Pervasive health monitoring is an eHealth service, which plays an important role ... (BS), which further forwards the data to the medical server over the Internet.
OPTIMAL RESOURCE ALLOCATION TO PROVIDE QOS GUARANTEE IN PERVASIVE HEALTH MONITORING SYSTEMS Yifeng He1 , Wenwu Zhu2 , and Ling Guan1 ECE Department, Ryerson University, Toronto, Canada1 Microsoft Research Asia, Beijing, China2 ABSTRACT Pervasive health monitoring is an eHealth service, which plays an important role in prevention and early detection of diseases. In health monitoring systems, a loss or an excessive delay of the critical data may cause a fatal accident. Therefore it is important to provide Quality of Service (QoS) guarantee for the delivery of data streams. In this paper, we formulate the QoS optimization problem, which jointly optimizes the transmission power and the transmission rate at each aggregator to provide QoS guarantee to data delivery. The optimization problem is converted into a convex optimization problem, which can be solved efficiently. We demonstrate in the simulations that the proposed optimized scheme improves the service quality of the health monitoring system in terms of delay and Packet Loss Rate (PLR). Index Terms— Pervasive health monitoring, eHealth, Quality of Service (QoS), resource allocation, Geometric Programming (GP), convex optimization, queueing model, wireless communications 1. INTRODUCTION The worldwide population aged above 65 is expected to increase from 6.9% in 2000 to 12.0% in 2030 [1]. The ageing population will lead to increased healthcare cost as care for the elderly is much more expensive than that of other age groups. Electronic Health (eHealth), which integrates information processing and communications technologies into traditional medical services, emerges as a promising approach to improve healthcare efficiency. Pervasive health monitoring is an eHealth service, which plays an important role in prevention and early detection of diseases [2]. A pervasive health monitoring system with Body Sensor Networks (BSNs) [3] is illustrated in Fig. 1. Each patient wears a BSN on his or her body. A BSN is a body-area wireless sensor network, which consists of multiple wireless body sensors and an aggregator. The body sensors continuously monitor the patient’s vital signs, and then transmit them to the aggregator via wireless channels. In this paper, we choose 3G cellular wireless networks as the Internet access networks. The aggregator on a patient’s body collects the data from body
Aggregator Base station
Internet
Medical server
Fig. 1. Illustration of a pervasive health monitoring system
sensors, and then transmits the data to the the Base Station (BS), which further forwards the data to the medical server over the Internet. As long as the patients are in a 3G coverage area, they will always be monitored even in the presence of mobility. One of the major challenges in pervasive health monitoring systems is Quality of Service (QoS) guarantee for the delivery of data streams. The data streams acquired by the sensors have different priorities. For example, heart activity readings (e.g., Electrocardiography (ECG) waveforms) are often considered more important than body temperature readings. In health monitoring systems, a loss or an excessive delay of the prioritized data may cause a fatal accident. However, it is quite challenging to ensure reliable data transmission in wireless communications, especially when the channel is impaired by interference or fading. QoS support in health monitoring systems have been studied in the recent literature. In [4], the authors proposed a rate control algorithm based on the Q-learning approach to meet medical QoS requirements in ultrasound video streaming over wireless systems. Zhou et al proposed BodyQoS, a radioagnostic QoS for BSNs based on a common virtual MAC abstraction [5]. A Distributed Queueing Body Area Network (DQBAN) MAC protocol was proposed in [6] to guarantee that all packets are served with their particular applicationdependant QoS requirements, without endangering the battery lifetime of the body sensors in BSNs. Most existing work
[5][6] studied the QoS of the body sensors in a BSN, while our work focuses on the QoS for the communications from the aggregators to the base station. In this paper, we jointly optimize the transmission power and the transmission rate at each aggregator in a 3G cell area to provide QoS guarantee to the delivery of data streams. The QoS optimization problem is converted into a convex optimization problem, which is then solved efficiently.
