4.2. Optimal Power Control Strategy over Links
In this subsection, an optimal power control strategy is proposed for energy efficiency maximization. To begin with, energy efficiency is introduced to assess the quality of consumed power over a link. Then end-to-end delay and jitter are explored as the representatives of QoS requirements in the WBAN. After that, the power control strategy is described to maximize energy efficiency while QoS requirements are fulfilled. Finally, the theoretical value of the optimal power is obtained by formula derivations, along with a proposition to prove its existence.
According to our power control strategy, transmission reliability and energy consumption are two critical factors. Hence, the measurement on transmission reliability is introduced first. In a real-world WBAN, real-time SNR is typically unknown to devices because of the severe signal attenuation near the human body. As a result, measures computed according to a specific SNR cannot be a suitable criterion of transmission reliability, e.g., packet error rate, whereas, packet outage probability (POP) is a meaningful assessment of transmission reliability in realistic channels, due to the fact it signifies the fading realization with a guaranteed packet error rate
. The POP of packets transmitted from node
to node
is [
7]:
where
,
,
,
, and
are the logogram of probability, the cumulative distribution function of the standard normal distribution, packet size, transmission bit rate, and channel bandwidth respectively.
is packet error rate, computed as [
18]:
Besides, the ratio between effective throughput and transmission power is widely used to measure energy efficiency for networks with limited energy [
12,
23]. Hence, provided that node’s relay is designated, consumed power
over the link from node
to node
can be evaluated by an efficiency function, which is expressed as:
The energy efficiency signifies the conservative number of bits successfully forwarded over the link for per Joule of transmitter’s consumed energy with a guaranteed packet error rate. Hence, Equations (5) and (7) can be considered as basic assessments of power. Nevertheless, in order to optimize transmission power, some QoS requirement of the WBAN should be fulfilled, e.g., end-to-end delay constraint and jitter constraint. Therefore, before formally presenting the power control strategy, we first explore the end-to-end delay and jitter over links in the following paragraphs.
In a GI/G/1 queue, average end-to-end delay and jitter depend on the inter-arrival time and service time distribution. Inter-arrival time of a node combines the inter-arrival time of packets generated by itself and that of packets received from other nodes. We denote the inter-arrival distribution of packets at node
and that of self-generating packets at
as
and
respectively. Besides, as per our system model,
is exponentially distributed. Therefore,
and
are
and
, respectively. Besides, based on the work done in [
12], expected value
and variance
of
can be obtained as follows:
where
refers to the service time distribution of packets transmitted over link
, and
is a factor of utilized time for service, denoted as
.
Similarly, the expected value and variance of the service time over link
are given as follows [
7]:
where
denotes the conservative probability of successful transmission over
, with considerations of lost packets in signal collision case and false transmission case, and is computed by:
is the collision probability of packets transmitted from node
, depicted in:
Note that in Equation (13), collisions re considered in the whole topology, as each node is located in the carrier sensing range of any other node because of the WBAN’s small scale. Besides, is also the probability with which packets are transmitted over link.
Based on the inter-arrival time and service time obtained above, average end-to-end delay
and jitter
over link
can be approximated as below [
32]:
Furthermore, in a multi-hop path, the accumulated delay and jitter are calculated as follows:
where
is the relay of node
, and
refers to the multi-hop path from node
to the hub.
Thus far, one assessment of transmission power and two measures of QoS has been obtained. Besides, in WBAN, the main target of power control, is to maximize the power while fulfilling the QoS requirements. Accordingly, for a given WBAN topology, the transmission power over link
can be decided according to the following optimization:
and
are prescribed upper bounds of end-to-end delay and jitter in the multi-hop path from node
to the hub, which corresponds to the QoS requirements in the network. As the results in the equations above suggest, delay and jitter can be expressed as functions of the average SNR in the channel, which means that minimizing them is equivalent to maximizing the average SNR. Moreover, the specific relation between power and average SNR is:
where
and
refer to the channel gain over link
and thermal noise spectral density, respectively.
Hence, the optimization of transmission power can be expressed as below:
In order to obtain the solution to the optimization above, the maximization of
should be explored with no restriction at first, whose expression accords to Equations (5)–(7) and (19) as a function of
:
where:
Taking the derivative of the function above and equating it to zero, we derive that
is maximized when
equals to
constrained by:
Note that in Equation (23), the left side is monotonically decreasing as rises, while the right side is monotonically increasing at the same time. Besides, the ranges of the two sides overlap. Hence, as to Equation (23), there is a unique root given to over link. When the derivative of will not exceed zero, indicating that , which proves that further power enhancement has disadvantages to .
Furthermore,
is derived in unconstrained conditions. However, as mentioned above, delay and jitter constraints with respect to the lower bound of
have to be satisfied during the optimization of power control, leading to a possibility that
can be infeasible. As a result, power should be selected to maintain the average SNR
for transmission over link
where:
Based on the description and analysis above, a formal proposition is provided to prove the existence of optimized power using aforementioned methods:
Proposition 1. If the condition is guaranteed for each link in the WBAN, then the optimal power can be obtained.
