**3.2. Full aggregation**

78 Wireless Sensor Networks – Technology and Protocols

channel waiting time and is obtained the formulation as follows:

data in the queue waiting for data transmission is shown as follows:

The λ'i is arrival data rate at node ni and ( ) *<sup>s</sup>*

aggregation, can be determined as follows:

*i*

1

Node ni communicates with only one neighbor node at a time. If a neighbor node is transmitting data, node ni has to wait until its neighbor node finishes transmission, due to the over hearing caused by the omni-directional antenna. This waiting time is defined as

1 1 () 2 () () *<sup>c</sup>*

Total delay *Dnon(N)* is derived as follows where the number of hops from robot *<sup>i</sup> n* to the

<sup>1</sup>

Node ni transmits data and relays arrival data from the upper nodes. Since the consumed energy is proportionate to the number of data transmissions, we can find the mean number of data *LQnon(i)* in the service queue at node ni according to Little's formula. The number of

> () \* () *Q s*

queue to data having been received by the next neighbor node at node ni, in case of non-

 

Here Pt and Pr denote the energy consumption for transmitting and receiving data.

all the generated data will get to the sink, thus the data accuracy approaches to 100%.

*N non t r i*

<sup>1</sup> () () () *<sup>s</sup> s c*

According to equations (8) and (9), we obtain the whole energy consumption in N hops

 <sup>Q</sup> non 1

In non-aggregation, data are not aggregated and the packet drop occurs with the transmitting in real system. However, for simplicity, we assume there is no packet drop and retransmission,

( ) L ( )\*( 2 )

( ( ) ( ) ( )) *<sup>N</sup> s c non non non non*

 

*non i non non i ii* (6)

*DN i i i* (7)

*non i non Li Ti* (8)

*non non non non Ti i i i* (9)

*E N iP P* (10)

*non T i* is time duration from data joining the

 

Channel waiting time

*3.1.1. Total delay* 

*3.1.2. Energy consumption* 

network as follows:

*3.1.3. Data accuracy* 

sink is *N*.

We define the full aggregation that the arrival data are sent to an adjacent lower node only after having been aggregated with local generated data at nodes. It means data transmission occurs only after a new local data generated a node. Hence, the waiting time for data aggregating at a node is decided by the data generation rate of the node. When there is local generated data, the node aggregates all the arrival data with generated data then waits for transmission at server. Data after aggregation undergo the same procedure as non-aggregation to detect the server and the channel for further transmission.

The analytical model of full aggregation is shown in figue3. Before explaining the model, we introduce queue A, queue B and "G." Queue A denotes the arrival data queue at a node that is waiting for local generated data for data aggregation. Data in Queue B are waiting for server; when the server is idle, data are transmitted to a neighbor node. The "G" is assumed as a virtual gate between queue A and queue B. Immediately after local generated data aggregate with the arrival data in queue A, the gate opens and lets the aggregated data join queue B.

**Figure 3.** Analytical model of full aggregation

In full aggregation, the data join queue A with arrival rate of *<sup>i</sup>* <sup>1</sup> and wait for new generated data. When an event occurs at local node, the node aggregates the generated data and all arrival data in queue A according to the aggregation factor Af. The size of aggregated data becomes *Sav* and the aggregated data join queue B with the rate of *<sup>i</sup>* to await further transmission. In full aggregation, the difference from non-aggregation is that we have to determine how long the arrival data wait for aggregation in queue A.

### *3.2.1. Event waiting time*

To determine the event waiting time, we apply the state transition rate diagram. We describe the state transition rate diagram in fig. 4. The basic idea of the analysis is that data waiting in queue A for exponential distribution have an average of 1/2λi. In the diagram, the state variable is the number of data waiting for an event.

