Power Management in SMAC-based Energy-Harvesting Wireless ...
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Power Management in SMAC-based Energy-Harvesting Wireless Sensor Networks Using Queuing Analysis Navid Tadayon, Student Member, IEEE, Sasan Khoshroo, Elaheh Askari, Honggang Wang, Member, IEEE, and Howard Michel, Member, IEEE
Abstract—one of the most important constraints in traditional wireless sensor networks is the limited amount of energy available at each sensor node. The energy consumption is mainly determined by the choice of media access mechanism. SMAC is a typical access mechanism that has drawn much attention in recent years. In WSNs, sensors are usually equipped with capacity-limited battery sources that can sustain longer or shorter period, depending on the energy usage pattern and the activeness level of sensor nodes. To extend the lifetime of the sensor networks, ambient energy resources have been recently exploited in WSNs. Even though solar radiation is known as the superior candidate, its density varies over time depending on many factors such as solar intensity and cloud states, which makes it difficult to predict and utilize the energy efficiently. As a result, how to design an efficient MAC in a solar energy harvesting based WSN becomes a challenging problem. In this paper, we firstly incorporate a solar energy-harvesting model into SMAC and conduct its performance analysis from a theoretical aspect. Our research works provides a fundamental guideline to design efficient MAC for energy harvesting based WSNs. Our major contribution includes three folders: firstly, we model solar energy harvesting in a photovoltaic cell and then derive the throughput of SMAC in the energy-harvesting based WSNs. Secondly, we develop a new model based on queuing theory to calculate the average number of energy packets in battery in terms of both duty cycle and throughput. Finally, we form an optimization problem to find a suitable range for duty cycle to satisfy both quality of service (QoS) and network lifetime requirements. Index Terms—Duty Cycle, Energy-Harvesting, Queuing model, SMAC, Solar Radiation Modeling, Throughput, WSN.
I. I NTRODUCTION IRELESS sensor networks consist of hundreds of spatially-distributed sensors, called nodes, to cooperatively monitor a specific quantity which can be the level of pollutants, temperature, sound, vibration, pressure, and so on. These networks have been utilized in a wide variety of applications, like civil, industry and military applications. They have also been widely used for healthcare monitoring, object tracking and assembly line sensing. WSN (Wireless Sensor Networks) is composed of different layers, first of which is MAC (Medium Access Control) layer, which grants access of the wireless channel to different nodes. Depending on the type of MAC protocol being utilized, wireless sensor networks are divided into two general categories, scheduled
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N. Tadayon, S. Khoshroo, E. Askari, H. Wang, and H. Michel are with the Department of Electrical and Computer Engineering, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA (e-mails: {ntadayon, easkari, skhoshroo, hwang1, hmichel}@umassd.edu)
networks and contention-based networks [1]. In scheduled networks, the wireless channel is divided into sub-channels in terms of either time (Time Division Multiple Access-TDMA, frequency (Frequency Division Multiple Access-FDMA), orthogonal codes (Code Division Multiple Access-CDMA), or a combination of them and each of these sub-channels are assigned to each node. However, each of these protocols has its own challenges such as time synchronization in the case of TDMA, frequency generation/filtering and bandwidth requirements in the case of FDMA, and power control in the case of CDMA. These requirements cannot be simply satisfied using tiny, incapable sensors that are usually located in a place with no replacement/maintenance possibility. As a result, contention-based access methods are more suitable due to their simple, autonomous and scalable nature. Here, nodes compete with each other to win the access to the shared medium. They are also flexible toward network topology changes, which is typical in wireless networks. Another important constraint in WSNs is the amount of energy available to each node. The power consumption should be uniform over the network to extend the network lifetime. Otherwise, there will be some portions of the network consisting of dead sensors that will degrade the overall QoS (Quality of Service) performance. As a result, in WSNs, power metric is more important than other QoS metrics or fairness of access. To circumvent this problem, the power consumption should be considered as a distributed parameter in the network instead of a point parameter in only one node, which demands careful considerations in MAC layer. Numerous works have been carried out to design energy-efficient MAC protocols. In fact, it has been shown in [2] that, the energy consumption using IEEE 802.11 MAC is very high when nodes are in the idle mode. Some of such recently proposed MAC protocols include PAMAS [3], SMAC [4], TMAC [5], and PMAC [6], among which, SMAC is of particular interest in this paper and is explained in details in next section. PAMAS is an improvement over MACA (Multiple Access Collision Avoidance) protocol by adding a separate signaling channel for exchanging the RTS/CTS packets, which enables the nodes to switch themselves off when they are not receiving or transmitting any packets. Therefore, this protocol is more efficient than the original MACA protocol. As the authors claim in [3], it could increase the power efficiency of most Ad-Hoc networks by 10% to 70%. In [7], a distributed Time-varying Opportunistic MAC Protocol (TOP) is proposed to maximize the network
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lifetime through transmission scheduling, based on Channel State Information (CSI) and Residual Energy Information (REI). In fact, a higher priority is assigned to the sensor node with better channel condition and higher residual energy. The authors show that this approach increases the network lifetime compared to other distributed MAC protocols like DPLM [8], and Pure Opportunistic [9]. In [10], the authors propose an Optimized MAC protocol to deal with the energy inefficiency and nodes latency. They prove, analytically and via extensive simulations that this scheme achieves high energy efficiency under a wide range of traffic loads and is able to adjust itself to improve the delay performance when the network traffic load is high. In SMAC [4] or sensor MAC protocol, nodes go to sleep and active states periodically in order to reduce power consumption. During the sleep mode, the node turns off its radio, and sets a timer to later awakening. The difference between PAMAS and SMAC is that SMAC uses in-channel signaling rather than using a separate channel for signaling, as in PAMAS. Being inspired by SMAC, TMAC [5] also uses the same periodic active/sleep scheme. However, the duty cycle is not fixed in TMAC by dynamically ending the active part of the cycle, which reduces the amount of energy wasted on idle listening. It has been shown in [5] that, in terms of energy efficiency, T-MAC outperforms S-MAC by a factor of 5. Recently, environmental energy resources like solar or wind power have been exploited in WSNs. An energy-harvesting node is defined as any system, which can absorb part or all of its energy from the environment [11]. An important difference between this kind of energy and that stored in the capacitylimited batteries is that ambient (and particularly solar) energy is potentially infinite. However, the energy generation rate at which this type of energy can be generated may be limited; for example, solar power is not available during night and cloudy conditions and the absorption rate continually changes over time. In this paper, our first contribution is to model the solar energy received at a photovoltaic cell. We are inspired by the research works in [12] and [13]. However, we consider different parameters (like cloud length and sunlight inclination) for modeling the absorbed energy rather than those considered in [12]. This kind of modeling is necessary for our second contribution, which is the modeling of the number of energy units (we call them energy packets from now on) in battery. Although power efficiency is the main concern in sensor networks, other QoS requirements also need to be satisfied. There are different methods to evaluate network’s performance. One of them is queuing analysis. In [14] an infinite queuing model has been proposed for contention-free sensor networks analysis and a more realistic finite queuing model has been used to analyze the tradeoff between energy consumption and QoS requirements in contention-based sensor networks in [1]. Our second and main contribution is the modeling of the number of energy packets in battery of a sensor that uses SMAC as a case study of its MAC protocol. Since the battery has an input (through which energy is absorbed) and an output (through which energy is consumed), queuing theory is
