Wireless Sensor Network, 2011, 3, 322-328
doi:10.4236/wsn.2011.39035 Published Online September 2011 (http://www.SciRP.org/journal/wsn)
An Energy-Based Stochastic Model for Wireless Sensor
Networks
Yuhong Zhang1, Wei Li2
1 Department of Engineering Technology, Texas Southern University, Houston, USA
2 Department of Computer Science, Texas Southern University, Houston, USA
Received August 2, 2011; revised August 25, 2011; accepted September 6, 2011
Abstract
We propose an energy-based stochastic model of wireless sensor networks (WSNs) where each sensor node
is randomly and alternatively in an active and a sleep mode. We first investigate the sensor model and derive
the formula of the steady-state probability when there are a number of data packets in different sensor modes.
We then determine important sensor’s performance measures in terms of energy consumptions, average data
delay and throughput. The novelty of this paper is in its development of a stochastic model in WSN with ac-
tive/sleep feature and the explicit results obtained for above mentioned energy consumption and performance
characteristics. These results are expected to be useful as the fundamental results in the theoretical analysis
and design of various hybrid WSNs with power mode consideration.
Keywords: Energy Efficiency, Wireless Sensor Networks, Energy-Based Stochastic Model
1. Introduction
Recently wireless sensor networks (WSNs) have re-
ceived more and more attention due to their potential in
civil and military applications as well as the advances in
micro-electromechanical systems (MEMS) technology
[1]. WSNs are composed of a large number of sensors
equipped with limited power and radio communication
capabilities. They can be deployed in extremely hostile
environments, such as battlefield target areas, earth-
quake disaster scenarios, and inaccessible spaces inside
a chemical plant or nuclear facility to monitor environ-
mental changes or other required information. There are
a number of recent survey publications [2-9] on WSNs
outlining several major directions in the area. Also,
there are several papers and references therein which
closely related to our current investigation. For example,
the paper [10] in 2006 introduced a QoS supporting
model and an optimal energy allocation in WSNs; the
paper [3] in 2009 discussed the main directions to en-
ergy conservation in WSNs; the paper [11] in 2011 in-
vestigated several characteristics of active/sleep model
in WSNs.
Once a WSN system has been designed, additional
energy savings can be achieved using dynamic power
management (DPM), which shuts down the sensor node
if no events occur. Every component in a node can be in
different states; for example, the status of each sensor
can be in active, idle, or sleep mode, so the sensor may
transmit packets to and receive messages from others.
Each node’s sleep status corresponds to a particular com-
bination of the component power modes. Mathematically
speaking, each sensor will have a finite number of dif-
ferent statuses and the state space of each status is also
different. The sensor node stays in each status or state for
a random time and then transmits into another status or
state and stays for another random duration. A special
case is that each sensor will have only two different sta-
tus, say active and sleep, similar as those in [12-14]. The
sensor node alternatively stays in active or sleep status
for a probability distributed duration. In this paper, we
will concentrate on determining the steady-state pro-
bability of data packets in a referenced sensor node and
then the sensor’s energy consumption and the sensor’s
performance characteristics.
The rest of this paper is organized as follows. Section
2 provides the description of the modeling under invest-
tigation. Section 3 concentrates on the study of the major
performance characteristics, and Section 4 analyzes our
numerical results. The final Section 5 provides a conclu-
sion for this paper.

Y. H. ZHANG ET AL.323
2. Description of the Model
We will consider a wireless sensor network (or part of it)
where each sensor may have different characteristics in
performance. A sensor may be used as a sink, sensor
head nodes or regular sensor node. Without loss of gen-
erality, we introduce the assumption in a specific node
(or head nose) and may temporally omit the node symbol
in this section. The detailed assumptions and notations
for a node under investigation of this sensor network are
as follows.
 Each node will have two major modes: active and
sleep. In an active state, the sensor node is fully
working—may generate data, process date (receive
and relay/transmit) and keep in idle; in a sleep mode,
the node cannot interact with the external world.
 Each node have a finite capacity, say size of C, to
store the data it generated or forwarded from other
nodes for relaying purpose.
 The sensor may stay in phase R of the active mode
for a random time with exponential distribution with
mean 1

. The sensor then may either move to
phase N if there is at least one data packet waiting for
processing or move to sleep mode when there is no
data packets waiting. In the phase N of the active
mode, the sensor node may only process (transmit or
relay) data packets with random exponential time
with mean 1

, and may not be able to generate data
or receive any data for relay from other sensors.
However, 1) in the phase R of the active mode if the
total number of data packet is less than a threshold
value K, the sensor node may
a) Generate packets according to a Poisson process
with rate λ;
b) Process (transmit or relay) data packets with ran-
dom exponential time with mean 1

