundamental purpose of this paper.
Horst Albach
Incentives and Regulation in Queues
Abstract
It is known that customers seeking service in a queuing system tend to overcrowd the facility when making their individual decisions based on a consideration of the benefits they derive from the service and the cost due to waiting at the system. To obtain an optimal utilization it is necessary to restrict entry by pricing. In this paper, the effect of applying the operating cost of the service center to its users is analyzed.
In a single class customers case it is shown that when service center cost is applied, the optimal arrival rate may be higher than the individual equilibrium rate and thus a subsidy to increase the arrival rate to the system may be required. For the multi-class case the class dominance property states that for optimal utilization, the system should be dominated by a single class of customers. It is shown here that when service center costs are absorbed by the users, there exists conditions for which the class 1 users may restrict their own arrival rate in order to allow class 2 customers to utilize the facility. The conditions for such are derived for the FIFO and nonpreemptive priority rules.
Kashi R. Balachandran
Two Models of Brand Switching
Abstract
It is well known that brand choice can be described by a Markov chain. In order to put some structure into the transition probabilities we model brand choice as a two-stage decision process: (1) whether to continue or whether to reconsider the last choice, (2) in the latter case which brand to choose. A distinction must then be made as to whether the last brand is ruled out (hypothesis II) or not (hypothesis I) giving rise to two different probability models. In the case of only two brands, the transition probabilities can always be modelled in either way.
The following problems are considered. How to test the existence of choice probabilities? How in the ergodic case the state probabilities, i.e., the long-run market shares are determined? What are the implications of zero-one probabilities of retention or choice? Under what conditions are market shares equalized? It is also suggested that the retention probabilities depend on product attributes while the choice probabilities respond to advertising.
Martin J. Beckmann
Stochastic Processes: Use and Limitations in Reliability Theory
Abstract
Stochastic processes are powerful tools for the investigation of the reliability and availability of repairable equipment and systems. Because of the involved models and in order to be mathematically tractable, these processes are generally confined to the class of regenerative stochastic processes with a finite state space, to which belong renewal processes, Markov processes, semi-Markov processes, and more general regenerative processes with only few (in the limit case only one) regeneration states. This contribution introduce briefly these processes and uses them to solve some reliability problems encountered in pratical applications. Investigations deal with different kinds of reliabilities and availabilities for one item, series, parallel, and series/ parallel structures. For series/parallel structures useful approximate expressions are developed.
A. Birolini
Stochastic Processes and Optimization Problems in Assemblage Systems
Abstract
Assemblage systems e.g. arise from production systems, where k different pieces are delivered by k parallel production lines. At some place this pieces are assembled in order to form the desired good, taking exactly one piece from every of the k single lines. According to random fluctuations of the production process parallel queues are formed by single pieces waiting to be processed at the assemblage station. If one neglects the time needed to transform a complete group of k different parts into the final assembled good, a pure assemblage system emerges. It is always possible, to separate an assemblage system into two stages: firstly a pure assemblage system, followed by a queueing system with a single waiting line of complete sets of k pieces. In this paper pure assemblage systems are considered. If such systems have unlimited waiting-room for each line, an equilibrium distribution of queue lenghts never exists. Therefore control measures such as limitation of the waiting-room or partial reduction of production speed are taken and give raise to various optimization problems. With few exceptions, such problems seem to be rather difficult for k > 2 production lines.
In this article the specific reasons of the difficulties of multiline systems should be pointed out and the following results should be given: Firstly, an algorithm in matrix form for the calculation of the equilibrium distribution in the case of three production lines; secondly, approximations for the distribution of the number of single parts waiting in front of the assemblage system; this approximations are needed for explicit handling of optimization problems.