**Figure 4.** State transition rate diagrams of full aggregation

According to calculation of state probability distribution and Little's formula, we determine the event waiting time as shown below; more details please read (Li, et. al, 2010).

$$\left(\pi\_{full}\right)^e \left(i\right) = \frac{\mathcal{X}\_{i+1}^{\prime}}{2\mathcal{X}\_i^2} \tag{11}$$

### *3.2.2. Total delay*

From the definition of full aggregation we know that the arrival data join queue B only if there is new generated data at local node, hence the data arrival rate to queue B is equal to data generation rate at the node. The data generation rate abides by Poisson distribution; therefore the arrival data rate to queue B is Poisson distribution. Since the data arrival rate involves only one server and the data transmission time for server is fixed, according to queuing theory we model the queue by means of M/D/1 queue. Similar to non-aggregation, we determine the total delay *Dful (N)* of the network in full aggregation is consists of event waiting time in queue A, server waiting time in queue B, channel waiting time and data transmission time at server.

$$D\_{full}\left(N\right) = \sum\_{i=1}^{N} \left(\tau\_{full}^{\epsilon}\left(i\right) + \tau\_{full}^{s}\left(i\right) + \tau\_{full}^{c}\left(i\right) + \tau\_{full}^{1}\left(i\right)\right) \tag{12}$$

### *3.2.3. Energy consumption*

The energy consumption is proportional to the number of data transmissions. ( ) *<sup>Q</sup> ful L i* is the number of data at a robot ni. The time duration from data joining Queue B to data having been received by the next neighbor node is *<sup>S</sup> ful T i* and it can be obtained as follows:

$$T\_{full}^{s}\left(\mathbf{i}\right) = \boldsymbol{\tau}\_{full}^{s}\left(\mathbf{i}\right) + \boldsymbol{\tau}\_{full}^{c}\left(\mathbf{i}\right) + \boldsymbol{\tau}\_{full}^{1}\left(\mathbf{i}\right) \tag{13}$$

According to Little's formula and equation (13), the amount of data in queue B is as follows:

$$L\_{full}^{Q} \left( \mathbf{i} \right) = \mathcal{A}\_{i} \; \text{\*} \; T\_{full}^{s} \left( \mathbf{i} \right) \tag{14}$$

Therefore, the whole energy consumption is determined as follows:

$$E\_{full}\left(N\right) = \sum\_{i=1}^{N} L\_{full}^{Q}\left(i\right) \* \left(p\_t + 2p\_r\right) \tag{15}$$

### *3.2.4. Data accuracy*

80 Wireless Sensor Networks – Technology and Protocols

**Figure 4.** State transition rate diagrams of full aggregation

been received by the next neighbor node is *<sup>S</sup>*

*i*

1

*3.2.2. Total delay* 

transmission time at server.

*3.2.3. Energy consumption* 

According to calculation of state probability distribution and Little's formula, we determine

( ( ) ( ) ( ) ( ))

*DN i i i i* (12)

*ful T i* and it can be obtained as follows:

*ful i ful L i Ti* (14)

*ful ful ful ful Ti i i i* (13)

<sup>1</sup>

*<sup>N</sup> esc ful ful ful ful ful*

The energy consumption is proportional to the number of data transmissions. ( ) *<sup>Q</sup>*

number of data at a robot ni. The time duration from data joining Queue B to data having

 <sup>1</sup> ( ) ( ) ( )) *s sc*

> \* () *Q s*

According to Little's formula and equation (13), the amount of data in queue B is as follows:

*i*

 <sup>1</sup> <sup>2</sup> 2 *e i*

*i* (11)

*ful L i* is the

From the definition of full aggregation we know that the arrival data join queue B only if there is new generated data at local node, hence the data arrival rate to queue B is equal to data generation rate at the node. The data generation rate abides by Poisson distribution; therefore the arrival data rate to queue B is Poisson distribution. Since the data arrival rate involves only one server and the data transmission time for server is fixed, according to queuing theory we model the queue by means of M/D/1 queue. Similar to non-aggregation, we determine the total delay *Dful (N)* of the network in full aggregation is consists of event waiting time in queue A, server waiting time in queue B, channel waiting time and data

the event waiting time as shown below; more details please read (Li, et. al, 2010).

*ful*

In full aggregation, the aggregation factor Af=1. Thus, we can get the data accuracy in N hops transmission as follow:

$$A\_{\phantom{e}} = \frac{1}{N} \tag{16}$$

### **3.3. Partial data aggregation**

According to previous analyses of non-aggregation and full aggregation, we find that nonaggregation sends all the generated data to sink node which results in large energy consumption. In case of full aggregation, the arrival data must wait for local generated data to aggregate, which causes the prolonged transmission delay and low data accuracy for the data that come from nodes far away from sink.