;
c) Relay packets coming from other sensors in accor-
dance with a Poisson process with rate λE.
2) in the phase R of the active mode, if the total num-
ber of data packet is more than the threshold value K, the
sensor node may only have the above function a) and b),
and will not be able to relay any packets from neighbor
sensor nodes.
 The duration of the sensor in a sleep mode is expo-
nentially distributed with mean 1

. When a sensor
is in a sleep mode, it may disconnect with external
world.
Power consumption models of the radio in embedded
devices must take both transceiver and start-up power
consumption into account along with an accurate model
of the amplifier. The latter actually becomes dominant
with small packet sizes and long transition times to re-
ceive mode because of frequency synthesizer settle down
time. In general, the energy consumed per bit in trans-
mission is given [10,15] in terms of the energy per bit
needed by the transmitter electronics (including the cost
of startup energy), the receiver electronics, the consump-
tion of the transmitting amplifier to send one bit over one
unit distance, the specific distance and the path loss fac-
tor etc. In this paper, we will consider the energy con-
sumption in terms of the number the packet transmitted
and the sensor mode status, and will use the following
notations:
tr : the transmitter power consumption per data packet
in phase R of the active mode;
e
tn : the transmitter power consumption per data
packet in phase N of the active mode;
e
as : the power consumption when sensor switches
from the active mode to the sleep mode;
e
s
a: the power consumption when sensor switches
from the sleep mode to the active mode.
e
3. Performance Characteristics
3.1. Distribution of the Number of Data Packets
in Sensor Node
The purpose of this section is to derive the formula for
the steady-state probability of the node when there are i
() packets (including the one being processing and
the others being waiting) in the sensor node. Denote by
0i
 )( n RP as the steady-state probability of the node
when there are n (Cn 

0) data packets in refer-
enced sensor node and the node is in phase R of ac-
tive mode;
 )( n NP as the steady-state probability of the node
when there are n (Cn 

0) data packets in refer-
enced sensor node and the node is in phase N of ac-
tive mode;
 )(SP as the steady-state probability of the node when
the node is in the sleep mode.
In order to reach our result, we need to introduce three
stochastic processes. One is the mode status of the node
at time t, say I(t). The space of this process consists of
phase R in active mode, phase N in active mode or sleep
mode. The second process is the number of packets in
phase R of the active mode node at time t, say XI(t). The
space of this process is from zero to the maximum ca-
pacity of sensor, C, when I(t) is at phase R of active
mode; The third process is the number of packets in
phase N of the active mode node at time t, say YI(t). The
space of this process is from one to the maximum capac-
ity of sensor when I(t) is at phase N of active mode. Based
on the description of the sensor node as proposed in the
previous section and by noting a similar but different
consideration in our papers [16,17], we will determine

Y. H. ZHANG ET AL. 324

that the joint process
 

,
II
X
tYt forms a multiple
dimensional Markov process with the transition rate dia-
gram as in the Figure 1. In this Figure, the cycle notation
with Ri means that the referenced sensor node is in phase
R of active mode and there are i data packets in the ref-
erenced sensor node; the cycle notation with Ni inside
means that the referenced sensor node is in phase N of
active mode and there are i data packets in the refer-
enced sensor node; the cycle notation with S means that
the referenced sensor node is in sleep mode. By noting
above transition rate diagram, the corresponding transi-
tion rate matrix, say Q, of the constructed two dimen-
sional Markov Process



,
II


X
tYt can now be
given by:
00
111
111
0 000
000
000
000 0
CCC
CC
EA
BEA
Q
BEA
BE














,
where the matrices i
A
, and are given as fol-
lows.
i
Bi E
 Matrix i
A

 refers to an arrival of
a data packet, through both sensor’s sensing and/or re-
laying from the neighbor nodes (for )
or only through sensor’s sensing
(for 1,,1), when there are already i
data packets in the sensor, and

1, 2, , 1iC
K C 
0,1, ,1iK
,iK
a) If , 0,1, ,1iK
i
A
is a 2 × 2 matrix given by
0
00
E
i
A
 




1
.
(1)
b) If ,
,1,,iKK C i
A
is a 2 × 2 matrix
given by
0
00
i
A






. (2)
 Matrix i B

C refers to a completion
of data packet transmission when there are i data
packets in the sensor node, and

1, 2, , i
0
0
i
B







. (3)
α
β
α α α
Figure 1. Transition rate diagram of the Markov Process
for sensor status.
 Matrix i E (