(t)
delay TP,m of the prioritized packets at aggregator m at time slot t is given by [8] (t)
(t)
TP,m =
1/um (t)
(t)
1 − λP,m /um
, ∀m ∈ M.
(2)
(t)
The average queueing delay Tm of all packets including prioritized packets and normal packets at aggregator m at time slot t is given by [8]
2. SYSTEM MODELS (t)
A major difference between body sensor networks and other wireless sensor networks is that BSNs may contain critical readings, which should be picked out and then treated with a higher priority. Therefore, we deploy a differentiated service at each sensor. If the physiological reading is in the normal range, the corresponding packet will be classified into a normal packet. On the other hand, if the physiological reading is in the abnormal range, the corresponding packet will be classified into a prioritized packet. At each sensor, the prioritized packets will be first transmitted to the aggregator before the normal packets. 2.1. Queueing Model Since the movement of the patients and the traffic from the sensors are dynamic, we study the QoS problem in a discretetime manner. The aggregator aggregates the packets from all the body sensors worn by a patient. We assume that the ar(t) rivals of the packets follow a Poisson process with a rate λm , the arrivals of the prioritized packets follow a Poisson process (t) with a rate λP,m , and the arrivals of the normal packets follow
(t) Tm =
1/um (t)
(t)
1 − λm /um
, ∀m ∈ M.
(3)
In the M/M/1 queueing model, the tail probability is defined as the probability that the number of packets in the system is larger than a threshold ρm [8]. Let the threshold ρm represent the length of the queue at aggregator m. The tail probability represents the packet drop probability due to c queue overflow. The tail probability PP,m for the prioritized packets at aggregator m at time slot t is given by [8] (t)
c (ρm +1) PP,m = Pr (NP,m > ρm ) = (λP,m /u(t) , ∀m ∈ M, m) (4) where NP,m is a random variable representing the number of the prioritized packets at aggregator m. The tail probability c Pm for all packets at aggregator m at time slot t is given by [8] c (t) (ρm +1) Pm = Pr (Nm > ρm ) = (λ(t) , ∀m ∈ M, (5) m /um )
where Nm is a random variable representing the number of the packets at aggregator m.
(t)
a Poisson process with a rate λN,m , respectively, at aggregator m during time slot t. Based on decomposition of Poisson (t) (t) (t) process [7], we have λm = λP,m +λN,m . We model each aggregator as an M/M/1 queueing system with preemptive priority service [8], in which a prioritized packet will always be scheduled for transmission once it arrives at the queue, and normal packets can be transmitted only after all of waiting prioritized packets have left the queue. The transmission rate at aggregator m at time slot t is de(t) (t) noted by Rm . The service rate um for the packets at ag(t) (t) gregator m at time slot t is given by um = Rm /Lm where Lm is the average packet length at aggregator m. The M/M/1 queueing system needs to satisfy the follow condition in order to be stable [8]: (t)
(t)
λN,m + λP,m ≤ u(t) m , ∀m ∈ M.
(1)
where M is the set of the aggregators in a cell area. In the queueing theory, the queueing delay of a packet is defined as the duration from the time when the packet arrives in the queue to the time when the packet is sent out of the queue. In the M/M/1 queueing model, the average queueing
2.2. CDMA Model Code-Division Multiple Access (CDMA) is a spread spectrum multiple access technique, which allows multiple users to transmit the data streams to the base station simultaneously over the same physical channel. CDMA has been widely used in existing 3G cellular wireless systems. In the CDMA model, the spread-spectrum bandwidth is denoted by W , the power spectrum density of the Additive White Gaussian Noise (AWGN) is denoted by N0 . The channel gain from aggregator m to the base station at time slot t is denoted by (t) hm . The received Bit-Energy-to-Interference-Density Ratio (BEIDR) at the base station from aggregator m at time slot t (t) is denoted by ym , which is given by [9] (t) ym =(
W
)( (t)
Rm
(t)
(t)
hm Pm (t)
(t)
δΣj∈M,j6=m hj Pj
+ N0 W
), ∀m ∈ M,
(6) where δ is the orthogonality factor representing Multiple Access Interference (MAI) from the imperfect orthogonal spreading codes.