To minimize these two shortcomings, we propose a partial data aggregation. The main idea of partial aggregation is that nodes process data aggregation and transmit data only if a) if there are new local generated data at a node or b) after waiting a holding time at a node; the inverse of the holding time we call random pushing rate λDi. The analytical model of partial aggregation is shown as follows.

**Figure 5.** Analytical model of partial aggregation

For the purpose of simplifying our analytical model, we assume the arrival data rate from adjacent upper node is approximated to Poisson distribution and the arrival data join event waiting queue A in fig. 5. Data generation rate λi is assumed to be Poisson distribution. Random pushing rate is λDi and assumed to be exponential distribution. If new generated data occur at a node or if holding time is over for arrival data, all the data are aggregated into one data, and the gate G opens and lets aggregated data join queue B. λ'i is data arrival rate to queue B in which data are waiting for service (data transmission).

### *3.3.1. Event waiting time*

Assume that a number of data are waiting for an event at robot *ni* in queue; we describe the state transition diagram as shown in Fig.6.

**Figure 6.** State transition rate diagram

Data are waiting in queue for the duration according to the exponential distribution of average 1/(2 ) *<sup>D</sup> i i* . Similar to full aggregation, the event waiting time of partial aggregation can be determined as follows:

$$
\pi\_{par}^e = \frac{\mathcal{\lambda}\_{i+1}^e}{(2\mathcal{\lambda}\_i + \mathcal{\lambda}\_i^D)^2} \tag{17}
$$

### *3.3.2. Arrival process to Queue B*

From the analytical process we find that arrival data rate λ' i is decided by the random pushing rate and data generation rate at a node. To determine the formulation of λ' i, we calculate the property distribution of λi and λDi. We define that λDi and λi are the independent distribution X and Y. Through proofing of the property Y is bigger than X, we determine the arrival process to Queue B as follows; the proof can be found in (Li, et. al, 2010).

$$
\lambda\_{i}"=\lambda\_{i}+\frac{\lambda\_{i+1}"}{\lambda\_{i+1}"+\lambda\_{i}"^{D}}\*\lambda\_{i}^{D} \tag{18}
$$

### *3.3.3. Total delay*

Since the data generation rate is Poisson distribution and the random pushing rate abides by exponential distribution, the data arrival rate to queue B approximates to Poisson distribution. Therefore, we can confirm that the queuing system approximates to M/D/1 model. With the same way of full aggregation, the server waiting time and channel waiting time can be determined easily. Therefore, the total delay of partial aggregation is as follows:

$$D\_{par}\left(N\right) = \sum\_{i=1}^{N} \left(\tau\_{par}\,^{c}\{i\} + \tau\_{par}\,^{c}\{i\} + \tau\_{par}^{1}\{i\} + \tau\_{par}^{s}\{i\}\right) \tag{19}$$

### *3.3.4. Total energy consumption*

82 Wireless Sensor Networks – Technology and Protocols

state transition diagram as shown in Fig.6.

**Figure 6.** State transition rate diagram

*3.3.2. Arrival process to Queue B* 

aggregation can be determined as follows:

average 1/(2 ) *<sup>D</sup>*

2010).

*3.3.3. Total delay* 

Assume that a number of data are waiting for an event at robot *ni* in queue; we describe the

Data are waiting in queue for the duration according to the exponential distribution of

 1 <sup>2</sup> (2 )

 

*e i par D i i*

pushing rate and data generation rate at a node. To determine the formulation of λ'

calculate the property distribution of λi and λDi. We define that λDi and λi are the independent distribution X and Y. Through proofing of the property Y is bigger than X, we determine the arrival process to Queue B as follows; the proof can be found in (Li, et. al,

*i i D i i i*

Since the data generation rate is Poisson distribution and the random pushing rate abides by exponential distribution, the data arrival rate to queue B approximates to Poisson distribution. Therefore, we can confirm that the queuing system approximates to M/D/1 model. With the same way of full aggregation, the server waiting time and channel waiting time can be determined easily. Therefore, the total delay of partial aggregation is as follows:

