Markov Chain-based Analytical Model of

Opportunistic Routing Protocol for Wireless Sensor

Networks

Mohd Ezanee Rusli, Richard Harris and Amal Punchihewa

School of Engineering and Advanced Technology

Massey University

Palmerston North, New Zealand

{m.e.rusli,r.harris,g.a.punchihewa}@massey.ac.nz

Abstract — In a resource constrained wireless sensor network

(WSN), communication tasks can be expensive in terms of

power consumption. Thus, when good communication is

required, a mechanism that can maximize the chances of a

successful transmission of information via wireless network is

very desirable. The Opportunistic Routing (OR) protocol [1, 2]

has been proposed as an alternative and efficient method for

exploiting the spatial and temporal characteristics of a wireless

network. In this paper, we propose an analytical framework

model based on Markov Chain of OR and M/D/1/K queue to

measure its performance in term of end-to-end delay and

reliability in wireless sensor network. The performance

predicted by the proposed framework is validated through

simulation and is in very good agreement. We also discuss the

enhancement strategy on OR to achieve better QoS guarantee

in term of delay constraint.

Keywords-Opportunistic Routing, Analytical Framework,

Wireless Sensor Networks, QoS

I. INTRODUCTION

Research in the area of Wireless Sensor Networks (WSNs) has increased in importance in recent years. In ad hoc wireless networks, resource limitations i.e. power and memory pose significant problems during their operation. Typical WSNs operate in a distributed and collaborative manner through interaction among nodes of homogenous/heterogeneous WSNs to process data cooperatively from nodes at the network’s edge to satisfy shared mission objectives

Routing is an important requirement in networking for data communication systems. There are three main elements needed for a routing protocol: a destination specification, routing objectives and routing strategies. Most routing protocols are based on a fixed destination specification, routing objectives and routing strategies tend to follow a layered scheme. In both general purpose wired networks and in wireless networks, a shortest path (or minimum cost) routing paradigm has typically been adopted, where a single shortest path between any source-destination pair is computed. Implicitly, this is referred to as deterministic routing. In WSNs, to account for and accommodate resource constraints and frequent disruption and node failures in a challenging environment, routing for WSNs must be

carefully designed and optimized - ideally with the ability to locally adapt to changes in data rates and network conditions. In addition, a typical communication pattern in WSNs, involves the implementation of sensor and sink nodes. Detailed surveys on routing protocols for wireless sensor networks can be found in [3]

Opportunistic Routing (OR) is an alternative routing approach proposed to overcome some of the deficiencies of conventional routing when it is applied in a wireless network environment [1, 2]. This concept implements a different approach from traditional routing techniques and has the objective of exploiting both the spatial and temporal diversity of wireless networks. In OR, the choice of the next relay node in a path is made in an online manner and after packets have been transmitted. It integrates the network and MAC layers where the network layer passes down a number of forwarding nodes from a Candidate Relay Set (CRS) and the MAC layer makes a final decision on the node to use for relaying, depending on current connectivity.

There have been a number of proposed variants of OR that have adopted the concept of opportunistic transmission aiming at the exploitation of the spatial and temporal diversity of wireless networks. In general, the two most common approaches that have been proposed fall into 2 categories:

i) OR with random and unlimited CRS [4-6]

ii) OR with predetermined and limited CRS [1, 7, 8]

The first type of OR is usually suitable in an environment where energy is not a major issue; whereas the second type tries to ensure that the potential relay nodes taking part in forming the source-destination path have the maximum potential return in terms of energy efficiency. The first requires only a little initialization time for choosing the potential relay node; whereas the second version does require initialization time to determine the set of relays that can be used by each node for each transmission. This will be the version that will be explored in detail in this paper due to its potential use in WSNs.

