By Sean Meyn

Energy grids, versatile production, mobile communications: interconnectedness has results. This notable ebook provides the instruments and philosophy you want to construct community versions distinct sufficient to seize crucial dynamics yet basic adequate to show the constitution of powerful regulate suggestions and to elucidate research. middle chapters imagine basically publicity to stochastic techniques and linear algebra on the undergraduate point; later chapters are for complex graduate scholars and researchers/practitioners. This slow improvement bridges classical concept with the state of the art. The workload version that's the foundation of conventional research of the one queue turns into a starting place for workload relaxations utilized in the remedy of complicated networks. Lyapunov capabilities and dynamic programming equations bring about the prestigious MaxWeight coverage in addition to many generalizations. different subject matters comprise tools for synthesizing hedging and security shares, balance idea for networks, and methods for speeded up simulation. Examples and figures all through make principles concrete. recommendations to end-of-chapter workouts to be had on a significant other web site.

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**Extra info for Control Techniques for Complex Networks**

**Example text**

We wish to construct a stochastic process X = {X(k) : k ≥ 0} that shares these asymptotic properties. d. nonnegative random variables. Given a sampling increment Ts > 0 we define the sampling times tk = kTs for k ≥ 0, and choose the distribution of E x so that the random variables R(kTs ) and X(k) have approximately the same distribution for k ≥ 1. Consideration of large k suggests the two restrictions E[E x (1)] = µTs and a V r [E x (1)] = σ 2 µ3 Ts . 3) Note that the random walk X is not an exact representation of R, but shares some of its asymptotic properties.

It is useful to outline some of this advanced material here since, for example, the Law of Large Numbers and the Central Limit Theorem for martingales and renewal processes serves as motivation for the idealized network models developed in Parts I and II. The starting point of probability theory is the probability space, defined as the triple (Ω, F, P) with Ω an abstract set of points, F a σ-field of subsets of Ω, and P a probability measure on F. A mapping X : Ω → X is called a random variable if X −1 {B} := {ω : X(ω) ∈ B} ∈ F for all sets B ∈ B(X): that is, if X is a measurable mapping from Ω to X.

6 (Martingale convergence theorem). Let M be a supermartingale, and suppose that sup E[|M (t)|] < ∞. t Then {M (t)} converges to a finite limit with probability one. 10), E[|M (t)|] = E[M (t)] ≤ E[M (0)] < ∞, t ∈ Z+ . Hence we have as a direct corollary to the Martingale Convergence Theorem. 7. A positive supermartingale converges to a finite limit with probability one. Since a positive supermartingale is convergent, it follows that its sample paths are bounded with probability one. The following result gives an upper bound on the magnitude of variation of the sample paths of both positive supermartingales and general martingales.