> 

<sup>1</sup>

 1 1

"

 

" '

 

 

*DN i i i i* (19)

( ( ) ( ) ( ) ( )) *<sup>N</sup> ec s par par par par par*

*i D*

*i*

1

From the analytical process we find that arrival data rate λ'

*i i* . Similar to full aggregation, the event waiting time of partial

(17)

i, we

i is decided by the random

(18)

*3.3.1. Event waiting time* 

In the *N* hops transmission in partial aggregation, total energy consumption *NE* )( *par* is the sum of transmission energy consumption, reception energy consumption and overhearing energy consumption. *Pt* and *Pr* are energy required for transmitting or receiving a data. The period of time that aggregated data wait in a queue for transmission can be determined as follows:

$$T\_{par}^{s}\left(\dot{\imath}\right) = \tau\_{par}^{s}\left(\dot{\imath}\right) + \tau\_{par}^{c}\left(\dot{\imath}\right) + \tau\_{par}^{1}\left(\dot{\imath}\right)\tag{20}$$

According to Little's formula and equation (25), we determine the amount of data in queue B at node *ni* as follows:

$$L\_{Par}^{Q}\left(\mathbf{i}\right) = \mathcal{X}\_{\mathbf{i}}^{\prime \ast} \, T\_{par}^{s}\left(\mathbf{i}\right) \tag{21}$$

Accordingly, we determine the total energy consumption for the network as follows:

$$E\_{par}\left(N\right) = \sum\_{i=1}^{N} L\_{par}^{\mathcal{Q}}\left(i\right) \* \left(P\_t + \mathcal{Q}P\_r\right) \tag{22}$$

### *3.3.5. Data accuracy*

The total generated data *Lpar(N)* in *N* hops network is obtained as follows:

$$L\_{par}\left(N\right) = \sum\_{i=1}^{N} \mathcal{A}\_i \, ^\*D\_{par}\left(N\right) \tag{23}$$

The amount of data received by sink *Lpar(S)* is as follows:

$$L\_{par}\left(\mathcal{S}\right) = \mathcal{X}\_1' \, ^\ast D\_{par}\left(\mathcal{N}\right) \tag{24}$$

According to the definition and above equations, we determine the data accuracy as follows:

$$A\_{\
u} = \frac{L\_{\mu r} \text{(S)}}{L\_{\mu r} \text{(N)}} \text{\* } AF \tag{25}$$

### **3.4. Evaluation**

Here we show the analytic results of the previous sections. The parameters are as below:

Transmission rate is 250[kbps], Data size is 4096 [bit], Energy consumption for data reception is 17.4 [mA] and for data transmission is 19.7 [mA]. In this section, we evaluate total delay, energy consumption and data accuracy when the aggregation factor is Af=1.

Fig. 7 to Fig. 9 show the total delay, energy consumption of whole network, robot energy consumption and data accuracy of five hops transmission where λi=λ. Partial-T1 and Partial-T2 are two sets of random pushing rate vectors in partial aggregation. We get the vectors randomly [1, 2, 3, 4, 5] and [5, 10, 15, 20, 25].

From figure 7, we find that when event generation rate is small, full aggregation has long transmission delay in comparison to non-aggregation. The reason for concaving up of delay of the full aggregation is that, when event generation rate is small, the received data has to wait for generated data longer duration. In addition at a robot near to the sink, the total delay increases because of the large waiting time due to the congestion around the sink. As long as total delay is concerned, non-aggregation is suitable for situation of small event generation rate. From the figure, we also find that the performances of partial-T1 and partial-T2 are between non-aggregation and full aggregation. If *D i* is infinite, it means fully non-aggregation, if *D <sup>i</sup>* is zero, it means fully aggregation.

Fig. 8 shows the energy consumption of the whole network. Obviously, non-aggregation consumes much more energy than full aggregation. Thus, full aggregation is suitable for energy consumption while non-aggregation is efficiency for transmission delay. The partial-T1 and partial-T2 has energy consumption between non-aggregation and full aggregation. In addition, the smaller random pushing rate vector set partial-T1 has less energy consumption than the set of partial-T2.

