Renewal Sequence Research Articles

APPLICATION OF RENEWAL REWARD PROCESSES IN HOMOGENEOUS DISCRETE MARKOV CHAIN

A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight into the performance measures in renewal process and sequence such as, the mean time between successive renewals, ; Laplace-Stiltjes transform (LST) of the mean time, ; the Laplace-Stieltjes transform (LST) of the mean time distribution function, ; Laplace-Stiltjes transform (LST) of fold convolution of distribution function, ; the time at which the renewal occurs, the average number of renewals per unit time over the interval (0, t], and expected reward, . Our quest is to analyse the distribution function of the renewal process and sequence using the concept of discrete time Markov chain to obtain the aforementioned performance measures. Some properties of the Erlang , exponential and geometric distributions are used with the help of some existing laws, theorems and formulas of Markov chain. We concluded our study through illustrative examples that, it is not possible for an infinite number of renewals to occur in a finite period of time; Also, the expected number of renewals increases linearly with time; and from the uniqueness property, we affirmed that, the Poisson process is the only renewal process with a linear mean-value function; and lastly, we obtained the optimal replacement policy for a manufacturing machine which showed that, the exponential property of the lifetime

Health evaluation of urban water supply pipe network using the Bayesian method based on triangular fuzzy number optimization

With the rapid increase of urbanization level in China, the accidents related to urban water supply pipe network occurred frequently due to the impacts of pipeline aging, environmental change and human factors, thus the health evaluation of urban water supply network has become one of the hot topics in recent years. Related researches mostly refer to qualitative and static analysis, some quantitative studies consider the influence of only one aspect on the status of water supply pipe network such as hydraulic, water quality and static structure, and cannot objectively reflect the health status of pipe network due to neglecting or simplifying the influence of internal hydraulic condition on the health of the pipe network. Comprehensively considering the influencing factors and the hydraulic mechanism during the operation process of pipe network, this study constructs a health evaluation index system for the water supply pipe network, and introduces the triangular fuzzy number theory into the traditional Bayesian water supply network health status evaluation model taking the uncertainty of model structure and data into account. Through applying the method to Yunhe county of Zhejiang Province China, some suggestions for the optimization and transformation of the regional water supply network are proposed, which can provide a basis for ensuring the safety of regional water supply and scientifically determining the renewal sequence of the pipe network.

On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence

We consider a Lévy driven continuous time moving average process X sampled at random times which follow a renewal structure independent of X. Asymptotic normality of the sample mean, the sample autocovariance, and the sample autocorrelation is established under certain conditions on the kernel and the random times. We compare our results to a classical non-random equidistant sampling method and give an application to parameter estimation of the Lévy driven Ornstein–Uhlenbeck process.

Properties of the stochastic ordering for discrete distributions and their applications to the renewal sequence generated by a nonhomogeneous Markov chain

Properties of the stochastic ordering for discrete distributions and their applications to the renewal sequence generated by a nonhomogeneous Markov chain

The random pinning model with correlated disorder given by a renewal set

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha$ > 0, when the correlated sequence is given by another independent renewal set with loop exponent $\alpha$ > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case $\alpha$ > 2 (summable correlations), disorder is irrelevant if $\alpha$ < 1/2 and relevant if $\alpha$ > 1/2, which extends the Harris criterion for independent disorder. The case $\alpha$ $\in$ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha$ > 1/ $\alpha$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\alpha$ $\in$ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.

Renewal sequences and record chains related to multiple zeta sums

For the random interval partition of [ 0 , 1 ] [0,1] generated by the uniform stick-breaking scheme known as GEM ( 1 ) (1) , let u k u_k be the probability that the first k k intervals created by the stick-breaking scheme are also the first k k intervals to be discovered in a process of uniform random sampling of points from [ 0 , 1 ] [0,1] . Then u k u_k is a renewal sequence. We prove that u k u_k is a rational linear combination of the real numbers 1 , ζ ( 2 ) , … , ζ ( k ) 1, \zeta (2), \ldots , \zeta (k) where ζ \zeta is the Riemann zeta function, and show that u k u_k has the limit 1 / 3 1/3 as k → ∞ k \rightarrow \infty . Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM ( θ ) (\theta ) model, with beta ( 1 , θ ) (1,\theta ) instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.

Occupation times for the finite buffer fluid queue with phase-type ON-times

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. Using an alternating renewal sequence, we determine the double transform of the occupation time; the occupation time for the finite buffer M/G/1 queue with phase-type jumps follows as a limiting case.

An estimate of the expectation of the excess of a renewal sequence generated by a time-inhomogeneous Markov chain if a square-integrable majorizing sequence exists

An estimate of the expectation of the excess of a renewal sequence generated by a time-inhomogeneous Markov chain if a square-integrable majorizing sequence exists

The distribution of some extremum on the risk process whose income depend on the current reserve.

This paper considers the distribution of some extremum on the risk process whose income depend on the current reserve. We first construct the defective renewal sequence and obtain the density function of them. By the presented renewal measure and the strong Markov property, the distribution of the first hitting time is obtained explicitly. Then, the ruin probability and the probability that the surplus process less than x is obtained. Furthermore, the distribution of supreme profits before ruin, the joint distributions of the supreme profit and the deficit before the time of the surplus process first up-crossing level zero after ruin, and the joint distributions of the supreme profit and the deficit before the surplus process leave zero ultimately are derived. Finally, the exact calculating results for them are obtained when the individual claim amounts in the compound Poisson risk model are exponentially distributed.