applicable to model non-data identities. These identities in our analysis are energy packets, which will be served (consumed) in FIFO order. We assume that there is no energy leakage from battery, and energy consumption is quantized so that one energy packet is consumed to transmit one data packet. The only point that should be noted is that energy packets are not distinguishable identities. This should not be a problem, since we are only interested in the number of energy packets in the queue (battery) which is irrelevant to serving policy, and each energy packet’s position in the queue. The used model in our analysis is G/G/1 queue with batch transmission (due to message passing technique used in SMAC). The input process to this queue (battery), a(n), is a point process that represents the number of energy packets (n) absorbed by the photovoltaic cell and delivered to the battery. The output process, b(t), represents the discrete time interval between two successful consecutive transmissions from a sensor node. We employ the approaches in [15] and [16] in our analysis to achieve two goals; firstly, to prolong the network lifetime by not letting the sensor nodes be out of power or exceed a critical battery level; secondly, to guarantee the network QoS by not letting the throughput fall below a critical threshold. Since there is tradeoff between the energy consumption and throughput, the two goals can be achieved through solving a constrained optimization problem. Consequently, the third contribution of this paper is modeling the SMAC throughput. To the best of our knowledge, very few works [17] have been done on finding the throughput of SMAC. However, none of them has explicitly extracted a closed form expression for the network throughput for energy harvesting based WSNs. In our analysis, the parameter of interest that shifts the benefits toward each of the abovementioned targets is the duty cycle, which is the ratio of the active time of a sensor to the whole cycle time. The rest of the paper is organized as follows. In section II, we give a brief introduction to SMAC [4]. In section III, we present our solar energy model that describes the process of feeding the number of solar energy units into the battery. The closed form expressions of SMAC throughput and service time in terms of network parameters are derived in section IV. Then, in section V, we establish our queuing model based on the model described in the previous sections to find the average number of energy packets in queue (battery) and form the optimization problem. Finally, we summarize our results in section VI. II. S MAC P ROTOCOL OVERVIEW In the SMAC, nodes go to periodic sleep and active states in order to reduce the energy consumption. As specified in [4], the active period of each cycle, which is determined by the MAC-layer’s contention window size, is assumed to be fixed. Thus, duty cycle depends only on the variable sleep period. Additionally, each active period is subdivided into two phases; the first one is reserved for the synchronization purpose among the nodes, and the second one is for RTS/CTS handshake as specified in [18]. In synchronization phase, neighboring sensors synchronize their transmissions by exchanging their sleep-active schedules [17] periodically. If a
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node wants to transmit a data packet, it needs to compete with other nodes for the medium access. This channel access contention takes place in the second phase. Like IEEE 802.11 MAC protocol, SMAC uses RTS/CTS/DATA/ACK packets in order to guarantee successful transmissions. It starts transmission as soon as the active period is over. All other nodes that have been involved in the contention must go to sleep if they fail to win the contention or if they encounter an RTS collision. On the other hand, the winner of the contention will not go back to sleep until the transmitted data packets are acknowledged. Another technique used in SMAC is message passing. In this technique, long messages are divided into many small fragments and these fragments are transmitted in bursts [4]. Expressing clearly, the medium is reserved for transmitting all the fragments by performing just one RTS/CTS handshake for the whole message. But still, to prevent the hidden terminal problem, an ACK is sent by the receiver to the sender for each and every fragment. In fact, after sending a fragment, the sender waits for its ACK from the receiver and if it does not receive it within a specific time, it will expand the reserved access time for one more fragment and transmits that fragment again [4]. If a new sensor node, which is only the neighbor of the receiver, joins the network in the middle of the transmission, it will not sense the transmitted fragments. It assumes that the medium is free and will start transmitting its own data. This transmission could interrupt the original transmission. Thus, the transmission of the ACK packets by the receiver will prevent this problem. This technique reduces the contention latency in the network and is suitable for the networks in which fairness is pursued at an application level rather than at the MAC layer. This protocol has been implemented on test-bed nodes in [4] and the experiment results well demonstrate the effectiveness of SMAC compared to IEEE 802.11 in terms of energy consumption in source and immediate nodes. III. S OLAR R ADIATION M ODEL AND A BSORBED E NERGY As stated before, ambient energy resources have recently been exploited in WSNs and an energy-harvesting node is defined as the one that can absorb part or all of its energy from the environment. Our contribution in this section is to model the amount of solar energy that will be absorbed by photovoltaic cells and delivered to rechargeable battery. We consider solar energy as the most effective and most available source of the energy that can be harvested by existing solar cells. Depending on the solar intensity, cloud states, the hour of the day, season of the year, spatial angles and so on, the amount of absorbed energy is different. Particularly, when the emission intensity is high, solar cell produces more amount of energy that is fed to the rechargeable battery. On the other hand, when the emission level is low (when long and bulky clouds obstruct solar emission during the nights or so) and the solar cell is incapable of producing enough amount of energy, the sensor will use up the reserved energy in battery. Fig. 1 illustrates this radiation process. As the rays of the sun penetrate into the atmosphere, they encounter different attenuating obstructs. The flux intensity that arrives just before
Fig. 1: Example model for solar radiation.
the clouds could be approximated by the equation presented in [19]. This equation expresses flux intensity in terms of hour of the day (t), latitude of the collector site (λ) and solar declination (η) as: G (ζ) = G0 e−0.357(sec(ζ))
0.678
,
sec (ζ) = 1/ (sin (λ) sin (η) + cos (λ) cos (η) cos (t)) ,
(1)
where t = (360/24)T (T indicates the number of hours in a solar noon). G0 indicates the solar flux intensity outside the atmosphere and its value is 1.35 KW/m2 [19]. This flux intensity when passed through the clouds, experiences severe attenuation. The main causes of this are cloud length and thickness. The lengthiness and thickness of each cloud varies over time and is different from others, so it should be considered as a random variable rather than a deterministic identity. In our analysis, the cloud length follows exponential distribution fci (xi ). In fact, the behavior of many of the events in nature can be modeled with this distribution. Now, assuming that cloud profile contains limited number of states (as illustrated in Fig. 1), the PDF (Probability Density Function) of the length of cloudy profile could be written as a hyper-exponential distribution since at each instance of time, only one cloud state could block the direct sunlight radiation with probability Pi . Then: fX (x) =
k ∑
Pi fci (x) ,
i=1
where, fci (xi ) =
1 − xc i e i ci
(xi > 1),
(2)
and, n ∑
Pi = 1.