Fig. 9 shows the data accuracy of different data aggregation. From fig. 9, we find that the data accuracy of partial aggregation is between non-aggregation and full aggregation. The partial aggregation with the larger random pushing rate achieves higher data accuracy.

**Figure 7.** Total delay

**Figure 8.** Energy consumption

randomly [1, 2, 3, 4, 5] and [5, 10, 15, 20, 25].

aggregation. If

accuracy.

**Figure 7.** Total delay

fully aggregation.

*D*

consumption than the set of partial-T2.

Fig. 7 to Fig. 9 show the total delay, energy consumption of whole network, robot energy consumption and data accuracy of five hops transmission where λi=λ. Partial-T1 and Partial-T2 are two sets of random pushing rate vectors in partial aggregation. We get the vectors

From figure 7, we find that when event generation rate is small, full aggregation has long transmission delay in comparison to non-aggregation. The reason for concaving up of delay of the full aggregation is that, when event generation rate is small, the received data has to wait for generated data longer duration. In addition at a robot near to the sink, the total delay increases because of the large waiting time due to the congestion around the sink. As long as total delay is concerned, non-aggregation is suitable for situation of small event generation rate. From the figure, we also find that the performances of partial-T1 and partial-T2 are between non-aggregation and full

*i* is infinite, it means fully non-aggregation, if

Fig. 8 shows the energy consumption of the whole network. Obviously, non-aggregation consumes much more energy than full aggregation. Thus, full aggregation is suitable for energy consumption while non-aggregation is efficiency for transmission delay. The partial-T1 and partial-T2 has energy consumption between non-aggregation and full aggregation. In addition, the smaller random pushing rate vector set partial-T1 has less energy

Fig. 9 shows the data accuracy of different data aggregation. From fig. 9, we find that the data accuracy of partial aggregation is between non-aggregation and full aggregation. The partial aggregation with the larger random pushing rate achieves higher data

*D*

*<sup>i</sup>* is zero, it means

**Figure 9.** Data accuracy

From above evaluations we find that the partial aggregation with random pushing rate vectors can control the energy, delay and data accuracy between non-aggregation and full aggregation. Hence, one can achieve desired MSRN by controlling the random pushing rate.

### **4. Tradeoffs among accuracy, energy and delay**

### **4.1. Trade off index TOI**

Previous section clearly shows partial aggregation with random pushing rate *D <sup>i</sup>* can control the energy consumption, transmission delay and data accuracy. In MRSN, according to applications, delay taken to collect data, energy consumed by each sensor node for communication and data accuracy of the collected data are critical concerns and are in tradeoff each other. Energy, delay and accuracy cannot reach full potential at the same time, but we can achieve the best possible tradeoff between them. To obtain the best trade-off value of practical application, we propose a Trade-Off Index (TOI). In the following subsections, we discuss energy, delay and accuracy of trade-offs in respect of TOI as criteria. Here E denotes total energy consumption, D denotes total delay, Ac denotes data accuracy. α, β, γ indicate the significance of accuracy, energy and delay and larger α, β, γ indicate more significance of energy, delay and accuracy. The smallest TOI value denotes the best data aggregation.

$$TOT = \frac{E^{\beta} \* D^{\gamma}}{A\_c^{\alpha}} \tag{26}$$

### **4.2. Applications of WSNs with different criteria**

In MRSN, according to the different applications and objectives, we need different significances for transmission delay, energy consumption and data accuracy. Some application areas need to save energy because it is impossible to replace or recharge the battery. In some applications not only the energy is significant, but also the data freshness, such as in military monitoring and disaster monitoring; however data accuracy is most important in medical utilization and in quality control. According to real application, we formulate some of the applications according to the significances of energy, data accuracy and transmission delay in table 1.


**Table 1.** Applications of MRSN

Here the "L" denotes large significance and "S" denotes small significance; the application is formed from left to right along a scale from smaller event generation rate to bigger generation rate. According to the table 1 we can decide the significant parameters of the application in order to perform our proposed TOI; we can achieve the best data aggregation corresponding to the applications.
