Transient renewal processes in the subexponential case

1987 ◽  
Vol 24 (1) ◽  
pp. 88-96
Author(s):  
Emily S. Murphree
Keyword(s):  
Renewal Process ◽  
Waiting Times ◽  
Renewal Processes ◽  
Conditional Moments ◽  
Subexponential Class

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Transient renewal processes in the subexponential case

1987 ◽  
Vol 24 (01) ◽  
pp. 88-96
Author(s):  
Emily S. Murphree
Keyword(s):  
Renewal Process ◽  
Waiting Times ◽  
Renewal Processes ◽  
Conditional Moments ◽  
Subexponential Class

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

scholarly journals Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 55
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Renewal Process ◽  
Markov Chain Model ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Chain Model ◽  
Markov Renewal Processes ◽  
Risky Bonds ◽  
Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Kyo-Shin Hwang ◽  
Wensheng Wang
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Anomalous Diffusion ◽  
Renewal Process ◽  
Continuous Time ◽  
Waiting Times ◽  
Continuous Time Random Walk ◽  
Domains Of Attraction ◽  
Iterated Logarithm ◽  
Continuous Time Random Walks

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

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.

Renewal processes with random numbers of delays: application to a conception and birth model

1978 ◽  
Vol 15 (02) ◽  
pp. 209-224
Author(s):  
Kenneth Lange ◽  
Norman J. Johnson
Keyword(s):  
Birth Control ◽  
Renewal Process ◽  
Random Number ◽  
Asymptotic Formulas ◽  
Random Numbers ◽  
Renewal Processes ◽  
Demographic Model ◽  
Two Stage ◽  
Live Births ◽  
Stieltjes Transforms

Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.

A STORAGE SYSTEM WITH SPORADIC AND CONTINUOUS CLEARINGS

2007 ◽  
Vol 21 (4) ◽  
pp. 539-549 ◽  
Author(s):  
Wolfgang Stadje
Keyword(s):  
Laplace Transform ◽  
Renewal Process ◽  
Storage System ◽  
Renewal Processes ◽  
Finite Capacity ◽  
Special Case ◽  
Independent And Identically Distributed

We study a cumulative storage system that is totally cleared sporadically at stationary renewal times and whenever a finite-capacity threshold is exceeded. The independent and identically distributed inputs occur at time epochs that also form a stationary renewal process. We determine the distribution of the interoverflow times. Although this distribution is quite intricate when both underlying renewal processes are general, in the special case of Poisson sporadic clearings we obtain a neat formula for its Laplace transform.

scholarly journals Some performance measures for vacation models with a batch Markovian arrival process

1994 ◽  
Vol 7 (2) ◽  
pp. 111-124 ◽  
Author(s):  
Sadrac K. Matendo
Keyword(s):  
Performance Measures ◽  
Waiting Times ◽  
Distribution Functions ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Service Discipline ◽  
Renewal Processes ◽  
Single Server ◽  
Batch Markovian Arrival Process ◽  
Interarrival Times

We consider a single server infinite capacity queueing system, where the arrival process is a batch Markovian arrival process (BMAP). Particular BMAPs are the batch Poisson arrival process, the Markovian arrival process (MAP), many batch arrival processes with correlated interarrival times and batch sizes, and superpositions of these processes. We note that the MAP includes phase-type (PH) renewal processes and non-renewal processes such as the Markov modulated Poisson process (MMPP).The server applies Kella's vacation scheme, i.e., a vacation policy where the decision of whether to take a new vacation or not, when the system is empty, depends on the number of vacations already taken in the current inactive phase. This exhaustive service discipline includes the single vacation T-policy, T(SV), and the multiple vacation T-policy, T(MV). The service times are i.i.d. random variables, independent of the interarrival times and the vacation durations. Some important performance measures such as the distribution functions and means of the virtual and the actual waiting times are given. Finally, a numerical example is presented.

The limit theorem for aperiodic discrete renewal processes

1964 ◽  
Vol 4 (1) ◽  
pp. 122-128
Author(s):  
P. D. Finch
Keyword(s):  
Limit Theorem ◽  
Real Number ◽  
Renewal Process ◽  
Positive Real Number ◽  
Random Variables ◽  
Renewal Processes ◽  
Positive Real ◽  
Image Position

A discrete renewal process is a sequence {X4} of independently and inentically distributed random variables which can take on only those values which are positive integral multiples of a positive real number δ. For notational convenience we take δ = 1 and write where .

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