## 1 Introduction
In these works on femtocells, the authors propose autonomous and coordinated resource allocation algorithms for interference mitigation based on cell transmit power minimization [14], the resource allocation scheme to maximize the capacity of delay-sensitive and delay-tolerant users subject to QoS and interference constraints [15], the cooperativeNash bargaining resource allocation scheme under the constraint of a cross-tier interference temperature limit to protect the primary macrocell and imperfect channel state information [16], spectrum efficient cognitive femtocell optimization with fairness and imperfect spectrum sensing [17] and a distributed and coordinated radio resource allocation algorithm which minimizes total downlink transmit power [18]. In a cognitive radio network, delay, hop count, power consumption, spectrum availability, route stability, interference, _etc._, are used for routing metrics [19]. Recently, the idea of using the unlicensed spectrum that is currently used by Wi-Fi for small cells has been proposed, too [20]. Femto BSs can turn off their radio interfaces without compromising the reachability of MSs and save energy when there is no MS under the coverage of the femto BSs or MSs do not transmit or receive data packet for a long time, especially late at night, as proposed in [21]. In [21], the authors use an additional low-power radio interface to wake up the radio interface of femto BSs in power saving mode [22] when MSs have any data to transmit via the femto BSs. By doing this, femto BSs can stay in a low-power state and serve MSs when the MSs request data transmission by waking up the radio interface of the femto BSs with the help of an additional low-power radio interface. Although the performance of the energy-efficient femto BSs was analyzed using a testbed in [21], a detailed analysis of the effect of various parameters on the performance based on the state transition model was not dealt with analytically, to the best of our knowledge. In this paper, we propose a state model for energy-efficient femto BSs in [21] consisting of active (data), idle, active (signaling), sleep entering, sleep and waking up states and analyze the performance of the femto BSs, from the aspects of energy consumption, cumulative average delay, cost and low-power radio signaling load. We believe that the detailed mathematical analysis conducted in this paper based on the proposed state model is an essential first step to design a better power saving scheme for femto BSs. We note that although the state model of energy-efficient femto BSs was introduced and analyzed simply in our preliminary work [23], the work in this paper contains the following significant extensions:  
* A new state of active (signaling) is defined to accommodate the processing of location registrations of MSs. * Two new states of sleep entering and waking up are defined to capture the energy consumptions of femto BSs when they are entering into the sleep state and waking up from the sleep state, which have higher energy consumptions than other states [24]. * A detailed analysis of the proposed state model is developed thoroughly, and a closed form of the steady-state probability of each state is derived completely. * The performance of the proposed state model is newly analyzed, from the aspect of cost, which is defined as a weighted sum of energy consumption and cumulative average delay. * The low-power radio signaling load of the proposed state model is newly analyzed, which is an overhead introduced in energy-efficient femto BSs. Finally, the main contributions of this paper are summarized as follows:  
* We proposed active (data), idle, active (signaling), sleep entering, sleep and waking up states for energy-efficient femto BS and formulated transition events between states. * We derived a closed form of the steady-state probability of each state in the proposed state model analytically using a semi-Markov process approach.