## 1 Introduction  
There are number of server farms equipped with hundreds of processors. The cost of energy used for cooling and running a machine for around three years surpasses the hardware cost of the machine [1]. Consequently, the major integrated chips manufacturers such as Intel and AMD are producing the dynamic speed scaling (DSS) enabled multiprocessor/multi-core machine and software such as Intel's SpeedStep [2], which support the operating system in managing the energy by varying the execution speed of processors. A founder chip maker Tilera forecasted that the numbers of processors/cores will be doubled every eighteen months [3], which will increase the energy demand to a great extent. Data centers consume 1.5% of total electricity usage in United States [4]. To avoid such critical circumstances, the current issue in the scheduling is to attain the good quality of service by generating an optimal schedule of jobs and to save the energy consumption, which is a conflicting and complicated problem [5]. The power \(P\) consumed by a processor running at speed \(s\) is \(sV^{2}\), where \(V\) is a voltage [6]. The traditional power function is \(P~{}=~{}s^{\alpha}\) (\(\alpha\geq 2\) for CMOS based chips [7; 8]). There are two types of speed models: the first unbounded speed model, in which the processor's speed range is, i.e., \([0,~{}\infty)\); the second bounded speed model, in which the speed of a processor can range from zero to some maximum speed, i.e., \([0,~{}\eta]\). This DSS plays a vital role in energy management, where in a processor can regulate its speed to save energy. A few qualities of service metrics are slowdown, throughput, makespan, flow time and weighted flow time. At low speed, the processor finishes jobs slower and save energy, whereas at high speed, the processor finishes jobs faster but consumes more energy, as shown in Figure 1. To get a better quality of service and low energy consumption the objective should be to minimize the sum of flow time and energy; in case, if the importance or priority is attached, the objective should be to minimize the sum of importance-based flow time and energy. The objective of minimizing the IBFt+Ehas a natural explanation, as it can be considered in monetary terms [9]. In the multiprocessor systems, there is a requirement of three different policies: the first policy is job selection, which decides the next job to be executed on every processor; the second policy is speed scaling, which decides every processor's execution speed at all time; the third policy is job assignment, which indicates that to which processor the new job should be assigned. In the \(c\)-competitive online scheduling algorithm, for each input the cost received is less than or equal to \(c\) times the cost of optimal offline algorithm [9]. Unlike non-clairvoyant scheduling, the size of job is unknown at arrival time, such as in UNIX operating system where jobs arrive with no information of processing requirement. Unlike online modes, in the offline mode, the whole job progression is known in advance. No online algorithm can attain a constant competitiveness with equal maximum speed to optimal offline algorithm [10]. [PERSON] et al. [10] commenced the study of the non-clairvoyant scheduling algorithms. [PERSON] et al. inducted the theoretical study of speed scaling scheduling algorithm [11]. [PERSON] et al. proposed an algorithm average rate heuristic (AVR) with a competitive ratio at most \(2^{\alpha-1}\alpha^{\alpha}\) using the traditional power function. [PERSON] et al. [12] presented an optimal online scheduling algorithm \(D^{\text{over}}\) for a overloaded uniprocessor system with competitive ratio-\(\left(\frac{1}{\left(1+\sqrt{k}\right)^{2}}\right)\) for the objective of  
Figure 1: Performance and speed curve. minimizing the throughput, where \(k\) is the importance ratio.