Analytical and Stochastic Modeling Techniques and by Dieter Claeys, Joris Walraevens, Koenraad Laevens, Bart

By Dieter Claeys, Joris Walraevens, Koenraad Laevens, Bart Steyaert, Herwig Bruneel (auth.), Khalid Al-Begain, Dieter Fiems, William J. Knottenbelt (eds.)

This booklet constitutes the refereed court cases of the seventeenth foreign convention on Analytical and Stochastic Modeling suggestions and purposes, ASMTA 2010, held in Cardiff, united kingdom, in June 2010. The 28 revised complete papers offered have been conscientiously reviewed and chosen from a number of submissions for inclusion within the publication. The papers are geared up in topical sections on queueing concept, specification languages and instruments, telecommunication platforms, estimation, prediction, and stochastic modelling.

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Extra info for Analytical and Stochastic Modeling Techniques and Applications: 17th International Conference, ASMTA 2010, Cardiff, UK, June 14-16, 2010. Proceedings

Example text

N−k) }. {˜ n, k, Lemma 2. The matrix Wi (s) is computed by formula where Wi (s) = (Z(s))i+1 , (11) Z(s) = (sI − Q0 − Q+ )−1 Q− . (12) Proof of the lemma consists of derivation of the recursion (sI − Q0 − Q+ )Wi (s) = Q− Wi−1 (s), i ≥ 0, (s) with initial condition W−1 (s) = I based on probabilistic reasonings. Corollary 1. The vector Wi e = − dWdsi (s) i Wi e = l=0 e is computed by s=0 (T Q− )l T e, i ≥ 0, (13) Queueing System M AP/P H/N with Propagated Failures where 25 T = −(Q0 + Q+ )−1 .

The accumulating phase sojourn time will then be T , unless the N th customer arrives at the queue sooner. This can be expressed as λq1,t,N −1 , N −1≤t≤T −1 , Prob[Φ1 = t] = (15) N −1 q , t=T , n=1 1,T,n with pgf T −1 Φ1 (z) E z Φ1 = λ N −1 q1,m,N −1 z m + m=N −1 q1,T,n z T . (16) n=1 We now introduce ω as T ω Prob[N customers have accumulated during Φ1 ] = λ q1,t,N −1 . t=N −1 (17) Serving phase. During the final phase the server is active and the customers get served. The phase lasts until the queue becomes empty and no more customers are present in the system.

It is just a name of items that arrived in the virtual stationary Poisson flow with intensity s. We can not avoid the use of the word catastrophe here because it is the standard notion in the method of additional events. It is easy to see that the LST w(s) is equal to probability that no one catastrophe arrives during the waiting time. To derive expression for the LST w(s), we need to introduce and compute auxiliary matrices. ,˜ η (˜ } having the following probabilistic sense. Such an entry defines probability that during the time interval starting at the moment of an arbitrary customer arrival to the buffer when the queue length is equal to i and finishing the moment when this customers begins the service two events occur: (i) catastrophe does not arrive and (ii) the finite components of the Markov chain χt , t ≥ 0, transit from the state {n, k, r, ν, ζ, ξ (1) , .

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