i=1
In (2), xi and ci represent instantaneous cloudy profile’s length and mean cloud’s length in state i (in km), respectively. From [13], the amount of flux intensity on the plates of
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( ) photovoltaic cell watt/m2 is equal to: g = (1 − rX) G (ζ) ,
(3)
where G (ζ) is given in (1), and r is a constant with nominal value in between 0.02 and 0.03 depending on the cloud profile in the collection site. Now the target is finding PGF (Probability Generation Function) of the number of energy packets (a (n) &A (z)) that is absorbed by solar cell and delivered to battery. This function can be easily found from PGF of g since we have a (n) = (gApanel )/Ebit Rdata where Apannel is the effective area of solar panel (less than 1cm3 ), Rdata is the transmission data( rate (19.2 Kbit/s), ) and Ebit is the quantized unit of energy 1.04·10−6 J/bit consumed for transmitting a single data packet as long as we assume in this paper that the data packet has a fixed length, this derivation for a (n) leads to a correct answer. By carrying out probabilistic operations, the PGF of a (n) (A (z)) is achieved To prevent the loquaciousness, details are not mentioned here, and only the results have been provided. For simplicity, we consider seven states for the cloud profile (k=7):
A (z) =
∞ ∑
z i fa(n) (i)
) Pi k=7 ∑ ci × = G (ζ) 1 i=1 rln (z) + (Ebit Rdata /Apanel ) ci i=1
(
(4)
From (5) we see that Dcycle is always upper-bounded with one. Using the abstraction M = Ta2 /Ta , and the fractional definition of throughput as: Fraction of a cycle in which pure data is transmitted = Total cycle Time T we get: T hroughput = (Ps Dcycle l · TP M ) Dcycle . (M ((1 − Pb ) δ + (Pb − Ps ) Tc + Ps Ts )) + Dcycle . ((1 − M ) Ta1 ) + (1 − Dcycle ) (T − Ta − Ts )
(6)
In (6), the numerator represents the fraction of a cycle time, which is spent for successful pure data transmissions. Primarily, transmissions can only happen in wake-up intervals. This fact is reflected by embedding Dcycle in numerator. After that, since the access mechanism in SMAC is contention-based, transmitted packet may or may not be received successfully. This reveals that only fraction of each transmission can be succeeded, which is reflected by factor Ps in numerator. Finally, l reflects the effect of burst length due to message passing mechanism which is specific to SMAC. The denominator in (6) is composed of all possible events that can happen during a cycle time of a sensor node. After replacing (5) into (6), and substituting Ta1 = (1 − M ) Ta , we get: T hroughput = (Ps Dcycle l · TP M ) Dcycle . (M ((1 − Pb ) δ)+ (Pb − Ps ) Tc + Ps Ts )) + (
G (ζ) (E R /Apanel ) . bit data z
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Table I shows the values of Pi and ci for the sample cloud profile we used in our numerical simulation. A (z) will be exploited in our next section’s analysis as the input process to the queue model. The inverse z-transform of this PGF is the PDF of the number of energy packets that enter into the entrance of the battery. TABLE I: Different Values Of Cloud Profile
i P C (km)
1 0.35 40
2 0.1 33
3 0.045 17
4 0.01 7
5 0.045 17
6 0.1 33
7 0.35 40
IV. SMAC T HROUGHPUT M ODEL From now on, Dcycle represents the ratio of the active period to the whole cycle time. This parameter varies as system jumps to the next cycle due to the nondeterministic nature of the sleep lengths. Fig. 2 illustrates different existing packets and intervals in S-MAC protocol. Depending on the value of the probability of success (Ps ), (5) yields this quantity. Dcycle = figure
(Ta + Ts ) Ta (1 − Ps ) + Ps . T T
(5)
Dcycle . (1 − M ) Ta + (Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) (1 − Dcycle ) , Dcycle
(7)
where M , l, and Tp represent the ratio of the RTS/CTS period to the total active period, the number of data packets in each burst (due to message passing feature of SMAC), and the MAC layer information length in each data packet, respectively. Furthermore, Ps , Pb , Ts , and Tc symbolize successful transmission probability, channel busyness probability, successful transmission length, and collision length, respectively. From the previous research done in [20], [21], [22], [23] on performance evaluation of CSMA/CA, the appropriate equations for Ps , Pb , Ts , and Tc are shown in (8) where n and τ represent the number of contending stations and transmission probability, respectively. Also THeader , ACK, CTS, RTS, SIFS, DIFS are the same parameters as in IEEE 802.11 and [20]. n−1
Ps = nτ (1 − τ ) , n Pb = 1 − (1 − τ ) , Ts = RT S + +CT S + l (Tp + THeader + ACK) + DIF S+ (2l + 1) SIF S, Tc = RT S + SIF S + ACK + DIF S. (8)
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Fig. 2: All possible packets and intervals in between any two successful consecutive transmissions from a sensor node running on S-MAC.
Table II shows the nominal values for parameters that are required for calculation of throughput. For these set of parameters, transmission probability τ is calculated and the value is 0.027. TABLE II: SMAC Parameters In Numerical Simulation
Parameter Rate (bit/s) Tp (Byte) THeader (Byte) CTS (bit) RTS (bit) ACK (bit)
Value 19200 100 8 304 352 304
Parameter Sifs (ms) δ (ms) Difs (ms) Ta (ms) n
Value 10 20 50 300 20
It should be mentioned that this approach on finding SMAC throughput is only valid for the case that all sensors contend for channel access in every cycle. In other words, the transmission queues of sensors have always packets waiting to be transmitted. With data rate of 19200 bit/s it is always guaranteed that sensors are in the above mentioned saturated condition. Fig. 3 depicts the throughput in terms of duty cycle for different values of M (shown in Fig. 3a) and l(shown in Fig. 3b). The trend in both figures shows that as duty cycle increases, throughput first improves (due to increased transmission probability) and then degrades (due to increased collision probability) which is very intuitive. On the other hand, for larger data bursts (larger l), less contention is required for transmitting specific number of data packets and higher throughput is achieved. V. P ERFORMANCE A NALYSIS BASED O N T HE Q UEUING M ODEL A. PGF of Successive Transmission Intervals In this section, we establish our queuing model for the number of energy packets in battery using queuing theory. Each classic queue is characterized by several features such as input process, output process, number of servers, scheduling policy, and so on. Depending on the characteristics of input and output processes, queues are classified into different categories the most general of which is G/G/1 category. In G/G/1 queues, the input and output processes can be any arbitrary random
(a) Burst length l is fixed at 4
(b) Synchronization ratio M is fixed at 0.8
Fig. 3: SMAC throughput vs. Duty cycle.
