**3.1. Non-aggregation**

The arrival data are transmitted to the adjacent lower node immediately after having been received; data neither wait for local generated data nor aggregate with any other data. The analytical model of non-aggregation is shown as follows in figure 2.

In the analytical model of node ni in fig. 2, the average arrival rate from the upper node is approximates to Poisson distribution. Generated data rate at a node is assumed to be Poisson distribution. The generated data and arrival data join the service queue and wait for transmission. There is one server for data transmission at each node. All data in the queue will be sent based on first in first out. *i* is the data rate upon exiting the server at node ni.

**Figure 2.** Analytical model of non-aggregation

Arrival process to the queue

According to the analytic model, we find that the arrival rate to the queue is

$$
\lambda\_i' = \lambda\_{i+1}'' + \lambda\_i \tag{2}
$$

Strictly speaking, arrival data from the upper node is not Poisson distribution. However, for the purpose of simplicity, we approximate the process as Poisson distribution. Since the arrival data rate and local generated data rate are independent Poisson distributions, the sum of the two is also a new Poisson distribution.

Service process

76 Wireless Sensor Networks – Technology and Protocols

**2.4. Transmission delay** 

lower robot. Channel waiting time

Event waiting time

**2.5. Energy consumption** 

sensed data at all the robots.

**3. Data aggregations** 

**3.1. Non-aggregation** 

will be sent based on first in first out.

**2.6. Data accuracy** 

AF times of the generated data size. AF=1 means that aggregated data have the same data

 Aggregated data size Generated data size

Total delay *D(N)* shows a time interval between the instance when event *Eij* occurred at

 Data transmission time Շ' is defined as a time interval between the instance that data are transmitted from robot and the instance that the data are received at the adjacent

the arrival data have to wait for local generated data to be aggregated together, hence

Total energy consumption *E(N)* is defined to be the sum of energy consumption of an event data that is generated at node nn and finally received by sink node in *N* hops networks.

We define the data accuracy as the proportion of collected data at sink and the amount of

In this part, we analyze and evaluate the data aggregation simply in terms of non-

The arrival data are transmitted to the adjacent lower node immediately after having been received; data neither wait for local generated data nor aggregate with any other data. The

In the analytical model of node ni in fig. 2, the average arrival rate from the upper node is approximates to Poisson distribution. Generated data rate at a node is assumed to be Poisson distribution. The generated data and arrival data join the service queue and wait for transmission. There is one server for data transmission at each node. All data in the queue

*<sup>i</sup>* is the data rate upon exiting the server at node ni.

*AF* (1)

*<sup>c</sup> i* : it is the time interval that data cannot utilize the channel.

( )*i* : In full aggregation, before a robot processes data aggregation,

size with generated data, and we assume there is one generated data at one time.

robot nn and the instance when the sink receives Dij in *N* hops networks.

( )

the waiting time of arrival data called event waiting time.

aggregation, full aggregation and partial data aggregation.

analytical model of non-aggregation is shown as follows in figure 2.

e

In our network model, each node has one server. The ACK packet transmission time is not considered. Data aggregating time is very short and negligible. Therefore the service time is one hop data transmission time. In our work, data transmission rate is *vc* and local generated data size is *Si*. Therefore the service time for each generated data is:

$$
\sigma^1 = \frac{s\_i}{v\_c} \tag{3}
$$

Since *vc* and *Si* are constant in non-aggregation, the service time for each data are fixed and constant.

From the above analysis we can determine that the queuing system approximates to M/D/1 model.

According to equation (4), the average data transmission time that we obtain at a node is:

$$\tau^1\_{\text{non}}\left(i\right) = \frac{\tau^1}{\left(1 \cdot \mathcal{X}\_i' \tau^1\right)}\tag{4}$$

According to queuing theory and equation (4), we determine the server waiting time as follows:

$$\tau\_{non}^{s}(i) = \frac{\lambda\_i'(\tau\_{non}^1(i))^2}{\Im(1 - \lambda\_i'\tau\_{non}^1(i))}\tag{5}$$

### Channel waiting time

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 channel waiting time and is obtained the formulation as follows:

$$
\pi\_{nom}^c(i) = \mathbb{Z}\boldsymbol{\lambda}\_i^\prime \ast \boldsymbol{\tau}\_{nom}^1(i) \ast \boldsymbol{\tau}\_{nom}^1(i) \tag{6}
$$

### *3.1.1. Total delay*

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

$$D\_{non}\left(N\right) = \sum\_{i=1}^{N} \left(\tau\_{non}^{1}\left(i\right) + \tau\_{non}^{s}\left(i\right) + \tau\_{non}^{c}\left(i\right)\right) \tag{7}$$

### *3.1.2. Energy consumption*

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

$$L\_{nom}^{Q}(\mathbf{i}) = \mathcal{X}\_{\mathbf{i}}^{\prime \ast} T\_{nom}^{s}(\mathbf{i}) \tag{8}$$

The λ'i is arrival data rate at node ni and ( ) *<sup>s</sup> non T i* is time duration from data joining the queue to data having been received by the next neighbor node at node ni, in case of nonaggregation, can be determined as follows:

$$T\_{nm}^{s}\left(\mathbf{i}\right) = \boldsymbol{\tau}\_{nm}^{1}\left(\mathbf{i}\right) + \boldsymbol{\tau}\_{nm}^{s}\left(\mathbf{i}\right) + \boldsymbol{\tau}\_{nm}^{c}\left(\mathbf{i}\right)\tag{9}$$

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

$$E\_{non}\text{(N)} = \sum\_{i=1}^{N} \mathcal{L}\_{non}^{\mathcal{Q}}(i)^{\*}(P\_{t} + \mathcal{Q}P\_{r})\tag{10}$$

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

### *3.1.3. Data accuracy*

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