By Michael J. Neely, Jean Walrand

This article offers a latest conception of research, regulate, and optimization for dynamic networks. Mathematical concepts of Lyapunov go with the flow and Lyapunov optimization are built and proven to allow limited optimization of time averages quite often stochastic platforms. the point of interest is on verbal exchange and queueing platforms, together with instant networks with time-varying channels, mobility, and randomly arriving site visitors. an easy drift-plus-penalty framework is used to optimize time averages equivalent to throughput, throughput-utility, energy, and distortion. specific performance-delay tradeoffs are supplied to demonstrate the price of imminent optimality. This idea can also be acceptable to difficulties in operations learn and economics, the place energy-efficient and profit-maximizing judgements has to be made with out figuring out the longer term. themes within the textual content contain the next: - Queue balance thought - Backpressure, max-weight, and digital queue equipment - Primal-dual equipment for non-convex stochastic application maximization - common scheduling concept for arbitrary pattern paths - Approximate and randomized scheduling thought - Optimization of renewal structures and Markov choice platforms exact examples and diverse challenge set questions are supplied to enhance the most techniques. desk of Contents: advent / advent to Queues / Dynamic Scheduling instance / Optimizing Time Averages / Optimizing capabilities of Time Averages / Approximate Scheduling / Optimization of Renewal structures / Conclusions

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**Additional resources for Stochastic Network Optimization with Application to Communication and Queueing Systems **

**Sample text**

Likewise, queue 2 is rate stable if and only if λ2 ≤ E b2∗ (t) . 10). 9) Define λ=(λ1 , λ2 ), and define max (λ) as the maximum value of in the above problem. It can be shown that the network capacity region is the set of all non-negative rate vectors λ for which max (λ) ≥ 0. The value of max represents a measure of the distance between the rate vector λ and the capacity region boundary. If the rate vector λ is interior to the capacity region , then max (λ) > 0. In this simple example, it is possible to compute the capacity region explicitly, and that is shown in Fig.

2 ˆ i=1 Qi (t)bi (α(t), S (t)) = Q2 (t)S2 (t) if we choose to transmit over channel 2. • 2 ˆ i=1 Qi (t)bi (α(t), S (t)) = 0 if we choose to remain idle. It follows that the max-weight algorithm chooses to transmit over the channel i with the largest (positive) value of Qi (t)Si (t), and remains idle if this value is 0 for both channels. This simple algorithm just makes decisions based on the current queue states and channel states, and it does not need knowledge of the arrival rates or channel probabilities.

1 SCHEDULING FOR STABILITY Consider a slotted system with two queues, as shown in Fig. 1(a). d. over slots, where A1 (t) and A2 (t) take integer units of packets. The arrival rates are given by λ1 =E {A1 (t)} and λ2 =E {A2 (t)}. The second moments E A21 =E A1 (t)2 and E A22 =E A2 (t)2 are assumed to be finite. The wireless channels are time varying, and every 30 3. DYNAMIC SCHEDULING EXAMPLE slot t we have a channel vector S (t) = (S1 (t), S2 (t)), where Si (t) is a non-negative integer that represents the number of packets that can be transmitted over channel i on slot t (for i ∈ {1, 2}), provided that the scheduler decides to transmit over that channel.