Proof. When , the definition of is meaningful. In this case, it is feasible for to be same as , so that the optimization of energy efficiency over each link is achieved. Hence, according to Equation (19), the corresponding transmission power is . □
4.3. Optimal Relay Decision Strategy for Sensor Nodes
In this subsection, an optimal relay decision strategy is proposed. In the beginning, the energy efficiency on route and energy consuming speed of a node are explored. Then, based on the efficiency and speed consumption, a candidate assessment function is introduced to select relays for sensor nodes. Afterwards, the detailed steps of the relay decision strategy are presented and a proposition is given to validate the effectiveness of the strategy. In the end, we summarize our strategy in the form of an algorithm.
In
Section 4.2, the optimal power control strategy is adequately explored for an established transmission link
, i.e., the strategy can only be implemented on sensor nodes that have already selected their relays. Therefore, the next step of the CLDO scheme is to determine the strategy for selecting relays for each sensor node, or to construct the topology of the WBAN.
When a sensor node chooses to transmit through a multi-hop path instead of direct transmission to the hub to avoid severe signal attenuation on the route, a relay or relays are required. The relays, however, lead to transmissions of the same packet in the WBAN. As a result, overall delay and jitter for transmission of a specific packet are expected to rise. Hence, a relay decision is vital for improving the performance of a WBAN. In order to resolve this problem, we consider enhancing the energy efficiency as much as possible while balancing the distribution of energy consumption through fully utilizing the remaining energy in nodes when the WBAN is declared dead.
In Equation (7), the evaluation function of energy efficiency has already been given, and it can be used here to help decide the relays, whereas due to the fact that the number of links in a multi-hop path is equivocal, the overall POP and total consumed power of the total path should be more appropriate than that of a single link when selecting a desired relay candidate.
As a result, a modified function that enables power efficiency to be evaluated in the path for relay decision is proposed, depicted as below:
where
refers to the overall power efficiency on the route
from sensor node
to relay node
, then to the hub, while
is the direct relay of sensor node
.
Note that in Equation (25), though the overall transmission path is taken into consideration, it is utilized to select the optimal relay of node , i.e., for relay candidates of node , their respective routes to the hub and the optimal powers consumed on the routes have been decided before calculating the power efficiency of node , which mainly determines the value of . Based on the description above, such a scene could happen if is taken as the only measure in our relay decision strategy.
Several sensor nodes, having not chosen their relays, share a candidate with a relatively high overall power efficiency value; in turn, all of them prefer to transmit packets through , leading to excessive packet accumulation in its transmitting queue. As a result, end-to-end delay and jitter are inevitably raised according to Equations (14) and (15). Besides, has no choice but to increase its energy consumption to transmit packets due to the growth of the total data size. Other candidates, however, are ignored even if their queues are not crowded. Therefore, they consume energy at a limited rate. Hence, this kind of relay decision will produce an unbalanced distribution of energy consumption in the WBAN. Consequently, the data size burden of candidates should be involved in any relay decision strategy to achieve a more balanced energy consumption distribution.
Since the transmission rate is constant, when
is selected to be the relay of node
, why are nodes sometimes italic font and other times not—pick and use a consistent style the energy consumption speed of candidate
for transmitting burden data can be approximated by the product of the optimized power
over link
and time spent on transmitting total data that arrives at
per second, which includes data generated by itself, the received data from node
, and the collected data from other sensor nodes. Therefore, the energy consuming speed is depicted as:
Combining the evaluation function of power efficiency on the transmission route and data size burden by candidates, a candidate assessment function is derived to assess the fitness of candidate
to be the relay
of sensor node
:
where
is the relative impact factor whose value reflects that the selection focus more on power efficiency or the equilibrium of consumed energy in WBANs.
The definition of
indicates that high power efficiency on the route and low burden data size of the candidate are welcome in relay decision. Hence, the selection is equivalent to choosing the crest value of
:
where
refers to the set of candidates in the relay decision of sensor node
. Due to the small scale of the WBAN, all sensor nodes except for itself have the potential to be its relay. However, there is no need for one sensor node to choose another node that is farther from the hub as its relay. Therefore, let
denote the physical distance between device
and
, and
can also be expressed as
where the distance constraint
is satisfied.
In the CLDO scheme, we aim to guarantee high energy efficiency and balanced consumed energy distribution in the WBAN. Hence, the specific procedure of topology formation can be described as follows:
- (1)
Initialize a star topology for the network, centered at the hub.
- (2)
Sort sensor nodes within the network into a queue according to the ascending order of their distance to the hub.
- (3)
Pick the head of the queue every time and implement a relay decision strategy for it until the queue is empty.
- (4)
Return to step (2) to start the next round of relay reselection.
The method of constructing topology described above is conducted in an iterative way. To make the method of selecting relay clearer, our relay decision (RD) scheme is described as Algorithm 1.