We assume Binary Phase Shift Keying (BPSK) modulation is used in the CDMA system. The Bit Error Rate (BER) of the data transmitted q from aggregator m to the base station (t)
is given by em = Q( 2ym ) where Q(x) is a Q-function [7]. If a packet is received in error, it will be dropped at the base station. We assume the bit errors occur independently in a packet. Therefore the Packet Loss Rate (PLR) due to transmission errors of the packets from aggregator q m is then
delays of the prioritized packets and all the packets, respectively. Mathematically, the problem is formulated as follows. P (t) minimize m∈M q Pm (t) subject to Q( 2ym ) ≤ eth , ∀m ∈ M, (t) um = 1/u(t) m
(t)
e given by Pm = 1 − (1 − em )Lm = 1 − (1 − Q( 2ym ))Lm where Lm is the average packet length in bits at aggregator m.
(t)
P LR PP,m
c e = 1 − (1 − PP,m )(1 − Pm ) (t)
q (t) − Q( 2ym ))Lm . (7) P LR The PLR of all packets at aggregator m, denoted by Pm , is given by = 1 − (1 − (
P LR Pm
λP,m (ρ +1) m )(1 (t) ) um
c e = 1 − (1 − Pm )(1 − Pm )
q (t) λ(t) (ρm +1) m = 1 − (1 − ( (t) ) )(1 − Q( 2ym ))Lm . um
(8) The delay of a packet consists of the queueing delay and the propagation delay. The propagation delay is small compared to the queueing delay, thus it is negligible. The delay of a packet is then equal to the queueing delay. The average queueing delay of the prioritized packets at aggregator m is given by Equation (2), and the average queueing delay of all packets at aggregator m is given by Equation (3). 3. QOS OPTIMIZATION PROBLEM
We optimize the resource allocation in a 3G cell area to provide QoS guarantee to the delivery of the data streams. The QoS optimization problem can be stated as: to minimize the sum of the transmission powers of all the aggregators by optimizing the transmission power and the transmission rate at each aggregator, subject to the power constraints, the requirements of the congestion PLR, the transmission BER, and the
), ∀m ∈ M,
≤ TP,th , ∀m ∈ M,
(t) (t) ≤ Tth , ∀m ∈ M, 1−λm /um (t) (t) (λP,m /um )(ρm +1) ≤ PP,th , ∀m ∈ M, (t) (t) (λm /um )(ρm +1) ≤ Pth , ∀m ∈ M, (t) (t) (t) λN,m + λP,m ≤ um , ∀m ∈ M, (t) 0 ≤ Pm ≤ Pmax , ∀m ∈ M, (t) Rm > 0, ∀m ∈ M,
(9) (t) where Pm is the transmission power at aggregator m at time (t) slot t, Rm is the transmission rate at aggregator m at time slot (t) t, eth is the threshold of BER, um is the service rate for the packets at aggregator m at time slot t, TP,th is the threshold of the queueing delay for the prioritized packets at an aggregator, Tth is the threshold of the queueing delay for all packets at an aggregator, PP,th is the threshold of congestion PLR for the prioritized packets at an aggregator, Pth is the threshold of congestion PLR for all packets at an aggregator, and Pmax is the maximum transmission power at an aggregator. Q function is a monotonically decreasing function. Thereq (t)
(t)
fore we convert the constraints, Q( 2ym ) ≤ eth and ym = (
W (t) )( Rm
(t) h(t) m Pm (t) (t) δΣj∈M,j6=m hj Pj +N0 W
(9), to an equivalent form, ( −1
(Q
(eth )) 2
) in the optimization problem
