By R. Srikant

ISBN-10: 1107036054

ISBN-13: 9781107036055

Presents a contemporary mathematical method of the layout of conversation networks for graduate scholars, mixing keep an eye on, optimization, and stochastic community theories. A extensive diversity of functionality research instruments are mentioned, together with very important complex themes which were made available to scholars for the 1st time. Taking a top-down method of community protocol layout, the authors start with the deterministic version and growth to extra refined versions. community algorithms and protocols are tied heavily to the speculation, illustrating the sensible engineering functions of every subject. The history at the back of the mathematical analyses is given sooner than the formal proofs and is supported through labored examples, permitting scholars to appreciate the massive photograph ahead of going into the distinctive conception. End-of-chapter difficulties hide various problems, with complicated difficulties damaged into numerous elements, and tricks to many difficulties are supplied to lead scholars. complete recommendations can be found on-line for teachers.

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**Additional info for Communication Networks: An Optimization, Control and Stochastic Networks Perspective**

**Sample text**

15) where x∗ is the maximizer of f (x). Thus, the optimal rates {xr∗ }, when U(xr ) = log xr , satisfy r xr − xr∗ ≤ 0, xr∗ where {xr } is any other set of feasible rates. An allocation with such a property is called proportionally fair. The reason for this terminology is as follows: if one of the source rates is increased by a certain amount, the sum of the fractions (also called proportions) by which the different users’ rates change is non-positive. A consequence of this observation is that, if the proportion by which one user’s rate changes is positive, there will be at least one other user whose proportional change will be negative.

If we fix the value of y1 , the optimization problem becomes yr , subject to yr = 1 − y1 . min 1 − yr r=1 r=1 The solution is given by yr = (1 − y1 )/(R − 1), so that PoA ≥ min y1 + 1 R ≤y1 ≤1 (1 − y1 )2 R−1 r=1 1 . 1 − yr Since 1 − yr = 1 − 1 − y1 R − 2 + y1 = , R−1 R−1 we have r=1 1 (R − 1)2 = 1 − yr R − 2 + y1 and PoA ≥ min y1 + 1 R ≤y1 ≤1 (R − 1)(1 − y1 )2 . 8 Summary and PoA ≥ min y1 + (1 − y1 )2 1 R ≤y1 ≤1 ≥ min y1 + (1 − y1 )2 0≤y1 ≤1 3 . , Ur (xr ) = wr xr . Assume the utility functions are U1 (x1 ) = 2x1 , and Ur (xr ) = xr for r = 2, .

L Show that V(k + 1) − V(k) ≤ K 2 + qr (k)(xr (k) − xˆ r ), r for some constant K > 0, where xˆ is an optimal solution to the utility maximization problem max x≥0 Ur (xr ), xr ≤ cl . subject to r r:l∈r Assume that Xmax > maxr xˆ r . (2) Next, show that V(k + 1) − V(k) ≤ K 2 + (Ur (xr ) − Ur (ˆxr )).

### Communication Networks: An Optimization, Control and Stochastic Networks Perspective by R. Srikant

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