## 1 Introduction  
Nowadays, with the rapid development of mobile communication technologies and mobile devices, wireless data traffic is experiencing an explosive increase, especially in terms of mobile video streaming, high definition (HD) video and video webcasting [1]. A recent Cisco report estimates that the global mobile data volume will grow nearly ten times in the next five years, and the world's mobile data traffic will reach 30.6 monthly exabytes by 2020 [2]. It has led to significant increases in user latency and imposed a heavy burden on backhaul links connecting local base stations (BSs) to the core network (CN). In addition, the rapid growth of mobile data traffic has been compelling mobile network operators (MNOs) to provide more and more network capacities and meet these pressing traffic demands, which are achieved by extending their network infrastructure and enhancing spectral efficiency. For example, it is still far from enough to satisfy the mobile data traffic demands although the capacity of cellular network which can be immensely increased by deploying a large amount of BSs [3]. In fact, since plenty of the available backhaul networks are of low capacity and often cannot catch up with the rate requirements, backhaul capabilities have been regarded as a bottleneck for mobile cellular networks. One promising solution to meet the demand is edge caching, which brings video contents closer to the users, reduces data traffic going through the backhaul links, the time required for content delivery, as well as help to smooth the traffic during peak hours. In wireless edge caching, highly sought-after videos are cached in the cellular BSs or wireless access points so that demands from users to the same content can be accommodated easily without duplicate transmissions from remote servers. Specifically, local caching can be more effective when a fraction of requested contents has high popularity. Recently, Mobile Edge Computing (MEC) [4; 5] has been introduced as an emerging paradigm enabling a capillary distribution of cloud storage capabilities to the edge of the cellular radio access network (RAN). In particular, the MECSs are implemented directly at the BSs using generic-computing platforms, which enable context-aware services and caching deployment in close-proximity to the mobile users. As a consequence of this, MECS presents a unique opportunity to not only implement edge caching but also perform caching placement strategy design. With the benefits of avoiding potential network congestion and alleviating the backhaul links burden, caching popular content at MECSs for backhaul capacity-limited mobile networks has emerged as a cost effective solution [6; 7]. Recently, a good deal of works have been focused on big data analysis strategies for edge caching [8; 9], context-aware caching deployment strategy design [10; 11], and decentralized coded caching strategies [12; 13]. Nevertheless, the cache allocation mechanism, more specifically, the energy efficiency (EE) cache deployment, has received less attention. When the actual budget is given, the cache size deployed at MECS will not be arbitrarily large. Caching more content requires activating more MECS, which results in more energy consumption. Moreover, providing grid power supply with fixed power uninterruptedly in support of MECS is costly and even infeasible, especially when the load changes dynamically over time. Hence, the cost energy of MECS should be carefully investigated, and the EE of MECS within the 5G cellular network should be optimized. As a result, the interplay between the EE and backhaul capacity is supposed to be intensively studied. Recently, the issue of energy efficiency has received a lot of attention in the MEC system [14]. In [15], user association and power allocation in millimeter-wave-based ultra-dense networks is considered with attention to load balance constraints, energy harvesting by base stations, user quality of service requirements, energy efficiency, and cross-tier interference limits.