(t) h(t) W m Pm (t) )( (t) (t) Rm δΣj∈M,j6=m hj Pj +N0 W
)≥
2
, where Q−1 (x) is the inverse Q-function [7]. After the conversion, the optimization problem (9) is changed to the following equivalent form. P (t) minimize m∈M Pm subject to
(
(t) h(t) W m Pm (t) )( (t) (t) Rm δΣj∈M,j6=m hj Pj +N0 W
)≥
(Q−1 (eth ))2 , 2
∀m ∈ M, (t) R(t) um = Lm , ∀m m 1/u(t) m (t)
3.1. Problem Formulation
(t)
1−λP,m /um 1/u(t) m
2.3. QoS Metrics We examine two QoS metrics, the PLR and the delay, of the prioritized packets and all the packets at the aggregator, respectively. The PLR and the delay of the prioritized packets at the aggregator indicate the delivery quality of the critical data, while the PLR and the delay of all packets at the aggregator indicate the overall transmission performance. The PLR consists of the congestion PLR due to queue overflow and the transmission PLR due to transmission errors. The PLR of the prioritized packets at aggregator m, denoted P LR by PP,m , is given by
(t) h(t) W m Pm (t) )( (t) (t) Rm δΣj∈M,j6=m hj Pj +N0 W (t) Rm /Lm , ∀m ∈ M,
(t)
ym = (
(t)
1−λP,m /um 1/u(t) m
∈ M,
≤ TP,th , ∀m ∈ M,
(t) (t) ≤ Tth , ∀m ∈ M, 1−λm /um (t) (t) (λP,m /um )(ρm +1) ≤ PP,th , ∀m ∈ M, (t) (t) (λm /um )(ρm +1) ≤ Pth , ∀m ∈ M, (t) (t) λm ≤ um , ∀m ∈ M, (t) 0 ≤ Pm ≤ Pmax , ∀m ∈ M, (t) Rm > 0, ∀m ∈ M,
(10)
2
−1
th )) Let γth = (Q (e , representing the threshold of the 2 received BEIDR. If the received BEIDR at the base station is larger than γth , the BER of the received signal will be less than eth . In the optimization problem (10), the objective is to minimize the sum of the transmission powers of all the aggrega-
tors. The first constraint, (
(t) h(t) W m Pm (t) )( (t) (t) Rm δΣj∈M,j6=m hj Pj +N0 W
)≥
(Q−1 (eth ))2 , 2
requires the received BEIDR at the base station from aggregator m to be no less than the threshold γth . The (t) (t) second constraint, um = Rm /Lm , represents the service rate for the packets at aggregator m at time slot t. The third constraint,
1/u(t) m (t) (t) 1−λP,m /um
≤ Tth , requires that all packets at aggregator m
at time slot t have a average queueing delay no larger than (t) (t) a threshold Tth . The fifth constraint, (λP,m /um )(ρm +1) ≤ PP,th , requires that the prioritized packets at aggregator m at time slot t have a congestion PLR no larger than a threshold (t) (t) PP,th . The sixth constraint, (λm /um )(ρm +1) ≤ Pth , requires that all the packets at aggregator m at time slot t have a congestion PLR no larger than a threshold Pth . The seventh (t) (t) constraint, λm ≤ um , requires that the service rate at aggregator m at time slot t should be no less than the arrival rate of all packets in order to maintain a stable M/M/1 queueing system. 3.2. Optimal Numerical Solution We convert the first constraint in the optimization problem (10) to an equivalent form (t) γth N0 Rm (t) ( (t) ) hm Pm
(t) δγth Rm (t)
(t)
(t)
hm Pm W
≤ 1. The constraint, (t)
(t)
Σj∈M,j6=m hj Pj 1/u(t) m (t)
(t)
1−λP,m /um
+
≤ TP,th ,
(t)
Lm +TP,th λP,m Lm . The constraint, TP,th (t) L +T λ(t) L equivalent to Rm ≥ m Tthth m m .
