Applied Stochastic Models and Data Analysis, Mar 1, 1993
A continuous time Markov‐renewal model is presented that generalizes the classical Young and Almo... more A continuous time Markov‐renewal model is presented that generalizes the classical Young and Almond model for manpower systems with given size. The construction is based on the associated Markov‐renewal replacement process and exploits the properties of the embedded replacement chain. The joint cumulant generating function of the grade sizes is derived and an asymptotic analysis provides conditions for these to converge in distribution to a multinominal random vector exponentially fast independently of the initial distribution, both for aperiodic and periodic embedded replacement chains. A regenerative approach to the wastage process is outlined and two numerical examples from the literature on manpower planning illustrate the theory.
We consider a finite, aperiodic, time homogeneous, absorbing Markov chain on a fuzzy partition, f... more We consider a finite, aperiodic, time homogeneous, absorbing Markov chain on a fuzzy partition, for which each time an absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The resulting process is a Markov replacement chain on a fuzzy partition. We study certain aspects of the aggregated process emerging from the classical theory on
ABSTRACT Several methods are considered for the generation of a complete set of order statistics ... more ABSTRACT Several methods are considered for the generation of a complete set of order statistics from a specified distribution. In the case of the uniform distribution, several methods in the literature are collected and reviewed. Three methods appropriate for general distributions are then described, with the normal and beta distributions considered as examples. The recommended method, which appears to be new, consists of dividing the range of the distribution into a large number of intervals and applying rejection sampling on each interval.
This paper is an exposition on the dynamic stochastic equilibria in multi-type time dependent Mar... more This paper is an exposition on the dynamic stochastic equilibria in multi-type time dependent Markov population processes for different types of environmental behaviour. The joint multivariate cumulant generating function of the various population sizes at any time is derived together with the covariance structure both with and without time lags. An asymptotic analysis shows that there are either one or several equilibria of dynamic character, depending critically on the behaviour of the system parameters as well as on the total population size. The possible stochastic equilibria are characterized as stable or unstable according to whether or not they are independent of initial conditions.
This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov sy... more This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
We consider a migration process whose singleton process is a time-dependent Markov replacement pr... more We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.
The asymptotic behaviour of the variances and covariances of the class sizes in closed and open m... more The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobility models (closed systems). Secondly, the non-homogeneous problem is studied and a theorem is given where under certain conditions it is proved that the vector sequences of means, variances and covariances converge. Finally, we relate our theoretical results to examples from the literature on manpower planning.
Proceedings of ... IEEE International Conference on Fuzzy Systems, Jun 1, 2007
We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition a... more We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition and for the resulting aggregated process we study two aspects emerging from the classical theory on hard partitions. The first aspect is lumpability, a technique for recovering from the large state space of a stochastic system. We provide necessary and sufficient conditions for strong lumpability on
ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is c... more ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.
We consider an absorbing semi-Markov chain for which each time absorption occurs there is a reset... more We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.
SIAM Journal on Matrix Analysis and Applications, 1992
This paper investigates the ergodic behaviour of the vector of means and the covariance matrix of... more This paper investigates the ergodic behaviour of the vector of means and the covariance matrix of the grade sizes for a nonhomogeneous Markov system that undergoes a cyclic behaviour, both in discrete and continuous time. It is shown that the first and second central moments converge to a cyclic family of multinomial type with the same period, independently of the initial distribution. The regions of cyclically ergodic distributions are determined as convex hulls of certain points as the recruitment distribution varies. The rate of convergence to the cyclic distribution is also examined, together with some transient aspects of the system concerning stability and quasi stationarity. Two numerical examples from the literature on manpower planning illustrate the theory.
Page 1. Mathematics and Its Applications Vladimir V. Kalashnikov Mathematical Methods in Queuing ... more Page 1. Mathematics and Its Applications Vladimir V. Kalashnikov Mathematical Methods in Queuing Theory Kluwer Academic Publishers Page 2. Page 3. Page 4. Page 5. Mathematical Methods in Queuing Theory "This One 5JDS-EZL-Y7JB Page 6. ...
This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov sy... more This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
In the present paper we study three aspects in the theory of non-homogeneous Markov systems under... more In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.
ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is c... more ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.
Applied Stochastic Models and Data Analysis, Mar 1, 1993
A continuous time Markov‐renewal model is presented that generalizes the classical Young and Almo... more A continuous time Markov‐renewal model is presented that generalizes the classical Young and Almond model for manpower systems with given size. The construction is based on the associated Markov‐renewal replacement process and exploits the properties of the embedded replacement chain. The joint cumulant generating function of the grade sizes is derived and an asymptotic analysis provides conditions for these to converge in distribution to a multinominal random vector exponentially fast independently of the initial distribution, both for aperiodic and periodic embedded replacement chains. A regenerative approach to the wastage process is outlined and two numerical examples from the literature on manpower planning illustrate the theory.