process (rather than uniform P oisson process), which makes it complicated to analyze when the Markovian property is not applicable anymore. In the queuing theory, the number of packets waiting in the queue (system occupancy) is of great importance. However, finding the closed form expression for PDF of this quantity
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in a G/G/1 queue is a cumbersome and sometimes impossible task. There are some research [15], [16], and [24] that have addressed this problem by finding its PGF or mean value. However, this is not sufficient for our case because G/G/1 formulations are unable to address burst/ bulk processing issue, which makes the analysis even more complicated. We found that the concept of batch processing in queue analysis exactly matches with burst transmission in networks. The analysis of system occupancy for a general input/ output queue with batch processing has been addressed in [16] and [25]. In our case, we resort to the same type of queue modeling since the input process a(n) (number of energy packets delivered to the battery) and output process b(t) (consumption interval) are both general distribution discrete processes and message passing feature of SMAC is nothing more than a batch processing in queue’s server. From [25], we know the PGF of system occupancy for a G/G/1 queue is equal to: U (z) = (1 − A′ (1) B ′ (1))
((z − 1) B (A (z))) z − B (A (z))
Given that Tsleep is a random variable in SMAC specification, Tf would be a random variable too. Assuming that Tsleep is uniformly distributed in the interval [0, T − Ta − Ts ], the probability mass function (pmf) of Tf would be equal to:
PTf
(Ta + Ts ) (T − Ta − Ts ) +1 Ps Ps (11) Ps z 1−z (Tf ) = × (T − Ta − Ts ) 1−z
Applying T = (Ta +Ts Ps ) /Dcycle into (11) and then taking its z-transform, the PGF of Tf would be equal to:
B (z) =
z × (Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) Ps
(9)
where, A(z) and B(z) are the PGFs of a(n) and b(t) as defined in the previous paragraph and ′ represents the derivation operator. In section III, we were able to find A(z) and now the only remaining task is finding B(z). As explained before, b(t) is the interval between every two consecutive energy packets’ consumption. Since the majority of energy packets in SMAC are only consumed on transmission of data packets and very little are consumed on idle listening, overhearing, and synchronizations, it is almost accurate to label b(t) as the interval between transmissions of two consecutive data packets as well. This last description of b(t) is the exact definition for service delay which has been symbolized by Tf in Fig. 2. Fig. 4 simply illustrates this convergence between two queues (data queue and energy queue) in a sensor node.
(Dcycle T + Ts (1 − Ps )) Ps
1−z
(Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) +1 Ps 1−z
(12)
The mean value of this service delay is achievable by taking the inverse z-transform of this PGF and evaluating it at z = 1. Fig. 5 shows this mean service time in terms of duty cycle for diverse burst lengths. Apparently, as the duty cycle increases, sensors find more transmission opportunities in less congested transmission intervals which leads to less service delay. On the other hand, bigger bursts take longer time to be transmitted but have higher throughput as shown in Fig. 3b.
Fig. 4: A simple schematic model for an energy harvesting sensor node with two queues: data, and energy.
Tf is made up of two parameters T (cycle length) and m (the average number of retransmissions of a specific packet until it is successfully transmitted) where Tf = T · m. Referring to [20] and [21] m= ¯ 1/Ps due to the fact that the number of transmission trials (m) is geometrically distributed with parameter Ps . As a result Tf can be approximated with: (Ta + Ts + Tsleep ) Ps (Dcycle T + Ts (1 − Ps ) + Tsleep ) = Ps
Tf = T · m ¯ =
(10)
Fig. 5: Duration Interval between two consecutive successful transmissions from a SMAC sensor node vs. duty cycle under different burst lengths.
B. Number of Energy Packets in Battery Incorporating the appropriate equations (4) and (12) into (9), gives us the PGF of number of energy packets in battery
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when the bulk transmission mechanism (message passing) is not used. Then by replacing this result inside equation (3.3) in [16] the exact average number of energy packets in the queue will be achieved as per following: ( ) ′ (l − 1) (13) + l L + U (1) + ρ 2 where ρ (server utilization factor) is equal to A′ (1) B ′ (1) and L is the initial number of energy packets (meaning that sensor nodes are launched with a nonzero battery level) in battery (L = 1500 as default), respectively. The strict stability condition in queuing theory is to have a utilization factor less than one so that packets do not experience unbearable loss or delay. In fact (9) is only valid for ρ < 1. However since the nature of defined energy packets are different from the data packets, energy packets are not treated separately and distinctively as data packets are treated. Fortunately, application of condition ρ < 1 does not impose huge restriction on the range of the values Dcycle can take and it should only be above a threshold level. Fig. 6, illustrates utilization factor ρ in terms of duty cycle and shows the mentioned threshold value (less than 0.1) above which utilization factor falls inside the legitimate bound. Nenergy =
Fig. 7: Number of energy packets in battery vs. duty cycle under different burst lengths.
tions, our targeted optimization problem leads to a solution which associates number of energy packets in battery and throughput. This relationship is shown in (14) and plotted in Fig. 8 for different values of synchronization ratio M (Fig. 8a) and burst length l (Fig. 8b). ( ) Nenergy = g f −1 (T hroughput)
Fig. 6: Server utilization ρ vs. duty cycle in a G/G/1 queue.
Plotting Nenergy in terms of Dcycle will result in the trend shown in Fig. 7. This rising/ falling trend is intuitively expected; with increasing duty cycle from the lower values, success probability increases up to a peak. However, after this peak, the collision probability increases drastically and the number of retransmission attempts increases so that larger and larger numbers of energy packets are required for transmitting specific number of data packets. On the other hand, with boosting the burst length up, collision probability diminishes and less number of energy packets is required for transmission of specific number of data packets. In addition the peak moves forward as the values of burst length (l) increases. C. Performance Evaluation By merging both functions Nenergy = g (Dcycle ) and Nenergy = g (Dcycle ) and eliminating Dcycle from both func-
(14)
As shown in Fig. 8, with the increment of throughput, the number of energy packets in the battery decreases. This means that we cannot have a single optimal point (duty cycle) in which both the throughput and number of energy packets are optimum. In fact, there is a tradeoff between throughput and remaining energy in the battery. Therefore, it depends on the network application to choose an upper/ lower bound for the duty cycle to meet the desired requirements. For example, depending on the application requirements, one can set a minimum threshold for the throughput and energy level that should remain in battery based on these trends, and come up with a minimum and maximum value for duty cycle. Of course, the minimum value of duty cycle meets the throughput requirement and the maximum duty cycle satisfies the minimum number of remaining energy packets in the battery. By observing these conditions, one can make sure that the QoS requirements of the network are satisfied on the one hand, and on the other hand, network lifetime will be improved dramatically. Further, as shown in Fig. 3, 5, 7,and 8, we could find out the ceaseless tradeoff between two important network’s parameters i.e. throughput and delay., In other words, there is not an optimal point in which both of these metrics are maximized. One should be sacrificed in the expense of having the other one on a desirable level. VI. C ONCLUSION In this paper, we developed a new model to analyze the energy harvesting based MAC (i.e., SMAC) throughput in
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[2] [3] [4] [5]
[6]
(a) Burst length l is fixed at 4
[7] [8] [9]
[10] [11] [12]
[13] [14] (b) Synchronization ratio M is fixed at 0.8
Fig. 8: Number of energy packets in battery vs. throughput.
[15] [16]
WSNs. Based on our analytical results, it was shown that the network throughput first boosts with the increase of the duty cycle and then degrades due to the increased collision probability. In addition, we modeled the received solar energy at the photovoltaic cells. Based on this model, we developed a queuing model and found a relationship between the average number of energy packets in battery and the duty cycle. It was observed that the average number of energy packets in the battery is a decreasing function with respect to the duty cycle. This discloses a tradeoff between the network throughput and the energy consumption. Finally, we concluded that, based on the application-level requirements of the network, we can set a minimum and maximum value for the duty cycle to meet the QoS and sensor lifetime requirements at the same time. Our research works will provide a fundamental guideline for the researchers to design an efficient MAC protocol in energyharvesting based WSNs.
