## 3 Numerical Examples  
In the numerical examples, we show the energy consumption, cumulative average delay, cost and low-power radio signaling load for varying the values of inactivity timer. Furthermore, the effect of the energy consumption of entering into the sleep state and waking up from the sleep state is analyzed by considering two different time duration sets for \(T_{E}\) and \(T_{W}\). Figure 2 shows the steady-state probability of the femto BS state by varying the values of inactivity timer \(T_{I}\) with \(\lambda_{s}\) = 10 (/h), \(\mu_{s}\) = 100 (/h), \(\lambda_{r}\) = 100 (/h), \(\mu_{r}\) = 2000 (/h), \(\psi_{1}\) = 12 W, \(\psi_{2}\) = 6 W, \(\psi_{3}\) = 1  
both idle and sleep states, which have the opposite probability distributions. The probability of the active (signaling) state increases as the values of the inactivity timer increase, similar to that of the idle state, since the transition into the active (signaling) state occurs from the idle state. The probabilities of sleep entering and waking up are the same and decrease as the values of the inactivity timer increases, similar to that of the sleep state, but they have much smaller probability values. From the results, we can see that the value of the inactivity timer has a significant effect on the steady-state probability of the femto BS state. Figure 3 shows the steady-state probability of the femto BS state by varying the values of the inactivity timer \(T_{I}\) with the same parameter values, except for \(T_{E}\) = 100/3600 (h) and \(T_{W}\) = 100/3600 (h). The figure shows both analysis and simulation results, and we can see that the analysis result is validated by simulation results. The distributions of probabilities of states in this figure are very similar to those in Figure 2. However, the probabilities of the sleep entering and waking up states are higher than those in Figure 2, since MSs stay in these states for a longer time duration. On the other hand, the probability of the sleep state decreases due to increased mean residence time, _i.e._, steady-state probability, in the sleep entering and waking up states. Figure 2: Steady-state probability by varying the session arrival rate (small values of \(T_{E}\) and \(T_{W}\)). Figure 4 and Figure5 show the energy consumption by varying the values of the inactivity timer \(T_{I}\) and the rate of data session arrival \(\lambda_{r}\), with the same parameter settings as those of Figure 2 and Figure 3, respectively. The energy consumption of Figure 4 increases as the values of the inactivity timer increase, since the probability of the idle state increases, which has a higher power consumption than the sleep state, and the effects of energy consumptions of the sleep entering and waking up states are not significant due to the small mean residence times, _i.e._, steady-state probabilities, of these states. On the other hand, the energy consumption of Figure 5 decreases as the values of the inactivity timer increase, since the effects of energy consumptions of the sleep entering and waking up states are significant due to the increased mean residence times, _i.e._, steady-state probabilities, of these states, although the probability of the idle state increases, similar to Figure 4. These results show that different inactivity timer values should be selected appropriately to reduce energy consumptions depending on the effect of the energy consumptions of the sleep entering and waking up states. For a fixed value of the inactivity timer, the energy consumption increases as the rate of session arrival increases, in both Figure 4 and Figure 5, since higher data session arrivals cause more transitions into the active (data) state. From the results, we can see that an appropriate value for the inactivity timer should be selected to reduce energy consumption, depending on the energy consumption in the sleep entering and waking up states. Figure 3: Steady-state probability by varying the session arrival rate (large values of \(T_{E}\) and \(T_{W}\)).