C refers to no change
in the total number of data packets in the sensor when
there are i data packets in the sensor, and
0,1, 2, , i
a) if i = 0, then


0
0
E
E
 










.
b) if 1,2,,1iK

, then


0
E
i
E

 









.
c) if ,1,,iKKC1

, then


0
i
E

 









.
d) if ,iC

then


0
C
E











.
Let




00
π,PR PS




 and



,.
nnn
PR PN 


For 1, 2,, nC


n
, by using Lemma 3 in [18], if we
denote for matrices , the
steady-state probability can be determined by
12
1
i
i
bbb b


n
i
12
,,,
n
bb b

1
01
1
ππ
n
ni
i
AD








, (4)
where
i
D


0, 1,, i
C
Care recursively calculated by
C
DE

and for 0,1,,1nC

,


1
1nnn nn
DEADB


 1
, (5)
and is determined by and
0
π00 π0D

1
01
11
π1
n C
ii
ni
I
AD e











. (6)
From above result, if denote by ,
and , we will have
1
1
0
e 
 2
0
1
e 

1
1
e 



1
π
nn
PR e for ; (7) 0,1, ,nC


02
π;PS e (8)


2
π
nn
PN e for . (9) 1,2, ,nC
3.2. Energy Consumption Measure of the Sensor
As long as the result in the formulas (4) and (6) of steady
probability is derived, we can find various energy con-
sumption measures in milliwatt-second (mW*s) of the

Y. H. ZHANG ET AL.325
sensor node. We will list the following results to demon-
strate how to utilize this formula to obtain the sensor
node measures.
 The average energy expended per unit time switch-
ing from active mode to sleep mode, denoted by EAS.
Since the sensor will consume eas milliwatt (mW) in
power each time when the sensor switches from the
active mode to the sleep model, and by noting the
expected switching number from active model
to sleep mode per unit time is

1
C
i
i
PR


, we will
have

1
11
π
CC
ASi asias
ii
EPRe ee





.
 The average energy expended per unit time switch-
ing from sleep mode to active mode, denoted by ESA.
Since the sensor will consume esa mW each time
when the sensor switches from the sleep mode to the
active model, and by noting the expected switching
number from sleep model to active mode per unit
time is

PS

, we will have

02
π
SAsa sa
EPSe ee

 .
 The average energy consumption when the sensor
node is in the phase R of active mode, denoted by
ETR. Since the sensor will consume etr mW for trans-
mitting each data packet in phase R of active mode,
and the expected number of data packets in the phase
R of active mode is , we will have
1
1
π
C
i
i
ie



1
11
π
CC
TRi tritr
ii
EiPReie



e
e
.
 The average energy consumption when the sensor
node is in the phase N, denoted by ETN. Since the
sensor will consume etn mW for transmitting each
data packet in the phase N, and the expected number
of data packets in the phase N is , we will
have

1
C
i
i
iP N



2
11
π
CC
TNi tnitn
ii
EiPNeie



.
3.3. Sensor Performance Metrics
We now discuss several major sensor performance met-
rics as follows.
 The average delay of a data packet in the sensor,
denoted by D. Since the rate of the sensor’s sensing
data is λ and the rate of sensors’ relay request from
other sensors is λE, by using the Little’s law [19], we
will have
 
 
1
1
11
1
1
11
ππ.
K
ii
i
E
C
ii
iK
KC
ii
iiK
E
DiPRP
iPR PN
ie ie


 



 

N

 







 If we denote the throughput of a sensor node as the
average number of the data packets transmitted from
the sensor per unit time, denoted by Tn, then
  
0
11
π1π
niii
ii
TPRPNee




 


.
Remark: Based on our above explicit results, we
would like to point out that
 As all above measures are determined by several 2-
dimensional vector or 2-dimensional matrix, one will
easily conduct the related numeral analysis.
 As the sensor node has to relay all data packets in the
node when the sensor’s mode is changed from phase
R to phase N, the probability that the sensor is in
sleep mode should be less than


 , and the
probability that the sensor is in active mode should be
bigger than



 .
3.4. Wireless Sensor Network Model
Now, we will briefly introduce our method for analyzing
the hybrid WSNs. The whole wireless sensor network
consists of all above mentioned sensor node with active
mode in R phase and in N phase, and sleep mode. But the
sensors’ characterization such as energy installed and the
period of active and sleep mode in a sensor may different.
We will not approximate this sensor network model as an
open queueing network as described in paper [20] or [21].
The key idea in our research is to introduce a trigger
strategy as we presented recently in papers [22-24]. Spe-
cifically, we will consider a protocol mechanism under
which a potential relayed message n to a sensor node
with phase N in active mode or sleep mode is either lost
from the network or is deemed as a relayed message to a
neighbor sensor node with certain probability. Most of
the routing protocols in mobile ad hoc networks follow
this idea. Under our proposed protocol in hybrid WSNs,
we may prove that the stationary distribution of the
wireless sensor network has a product form. This allows
us to derive explicit expressions for relay message rate
from a sensor node (in N phase and sleep model) to an-
other node, and the message lost probability for a gener-
ated message and relayed message.