[17]
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[24]
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Power Management in SMAC-based Energy-Harvesting Wireless Sensor Networks Using Queuing Analysis Navid Tadayon, Student Member, IEEE, Sasan Khoshroo, Elaheh Askari, Honggang Wang, Member, IEEE, and Howard Michel, Member, IEEE
Abstract—one of the most important constraints in traditional wireless sensor networks is the limited amount of energy available at each sensor node. The energy consumption is mainly determined by the choice of media access mechanism. SMAC is a typical access mechanism that has drawn much attention in recent years. In WSNs, sensors are usually equipped with capacity-limited battery sources that can sustain longer or shorter period, depending on the energy usage pattern and the activeness level of sensor nodes. To extend the lifetime of the sensor networks, ambient energy resources have been recently exploited in WSNs. Even though solar radiation is known as the superior candidate, its density varies over time depending on many factors such as solar intensity and cloud states, which makes it difficult to predict and utilize the energy efficiently. As a result, how to design an efficient MAC in a solar energy harvesting based WSN becomes a challenging problem. In this paper, we firstly incorporate a solar energy-harvesting model into SMAC and conduct its performance analysis from a theoretical aspect. Our research works provides a fundamental guideline to design efficient MAC for energy harvesting based WSNs. Our major contribution includes three folders: firstly, we model solar energy harvesting in a photovoltaic cell and then derive the throughput of SMAC in the energy-harvesting based WSNs. Secondly, we develop a new model based on queuing theory to calculate the average number of energy packets in battery in terms of both duty cycle and throughput. Finally, we form an optimization problem to find a suitable range for duty cycle to satisfy both quality of service (QoS) and network lifetime requirements. Index Terms—Duty Cycle, Energy-Harvesting, Queuing model, SMAC, Solar Radiation Modeling, Throughput, WSN.
I. I NTRODUCTION IRELESS sensor networks consist of hundreds of spatially-distributed sensors, called nodes, to cooperatively monitor a specific quantity which can be the level of pollutants, temperature, sound, vibration, pressure, and so on. These networks have been utilized in a wide variety of applications, like civil, industry and military applications. They have also been widely used for healthcare monitoring, object tracking and assembly line sensing. WSN (Wireless Sensor Networks) is composed of different layers, first of which is MAC (Medium Access Control) layer, which grants access of the wireless channel to different nodes. Depending on the type of MAC protocol being utilized, wireless sensor networks are divided into two general categories, scheduled
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N. Tadayon, S. Khoshroo, E. Askari, H. Wang, and H. Michel are with the Department of Electrical and Computer Engineering, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA (e-mails: {ntadayon, easkari, skhoshroo, hwang1, hmichel}@umassd.edu)
networks and contention-based networks [1]. In scheduled networks, the wireless channel is divided into sub-channels in terms of either time (Time Division Multiple Access-TDMA, frequency (Frequency Division Multiple Access-FDMA), orthogonal codes (Code Division Multiple Access-CDMA), or a combination of them and each of these sub-channels are assigned to each node. However, each of these protocols has its own challenges such as time synchronization in the case of TDMA, frequency generation/filtering and bandwidth requirements in the case of FDMA, and power control in the case of CDMA. These requirements cannot be simply satisfied using tiny, incapable sensors that are usually located in a place with no replacement/maintenance possibility. As a result, contention-based access methods are more suitable due to their simple, autonomous and scalable nature. Here, nodes compete with each other to win the access to the shared medium. They are also flexible toward network topology changes, which is typical in wireless networks. Another important constraint in WSNs is the amount of energy available to each node. The power consumption should be uniform over the network to extend the network lifetime. Otherwise, there will be some portions of the network consisting of dead sensors that will degrade the overall QoS (Quality of Service) performance. As a result, in WSNs, power metric is more important than other QoS metrics or fairness of access. To circumvent this problem, the power consumption should be considered as a distributed parameter in the network instead of a point parameter in only one node, which demands careful considerations in MAC layer. Numerous works have been carried out to design energy-efficient MAC protocols. In fact, it has been shown in [2] that, the energy consumption using IEEE 802.11 MAC is very high when nodes are in the idle mode. Some of such recently proposed MAC protocols include PAMAS [3], SMAC [4], TMAC [5], and PMAC [6], among which, SMAC is of particular interest in this paper and is explained in details in next section. PAMAS is an improvement over MACA (Multiple Access Collision Avoidance) protocol by adding a separate signaling channel for exchanging the RTS/CTS packets, which enables the nodes to switch themselves off when they are not receiving or transmitting any packets. Therefore, this protocol is more efficient than the original MACA protocol. As the authors claim in [3], it could increase the power efficiency of most Ad-Hoc networks by 10% to 70%. In [7], a distributed Time-varying Opportunistic MAC Protocol (TOP) is proposed to maximize the network
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lifetime through transmission scheduling, based on Channel State Information (CSI) and Residual Energy Information (REI). In fact, a higher priority is assigned to the sensor node with better channel condition and higher residual energy. The authors show that this approach increases the network lifetime compared to other distributed MAC protocols like DPLM [8], and Pure Opportunistic [9]. In [10], the authors propose an Optimized MAC protocol to deal with the energy inefficiency and nodes latency. They prove, analytically and via extensive simulations that this scheme achieves high energy efficiency under a wide range of traffic loads and is able to adjust itself to improve the delay performance when the network traffic load is high. In SMAC [4] or sensor MAC protocol, nodes go to sleep and active states periodically in order to reduce power consumption. During the sleep mode, the node turns off its radio, and sets a timer to later awakening. The difference between PAMAS and SMAC is that SMAC uses in-channel signaling rather than using a separate channel for signaling, as in PAMAS. Being inspired by SMAC, TMAC [5] also uses the same periodic active/sleep scheme. However, the duty cycle is not fixed in TMAC by dynamically ending the active part of the cycle, which reduces the amount of energy wasted on idle listening. It has been shown in [5] that, in terms of energy efficiency, T-MAC outperforms S-MAC by a factor of 5. Recently, environmental energy resources like solar or wind power have been exploited in WSNs. An energy-harvesting node is defined as any system, which can absorb part or all of its energy from the environment [11]. An important difference between this kind of energy and that stored in the capacitylimited batteries is that ambient (and particularly solar) energy is potentially infinite. However, the energy generation rate at which this type of energy can be generated may be limited; for example, solar power is not available during night and cloudy conditions and the absorption rate continually changes over time. In this paper, our first contribution is to model the solar energy received at a photovoltaic cell. We are inspired by the research works in [12] and [13]. However, we consider different parameters (like cloud length and sunlight inclination) for modeling the absorbed energy rather than those considered in [12]. This kind of modeling is necessary for our second contribution, which is the modeling of the number of energy units (we call them energy packets from now on) in battery. Although power efficiency is the main concern in sensor networks, other QoS requirements also need to be satisfied. There are different methods to evaluate network’s performance. One of them is queuing analysis. In [14] an infinite queuing model has been proposed for contention-free sensor networks analysis and a more realistic finite queuing model has been used to analyze the tradeoff between energy consumption and QoS requirements in contention-based sensor networks in [1]. Our second and main contribution is the modeling of the number of energy packets in battery of a sensor that uses SMAC as a case study of its MAC protocol. Since the battery has an input (through which energy is absorbed) and an output (through which energy is consumed), queuing theory is