Algorithm 1 Relay Decision (RD) Algorithm for Forming Topology |
1: input one value as the iteration time |
2: For |
3: ; //initialize a star network topology |
4: ; //obtain relay candidates |
5: End |
6: While |
7: sort elements in into a queue in an ascending order of distance |
8: While is not empty |
9: pull out the head of ; |
10: For |
11: assume an established link for transmission; |
12: compute the optimized power ; |
13: compute the candidate assessment function ; |
14: End |
15: choose node as the relay of that satisfies where |
16: End |
17: ; |
18: End |
4.4. Power Rearrangement Strategy and Packet Size Choice Strategy
In
Section 4.4, a power rearrangement strategy and packet size choice strategy are proposed. Besides, our CLDO scheme is proposed here. Firstly, the power rearrangement strategy is obtained to fully utilize the energy in sensor nodes and another proposition is given to show that QoS requirements are still fulfilled. After that, the impact of packet size on the performance of the WBAN is introduced. In the end, an algorithm is proposed to illustrate the CLDO scheme that achieves a comprehensive optimization of energy efficiency and energy balance in the WBAN.
In
Section 4.2 and
Section 4.3, a power control strategy based on a single transmission link and a relay decision strategy for determining the topology of the network have been proposed. Nevertheless, since the network lifetime mainly depends on the first dead node, there still remains much energy in sensor nodes when the network is declared dead, especially in those nodes that just transmit their own sensed data, i.e., the leaf nodes in the topology.
Hence, it is reasonable to further enhance the optimized power of leaf nodes, since fully utilizing the remaining energy enables the network to fulfill higher QoS requirements, such as delay, jitter, and transmission reliability. Besides, as leaf nodes will not act as relays of any other node, they are able to rearrange power while avoiding influencing other nodes.
In order to meet the requirement above, the energy consumption speed of leaf nodes should be rearranged as close as possible to the maximum consumption speed within the WBAN. We denote the energy consumption speed as
when the rearranged power
is adopted over the link
. According to the energy consumption speed given in Equation (26), for one leaf node
, the power rearrangement strategy over link
can be simply expressed as the problem of resolving the following optimization problem:
Note that delay and jitter constraints are not mentioned in the power rearrangement strategy, in fact, they are implicit in the consumption speed constraint , as indicates the enhancement of transmission power, which ensures less delay and jitter.
So far, a power control strategy, relay decision strategy and power rearrangement strategy have been introduced to optimize the performance of a WBAN. All the strategies above are based on the assumption that the packet size is given. As we know, sensor nodes tend to add a header to include essential information for smoothing the communication of each packet. Moreover, since transmissions are standardized by transport protocols employed in layers, the header size is fixed no matter how large the packet size is. As a result, the packet size determines the percent of sensed data size in a packet, whose variation has remarkable impacts on delay and jitter. Besides, packet size will influence the packet error rate in light of Equation (6), specifically, the packet error rate rises with the growth of packet size. Based on the description above, the optimal packet size can be obtained for improving the performance of a WBAN.
Due to the small scale of a WBAN, simultaneous transmission of any two nodes will cause communication collisions. Moreover, if their packet sizes are different, there will be difficulty in maintaining time synchronization of nodes in ALOHA. Hence, in order to ensure uniformity, our choice of packet size takes the overall WBAN into consideration. Assume that there is a set of packet sizes to choose from, the we denote the chosen size as in following parts.
To explore the specific impact of packet size on WBAN’s performance, firstly, we discuss its impact on the percent of sensed data in packets. For each packet, the context within it consists of two parts: the sensed data from body environment and header data, expressed as:
where
refers to the data size in a packet when the packet size is set to
, and
is the header’s data size.
Hence, in light of Equation (30), the percent of sensed data in a packet can be computed according to the following formula:
Note that
is a constant. Therefore, the bigger the packet size is, the larger the percentage that the sensed data occupies. Hence, the method of computing the percentage has been obtained. It can be used in turn to derive the generation speed of sensed data. Since each node senses data at an individual speed that is irrelevant of packet size, the average packet generation speed of a node should be modified as per the sensed data consistency principle, that is:
and converting it to the following form:
where
is the modified average packet generation speed when the packet size is set to
. The new performance of the WBAN with packet size variations can be derived through using
and
to re-compute formulas based on the default packet size. Therefore, our packet size can be optimized to achieve the maximization of average energy efficiency
on all transmission links in the WBAN. The optimization problem is expressed as:
is calculated as:
where
refers to the energy efficiency over link
when the given packet size is
.
Based on the aforementioned description and the analysis related to power rearrangement strategy and packet size choice strategy, Algorithm 2 is proposed to illustrate a cross layer design optimization (CLDO) scheme, which is designed as below.
Algorithm 2 Cross Layer Design Optimization (CLDO) Algorithm |
1: | For |
2: | employ RD scheme to form network topology and arrange consumed power; |
3: | employ power rearrangement strategy to fully utilize remaining energy; |
4: | compute average energy efficiency ; |
5: | End |
6: | choose the packet size that corresponds to the crest value of where ; |
7: | implement the optimized power, the optimized topology obtained in step 1–4 that corresponds to the chosen packet size in WBAN. |
So far, our CLDO scheme is presented to optimize energy efficiency and energy balance in WBANs based on the power control strategy, relay decision strategy, power rearrangement strategy, and packet size choice strategy.