By Michel Mandjes
In recent times the importance of Gaussian techniques to conversation networks has grown significantly. The inherent flexibility of the Gaussian site visitors version permits the research, in one mathematical framework, of structures with either long-range and short-range established enter streams.
huge Deviations for Gaussian Queues demonstrates how the Gaussian site visitors version arises certainly, and the way the research of the corresponding queuing version should be played. The textual content offers a normal creation to Gaussian queues, and surveys fresh examine into the modelling of communications networks. assurance contains:
- Discussion of the theoretical innovations and functional facets with regards to Gaussian site visitors versions.
- Analysis of contemporary learn asymptotic effects for Gaussian queues, either within the large-buffer and many-sources regime.
- An emphasis on rare-event research, counting on a number of asymptotic options.
- Examination of single-node FIFO queuing platforms, in addition to queues working below extra complicated scheduling disciplines, and queuing networks.
- A set of illustrative examples that without delay relate to special functional difficulties in communique networking.
- A huge selection of instructive routines and accompanying recommendations.
huge Deviations for Gaussian Queues assumes minimum previous wisdom. it's perfect for postgraduate scholars in utilized chance, operations examine, computing device technological know-how and electric engineering. The book’s self-contained sort makes it excellent for practitioners within the communications networking and for researchers in comparable parts.
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Extra resources for Large Deviations for Gaussian Queues: Modelling Communication Networks
Sample text
Convergence of on-off traffic to a Gaussian process The proofs of weak convergence results on the process level are usually rather technical. In general, a three-step procedure is followed to show that Xn (·) → X(·): (i) the convergence of Xn (t) to X(t) for any t, (ii) the convergence of the finite-dimensional distributions, (iii) the convergence Xn (·) → X(·) using a tightness argumentation. For the general theory, we refer to, for instance, [286]; in [64, 67] it is demonstrated for specific situations how the three-step procedure works.
As some random variable Y (t)), independently of Nt . d. (as a random variable Z), and N Poisson with mean ν (independent of the Zi ), we have that N Var Zi = ν VarZ, i=1 we find that we are left with finding VarX(t) and VarY (t). To this end, we wonder how X(t) and Y (t) are distributed. Let fD (·) (FD (·)) be the density (distribution function) of D. In the sequence we need the notion of 17 The Gaussian source model the so-called integrated tail (or residual lifetime) of D, which is a random variable that we denote by D r .
If this remaining time is s < t, then X(t) = s’ otherwise it is t. In other words, the kth moment of X(t) equals t E(X(t))k = s k fD r (s) ds + t k (1 − FD r (t)). 0 This immediately yields a formula for VarX(t). Then focus on Y (t). It is well known that the epoch the job arrives is uniformly distributed on (0, t). With a similar reasoning as above, where the variable u corresponds to the epoch the job arrives, we find that t E(Y (t))k = 0 1 (t − u)k (1 − FD (t − u)) + t t−u s k fD (s) ds du. 0 Again, this enables computation of VarY (t).