is equivalent to Rm ≥ 1/u(t) m (t) (t) 1−λm /um
≤ Tth , is
The constraint, (t)
Rm ≥
(t) (t) (λP,m /um )(ρm +1)
λP,m Lm 1/(ρ
PP,th m
+1)
(t)
(t)
(t)
λ(t) m Lm 1/(ρm +1)
Pth
(t)
(t)
um , is equivalent to Rm ≥ λm Lm . Therefore, the optimization problem (10) is then converted to the following equivalent form. P (t) minimize m∈M Pm δγth R(t) m
(t) (t) Σj∈M,j6=m hj Pj (t) (t) hm Pm W
(12)
(1)
a
(2)
(n)
a
a
monomials, given by gl (x) = dl x1 l x2 l ...xnl where (j) dl > 0 and al ∈ R for j = 1, 2, ..., n. In the optimization problem (11), the objective function is a posynomial, and the left side of the first constraint (t) (t) (t) δγth R(t) N0 Rm m Σj∈M,j6=m hj Pj + γth(t) ( (t) ) is also a posyn(t) (t) hm Pm W hm Pm omial. Therefore, the optimization problem (11) is a GP. The GP problem in (11) is not a convex optimization problem, because the inequality constraint function, (t) (t) (t) δγth R(t) N0 Rm m Σj∈M,j6=m hj Pj + γth(t) ( (t) ), is not a con(t) (t) hm Pm W hm Pm vex function. However, with a logarithmic change of the variables, the GP in (11) can be converted into a convex opti(t) (t) (t) (t) mization problem. Let zm = ln(Rm ) and vm = ln(Pm ). Taking the natural logarithms of both the objective function and the constraints, the optimization problem (11) is converted into the following problem: P (t) minimize ln m∈M exp(vm ) subject to
(t)
(t)
(t)
ln(Σj∈M,j6=m exp(zm + vj − vm + ln( (t) + exp(zm
−
(t) vm
(t)
δγth hj
(t) hm W
))
N0 + ln( γth(t) ))) ≤ 0, ∀m ∈ M, hm
(t)
vm ≤ ln(Pmax ), ∀m ∈ M, (t) LB zm ≥ ln(Rm ), ∀m ∈ M, Pn (13) Since the log-sum-exponential function f (x) = log( i=1 exp(xi )) is always a convex function [11], the objective function, P (t) ln m∈M exp(vm ), and the inequality constraint function, (t)
(t)
(t)
ln[Σj∈M,j6=m exp(zm +vj −vm +ln (t)
(t)
δγth hj
(t) hm W
(t)
)+exp(zm −
N0 vm + ln γth(t) )], in the optimization problem (13), are both hm ≤ 1, convex functions. Therefore, the optimization problem (13) is a convex optimization problem. ♦ The optimization problem (11) is a geometric program(11) ming [10] that can be transformed into a convex optimiza-
(t) γth N0 Rm (t) ( (t) ) hm Pm
+ ∀m ∈ M, (t) 0 ≤ Pm ≤ Pmax , ∀m ∈ M, (t) LB Rm ≥ Rm , ∀m ∈ M,
f0 (x) fi (x) ≤ 1, i = 1, 2, ..., m, gl (x) = 1, l = 1, 2, ..., p,
where fi (x), i = 0, 1, 2, ..., m, are posynomials, given by (1) (2) (n) P Ki aik aik aik fi (x) = where dik > 0 and k=1 dik x1 x2 ...xn (j) aik ∈ R for j = 1, 2, ..., n, and gl (x), l = 1, 2, ..., p, are
(t)
. The constraint, λm ≤
(t)
, λm Lm }.
minimize(x) subject to
≤ PP,th , is equivalent to (t)
(t)
Lm +TP,th λP,m Lm Lm +Tth λ(t) λP,m Lm m Lm , , 1/(ρ +1) , TP,th Tth PP,th m
Theorem 1: The optimization problem (11) is a Geometric Programming (GP) that can be transformed into a convex optimization problem. Proof : A GP problem is an optimization problem of the form as follows [10].
