By Laurent Decreusefond

ISBN-10: 1848212380

ISBN-13: 9781848212381

This ebook addresses the stochastic modeling of telecommunication networks, introducing the most mathematical instruments for that objective, reminiscent of Markov approaches, actual and spatial aspect methods and stochastic recursions, and proposing a large checklist of effects on balance, performances and comparability of platforms. utilizing those simple instruments, balance standards, functionality measures and comparability ideas are acquired for a large classification of versions, from the canonical M/M/1 and G/G/1 queues to extra refined platforms, together with the present “hot subject matters” of spatial radio networking, OFDMA and real-time networks.

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**Example text**

A state x is called recurrent when Px τx1 < ∞ = 1. Otherwise, x is said to be transient. The X chain is called recurrent (respectively transient) if all its states are recurrent (respectively transient). – For any (x, y) ∈ E × E, the following equality holds true, Py (τxk < ∞) = Px (τx1 < ∞)k−1 Py (τx1 < ∞). 7] In particular, if x = y, Px τxk < ∞ = Px (τx1 < ∞)k . Moreover, Ey [Nx ] = Py (τx1 < ∞) = 1 − Px (τx1 < ∞) p(n) (y, x). 8] n≥1 By taking the current time as that of the kth visit to the state x, according to the strong Markov property, the past and the future given this visit are independent.

A set of N cards is mixed by cutting it into two parts which are then interchanged. Each mixture of the set is represented by a permutation of {1, . . , N }. If N = 3 and the mixture is represented by (3, 2, 1), meaning that the card 3 is in position 1, the card 2 in position 2 and the card 1 in position 3. Xn denotes the state of the deck of cards after the nth mixing. The state space is hence the group of permutations of {1, . . , N }, denoted by SN . If X0 = (3, 2, 1) and the cut is between the ﬁrst and second card, we have X1 = (2, 1, 3).

A probability P on Ω is said stationary if for any A ∈ B(F Z ), P(A) = P(θ−1 A). In an equivalent manner, we have θ∗ P = P. Particularly, if one considers events of the form A = (Xk1 ∈ A1 , · · · , Xkn ∈ An ), we deduce that P(Xk1 ∈ A1 , · · · , Xkn ∈ An ) = P(Xk1 −1 ∈ A1 , · · · , Xkn −1 ∈ An ). 3] for any n, all k1 , · · · , kn and all A1 , · · · , An ∈ B(F ), will be said to be stationary. In particular, if the canonical space Ω = F Z is equipped with a stationary probability, the sequence of “coordinate” maps is a stationary sequence of random variables.

### Stochastic Modeling and Analysis of Telecoms Networks by Laurent Decreusefond

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