Intuitive approximations in discrete renewal theory, Part 1: Regularly varying case

It is usually impossible to find explicit expressions for the renewal sequence. This paper presents a simple method to approximate the renewal sequence, which covers many of the known approximations. The paper uses the ideas of Mitov and Omey (2014).

The Kendall theorem and its application to the geometric ergodicity of Markov chains

In this paper we prove an improved quantitative version of the Kendall’s Theorem. The Kendal Theorem states that under mild conditions imposed on a probability distribution on positive integers (i.e. probabilistic sequence) one can prove convergence of its renewal sequence. Due to the well-known property the first entrance last exit decomposition such results are of interest in the stability theory of time homogeneous Markov chains. In particular the approach may be used to measure rates of convergence of geometrically ergodic Markov chains and consequently implies estimates on convergence of MCMC estimators.

Ruin probability and joint distributions of some actuarial random vectors in the compound pascal model

The compound negative binomial model, introduced in this paper, is a discrete time version. We discuss the Markov properties of the surplus process, and study the ruin probability and the joint distributions of actuarial random vectors in this model. By the strong Markov property and the mass function of a defective renewal sequence, we obtain the explicit expressions of the ruin probability, the finite-horizon ruin probability, the joint distributions of T, U(T − 1), |U(T)| and \( \mathop {\inf }\limits_{0 \leqslant n < \tau _1 } \)U(n) (i.e., the time of ruin, the surplus immediately before ruin, the deficit at ruin and maximal deficit from ruin to recovery) and the distributions of some actuarial random vectors.

Inference in binomial [formula omitted] models

This paper studies inference methods for stationary time series with binomial distributions. Such series describe, for example, the number of rainy days in consecutive weeks. First, we formulate the renewal sequence version of the model that seemingly generates a new class of stationary binomial series. The model is shown to obey an AR(1) recursion in cases where the renewal lifetime has a constant hazard rate past lag one. Explicit asymptotic variances of the parameter estimators in the AR(1) case are derived from conditional least squares methods; likelihood techniques are also considered.

Total duration of negative surplus for the dual model

AbstractIn the dual model, we allow the surplus process to continue if the surplus falls below zero. By introducing the renewal measure of the defective renewal sequence constituted by the zero points of the surplus process, we obtain the probability of hitting the zero point. Further, we derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus and present some examples with an exponential individual jump amount distribution. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Characterization of equilibrium measures for critical reversible Nearest Particle Systems

We show that for critical reversible attractive Nearest Particle Systems all equilibrium measures are convex combinations of the upper invariant equilibrium measure and the point mass at all zeros, provided the underlying renewal sequence possesses moments of order strictly greater than \( \frac{{7 + \sqrt {41} }} {2} \) and obeys some natural regularity conditions.

Total Duration of Negative Surplus for the Risk Process with Constant Interest Force

Assuming the compound Poisson risk process with constant interest force, we consider the process as continuing if ruin occurs. We introduce the renewal measure of the defective renewal sequence constituted by the zero points of the surplus process. The density functions of this renewal measure and the first-hitting time of the zero point are derived. By these density functions together with the strong Markov property of the surplus process, we obtain the explicit expression for the total duration of negative surplus.

Correlation Lengths for Random Polymer Models and for Some Renewal Sequences

We consider models of directed polymers interacting with a one-dimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on $Z$ and gives a random (site-dependent) reward or penalty to the occurrence of a renewal at any given point of $Z$. These models are known to undergo a delocalization-localization transition, and the free energy $F$ vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length $\xi$, defined as the inverse of the rate of exponential decay of the two-point function, does not diverge faster than $1/F$. We prove also an exponentially decaying upper bound for the disorder-averaged two-point function, with a good control of the sub-exponential prefactor. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.

Joint distributions of some actuarial random vectors in the compound binomial model

The compound binomial model is first proposed by Gerber [Gerber, H.U., 1988. Mathematical fun with compound binomial process. Astin Bull. 18, 161–168]. In this paper, we introduce a renewal mass function of a defective renewal sequence constituted by the up-crossing zero points of the model and get its explicit expression. By the mass function together with the strong Markov property of the surplus process { X ( n ) } , we obtain the explicit expressions of the ruin probability, the joint distribution of T , X ( T − 1 ) and | X ( T ) | (i.e., the time of ruin, the surplus immediately before ruin and the deficit at ruin) and distributions of some actuarial random vectors containing more than three variables.

Joint distributions of some actuarial random vectors containing the time of ruin

In this paper, we introduce the renewal measure of the defective renewal sequence constituted by the zero points of the classical risk model U( t), t≥0. The density function of this renewal measure is derived. By this density function together with the strong Markov property of the surplus process, we obtain the explicit expressions for the ruin probability and the joint distributions of actuarial random vectors ( T, U( T −),| U( T)|) and (T,U(T −),|U(T)|, sup 0≤t<T U(t), sup T≤t<L U(t), inf 0≤t<L U(t)) , where T represents the time of ruin and L the time of the surplus process leaving zero ultimately. Finally, a special case with the claim amount being exponentially distributed is considered.

Functional Renewal of Behavior of Systems: Numerical Methods

The principles of functional renewal of the behavior of technical systems based on the functional capabilities in the design are stated. A finite deterministic automaton is used as the model. The functional renewal problem is solved within the framework of the theory of universal automata. A class of automata admitting modeling by a family of polynomials is described. The numerical model of automaton behavior is helpful in applying algebraic tools to solve the functional behavior renewal problem. For this automaton class, a universal enumerator is designed and analyzed, and a method of constructing a renewal sequence is described.