. The constraint, (λm /um )(ρm +1) ≤ Pth ,
is equivalent to Rm ≥
subject to
(t) λm Lm 1/(ρm +1)
Pth
≤ TP,th , requires that the prioritized
packets at aggregator m at time slot t have a average queueing delay no larger than a threshold TP,th . The fourth constraint, 1/u(t) m (t) (t) 1−λm /um
(t)
LB where Rm = max{
Transmission rate [Kbps]
Transmission power [mW]
150
100
50
0
1
2
3
4
5
6
7
8
9 10
20
15
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5
0
1
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Aggregator no.
(a)
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5
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(b)
Fig. 2. Optimal results obtained by solving the QoS optimization problem: (a) transmission powers, and (b) transmission rates tion problem, which can then be solved efficiently using the primal-dual interior-point method [11]. The globally optimal solution to the original optimization problem (9) will be ob(t)∗ (t)∗ (t)∗ (t)∗ tained by Pm = exp(vm ) and Rm = exp(zm ), ∀m ∈ (t)∗ (t)∗ M, where vm and zm are the optimal solution to the convex optimization problem (13). 4. SIMULATIONS In the simulations, the number of the patients in a 3G cell area is set to 10 if not specified particularly, the distance from the patient to the base station is uniformly distributed between 50 m and 200 m. The channel gain from aggregator m to the base station is given by hm = 200/d4m, where dm is the distance from aggregator m to the base station. The mean arrival rate of the packets at an aggregator is uniformly distributed between 50 and 80 packets/second. The mean arrival rate of the prioritized packets at an aggregator is uniformly distributed between 5 and 10 packets/second. In the CDMA model, we set W = 500KHz, δ = 0.1, and N0 = 10−13 W/Hz. The average packet length is 200 bits for all aggregators. The length of the queue at the aggregator is set to 10 packets. In the setting of QoS thresholds, we set the threshold of transmission BER to 10−5 , the threshold of the congestion PLR for the prioritized packets at an aggregator to 0.01, the threshold of the congestion PLR for all the packets at an aggregator to 0.05, the threshold of the queueing delay for the prioritized packets at an aggregator to 0.1 s, and the threshold of the queueing delay for all the packets at an aggregator to 0.3 s. The maximum transmission power is set to 1.0 W for all aggregators. We optimally allocate both the transmission power and the transmission rate at each aggregator to ensure that the data delivery from each aggreator meets the QoS requirement. The optimal values of the transmission powers and the transmission rates, as shown in Fig. 2, are obtained by solving the optimization problem (9). From Fig. 2(a), we see that the transmission powers vary greatly among different aggregators. The aggregators close to the base station (e.g., aggregators 1, 2, and 7) apply a lower transmission power than those
far away from the base station (e.g., aggregators 3 and 6). From Fig. 2(b), we see that the transmission rates vary little among different aggregators. In Fig. 3, we compare the PLR and the delay of the prioritized packets and all the packets at the aggregator, respectively, among three schemes: 1) the scheme with Jointly Optimized Power and Rate (JOPR), which is the proposed solution to the optimization problem (9), 2) the scheme with Equal Power and Equal Rate (EPER), in which the transmission powers and the transmission rates are equally allocated among the aggregators, and 3) the scheme with Proportional Power and Equal Rate (PPER), in which the transmission power at an aggregator is proportional to the distance from the aggregator to the base station, and the transmission rates are equally allocated among the aggregators. In order for fair comparison, the sum of the transmission powers and the sum of the transmission rates of all aggregators are equal among the three schemes. The proposed scheme optimizes the resources to provide QoS guarantee. The PLR for the prioritized packets at each aggregator is equal to 0.002, as shown in Fig. 3(a), and the PLR for all the packets at each aggregator is equal to 0.052, as