We consider a finite, aperiodic, time homogeneous, absorbing Markov chain on a fuzzy partition, f... more We consider a finite, aperiodic, time homogeneous, absorbing Markov chain on a fuzzy partition, for which each time an absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The resulting process is a Markov replacement chain on a fuzzy partition. We study certain aspects of the aggregated process emerging from the classical theory on
ABSTRACT Several methods are considered for the generation of a complete set of order statistics ... more ABSTRACT Several methods are considered for the generation of a complete set of order statistics from a specified distribution. In the case of the uniform distribution, several methods in the literature are collected and reviewed. Three methods appropriate for general distributions are then described, with the normal and beta distributions considered as examples. The recommended method, which appears to be new, consists of dividing the range of the distribution into a large number of intervals and applying rejection sampling on each interval.
This paper is an exposition on the dynamic stochastic equilibria in multi-type time dependent Mar... more This paper is an exposition on the dynamic stochastic equilibria in multi-type time dependent Markov population processes for different types of environmental behaviour. The joint multivariate cumulant generating function of the various population sizes at any time is derived together with the covariance structure both with and without time lags. An asymptotic analysis shows that there are either one or several equilibria of dynamic character, depending critically on the behaviour of the system parameters as well as on the total population size. The possible stochastic equilibria are characterized as stable or unstable according to whether or not they are independent of initial conditions.
This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov sy... more This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
We consider a migration process whose singleton process is a time-dependent Markov replacement pr... more We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.
The asymptotic behaviour of the variances and covariances of the class sizes in closed and open m... more The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobility models (closed systems). Secondly, the non-homogeneous problem is studied and a theorem is given where under certain conditions it is proved that the vector sequences of means, variances and covariances converge. Finally, we relate our theoretical results to examples from the literature on manpower planning.
Proceedings of ... IEEE International Conference on Fuzzy Systems, Jun 1, 2007
We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition a... more We consider a finite, irreducible, aperiodic, time homogenous Markov chain on a fuzzy partition and for the resulting aggregated process we study two aspects emerging from the classical theory on hard partitions. The first aspect is lumpability, a technique for recovering from the large state space of a stochastic system. We provide necessary and sufficient conditions for strong lumpability on
ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is c... more ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.
We consider an absorbing semi-Markov chain for which each time absorption occurs there is a reset... more We consider an absorbing semi-Markov chain for which each time absorption occurs there is a resetting of the chain according to some initial (replacement) distribution. The new process is a semi-Markov replacement chain and we study its properties in terms of those of the imbedded Markov replacement chain. A time-dependent version of the model is also defined and analysed asymptotically for two types of environmental behaviour, i.e. either convergent or cyclic. The results contribute to the control theory of semi-Markov chains and extend in a natural manner a wide variety of applied probability models. An application to the modelling of populations with semi-Markovian replacements is also presented.
SIAM Journal on Matrix Analysis and Applications, 1992
This paper investigates the ergodic behaviour of the vector of means and the covariance matrix of... more This paper investigates the ergodic behaviour of the vector of means and the covariance matrix of the grade sizes for a nonhomogeneous Markov system that undergoes a cyclic behaviour, both in discrete and continuous time. It is shown that the first and second central moments converge to a cyclic family of multinomial type with the same period, independently of the initial distribution. The regions of cyclically ergodic distributions are determined as convex hulls of certain points as the recruitment distribution varies. The rate of convergence to the cyclic distribution is also examined, together with some transient aspects of the system concerning stability and quasi stationarity. Two numerical examples from the literature on manpower planning illustrate the theory.
Page 1. Mathematics and Its Applications Vladimir V. Kalashnikov Mathematical Methods in Queuing ... more Page 1. Mathematics and Its Applications Vladimir V. Kalashnikov Mathematical Methods in Queuing Theory Kluwer Academic Publishers Page 2. Page 3. Page 4. Page 5. Mathematical Methods in Queuing Theory "This One 5JDS-EZL-Y7JB Page 6. ...
This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov sy... more This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.
In the present paper we study three aspects in the theory of non-homogeneous Markov systems under... more In the present paper we study three aspects in the theory of non-homogeneous Markov systems under the continuous-time formulation. Firstly, the relationship between stability and quasi-stationarity is investigated and conditions are provided for a quasi-stationary structure to be stable. Secondly, the concept of asymptotic attainability is studied and the possible regions of asymptotically attainable structures are determined. Finally, the cyclic case is considered, where it is shown that for a system in a periodic environment, the relative structure converges to a periodic vector, independently of the initial distribution. Two numerical examples illustrate the above theoretical results.
ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is c... more ABSTRACT The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.
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