“…Freiberger & Grenander (1971) demonstrate a similar approach to obtaining the renewal equation numerical solution for the homogeneous case, but there is no justification of the method. Many other papers (see section 1) deal with the same problem in the homogeneous case, but, as far as the authors are aware, the relationship between the discrete time and continuous time-renewal process has only been justified in the book by Janssen & Manca (2006), as they are in this paper in the non-homogeneous case. It is, however, the first time that the numerical treatment of the non-homogeneous renewal processes is presented.…”
Section: Solving the Non-homogeneous Discrete Time Evolution Equationmentioning
confidence: 95%
“…The first paper to state this relation in the homogeneous case in a Markov renewal environment is Corradi et al (2004). In the book by Janssen & Manca (2006), the same relation for the homogeneous renewal process case is presented for the first time. Regarding non-homogeneity, Janssen & Manca (2001) gave these results in a non-homogeneous semi-Markov environment.…”
Section: Introductionmentioning
confidence: 97%
“…If the cumulating d.f. of the renewal process is of a negative exponential type, then the renewal process becomes a Poisson’s process (see Janssen & Manca, 2006; Mikosch, 2009) and the related integral equation obtained for the calculation of the mean number of renewals in a time t , also in the non-homogeneous case, can be solved analytically (Çynlar, 1975). This is a very particular case.…”
Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution.\ud
Renewal theory has great relevance in general in economics and in particular in actuarial science, however most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are then dealt with and a real data application to an aspect of motorcar insurance is proposed
“…Freiberger & Grenander (1971) demonstrate a similar approach to obtaining the renewal equation numerical solution for the homogeneous case, but there is no justification of the method. Many other papers (see section 1) deal with the same problem in the homogeneous case, but, as far as the authors are aware, the relationship between the discrete time and continuous time-renewal process has only been justified in the book by Janssen & Manca (2006), as they are in this paper in the non-homogeneous case. It is, however, the first time that the numerical treatment of the non-homogeneous renewal processes is presented.…”
Section: Solving the Non-homogeneous Discrete Time Evolution Equationmentioning
confidence: 95%
“…The first paper to state this relation in the homogeneous case in a Markov renewal environment is Corradi et al (2004). In the book by Janssen & Manca (2006), the same relation for the homogeneous renewal process case is presented for the first time. Regarding non-homogeneity, Janssen & Manca (2001) gave these results in a non-homogeneous semi-Markov environment.…”
Section: Introductionmentioning
confidence: 97%
“…If the cumulating d.f. of the renewal process is of a negative exponential type, then the renewal process becomes a Poisson’s process (see Janssen & Manca, 2006; Mikosch, 2009) and the related integral equation obtained for the calculation of the mean number of renewals in a time t , also in the non-homogeneous case, can be solved analytically (Çynlar, 1975). This is a very particular case.…”
Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution.\ud
Renewal theory has great relevance in general in economics and in particular in actuarial science, however most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are then dealt with and a real data application to an aspect of motorcar insurance is proposed
“…Time of residence PMN at places , …, , …, is defined as the time of wandering through semi-Markov processes [ 9 , 10 ] , , which are abstract analogues of swarm unit onboard computers operation, and are defined as follows (Fig. 2 ): Fig.…”
Section: Petri-markov Model Of Synchronized Operationmentioning
Physical swarm system, including number of units, operated in physical time according to corporative algorithm, is considered. It is shown, that for proper corporative algorithm interpretation it is necessary to synchronize computational processes in units. Structural-parametric model of synchronized swarm operation, based on Petri-Markov nets apparatus, is worked out. In the Petri-Markov net transitions are abstract analogues of synchronization procedure, while places simulate corporative algorithm parts interpretation by swarm units. Primary Petri-Markov model is transformed into complex semi-Markov process. Formulae for calculation of stochastic and time characteristics of the process are obtained. It is shown, that after transformation all methods of ordinary semi-Markov processes investigation may be used for synchronized systems. With use the concept of distributed forfeit effectiveness of synchronization is evaluated.
“…In this case, the transitions of such a system are not merely described by a typical Markov chain procedure and Semi-Markov models are introduced as the stochastic tools that provide a more rigorous framework accommodating a greater variety of applied probability models [3][4][5]. Various applications of semi-Markov processes include manpower planning, credit risk, word sequencing and DNA analysis [6][7][8][9][10][11][12][13][14].…”
Semi-Markov processes generalize the Markov chains framework by utilizing abstract sojourn time distributions. They are widely known for offering enhanced accuracy in modeling stochastic phenomena. The aim of this paper is to provide closed analytic forms for three types of probabilities which describe attributes of considerable research interest in semi-Markov modeling: (a) the number of transitions to a state through time (Occupancy), (b) the number of transitions or the amount of time required to observe the first passage to a state (First passage time) and (c) the number of transitions or the amount of time required after a state is entered before the first real transition is made to another state (Duration). The non-homogeneous in time recursive relations of the above probabilities are developed and a description of the corresponding geometric transforms is produced. By applying appropriate properties, the closed analytic forms of the above probabilities are provided. Finally, data from human DNA sequences are used to illustrate the theoretical results of the paper.
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