applicable to model non-data identities. These identities in our analysis are energy packets, which will be served (consumed) in FIFO order. We assume that there is no energy leakage from battery, and energy consumption is quantized so that one energy packet is consumed to transmit one data packet. The only point that should be noted is that energy packets are not distinguishable identities. This should not be a problem, since we are only interested in the number of energy packets in the queue (battery) which is irrelevant to serving policy, and each energy packet’s position in the queue. The used model in our analysis is G/G/1 queue with batch transmission (due to message passing technique used in SMAC). The input process to this queue (battery), a(n), is a point process that represents the number of energy packets (n) absorbed by the photovoltaic cell and delivered to the battery. The output process, b(t), represents the discrete time interval between two successful consecutive transmissions from a sensor node. We employ the approaches in [15] and [16] in our analysis to achieve two goals; firstly, to prolong the network lifetime by not letting the sensor nodes be out of power or exceed a critical battery level; secondly, to guarantee the network QoS by not letting the throughput fall below a critical threshold. Since there is tradeoff between the energy consumption and throughput, the two goals can be achieved through solving a constrained optimization problem. Consequently, the third contribution of this paper is modeling the SMAC throughput. To the best of our knowledge, very few works [17] have been done on finding the throughput of SMAC. However, none of them has explicitly extracted a closed form expression for the network throughput for energy harvesting based WSNs. In our analysis, the parameter of interest that shifts the benefits toward each of the abovementioned targets is the duty cycle, which is the ratio of the active time of a sensor to the whole cycle time. The rest of the paper is organized as follows. In section II, we give a brief introduction to SMAC [4]. In section III, we present our solar energy model that describes the process of feeding the number of solar energy units into the battery. The closed form expressions of SMAC throughput and service time in terms of network parameters are derived in section IV. Then, in section V, we establish our queuing model based on the model described in the previous sections to find the average number of energy packets in queue (battery) and form the optimization problem. Finally, we summarize our results in section VI. II. S MAC P ROTOCOL OVERVIEW In the SMAC, nodes go to periodic sleep and active states in order to reduce the energy consumption. As specified in [4], the active period of each cycle, which is determined by the MAC-layer’s contention window size, is assumed to be fixed. Thus, duty cycle depends only on the variable sleep period. Additionally, each active period is subdivided into two phases; the first one is reserved for the synchronization purpose among the nodes, and the second one is for RTS/CTS handshake as specified in [18]. In synchronization phase, neighboring sensors synchronize their transmissions by exchanging their sleep-active schedules [17] periodically. If a
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node wants to transmit a data packet, it needs to compete with other nodes for the medium access. This channel access contention takes place in the second phase. Like IEEE 802.11 MAC protocol, SMAC uses RTS/CTS/DATA/ACK packets in order to guarantee successful transmissions. It starts transmission as soon as the active period is over. All other nodes that have been involved in the contention must go to sleep if they fail to win the contention or if they encounter an RTS collision. On the other hand, the winner of the contention will not go back to sleep until the transmitted data packets are acknowledged. Another technique used in SMAC is message passing. In this technique, long messages are divided into many small fragments and these fragments are transmitted in bursts [4]. Expressing clearly, the medium is reserved for transmitting all the fragments by performing just one RTS/CTS handshake for the whole message. But still, to prevent the hidden terminal problem, an ACK is sent by the receiver to the sender for each and every fragment. In fact, after sending a fragment, the sender waits for its ACK from the receiver and if it does not receive it within a specific time, it will expand the reserved access time for one more fragment and transmits that fragment again [4]. If a new sensor node, which is only the neighbor of the receiver, joins the network in the middle of the transmission, it will not sense the transmitted fragments. It assumes that the medium is free and will start transmitting its own data. This transmission could interrupt the original transmission. Thus, the transmission of the ACK packets by the receiver will prevent this problem. This technique reduces the contention latency in the network and is suitable for the networks in which fairness is pursued at an application level rather than at the MAC layer. This protocol has been implemented on test-bed nodes in [4] and the experiment results well demonstrate the effectiveness of SMAC compared to IEEE 802.11 in terms of energy consumption in source and immediate nodes. III. S OLAR R ADIATION M ODEL AND A BSORBED E NERGY As stated before, ambient energy resources have recently been exploited in WSNs and an energy-harvesting node is defined as the one that can absorb part or all of its energy from the environment. Our contribution in this section is to model the amount of solar energy that will be absorbed by photovoltaic cells and delivered to rechargeable battery. We consider solar energy as the most effective and most available source of the energy that can be harvested by existing solar cells. Depending on the solar intensity, cloud states, the hour of the day, season of the year, spatial angles and so on, the amount of absorbed energy is different. Particularly, when the emission intensity is high, solar cell produces more amount of energy that is fed to the rechargeable battery. On the other hand, when the emission level is low (when long and bulky clouds obstruct solar emission during the nights or so) and the solar cell is incapable of producing enough amount of energy, the sensor will use up the reserved energy in battery. Fig. 1 illustrates this radiation process. As the rays of the sun penetrate into the atmosphere, they encounter different attenuating obstructs. The flux intensity that arrives just before
Fig. 1: Example model for solar radiation.
the clouds could be approximated by the equation presented in [19]. This equation expresses flux intensity in terms of hour of the day (t), latitude of the collector site (λ) and solar declination (η) as: G (ζ) = G0 e−0.357(sec(ζ))
0.678
,
sec (ζ) = 1/ (sin (λ) sin (η) + cos (λ) cos (η) cos (t)) ,
(1)
where t = (360/24)T (T indicates the number of hours in a solar noon). G0 indicates the solar flux intensity outside the atmosphere and its value is 1.35 KW/m2 [19]. This flux intensity when passed through the clouds, experiences severe attenuation. The main causes of this are cloud length and thickness. The lengthiness and thickness of each cloud varies over time and is different from others, so it should be considered as a random variable rather than a deterministic identity. In our analysis, the cloud length follows exponential distribution fci (xi ). In fact, the behavior of many of the events in nature can be modeled with this distribution. Now, assuming that cloud profile contains limited number of states (as illustrated in Fig. 1), the PDF (Probability Density Function) of the length of cloudy profile could be written as a hyper-exponential distribution since at each instance of time, only one cloud state could block the direct sunlight radiation with probability Pi . Then: fX (x) =
k ∑
Pi fci (x) ,
i=1
where, fci (xi ) =
1 − xc i e i ci
(xi > 1),
(2)
and, n ∑
Pi = 1.