Y. H. ZHANG ET AL. 326
4. Numerical Analysis
To verify the validity of the model and our analytical
expressions obtained in the previous section, we have
done some further numerical analysis in this section. The
numerical results for a set of specific parameters for this
network are presented in this section. The performance
measures considered here are several energy consump-
tions, package time delay, and the throughput. We as-
sume the data storage capacity of the sensor network is
20 (C = 20) and the threshold value K is 5, 10 and 15
respectively. We will observe and compare the effects of
various performance measures in the three cases (K = 5,
10 and 15) versa the rate λ of the active sensor node on
generating message, and allow the λ changes from 0.05
to 0.5. The other parameters for this sensor network are
listed in Table 1.
Figure 2 shows the energy consumption when the sen-
sor node switches from the active mode to the sleep
mode. It is clear that the sensor may consume more en-
ergy when the generating rate is increase. Also, from this
figure, we know that the more the threshold of reserved
capacity in a sensor node, the more the energy the sensor
node may consume, although the energy assumption
does not change too much. Thus, from the viewpoint of
minimizing the energy consumption for switching from
active mode to sleep mode, it is optimal to minimize the
number of data packets which may need relay through
the sensor node under investigation. By considering the
relay is a vital property for wireless sensor network, it is
imperative to find an optimal threshold value of K to
leverage the energy consumption.
Figure 3 shows the energy consumption when the sen-
sor node switches from the sleep mode to the active
mode. In this case, the energy consumption is not in-
creasing with the sensor’s generating rate but decreases a
little bit. This is because that the increased generating
rate may increase the average staying time of the sensor
in state R and N, and reduce the number of the sleep over
a unit observation time and then reduce the energy con-
sumption of switching from sleep mode to active mode.
In addition, this energy consumption is increasing as the
threshold of K is increasing. Thus, similar as in the dis-
cussion for Figure 1, we also need to find an optimal
threshold value of K to leverage the energy consumption.
The curves in Figure 4 and Figure 5 show that with
the increase of the generating rate λ, both the energy as-
sumptions in phase N and R will increase because that
Table 1. The specific value of the parameters.
0.2
E
  0.51

 0.05


0.1
 
31 μW
tr
e 11 μW
tn
e 0.01 μW
as
e 0.5 μW
sa
e
Figure 2. Energy consumption for switching from active to
sleep mode.
Figure 3. Energy consumption for switching from sleep to
active mode.
Figure 4. Energy consumption for transmitting in phase N.

Y. H. ZHANG ET AL.327
Figure 5. Energy consumption for transmitting in phase R.
transmitting more package will spend more energy. Also,
these energy consumptions are increased as the sensor
generates more data.
The average data delay is depicted in Figure 6 when
the sensor generates more data. This delay is obviously
increased when either the sensor’s generate rate is in-
creased or more relayed message would be processed.
From the Figure 7 on the throughput of the sensor, we
know that the throughput will increase when either the
sensor’s generating rate increases or more relayed mes-
sage is processed. Based on above analysis, it is clear
that a threshold value of K should be identified to lever-
age the energy consumption, which is in the list of our
research agenda.
5. Conclusions
In this paper, we investigated several characteristics of
Figure 6. Average data delay vs the increase of sensor
node’s generating rate.
Figure 7. Throughput vs the increase of sensor node’s gen-
erating rate.
active/sleep mode in wireless sensor network (WSN).
We first derived the steady-state probability distribution
when the sensor is in different modes and then provided
sensor’s energy consumption measures and performance
characteristics. These technical results may have a great
intention to the theoretical analysis of various WSNs
with consideration of active and sleep features. We note
that the analytic result in the research of WSN is impor-
tant but often hard to derive. With the analytical method
from this paper, we foresee other related results in the
future.
6. Acknowledgements
This work was supported in part by NSF under grants
CNS-1059116 to Yuhong Zhang and HRD-1137732 to
Wei Li, and by AFOSR under grant FA-9550-10-1-0128
to Wei Li.
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