Although many opportunistic routing algorithms and protocols have been proposed in recent years, most of the

978-1-4244-6890-4/10$26.00 ©2010 IEEE TENCON 2010

257

performance analysis was done through simulation with few works opting to use an analytical approach to provide guidelines for future protocol design. Namely, in [9], the authors proposed a general analytical framework with specific closed-form expression for the average number of transmissions. An analysis of the upper bound speed that opportunistic routing can gain is studied by the authors in [10]. The performance of OR under the effects of a channel fading propagation model on the packet reception rate or link quality was investigated with an optimistic assumption that the coordination cost between network nodes is negligible [11]. In this paper, we propose a Markov chain-based with queuing model analytical framework for modeling the performance of OR in low power wireless sensor networks. In addition, our proposed analytical model incorporate coordination cost in term of coordination time rather than assuming it to be negligible which is deemed to be to idealistic and not practical. Based on the performance analysis of the standard OR using this framework, the OR protocol can be systematically tuned to increase its performance. The requirement to enhance its performance arises from the need to have a QoS requirement for WSN applications.

We also propose a simple mechanism to increase the chances of meeting the QoS delay requirement by adaptively and dynamically changing the delay time during the coordination procedure that takes into account the quality score points of candidate relay nodes.

The rest of this paper is organized as follows. Section II explains the procedure involves in Opportunistic Routing protocol. In Section III, the system model will be presented which is then followed with the detailed explanation of the proposed analytical framework in Section IV. The protocol implementation and the numerical results are discussed in Section V and VI. A brief discussion regarding the QoS requirement in OR and mechanism to incorporate it is covered in Section VII. Finally, Section VIII concludes the paper and proposes future direction for the research.

II. OPPORTUNISTIC ROUTING PROTOCOL DESCRIPTION

In location-based OR, its performance depends heavily on

several important factors:

• Selection of the forwarding candidate • Prioritization of the candidate

• Candidate coordination

A. Selection of the Forwarding Candidate

Due to the nature of wireless networks, every transmission

is, by default, a broadcast operation. All nodes within

transmission range can listen to messages and achieve a

different packet reception rate. In addition, recent studies

have shown that besides there being “connected” and

“disconnected” regions, many links are within an area

known as a “transitional region” and they may actually have

the highest energy efficiency [12-14]. However, in order to

ensure that each transmission is efficient in terms of energy,

reliability and accuracy, a proper selection metric must be

devised. This metric must make sure that only relevant

nodes are involved or considered during each transmission.

Ideally, the metrics should jointly take into account several

factors such as the packet reception ratio, distance, packet

forwarding time and hop advancement.

The main objective of the selection procedure is to

determine the set of relay candidates that give an optimal

packet forwarding efficiency. The trade-off involving the

number of candidates in the set needs to be carefully

analysed. Having many candidate relays in the set often

results in a decrease in the forwarding costs. Nevertheless,

some neighbours do not make progress as expected towards

the destination. Furthermore, in a dense network, increasing

the number of candidates can potentially increase the

overhead due to redundant transmissions as well as leading

to inefficient energy utilisation.

B. Prioritization of the Candidates

Having decided upon the number of nodes in the

Candidate Relay Set (CRS), the next challenge is to rank

these nodes according to a priority order. This ranking is

important for maximizing or minimizing the objective

criteria for each transmission. Examples of objective criteria

are lifetime, delay and throughput. This ranking is also

important for the coordination between selected nodes in the

CRS to improve the reliability of each transmission. The

properties of metrics that will be used in calculating the

priority of each candidate must ideally reflect the condition

of the links, distances, and energy levels, etc.

C. Candidate Coordination

With the potential for needing to deal with multiple nodes

in the CRS, coordination between these nodes will be

important. Basically, this coordination is conducted at the

MAC layer. The main objective of coordinating these nodes

is to prevent duplicate packets from being forwarded to the

destination and also to acknowledge successful packet

reception for reliability purposes. One of the coordination

approaches for these nodes involves appending priority

information for each CRS candidate and then transmitting it

as an overhead. By doing this, each node in the CRS will

know about the priority of the other nodes. Based on this

information, scheduling for the candidate nodes can be

performed and an acknowledgement packet can be sent to

the sender with the aim of avoiding collisions and reducing

congestion. The overall implementation of the coordination

mechanism is also challenging because the acknowledgment

procedure is also affected by wireless network conditions. A

robust mechanism is favoured in order to ensure the gain in

adopting Opportunistic Routing is high for WSNs.

