A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain

J. P. Lehoczky

doi:10.2307/3211898

A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain

Journal of Applied Probability ◽

10.1017/s0021900200035294 ◽

1971 ◽

Vol 8 (02) ◽

pp. 276-284 ◽

Cited By ~ 5

Author(s):

J. P. Lehoczky

Keyword(s):

Markov Chain ◽

Generating Functions ◽

Sample Path ◽

Exact Distribution ◽

Total Delay ◽

Traffic Flow Theory ◽

Second Moments ◽

Close Relationship ◽

First Moment ◽

Infinite Reservoir

Summary The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized. The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.

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On the limit behavior of a multicompartment storage model with an underlying Markov chain

Advances in Applied Probability ◽

10.2307/1427276 ◽

1988 ◽

Vol 20 (1) ◽

pp. 208-227

Author(s):

Eric S. Tollar

Keyword(s):

Brownian Motion ◽

Markov Chain ◽

Limit Behavior ◽

Moment Conditions ◽

Storage Model ◽

Second Moments ◽

First Moment ◽

Inputs And Outputs

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

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On the limit behavior of a multicompartment storage model with an underlying Markov chain

Advances in Applied Probability ◽

10.1017/s0001867800018000 ◽

1988 ◽

Vol 20 (01) ◽

pp. 208-227

Author(s):

Eric S. Tollar

Keyword(s):

Brownian Motion ◽

Markov Chain ◽

Limit Behavior ◽

Moment Conditions ◽

Storage Model ◽

Second Moments ◽

First Moment ◽

Inputs And Outputs

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

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Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

Yugoslav journal of operations research ◽

10.2298/yjor180418031k ◽

2019 ◽

Vol 29 (1) ◽

pp. 135-144

Author(s):

James Kim ◽

Mohan Chaudhry ◽

Abdalla Mansur

Keyword(s):

Discrete Time ◽

Renewal Process ◽

Generating Functions ◽

Continuous Time ◽

Constant Term ◽

Numerical Results ◽

Asymptotic Results ◽

Numerical Computations ◽

Second Moments ◽

Simplified Solution

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results.

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First-emptiness problems in applied probability under first-order dependent input and general output conditions

Journal of Applied Probability ◽

10.1017/s0021900200095334 ◽

1973 ◽

Vol 10 (02) ◽

pp. 330-342 ◽

Cited By ~ 1

Author(s):

J. P. Lehoczky

Keyword(s):

Markov Chain ◽

Integral Functional ◽

Applied Probability ◽

Input Process ◽

First Order ◽

Infinite Reservoir

Results for the first-emptiness time of a semi-infinite reservoir and the integral functional of the process up to first-emptiness time are derived under Markov chain input conditions and general output conditions. The results are further extended to allow an input process which is the sum of k consecutive elements of the Markov chain, k ≧ 1.

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A sample path analysis of the M/M/1 queue

Journal of Applied Probability ◽

10.1017/s002190020002742x ◽

1989 ◽

Vol 26 (02) ◽

pp. 418-422 ◽

Cited By ~ 2

Author(s):

Francois Baccelli ◽

William A. Massey

Keyword(s):

Generating Functions ◽

Queueing Networks ◽

Transient Analysis ◽

Bessel Functions ◽

Sample Path ◽

Laplace Transforms ◽

Sample Path Analysis ◽

Transient Distribution ◽

The Past ◽

Novel Approach

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

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Ergodicity of age structure in populations with Markovian vital rates. II. General states

Advances in Applied Probability ◽

10.1017/s0001867800043093 ◽

1977 ◽

Vol 9 (01) ◽

pp. 18-37 ◽

Cited By ~ 11

Author(s):

Joel E. Cohen

Keyword(s):

Markov Chain ◽

Age Structure ◽

Joint Distribution ◽

Sample Path ◽

Initial Distribution ◽

Transition Function ◽

Leslie Matrix ◽

Vital Rates ◽

Closed Population ◽

Leslie Matrices

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

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The stratum of random mapping classes

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.132 ◽

2017 ◽

Vol 38 (7) ◽

pp. 2666-2682 ◽

Cited By ~ 2

Author(s):

VAIBHAV GADRE ◽

JOSEPH MAHER

Keyword(s):

Random Walks ◽

Mapping Class Group ◽

Class Group ◽

Sample Path ◽

Mapping Class ◽

Elementary Subgroup ◽

Random Mapping ◽

Mapping Classes ◽

First Moment

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

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Convergence of Conditional Metropolis-Hastings Samplers

Advances in Applied Probability ◽

10.1239/aap/1401369701 ◽

2014 ◽

Vol 46 (2) ◽

pp. 422-445 ◽

Cited By ~ 6

Author(s):

Galin L. Jones ◽

Gareth O. Roberts ◽

Jeffrey S. Rosenthal

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Diffusion Process ◽

Sample Path ◽

Discrete Observations ◽

Entire Sample ◽

Bayesian Inferences ◽

Monte Carlo Algorithms

We consider Markov chain Monte Carlo algorithms which combine Gibbs updates with Metropolis-Hastings updates, resulting in a conditional Metropolis-Hastings sampler (CMH sampler). We develop conditions under which the CMH sampler will be geometrically or uniformly ergodic. We illustrate our results by analysing a CMH sampler used for drawing Bayesian inferences about the entire sample path of a diffusion process, based only upon discrete observations.

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On a Markov chain approach for the study of reliability structures

Journal of Applied Probability ◽

10.1017/s0021900200099770 ◽

1996 ◽

Vol 33 (02) ◽

pp. 357-367 ◽

Cited By ~ 4

Author(s):

M. V. Koutras

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Generating Functions ◽

General Framework ◽

Reliability Evaluation ◽

Markov Chain Approach ◽

Markov Chain Imbedding ◽

Finite Markov Chains

In this paper we consider a class of reliability structures which can be efficiently described through (imbedded in) finite Markov chains. Some general results are provided for the reliability evaluation and generating functions of such systems. Finally, it is shown that a great variety of well known reliability structures can be accommodated in this general framework, and certain properties of those structures are obtained on using their Markov chain imbedding description.

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