i=1
In (2), xi and ci represent instantaneous cloudy profile’s length and mean cloud’s length in state i (in km), respectively. From [13], the amount of flux intensity on the plates of
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( ) photovoltaic cell watt/m2 is equal to: g = (1 − rX) G (ζ) ,
(3)
where G (ζ) is given in (1), and r is a constant with nominal value in between 0.02 and 0.03 depending on the cloud profile in the collection site. Now the target is finding PGF (Probability Generation Function) of the number of energy packets (a (n) &A (z)) that is absorbed by solar cell and delivered to battery. This function can be easily found from PGF of g since we have a (n) = (gApanel )/Ebit Rdata where Apannel is the effective area of solar panel (less than 1cm3 ), Rdata is the transmission data( rate (19.2 Kbit/s), ) and Ebit is the quantized unit of energy 1.04·10−6 J/bit consumed for transmitting a single data packet as long as we assume in this paper that the data packet has a fixed length, this derivation for a (n) leads to a correct answer. By carrying out probabilistic operations, the PGF of a (n) (A (z)) is achieved To prevent the loquaciousness, details are not mentioned here, and only the results have been provided. For simplicity, we consider seven states for the cloud profile (k=7):
A (z) =
∞ ∑
z i fa(n) (i)
) Pi k=7 ∑ ci × = G (ζ) 1 i=1 rln (z) + (Ebit Rdata /Apanel ) ci i=1
(
(4)
From (5) we see that Dcycle is always upper-bounded with one. Using the abstraction M = Ta2 /Ta , and the fractional definition of throughput as: Fraction of a cycle in which pure data is transmitted = Total cycle Time T we get: T hroughput = (Ps Dcycle l · TP M ) Dcycle . (M ((1 − Pb ) δ + (Pb − Ps ) Tc + Ps Ts )) + Dcycle . ((1 − M ) Ta1 ) + (1 − Dcycle ) (T − Ta − Ts )
(6)
In (6), the numerator represents the fraction of a cycle time, which is spent for successful pure data transmissions. Primarily, transmissions can only happen in wake-up intervals. This fact is reflected by embedding Dcycle in numerator. After that, since the access mechanism in SMAC is contention-based, transmitted packet may or may not be received successfully. This reveals that only fraction of each transmission can be succeeded, which is reflected by factor Ps in numerator. Finally, l reflects the effect of burst length due to message passing mechanism which is specific to SMAC. The denominator in (6) is composed of all possible events that can happen during a cycle time of a sensor node. After replacing (5) into (6), and substituting Ta1 = (1 − M ) Ta , we get: T hroughput = (Ps Dcycle l · TP M ) Dcycle . (M ((1 − Pb ) δ)+ (Pb − Ps ) Tc + Ps Ts )) + (
G (ζ) (E R /Apanel ) . bit data z
2
Table I shows the values of Pi and ci for the sample cloud profile we used in our numerical simulation. A (z) will be exploited in our next section’s analysis as the input process to the queue model. The inverse z-transform of this PGF is the PDF of the number of energy packets that enter into the entrance of the battery. TABLE I: Different Values Of Cloud Profile
i P C (km)
1 0.35 40
2 0.1 33
3 0.045 17
4 0.01 7
5 0.045 17
6 0.1 33
7 0.35 40
IV. SMAC T HROUGHPUT M ODEL From now on, Dcycle represents the ratio of the active period to the whole cycle time. This parameter varies as system jumps to the next cycle due to the nondeterministic nature of the sleep lengths. Fig. 2 illustrates different existing packets and intervals in S-MAC protocol. Depending on the value of the probability of success (Ps ), (5) yields this quantity. Dcycle = figure
(Ta + Ts ) Ta (1 − Ps ) + Ps . T T
(5)
Dcycle . (1 − M ) Ta + (Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) (1 − Dcycle ) , Dcycle
(7)
where M , l, and Tp represent the ratio of the RTS/CTS period to the total active period, the number of data packets in each burst (due to message passing feature of SMAC), and the MAC layer information length in each data packet, respectively. Furthermore, Ps , Pb , Ts , and Tc symbolize successful transmission probability, channel busyness probability, successful transmission length, and collision length, respectively. From the previous research done in [20], [21], [22], [23] on performance evaluation of CSMA/CA, the appropriate equations for Ps , Pb , Ts , and Tc are shown in (8) where n and τ represent the number of contending stations and transmission probability, respectively. Also THeader , ACK, CTS, RTS, SIFS, DIFS are the same parameters as in IEEE 802.11 and [20]. n−1
Ps = nτ (1 − τ ) , n Pb = 1 − (1 − τ ) , Ts = RT S + +CT S + l (Tp + THeader + ACK) + DIF S+ (2l + 1) SIF S, Tc = RT S + SIF S + ACK + DIF S. (8)
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Fig. 2: All possible packets and intervals in between any two successful consecutive transmissions from a sensor node running on S-MAC.
Table II shows the nominal values for parameters that are required for calculation of throughput. For these set of parameters, transmission probability τ is calculated and the value is 0.027. TABLE II: SMAC Parameters In Numerical Simulation
Parameter Rate (bit/s) Tp (Byte) THeader (Byte) CTS (bit) RTS (bit) ACK (bit)
Value 19200 100 8 304 352 304
Parameter Sifs (ms) δ (ms) Difs (ms) Ta (ms) n
Value 10 20 50 300 20
It should be mentioned that this approach on finding SMAC throughput is only valid for the case that all sensors contend for channel access in every cycle. In other words, the transmission queues of sensors have always packets waiting to be transmitted. With data rate of 19200 bit/s it is always guaranteed that sensors are in the above mentioned saturated condition. Fig. 3 depicts the throughput in terms of duty cycle for different values of M (shown in Fig. 3a) and l(shown in Fig. 3b). The trend in both figures shows that as duty cycle increases, throughput first improves (due to increased transmission probability) and then degrades (due to increased collision probability) which is very intuitive. On the other hand, for larger data bursts (larger l), less contention is required for transmitting specific number of data packets and higher throughput is achieved. V. P ERFORMANCE A NALYSIS BASED O N T HE Q UEUING M ODEL A. PGF of Successive Transmission Intervals In this section, we establish our queuing model for the number of energy packets in battery using queuing theory. Each classic queue is characterized by several features such as input process, output process, number of servers, scheduling policy, and so on. Depending on the characteristics of input and output processes, queues are classified into different categories the most general of which is G/G/1 category. In G/G/1 queues, the input and output processes can be any arbitrary random
(a) Burst length l is fixed at 4
(b) Synchronization ratio M is fixed at 0.8
Fig. 3: SMAC throughput vs. Duty cycle.
