The multi-type galton-watson process with immigration
TL;DR: In this article, conditions for ergodicity are found for a subclass of k-type Galton-Watson processes and expressions for the first two moments of the nth generation (by way of a recurrence relation) and for the asymptotic moments, in a manner which to some extent generalises previous results.
Abstract: We consider the limiting behaviour of a k-type (k < ∞) Galton-Watson process which is augmented at each generation by a stochastic immigration component. In Section 2, conditions for ergodicity are found for a subclass of such processes. In Section 3, expressions are derived for the first two moments of the nth generation (by way of a recurrence relation) and for the first two asymptotic moments, in a manner which to some extent generalises previous results.
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TL;DR: In this paper, an estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectations is developed, and the estimators and their limiting covariance matrix are worked out in detail for a subcritical branching process with immigration.
Abstract: An estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectations is developed. Strong consistency, asymptotic joint normality and an iterated logarithm rate of convergence are shown to hold for the estimators under a variety of conditions. Special attention is given to the widely studied cases of stationary ergodic processes and Markov processes with are asymptotically stationary and ergodic. The estimators and their limiting covariance matrix are worked out in detail for a subcritical branching process with immigration. A brief Monte Carlo study of the performance of the estimators is presented.
429 citations
TL;DR: In this paper, the joint queue length process at polling instants of a fixed queue is shown to be a multitype branching process (MTBP) with immigration, and sufficient conditions for ergodicity and moment calculations are given.
Abstract: The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.
226 citations
TL;DR: This book discusses Maximization, Minimization, and Motivation, which is concerned with the optimization of Symmetric Matrices, and its applications in Programming and Mathematical Economics.
Abstract: Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.
65 citations
TL;DR: In this article, the authors considered a branching process with immigration, where each individual reproduces independently of all others and has probability aj (j = 0, 1, ···) of giving rise to j progeny in the following generation.
Abstract: Consider a branching process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, ···) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where, with probability bj (j = 0, 1, ···) j objects enter the population at each generation. Then letting Xn (n = 0, 1, ···) be the population size of the nth generation, it is known (Heathcote (1965), (1966)) that {Xn } defines a Markov chain on the non-negative integers and it is called a branching process with immigration (b.p.i.). We shall call the process sub-critical or super-critical according as the mean number of offspring of an individual, , satisfies α 1, respectively. Unless stated specifically to the contrary, we assume that the following condition holds.
61 citations
01 Jan 2008
TL;DR: A single-server cyclic polling system between visits to successive queues, the server is delayed by a random switch-over time, and the order in which customers are served in each queue is determined by a priority level that is assigned to each customer at his arrival.
Abstract: In this paper we consider a single-server cyclic polling system. Between visits to successive queues, the server is delayed by a random switch-over time. The order in which customers are served in each queue is determined by a priority level that is assigned to each customer at his arrival. For this situation the following service disciplines are considered: gated, exhaustive, and globally gated. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type.
30 citations
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16,450 citations
Book•
01 Jan 1966
TL;DR: In this paper, the Basic Limit Theorem of Markov Chains and its applications are discussed and examples of continuous time Markov chains are presented. But they do not cover the application of continuous-time Markov chain in matrix analysis.
Abstract: Preface. Elements of Stochastic Processes. Markov Chains. The Basic Limit Theorem of Markov Chains and Applications. Classical Examples of Continuous Time Markov Chains. Renewal Processes. Martingales. Brownian Motion. Branching Processes. Stationary Processes. Review of Matrix Analysis. Index.
3,881 citations
Book•
01 Jan 1960
TL;DR: In this article, the Second Edition Preface is presented, where Maximization, Minimization, and Motivation are discussed, as well as a method of Hermite and Quadratic Form Index.
Abstract: Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.
3,500 citations
Book•
01 Dec 1963
TL;DR: A review of the Galton and Watson mathematical model that applies probability theory to the effects of chance on the development of populations is given in this article, followed by a systematic development of branching processes, and a brief description of some of the important applications.
Abstract: A review of the Galton and Watson mathematical model that applies probability theory to the effects of chance on the development of populations, followed by a systematic development of branching processes (one of the generalizations from the Galton-Watson model), and a brief description of some of the important applications.
2,596 citations