Figure 2 shows a modified state transition model of BECA, where active state in original state transition model of BECA is divided into four sub-states; active-transmit, active-receive, active-forward, and active-idle states, for ease of mathematical derivation. In active-transmit state, a sensor node transmits its locally generated sensing data to a sink node. In active-forward state, on the other hand, a sensor node relays sensing data from other sensor nodes to neighbor sensor nodes. In active-receive state, a sensor node receives sensing data from other sensor nodes. In active-idle state, the sensor node does not receive or transmit any sensing data, and if there is no further data to be transmitted or received until the expiration of active timer, it moves to sleep state. We note that we differentiate active-forward state from active-transmit state because traffic characteristic for a sensor node’s locally generated sensing data traffic is different from aggregated sensing data generated by other sensor nodes, and thus, we need to consider them separately for an accurate modeling [16].

For mathematical derivation, we need to assume traffic model of sensor networks, which depends on the data reporting process of considered sensor network applications, i.e., event-driven, time-driven, and query-driven [17, 18]. In event-driven case, data are generated when an event occurs. In time-driven case, data are generated periodically. In query-driven case, data are generated by the request of a sink node when needed. In this paper, we mainly focus on event-driven case and it is proved that the data generation at an individual sensor node follows a Poisson process, based on the assumptions that the event is uniformly distributed in sensor networks and every event is independent of other events [17]. Regarding the service time of data packet, we use an exponential distribution model for more general consideration of service time, although simple constant bit rate (CBR) model is commonly used. The use of Poisson and exponential distributions allows us to develop a tractable analytical model for deriving steady state probability of sensor node states. The developed model based on Poisson and exponential distributions can be easily extended to other cases with different traffic model.

Based on the above discussions, we have made the following assumptions regarding the density functions of random variables:

Transmitting, receiving, and forwarding data packets at a sensor node occur according to a Poisson process with parameters λ_t, λ_r, and λ_f, respectively;

We denote sleep, listen, active-transmit, active-receive, active-forward, and active-idle states as states 1, 2, 3, 4, 5, and 6, respectively, for notational convenience. Since the residence times of a sensor node in sleep, listen, and active-idle states do not follow an exponential distribution, the sensor node state transition behavior is analyzed using a semi-Markov process approach [19].

The steady state probability of each sensor node state can be obtained as [19]: (1) P k = π k t k ¯ ∑ i = 1 6 π i t i ¯ ,               k = 1 , 2 , 3 , 4 , 5 ,   and   6where π_k denotes the stationary probability of state k and t k ¯ is the mean residence time of the sensor node in state k. The stationary probability is obtained by solving the following balancing equations [19]: (2) π j = ∑ k = 1 6 π k P kj ,   j = 1 , 2 , 3 , 4 , 5 ,   and   6 (3) 1 = ∑ k = 1 6 π k  where P_kj represents the state transition probability from state k to state j. The state transition probability matrix P = [p_kj] of the state transition model is given by: P = ( 0 P 12 P 13 0 0 0 P 21 0 P 23 P 24 P 25 0 0 0 0 0 0 P 36 0 0 0 0 0 P 46 0 0 0 0 0 P 56 P 61 0 P 63 P 64 P 65 0 )Then, stationary probabilities can be solved as: (4) π 1 = 1 D (5) π 2 = P 12 π 1 = P 12 D (6) π 3 = P 61 P 13 + P 61 P 12 P 23 + P 63 ( 1 − P 12 P 21 ) P 61 π 1 = P 61 P 13 + P 61 P 12 P 23 + P 63 ( 1 − P 12 P 21 ) P 61 D (7) π 4 = P 61 P 12 P 24 + P 64 ( 1 − P 12 P 21 ) P 61 π 1 = P 61 P 12 P 24 + P 64 ( 1 − P 12 P 21 ) P 61 D (8) π 5 = P 61 P 12 P 25 + P 65 ( 1 − P 12 P 21 ) P 61 π 1 = P 61 P 12 P 25 + P 65 ( 1 − P 12 P 21 ) P 61 D (9) π 6 = 1 − P 12 P 21 P 61 π 1 = 1 − P 12 P 21 P 61 Dwhere D is obtained by: (10) D = 1 + P 12 ( 1 + P 23 + P 24 + P 25 ) + P 13 + 1 + P 63 + P 64 + P 65 − P 12 P 21 ( 1 + P 63 + P 64 + P 65 ) P 61

