We are given a set of @math jobs and a single processor that can vary its speed dynamically. Each job @math is characterized by its processing requirement (work) @math , its release date @math and its deadline @math . We are also given a budget of energy @math and we study the scheduling problem of maximizing the throughput (i.e. the number of jobs which are completed on time). We propose a dynamic programming algorithm that solves the preemptive case of the problem, i.e. when the execution of the jobs may be interrupted and resumed later, in pseudo-polynomial time. Our algorithm can be adapted for solving the weighted version of the problem where every job is associated with a weight @math and the objective is the maximization of the sum of the weights of the jobs that are completed on time. Moreover, we provide a strongly polynomial time algorithm to solve the non-preemptive unweighed case when the jobs have the same processing requirements. For the weighted case, our algorithm can be adapted for solving the non-preemptive version of the problem in pseudo-polynomial time.
Different variants of throughput maximization in the speed scaling context have been studied in the literature (see @cite_3 @cite_12 @cite_1 @cite_6 @cite_4 @cite_7 ), but in what follows we focus on the complexity status of the offline case.
@cite_4 @cite_7 @cite_1 @cite_3 @cite_6 @cite_12
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Dynamic Voltage Scaling techniques allow the processor to set its speed dynamically in order to reduce energy consumption. It was shown that if the processor can run at arbitrary speeds and uses power s^@a when running at speed s, the online heuristic AVR has a competitive ratio (2@a)^@a 2. In this paper we first study the online heuristics for the discrete model where the processor can only run at d given speeds. We propose a method to transform online heuristic AVR to an online heuristic for the discrete model and prove a competitive ratio 2^@a^-^1(@a-1)^@a^-^1(@d^@a-1)^@a(@d-1)(@d^@a-@d)^@a^-^1+1, where @d is the maximum ratio between adjacent non-zero speed levels. We also prove that the analysis holds for a class of heuristics that satisfy certain natural properties. We further study the throughput maximization problem when there is an upper bound for the maximum speed. We propose a greedy algorithm with running time O(n^2logn) and prove that the output schedule is a 3-approximation of the throughput and a (@a-1)^@a^-^1(3^@a-1)^@a2@a^@a(3^@a^-^1-1)^@a^-^1-approximation of the energy consumption. We study network optimization that considers power minimization as an objective. Studies have shown that mechanisms such as speed scaling can significantly reduce the power consumption of telecommunication networks by matching the consumption of each network element to the amount of processing required for its carried traffic. Most existing research on speed scaling focuses on a single network element in isolation. We aim for a network-wide optimization. Specifically, we study a routing problem with the objective of provisioning guaranteed speed bandwidth for a given demand matrix while minimizing power consumption. Optimizing the routes critically relies on the characteristic of the speed-power curve f(s), which is how power is consumed as a function of the processing speed s. If f is superadditive, we show that there is no bounded approximation in general for integral routing, i.e., each traffic demand follows a single path. This contrasts with the well-known logarithmic approximation for subadditive functions. However, for common speed-power curves such as polynomials f(s) = μsα, we are able to show a constant approximation via a simple scheme of randomized rounding. We also generalize this rounding approach to handle the case in which a nonzero startup cost σ appears in the speed-power curve, i.e., f(s) = σ + μsα, if s >; 0; 0, if s = 0. We present an O((σ μ)1 α)-approximation, and we discuss why coming up with an approximation ratio independent of the startup cost may be hard. Finally, we provide simulation results to validate our algorithmic approaches. We consider the following offline variant of the speed scaling problem introduced by We are given a set of jobs and we have a variable-speed processor to process them. The higher the processor speed, the higher the energy consumption. Each job is associated with its own release time, deadline, and processing volume. The objective is to find a feasible schedule that minimizes the energy consumption. In contrast to , no preemption of jobs is allowed. Unlike the preemptive version that is known to be in P, the non-preemptive version of speed scaling is strongly NP-hard. In this work, we present a constant factor approximation algorithm for it. The main technical idea is to transform the problem into the unrelated machine scheduling problem with @math -norm objective. We analyze throughput-delay scaling laws of mobile ad hoc networks under a content-centric traffic scenario, where users are mainly interested in retrieving contents cached by other nodes. We assume limited buffer size available at each node and Zipf-like content popularity. We consider nodes uniformly visiting the network area according to a random-walk mobility model, whose flight size varies from the typical distance among the nodes (quasi-static case) up to the edge length of the network area (reshuffling mobility model). Our main findings are: 1) the best throughput-delay tradeoffs are achieved in the quasi-static case: increasing the mobility degree of nodes leads to worse and worse performance; ii) the best throughput-delay tradeoffs can be recovered by power control (i.e., by adapting the transmission range to the content) even in the complete reshuffling case.