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

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.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (02) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

Some passage-time generating functions for discrete-time and continuous-time finite Markov chains

1967 ◽  
Vol 4 (03) ◽  
pp. 496-507 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Finite Markov Chain ◽  
Finite Markov Chains

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk , of Yj + k — Yj and of Yj + k — Yj in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.

On the parity of individuals in a branching process

1976 ◽  
Vol 13 (2) ◽  
pp. 219-230 ◽  
Author(s):  
J. Gani ◽  
I. W. Saunders
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Branching Process ◽  
Yeast Cells ◽  
Death Process ◽  
Second Moments ◽  
The Mean ◽  
Watson Process ◽  
Number Of Cells ◽  
Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

Some passage-time generating functions for discrete-time and continuous-time finite Markov chains

1967 ◽  
Vol 4 (3) ◽  
pp. 496-507 ◽  
Author(s):  
J. N. Darroch ◽  
K. W. Morris
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Passage Time ◽  
Finite Markov Chain ◽  
Finite Markov Chains

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Yk denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Yk, of Yj + k — Yj and of Yj + k — Yj in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (4) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

scholarly journals A discrete-time system with service control and repairs

2014 ◽  
Vol 24 (3) ◽  
pp. 471-484 ◽  
Author(s):  
Ivan Atencia
Keyword(s):  
Performance Measures ◽  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Queueing System ◽  
Discrete Time System ◽  
Time System ◽  
The Third ◽  
Special Cases ◽  
Number Of Customers

Abstract This paper discusses a discrete-time queueing system with starting failures in which an arriving customer follows three different strategies. Two of them correspond to the LCFS (Last Come First Served) discipline, in which displacements or expulsions of customers occur. The third strategy acts as a signal, that is, it becomes a negative customer. Also examined is the possibility of failures at each service commencement epoch. We carry out a thorough study of the model, deriving analytical results for the stationary distribution. We obtain the generating functions of the number of customers in the queue and in the system. The generating functions of the busy period as well as the sojourn times of a customer at the server, in the queue and in the system, are also provided. We present the main performance measures of the model. The versatility of this model allows us to mention several special cases of interest. Finally, we prove the convergence to the continuous-time counterpart and give some numerical results that show the behavior of some performance measures with respect to the most significant parameters of the system

Further approaches to computing fundamental characteristics of birth-death processes

2001 ◽  
Vol 38 (04) ◽  
pp. 995-1005 ◽  
Author(s):  
Frank Ball ◽  
Valeri T. Stefanov
Keyword(s):  
Discrete Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Exponential Family ◽  
Death Process ◽  
Random Sum ◽  
First Passage ◽  
Reward Functions ◽  
Passage Times ◽  
Birth Death

General and unifying approaches are discussed for computing fundamental characteristics of both continuous-time and discrete-time birth-death processes. In particular, an exponential family framework is used to derive explicit expressions, in terms of continued fractions, for joint generating functions of first-passage times and a whole collection of associated random quantities, and a random sum representation is used to obtain formulae for means, variances and covariances of stopped reward functions defined on a birth-death process.

Design of Programmable Wideband Low Pass Filter with Continuous-Time/Discrete-Time Hybrid Architecture

2017 ◽  
Vol E100.C (10) ◽  
pp. 858-865 ◽  
Author(s):  
Yohei MORISHITA ◽  
Koichi MIZUNO ◽  
Junji SATO ◽  
Koji TAKINAMI ◽  
Kazuaki TAKAHASHI
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Architecture ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass
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