State transition probability P_kj can be derived based on the distribution of time from states k to j, T_kj. Exit from the sleep state is caused by any of the following events:

Sleep timer expiration (T₁₂);

Exit from the listen state is caused by any of the following events:

Listen timer expiration (T₂₁);

Exit from the active-idle state is caused by any of the following events:

Active timer expiration (T₆₁);

Based on the values of π_k and t k ¯ obtained from Equations (4)–(9) and Equations (22)–(27), respectively, we can finally derive the steady state probability of each sensor node state using Equation (1) [19]. Now, we can obtain the energy consumption of a sensor node per unit time by using the steady state probability as: (28) E = ∑ k = 1 6 ψ k P kwhere ψ_k is the power consumption in state k.R

In this section, numerical examples are presented in order to show the effect of various parameters on the steady state probability and energy consumption. In sensor network, since sensor nodes transmit locally generated sensing data to a sink node and the locally generated data traffic from sensor nodes contributes to forwarding data traffic of a sensor node, we relate local transmitting data traffic and forwarding data traffic of a sensor node by λ_f = w_fλ_t, where w_f is the weighting factor for forwarding data traffic to local transmitting data traffic and it is proportional to the number of sensor nodes in sensor network. Also, since forwarding traffic should be received first before forwarding [16], the relationship λ_r = λ_f is assumed. The relationship 1/μ_t = 1/μ_r = 1/μ_f is also assumed since each sensor node normally generates sensing data with the same size. Based on these relationships, we define activity of a sensor node as follows: (29) ρ = λ t + λ r + λ f μ t = ( 1 + 2 w f ) λ t μ t

The default parameters for timers and traffic characteristic are given by Table 1. Parts of the parameters in Table 1 vary in the following numerical examples in order to analyze the effect of the parameters on the performance. Power consumption in each state is assumed, as the ones depicted in Table 2, based on the energy model described in [6, 11, 12].

Figure 3 shows the effect of sleep timer on the steady state probability of sleep, listen, and active states. We note that instead of showing the probabilities of four sub-states; active-transmit, active-receive, active-forward, and active-idle states which have a very similar pattern, we just show the probability of active state collectively, in order to avoid confusion and strengthen the result. To validate the analytical methodology developed in this paper, we conducted a discrete-event simulation for steady state probabilities, using C++ language. The simulation results were obtained by running 10⁴ h of simulation time after 10³ hours of warm-up time. From the results, it is shown that the validity of the proposed analytical model is proved since analysis results are well matched with simulation results. As the value of sleep timer increases, the probability of sleep state decreases, since it is more likely that a sensor node stays in sleep state more for higher value of sleep timer. On the other hand, the probabilities of listen state and active state decrease, because the chance of staying in either listen state or active state decreases due to increased probability of sleep timer.

Figure 4 shows the effect of sleep timer on the energy consumption for varying activity, where the values of λ_t, λ_r, and λ_f are changed with 1 μ t = 1 μ r = 1 μ f = 1 3600 h. We note that simulation results only for steady state probabilities are presented but those for energy consumption are not presented in this paper since energy consumption is just a weighted sum of steady state probabilities, and thus, the validity of the analytical model for energy consumption is proved by just showing that for steady state probabilities. As can be expected from the result in Figure 3, the energy consumption decreases as the value of sleep timer increases, since the effect of sleep state becomes dominant, which has low power consumption. For a fixed value of sleep timer, energy consumption increases as the activity increases due to more transitions to active state with higher power consumption. We note that although the energy consumption decreases as the value of sleep timer increases, this also increases wakeup delay from sleep state to listen state. Therefore, an appropriate selection of sleep timer is needed to tradeoff between power consumption and wakeup delay.