Franz Ferschl
A Software Package Tool for Markovian Computing Models with Many States: Principles and its Applications
Abstract
A computing system can be formulated by modeling a continuous-time Markov chain with many states, and be evaluated by using the reliability/performance measures. The randomization technique is discussed to derive the transient solution for a Markov chain. A software package tool is implemented by using the randomization technique and introducing a new idea of identifying when the transient solution converges to the steady-state solution in advance. Numerical examples are illustrated by using our software package tool to evaluate the optimal maintenance policies for computing systems. Some interesting maintenance policies are suggested from the numerical examples.
Satoshi Fukumoto, Shunji Osaki
Reliability Assessment Measurement for Redundant Software Systems
Abstract
It is important to make a software system more reliable. We discuss reliability assessment for redundant software systems developed by multiversion programming, i.e., N-version programming and recovery blocks systems, during the operation phase. In this paper, the reliability assessment measures such as software reliability function, hazard rate function, and MTTF (mean time to failure) are obtained by using two software reliability growth models, i.e., Jelinski-Moranda and NHPP models. Also the results of reliability assessment based on the two software reliability growth models are numerically compared for each redundant software system.
Jun Hishitani, Shigeru Yamada, Shunji Osaki
A Lost Sales Inventory System with Multiple Reorder Levels
Abstract
In the analysis of continuous review inventory systems with lost sales, all the models studied so far impose the restriction s < S-s or s = S-l to make the analysis tractable.This paper deals with a general lost sales inventory system with renewal demands and exponential lead times without any condition on the values of s and S.The inventory level distribution and the mean reorder and shortage rates are obtained. A computational procedure to calculate the various limiting values along with numerical illustration is provided.
S. Kalpakam, G. Arivarignan
Queueing Models in Hierachical Planning Systems
Abstract
Frequently, modern systems of production management are characterized by a hierarchical structure: Decisions on a higher level set targets to be considered by production planning and carried out by production control on subordinate levels. Setting targets on the superior level requires information about the performance, such as net capacities available on the subordinate level. As its performance depends on inventory planning and scheduling, decisions on the subordinate level which, in turn, depend on the targets to be fulfilled, this information is not available for planning on the superior level. Capacities available and other measures of performance may, however, be derived using appropriate mathematical models of the planning process.
In this paper, several queueing theoretical approaches estimating the capacity and the throughput of subordinate manufacturing units in a hierarchical production planning surrounding are presented. Considering the assumptions necessary to find operable solutions, e.g. appropriate Markovian properties, it can be shown that models of open and closed queueing networks may be applied in the case of single item manufacturing and production of small batches. Whereas hybrid approach with a continuous flow of production and rates switching according to a semi-Markov process may be applied in the case of large batches.
Klaus-Peter Kistner
Reliability Analysis of a Complex System Using Boolean Function Technique
Abstract
This paper deals with reliability analysis of a complex system consisting of two subsystems, connected in parallel. Each subsystem consists of a generator, one main switchboard and a given number of auxiliary switchboards. Two models of this system are studied. The reliability of the system with its imposed conditions is then calculated using the Boolean function technique and combinatorial methods. Some special cases are studied and a numerical example is provided.
Yadavalli V. S. Sarma, Howard P. Hines
The Second Moment of the Markovian Reward Process
Abstract
In this note an m-state aperiodic irreducible Markov Chain is considered with an associated reward matrix. Expressions for the second moment of Sn, the accumulated reward in n transitions, have been obtained using elementary methods. These provide an alternative computational method compared to methods using the eigen value structure. Asmyptotic forms are also considered.
S. Subba Rao
Correlation Functions in Reliability Theory
Abstract
The multivariate point process induced by the stochastic behaviour of a two-unit warm standby redundant repairable system is studied. Expressions for the product densities of the events corresponding to the entry into each of the states and the interval reliability are obtained. The reliability and availability are deduced as special cases.
R. Subramanian, N. Ravichandran
Backmatter
Metadaten
Titel
Stochastic Processes and their Applications
herausgegeben von
M. J. Beckmann
M. N. Gopalan
R. Subramanian
Copyright-Jahr
1991
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-58201-1
Print ISBN
978-3-540-54635-1
DOI
https://doi.org/10.1007/978-3-642-58201-1