RENEWAL PROCESS FOR FUZZY VARIABLES

2007 ◽  
Vol 15 (04) ◽  
pp. 493-501 ◽  
Author(s):  
DUG HUN HONG
Keyword(s):  
Renewal Process ◽  
Renewal Theorem ◽  
Arrival Times ◽  
Fuzzy Variables ◽  
Continuity Assumption

Recently, Zhao and Liu [IJUFKS 11 (2003) 573–586] proposed a "fuzzy elementary renewal theorem" and "fuzzy renewal rewards theorem" for a renewal process in which the inter-arrival times and rewards are characterized as continuous fuzzy variables. The continuity assumption is restrictive. In this note, we prove the same results without the assumption of continuity of the inter-arrival times and rewards.

scholarly journals Delayed Renewal Process with Uncertain Random Inter-Arrival Times

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1943
Author(s):  
Xiaoli Wang ◽  
Gang Shi ◽  
Yuhong Sheng
Keyword(s):  
Random Process ◽  
Renewal Process ◽  
Random Variable ◽  
Random Delay ◽  
Renewal Theorem ◽  
Arrival Times ◽  
Chance Distribution ◽  
Renewal Rate ◽  
First Arrival

An uncertain random variable is a tool used to research indeterminacy quantities involving randomness and uncertainty. The concepts of an ’uncertain random process’ and an ’uncertain random renewal process’ have been proposed in order to model the evolution of an uncertain random phenomena. This paper designs a new uncertain random process, called the uncertain random delayed renewal process. It is a special type of uncertain random renewal process, in which the first arrival interval is different from the subsequent arrival interval. We discuss the chance distribution of the uncertain random delayed renewal process. Furthermore, an uncertain random delay renewal theorem is derived, and the chance distribution limit of long-term expected renewal rate of the uncertain random delay renewal system is proved. Then its average uncertain random delay renewal rate is obtained, and it is proved that it is convergent in the chance distribution. Finally, we provide several examples to illustrate the consistency with the existing conclusions.

RENEWAL PROCESS WITH FUZZY INTERARRIVAL TIMES AND REWARDS

2003 ◽  
Vol 11 (05) ◽  
pp. 573-586 ◽  
Author(s):  
RUIQING ZHAO ◽  
BAODING LIU
Keyword(s):  
Renewal Process ◽  
Interarrival Time ◽  
Renewal Theorem ◽  
Expected Number ◽  
Fuzzy Variables ◽  
Numerical Example ◽  
Interarrival Times ◽  
Renewal Reward Theorem

This paper considers a renewal process in which the interarrival times and rewards are characterized as fuzzy variables. A fuzzy elementary renewal theorem shows that the expected number of renewals per unit time is just the expected reciprocal of the interarrival time. Furthermore, the expected reward per unit time is provided by a fuzzy renewal reward theorem. Finally, a numerical example is presented for illustrating the theorems introduced in the paper.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (03) ◽  
pp. 752-756
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

PRESERVATION OF RELIABILITY CLASSES UNDER MIXTURES OF RENEWAL PROCESSES

2007 ◽  
Vol 22 (1) ◽  
pp. 1-17 ◽  
Author(s):  
F. G. Badía ◽  
C. Sangüesa
Keyword(s):  
Renewal Process ◽  
Sufficient Conditions ◽  
Renewal Processes ◽  
Arrival Times ◽  
Aging Properties

In this work we provide sufficient conditions for the arrival times of a renewal process so that the number of its events occurring before a randomly distributed time, T, independent of the process preserves the aging properties of T.

scholarly journals On randomly spaced observations and continuous-time random walks

2016 ◽  
Vol 53 (3) ◽  
pp. 888-898
Author(s):  
Bojan Basrak ◽  
Drago Špoljarić
Keyword(s):  
Random Walks ◽  
Asymptotic Distribution ◽  
Renewal Process ◽  
Continuous Time ◽  
Random Variables ◽  
Limiting Behavior ◽  
Arrival Times ◽  
Sojourn Times ◽  
Continuous Time Random Walks ◽  
Heavy Tailed

AbstractWe consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy-tailed steps, the limiting behavior of extreme observations until a given time t tends to be rather involved. We describe the asymptotics and extend several partial results which appeared in this setting. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous-time random walks.

A Sequential Assignment Match Process with General Renewal Arrival Times

1995 ◽  
Vol 9 (3) ◽  
pp. 475-492 ◽  
Author(s):  
Israel David
Keyword(s):  
Dynamic Programming ◽  
Failure Rate ◽  
Simple Form ◽  
Renewal Process ◽  
Optimal Policy ◽  
Functional Equations ◽  
Solution Algorithm ◽  
Sequential Assignment ◽  
Discount Function ◽  
Arrival Times

This work studies sequential assignment match processes, in which random offers arrive sequentially according to a renewal process, and when an offer arrives it must be assigned to one of given waiting candidates or rejected. Each candidate as well as each offer is characterized by an attribute. If the offer is assigned to a candidate that it matches, a reward R is received; if it is assigned to a candidate that it does not match, a reward r ≤ R is received; and if it is rejected, there is no reward. There is an arbitrary discount function, which corresponds to the process terminating after a random lifetime. Using continuoustime dynamic programming, we show that if this lifetime is decreasing in failure rate and candidates have distinct attributes, then the policy that maximizes total expected discounted reward is of a very simple form that is easily determined from the optimal single-candidate policy. If the lifetime is increasing in failure rate, the optimal policy can be recursively determined: a solution algorithm is presented that involves scalar rather than functional equations. The model originated in the study of optimal donor-recipient assignment in live-organ transplants. Some other applications are mentioned as well.

scholarly journals A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process

2019 ◽  
Vol 56 (2) ◽  
pp. 602-623
Author(s):  
Daryl J. Daley ◽  
Masakiyo Miyazawa
Keyword(s):  
Asymptotic Behaviour ◽  
Renewal Process ◽  
Stochastic Analysis ◽  
Counting Process ◽  
Intensity Function ◽  
Counting Processes ◽  
Renewal Theorem ◽  
Basic Tool ◽  
Semimartingale Representation ◽  
Blackwell's Renewal Theorem

AbstractMartingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and give extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell’s renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell’s renewal theorem holds.

A note on filtered Markov renewal processes

1981 ◽  
Vol 18 (3) ◽  
pp. 752-756 ◽  
Author(s):  
Per Kragh Andersen
Keyword(s):  
Renewal Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
The Moment

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.

scholarly journals Renewal function asymptotics refined à la Feller

2018 ◽  
Vol 37 (2) ◽  
pp. 291-298
Author(s):  
Daryl J. Daley
Keyword(s):  
Recurrence Relation ◽  
Renewal Process ◽  
Higher Order ◽  
Renewal Equation ◽  
Asymptotic Result ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Higher Order Moments ◽  
Second Moment ◽  
Key Renewal Theorem

RENEWAL FUNCTION ASYMPTOTICS REFINED À LA FELLERFeller’s volume 2 shows how to use the Key Renewal Theorem to prove that in the limit x!1, the renewal function Ux of a renewal process with nonarithmetic generic lifetime X with finite mean EX=1=and second moment differs from its linear asymptote x by the quantity 122EX2. His first edition 1966 but not the second in 1971 asserted that a similar approach would refine this asymptotic result when X has finite higher order moments. The paper shows how higher order moments may justify drawing conclusions from a recurrence relation that exploits a general renewal equation and further appeal to the Key Renewal Theorem.

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