process (rather than uniform P oisson process), which makes it complicated to analyze when the Markovian property is not applicable anymore. In the queuing theory, the number of packets waiting in the queue (system occupancy) is of great importance. However, finding the closed form expression for PDF of this quantity
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in a G/G/1 queue is a cumbersome and sometimes impossible task. There are some research [15], [16], and [24] that have addressed this problem by finding its PGF or mean value. However, this is not sufficient for our case because G/G/1 formulations are unable to address burst/ bulk processing issue, which makes the analysis even more complicated. We found that the concept of batch processing in queue analysis exactly matches with burst transmission in networks. The analysis of system occupancy for a general input/ output queue with batch processing has been addressed in [16] and [25]. In our case, we resort to the same type of queue modeling since the input process a(n) (number of energy packets delivered to the battery) and output process b(t) (consumption interval) are both general distribution discrete processes and message passing feature of SMAC is nothing more than a batch processing in queue’s server. From [25], we know the PGF of system occupancy for a G/G/1 queue is equal to: U (z) = (1 − A′ (1) B ′ (1))
((z − 1) B (A (z))) z − B (A (z))
Given that Tsleep is a random variable in SMAC specification, Tf would be a random variable too. Assuming that Tsleep is uniformly distributed in the interval [0, T − Ta − Ts ], the probability mass function (pmf) of Tf would be equal to:
PTf
(Ta + Ts ) (T − Ta − Ts ) +1 Ps Ps (11) Ps z 1−z (Tf ) = × (T − Ta − Ts ) 1−z
Applying T = (Ta +Ts Ps ) /Dcycle into (11) and then taking its z-transform, the PGF of Tf would be equal to:
B (z) =
z × (Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) Ps
(9)
where, A(z) and B(z) are the PGFs of a(n) and b(t) as defined in the previous paragraph and ′ represents the derivation operator. In section III, we were able to find A(z) and now the only remaining task is finding B(z). As explained before, b(t) is the interval between every two consecutive energy packets’ consumption. Since the majority of energy packets in SMAC are only consumed on transmission of data packets and very little are consumed on idle listening, overhearing, and synchronizations, it is almost accurate to label b(t) as the interval between transmissions of two consecutive data packets as well. This last description of b(t) is the exact definition for service delay which has been symbolized by Tf in Fig. 2. Fig. 4 simply illustrates this convergence between two queues (data queue and energy queue) in a sensor node.
(Dcycle T + Ts (1 − Ps )) Ps
1−z
(Ta (1 − Dcycle ) + Ts (Ps − Dcycle )) +1 Ps 1−z
(12)
The mean value of this service delay is achievable by taking the inverse z-transform of this PGF and evaluating it at z = 1. Fig. 5 shows this mean service time in terms of duty cycle for diverse burst lengths. Apparently, as the duty cycle increases, sensors find more transmission opportunities in less congested transmission intervals which leads to less service delay. On the other hand, bigger bursts take longer time to be transmitted but have higher throughput as shown in Fig. 3b.
Fig. 4: A simple schematic model for an energy harvesting sensor node with two queues: data, and energy.
Tf is made up of two parameters T (cycle length) and m (the average number of retransmissions of a specific packet until it is successfully transmitted) where Tf = T · m. Referring to [20] and [21] m= ¯ 1/Ps due to the fact that the number of transmission trials (m) is geometrically distributed with parameter Ps . As a result Tf can be approximated with: (Ta + Ts + Tsleep ) Ps (Dcycle T + Ts (1 − Ps ) + Tsleep ) = Ps
Tf = T · m ¯ =
(10)
Fig. 5: Duration Interval between two consecutive successful transmissions from a SMAC sensor node vs. duty cycle under different burst lengths.
B. Number of Energy Packets in Battery Incorporating the appropriate equations (4) and (12) into (9), gives us the PGF of number of energy packets in battery
7
when the bulk transmission mechanism (message passing) is not used. Then by replacing this result inside equation (3.3) in [16] the exact average number of energy packets in the queue will be achieved as per following: ( ) ′ (l − 1) (13) + l L + U (1) + ρ 2 where ρ (server utilization factor) is equal to A′ (1) B ′ (1) and L is the initial number of energy packets (meaning that sensor nodes are launched with a nonzero battery level) in battery (L = 1500 as default), respectively. The strict stability condition in queuing theory is to have a utilization factor less than one so that packets do not experience unbearable loss or delay. In fact (9) is only valid for ρ < 1. However since the nature of defined energy packets are different from the data packets, energy packets are not treated separately and distinctively as data packets are treated. Fortunately, application of condition ρ < 1 does not impose huge restriction on the range of the values Dcycle can take and it should only be above a threshold level. Fig. 6, illustrates utilization factor ρ in terms of duty cycle and shows the mentioned threshold value (less than 0.1) above which utilization factor falls inside the legitimate bound. Nenergy =
Fig. 7: Number of energy packets in battery vs. duty cycle under different burst lengths.
tions, our targeted optimization problem leads to a solution which associates number of energy packets in battery and throughput. This relationship is shown in (14) and plotted in Fig. 8 for different values of synchronization ratio M (Fig. 8a) and burst length l (Fig. 8b). ( ) Nenergy = g f −1 (T hroughput)
Fig. 6: Server utilization ρ vs. duty cycle in a G/G/1 queue.
Plotting Nenergy in terms of Dcycle will result in the trend shown in Fig. 7. This rising/ falling trend is intuitively expected; with increasing duty cycle from the lower values, success probability increases up to a peak. However, after this peak, the collision probability increases drastically and the number of retransmission attempts increases so that larger and larger numbers of energy packets are required for transmitting specific number of data packets. On the other hand, with boosting the burst length up, collision probability diminishes and less number of energy packets is required for transmission of specific number of data packets. In addition the peak moves forward as the values of burst length (l) increases. C. Performance Evaluation By merging both functions Nenergy = g (Dcycle ) and Nenergy = g (Dcycle ) and eliminating Dcycle from both func-
(14)
As shown in Fig. 8, with the increment of throughput, the number of energy packets in the battery decreases. This means that we cannot have a single optimal point (duty cycle) in which both the throughput and number of energy packets are optimum. In fact, there is a tradeoff between throughput and remaining energy in the battery. Therefore, it depends on the network application to choose an upper/ lower bound for the duty cycle to meet the desired requirements. For example, depending on the application requirements, one can set a minimum threshold for the throughput and energy level that should remain in battery based on these trends, and come up with a minimum and maximum value for duty cycle. Of course, the minimum value of duty cycle meets the throughput requirement and the maximum duty cycle satisfies the minimum number of remaining energy packets in the battery. By observing these conditions, one can make sure that the QoS requirements of the network are satisfied on the one hand, and on the other hand, network lifetime will be improved dramatically. Further, as shown in Fig. 3, 5, 7,and 8, we could find out the ceaseless tradeoff between two important network’s parameters i.e. throughput and delay., In other words, there is not an optimal point in which both of these metrics are maximized. One should be sacrificed in the expense of having the other one on a desirable level. VI. C ONCLUSION In this paper, we developed a new model to analyze the energy harvesting based MAC (i.e., SMAC) throughput in
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[2] [3] [4] [5]
[6]
(a) Burst length l is fixed at 4
[7] [8] [9]
[10] [11] [12]
[13] [14] (b) Synchronization ratio M is fixed at 0.8
Fig. 8: Number of energy packets in battery vs. throughput.
[15] [16]
WSNs. Based on our analytical results, it was shown that the network throughput first boosts with the increase of the duty cycle and then degrades due to the increased collision probability. In addition, we modeled the received solar energy at the photovoltaic cells. Based on this model, we developed a queuing model and found a relationship between the average number of energy packets in battery and the duty cycle. It was observed that the average number of energy packets in the battery is a decreasing function with respect to the duty cycle. This discloses a tradeoff between the network throughput and the energy consumption. Finally, we concluded that, based on the application-level requirements of the network, we can set a minimum and maximum value for the duty cycle to meet the QoS and sensor lifetime requirements at the same time. Our research works will provide a fundamental guideline for the researchers to design an efficient MAC protocol in energyharvesting based WSNs.
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