International Journal of Antennas and Propagation
sensors. Assume that a block-fading channel of seconds
with channel gain expressed as
𝑖
=
𝑖
2
()
keeps constant within each time slot. Here
𝑖
is exponen-
tially distributed in the condition of independent Rayleigh
fading. e AP broadcasts beacon signals to initiate the
data collection process and each sensor node estimates their
channel condition. Since the AP is usually a mobile phone or
another personal digital assistant (PDA) containing enough
energy, the energy consumption of the AP is not considered
in this work. We suppose that sensors can ensure satisfying
transmission and reduce unnecessary energy consumption
by adjusting their transmission power according to channel
conditions. In practical applications, sensors can only trans-
mitatanitenumberofpowerlevelsaccordingtohardware
limitations []. Let be the number of power levels and
1
,
2
,...,
𝑚
,...,and
𝑀
denote the power scaling factors of
atransmitter,where0
1
<⋅⋅⋅<
𝑀
1.epowerlevel
is then restricted to a nite set shown as
𝑖
=
𝑘
max
𝑀
𝑘=1
,
()
where
𝑖
is the transmission power available for sensor
transmitting a data packet to the AP if it is scheduled and
max
is the maximum transmission power that transmitter can
achieve. For simplicity, based on the Shannon theorem, the
transmission rate of sensor , denoted by V
𝑖
, can be expressed
as
V
𝑖
=log
2
1+
𝑖
0
≤log
2
1+
𝑖
𝑖
0
=log
2
1+
𝑖
𝑘
max
0
,
()
where is the desired value of transmission power in theory
and is the bandwidth. In (),
𝑘
adopts the minimum value
for the sake of matching with the inequality. Since
𝑘
depends
on the current channel gain and the transmission rate related
to the sensor , the energy consumption for data transmitting
of sensor in data collection can be written as
tx𝑖
=
𝑘
max
⋅+
𝑐
,
()
where
𝑐
is the energy consumption of the transmitter circuit
and it is identical for all sensor nodes.
2.2. Formulation of Lifetime in WBAN. A general formula of
lifetime in WSN is described in []. We adopt this lifetime
concept in WBAN, which expresses WBAN lifetime as
{
}
=
⋅
in
−
𝑤
⋅+
tx
. ()
In (),
in
is the initial energy of sensor nodes,
tx
is
the expected transmission energy consumed in one round of
data collection,
𝑤
is the expected wasted energy, and is
the energy required by a sensor for CSI acquisition. e
wasted energy is set to be the total unused energy when the
lifetime completes. It can be expressed as
𝑤
=
𝑁
𝑖=1
𝑤𝑖
,
()
where
𝑤𝑖
is the wasted energy of sensor .Asensornodeis
supposed to be dead when its residual energy is lower than the
transmitter circuit consumption; that is, under any channel
condition it has no enough energy to transmit. A WBAN is
considered to be dead when any sensor node in this network
is dead. In this paper, we express the lifetime of a WBAN as
the number of data allocations before the network dies.
3. Optimal Transmission Scheduling
In each time slot, only one sensor node is scheduled to
transmit its measurements directly to the AP through the
fading channel. We assume that the instantaneous CSI of all
sensorsisavailabletotheAP.Inthissection,weformulate
the problem of dynamically choosing which sensor should
communicate with the AP to maximize network lifetime
under the constraint of fairness as a CMDP. We propose
a centralized transmission scheduling algorithm that maxi-
mizes network lifetime under dierent constraint of fairness.
e optimal lifetime and optimal policy are achieved by
Bellman equation in dynamic programming. e optimal
policy using global CSI denes the limiting performance in
network lifetime for the model specied in Section .
3.1. Fairness Index. Fairness is in general a critical factor in
performance studies. Particularly in distributed networks
where resources are shared by a number of users, fair
allocation is extremely important and fairness is considered
as an important criterion in the design of a WBAN.
In the MAC layer of IEEE .. specication, time
is divided into superframes, each with equal length. e
superframes consist of four periods: control period, conten-
tion access period (CAP), contention-free period (CFP), and
inactive period. e CFP is further divided into a number
of time slots. We focus on the time division multiple access-
(TDMA-) based protocol, in which data packets are mainly
transmitted in the CFP. erefore, this is a time-slotted net-
work, where time is the resource to be allocated among the
sensor nodes.
In the literature, Jain’s fairness index [] has been widely
used as a measure of network-wise fairness performance. Let
𝑖
denote the actual transmission time of sensor and let
denote the total transmission times.
𝑖
indicates the weighting
factor, which expresses the degree of importance of sensor .
en the normalized time allocation of sensor can be given
as
𝑖
=
𝑖
𝑖
.
()