By Jacques Janssen, Raimondo Manca

ISBN-10: 038729547X

ISBN-13: 9780387295473

ISBN-10: 0387295488

ISBN-13: 9780387295480

Aims to offer to the reader the instruments essential to observe semi-Markov approaches in real-life problems.

The ebook is self-contained and, ranging from a low point of chance ideas, steadily brings the reader to a deep wisdom of semi-Markov processes.

Presents homogeneous and non-homogeneous semi-Markov techniques, in addition to Markov and semi-Markov rewards processes.

The innovations are basic for lots of functions, yet they aren't as completely offered in different books at the topic as they're here.

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**Extra resources for Applied Semi-Markov Processes**

**Example text**

S. This result is interesting for the concept of stopped process. 5 Let X be a stochastic process and T a stopping time. 15) where: Probability Tools withtAT 43 = mf{t,T]. From this definition, it follows that if the process X is adapted and cadlag, then so is the stopped process X^. This is due to the fact that / A T is also a stopping time and moreover: This leads to the last result we want to mention. 4 Let X be a right continuous uniformly integrable martingale: then the stopped process ^ =(^/Ar'^^[0'^Jj has the same properties with respect to the filtration (3^,^ ^ [^'°^])- 9 BROWNIAN MOTION There are a lot of particular stochastic processes and some of them will be extensively studied in the sequel, such as renewal processes, random walks, Markov chains, semi-Markov and Markov processes and their main extensions.

4): i/(0-XFW(0. 9) In several cases, it is useful to consider the initial renewal and to define at time t the random variable N^t) as being the total number of renewals on [ 0 , / ] . 6): R(t) = f^F^"\t). 14) R(t) = Uo(t)-^Hit). The classification of a renewal process is based on three concepts: recurrence, transience ^nd periodicity. 1 (i) A renewal process (T^, n>l) is recurrent if X^ < oo for all n; otherwise it is called transient. }, and 5 is the largest such number. Otherwise {that is, if there is no such strictly positive S ), the renewal process is aperiodic.

2 MAIN DEFINITIONS Let {X„, « > 1) be a sequence of non-negative, independent and identically distributed random variables defined on the probability space ( Q , 3 , P ) . , r, = X , + . . + Jr„ n>l, is called a renewal sequence or renewal process. v. v. X„,n>l interarrival times. 1 1) In the first section, we give an example in reliability theory. 2) Another important example is queueing theory. Let us consider a queueing system composed of a server, a process of customer arrivals, a process of service times and a discipline rule of the type "first in, first out" (FIFO), which means the first customer present in the system is the first served.

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