shown in Fig. 3(b). The EPER scheme or PPER scheme does not allocate the resources appropriately, thus some far-way aggregators (e.g., aggregators 3 and 6) suffer from a larger PLR, as shown in Fig. 3(a) and Fig. 3(b). From Fig. 3(c) and Fig. 3(d), we can see that the difference of the queueing delays among the three schemes is very small. This is because the transmission rates, which determine the queueing delays, are close in the three schemes. At each aggregator, the prioritized packets are transmitted in priority, thus having a much smaller delay than the other packets. We vary the number of the patients from 5 to 20 in a cell area, and then compare the PLRs. As shown in Fig. 4, the average PLRs of the prioritized packets and all the packets at the aggregator in the proposed JOPR scheme are 0.002 and 0.052, respectively, much lower than those in the EPER scheme or the PPER scheme. 5. CONCLUSION QoS is an important issue in pervasive health monitoring systems. In this paper, we formulate and solve the QoS optimization problem, in which we jointly optimize the transmission power and the transmission rate at each aggregator to provide QoS guarantee to data delivery. The simulation results demonstrate that the optimal resource allocations improve the service quality of the health monitoring system in terms of delay and PLR. 6. REFERENCES [1] K. Kinsella, and V. Velkoff, An Aging World: 2001, U.S. Census Bureau, 2001.
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EPER
PPER
JOPR
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PPER
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(a)
PPER
JOPR
0.4 0.2 0
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EPER
0.6
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Number of patients
(b)
0.8 0.6
Fig. 4. Comparison of PLR with different number of patients in a cell area: (a) average PLR of prioritized packets, and (b) average PLR of all packets
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[2] F. Chiti, R. Fantacci, F. Archetti, E. Messina, and D. Toscani, “An Integrated Communications Framework for Context Aware Continuous Monitoring with Body Sensor Networks,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 4, pp. 379-386, May 2009.
(a)
PLR of all packets
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PPER
JOPR
1 0.8 0.6
[3] G.Z. Yang, Body Sensor Networks, Springer Press, 2006.
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[4] R. Istepanian, N. Philip, and M. Martini, “Medical QoS Provision Based on Reinforcement Learning in Ultrasound Streaming over 3.5G Wireless Systems,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 4, pp. 566-574, May, 2009.
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[5] G. Zhou, J. Liu, C. Wan, M. Yarvis, and J. Stankovic, “BodyQoS: Adaptive and Radio-Agnostic QoS for Body Sensor Networks,” in Proc. of IEEE INFOCOM, pp. 565573, Apr. 2008.
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[7] R.D. Yates and D. J. Goodman, Probability and Stochastic Processes, a friendly introduction for electrical and computer engineering, 2nd Edition, John Wiley & Sons Inc., 2004. [8] N. U. Prabhu, Foundations of Queueing Theory, Springer Press, 1997.
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[6] B. Otal, L. Alonso, and C. Verikoukis, “Highly Reliable Energy-Saving MAC for Wireless Body Sensor Networks in Healthcare Systems,” IEEE J. on Selected Areas in Communications, vol. 27, no. 4, pp. 553-565, May, 2009.
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Aggregator no. (d)
Fig. 3. Comparison of QoS metrics: (a) PLR of prioritized packets, (b) PLR of all packets, (c) queueing delay of prioritized packets, and (d) queueing delay of all packets
[9] T. Shu, M. Krunz, and S. Vrudhula, “Joint Optimization of Transmit Power-Time and Bit Energy Efficiency in CDMA Wireless Sensor Networks,” IEEE Transactions on Wireless Communications, vol. 5, no. 11, pp. 31093118, Nov. 2006. [10] S. Boyd, S.-J. Kim, L. Vandenberghe, and A. Hassibi, “A Tutorial on Geometric Programming,” Optimization and Engineering, vol. 8, no. 1, pp. 67-127, Apr. 2007. [11] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