Figure 5 shows the effect of sleep timer on the energy consumption for varying service time, where the value of activity is fixed as 0.1. Although the value of activity is the same, smaller service time, i.e., higher data arrival, results in more state changes into active state. Therefore, the energy consumption increases as the service time for transmitting, receiving, or forwarding data decreases. This is because although the total service time for either transmitting, receiving, or forwarding data is the same for the same activity, the sensor node with higher data arrival rate has higher probability of active state, since the sensor node stays in active state additionally until the expiration of active timer, where the number of the expirations of active timer is higher for higher rate of data arrival.

Figure 6 shows the effect of listen timer on the steady state probability using both analysis and simulation. Similar to Figure 3, analysis results are well matched with simulation results. The probability of sleep state decreases, as the value of listen timer increases, since it is more likely that a sensor node stays in listen state more for higher value of listen timer, which is evident from the definition of listen timer. The probability of active state increases since it has more chances to make transition from listen state to active state due to data arrival, as the value of listen timer increases.

Figure 7 shows the effect of listen timer on the energy consumption for varying activity, where the values of λ_t, λ_r, and λ_f are changed with 1 μ t = 1 μ r = 1 μ f = 1 3600 h. As can be expected from the result in Figure 6, the energy consumption increases as the value of listen timer increases, since the probabilities of listen and active states, which have higher power consumption, increase. For a fixed value of listen timer, energy consumption increases as the activity increases due to more transitions into active state with higher power consumption. We note that although smaller value of listen timer is better, from the aspect of energy consumption, the smaller value of listen timer results in more transitions into sleep state. Since more transitions into sleep state result in higher wakeup delay, an appropriate selection of listen timer is needed to tradeoff between power consumption and wakeup delay.

Figure 8 shows the effect of listen timer on the energy consumption for varying service time, where the value of activity is fixed to 0.1. Although the value of activity is the same, the energy consumption increases as the service time for transmitting, receiving, or forwarding data decreases due to the similar rationale, as in Figure 5.

Figure 9 shows the effect of active timer on the steady state probability using both analysis and simulation. Similar to previous figures on steady state probabilities, analysis results are well matched with simulation results, as well. As can be expected, the probability of active state increases, as the value of active timer increases. On the other hand, the probability of sleep and listen states decreases, as the value of active timer increases. This is due to the fact that higher value of active timer results in less transitions to sleep state, and thus, the transition from sleep state to listen state is also less likely to occur.

Figure 10 shows the effect of active timer on the energy consumption for varying activity, where the values of λ_t, λ_r, and λ_f are changed with 1 μ t = 1 μ r = 1 μ f = 1 3,600 h. Since the probability of active state increases as the value of active timer increases, the energy consumption also increases since active state has more power consumption. For a fixed value of active timer, energy consumption generally increases as the activity increases, due to more transitions into active state with higher power consumption. On the other hand, for very small values of active timer, energy consumption decreases, even though the activity increases, because more transitions into active state due to higher activity also results in more transitions into listen state with very low power consumption from active state, after the fast expiration of small active timer.

Figure 11 shows the effect of active timer on the energy consumption for varying service time, where the value of activity is fixed as 0.1. Although the value of activity is the same, the energy consumption generally increases as the service time for transmitting, receiving, or forwarding data decreases due to the similar rationale, as in Figure 8. On the other hand, for very small values of active timer, energy consumption decreases as the service time increases, due to the similar rationale, as in Figure 10.

Figure 12 shows the effect of λ_t on the steady state probability for λ_f = λ_r = 10/h using both analysis and simulation. Similar to previous figures on steady state probabilities, analysis results are well matched with simulation results, too. The steady state probability of active state sharply increases, as the value of λ_t increases, which also increases the values of λ_r and λ_f. On the other hand, the steady state probabilities of both sleep and listen states decrease, as the value of λ_t increases, due to more transitions to active state from these states.

Figure 13 shows the effect of λ_t on the energy consumption for varying service time, where the value of activity increases as the service time increases. As can be expected from Figure 12, the energy consumption also sharply increases, as the value of λ_t increases, with the dominant effect of active state having higher power consumption. For a fixed value of λ_t, power consumption also increases as the service time increases due to higher activity.