III. SYSTEM MODEL

Fig. 1 illustrates a typical scenario for a wireless sensor

network modeled in our study using a source node S, which

transmits information to destination node D via

258

Sink

Source

neighbouring nodes. The information that is stored at each

node regarding its neighbouring nodes is in the form of a

tuple (dj, pj), where dj is the Euclidian distance of the

neighbour node j from node i, and pj is the packet reception

ratio (PRR) that reflects the estimated link quality between

nodes i and j. This information (dj, pj), can be determined

using a localization operation and probe messages

respectively. Additional information known as quality score

(qs), that adaptively estimates the level of quality of the

relay nodes based on the previous communication history is

also stored on each node.

Fig. 1: A typical scenario in randomly populated low power wireless

sensor network with source node, S, its neighbours (i1 – i7) and destination

node, D.

In this model, we have a set V of Vn = wireless sensor

node deployed randomly in a given area. Each node is

assigned a unique identifier { }ni

vvvvv ,...,,321

∈ and is

assumed to be attached to an omni-directional antenna and

having a finite queue buffer. Furthermore, we assume that

each node i is aware of its own location, its neighbouring

nodes and the required destination node for a message.

The wireless sensor network can be seen as a probabilistic

directed graph G(V, E, P) where Vvi∈ denotes a node and

an edge Eeji∈

, represents the communication link

between node i

v and j

v with packet reception rate (PRR)

determined by Ppji∈

,which represents a successful packet

transmission or the link quality through the radio channel

between node i and j. The PRR value is evaluated based on

a radio communication model derived using a log-normal

shadowing path loss model that takes into account the path

loss exponent and the distance between the nodes. In our

study, we assume that the WSN is deployed at isolated

places with no interference from other nodes with different

types of communication channel (i.e. 802.11 vs. 802.15.4).

Hence, links between each node and its neighbours are

assumed to be statistically independent channels [15]. The

PRR value is also assumed to be constant during the packet

delivery procedure and the consistency times of the channels

are large enough compared with packet delivery times. We

also adopted a persistent CSMA/CA procedure as our MAC

in this study with the probability that the channel is sensed

idle, and therefore leads to the successful channel contention

for a transmission between all active neighbouring nodes

NTactive. This probability is given as,

Pr{success| NTactive, M sensing slots}, 1

1

11

1

−

=

−

=∑

activeNTM

i

active

CSMAM

i

M

NTp (1)

In the persistent CSMA, number of sensing slot can be

large which will increase the probability of a node to access

the channel. Therefore, by approximating the wait time

based on a geometric distribution, (2) gives the expected

mean number of slots before a transmission can occur where

ptx is the probability of a successful channel transmission.

Hence, we can let txCSMAsucc

ppp .= to be the successful

probability of overall channel contention and transmission.

[ ]succ

pslotsE

1= (2)

In OR, each node will select and prioritize its candidate relay nodes to forward any packet towards its destination node. The nodes in the CRS are selected from its neighbors that are closer to the destination. Hence, these nodes should be within the overlapping area between the two circles as shown in Fig 2. In our model, we also assume that for any node in the WSN network,

}1,..2,1,,{),( −=∈ niVdsdsii

is the pair of nodes

representing the source and destination of a packet transmission respectively. Moreover, we define the set of ordered relay nodes Vr ∈ in CRS for each node according

to their priority. To simplify the prioritization procedure, the priority can is determined based on the distance to the destination from a given node in the CRS. Typically, the node in the CRS with the least distance to the destination will be assigned with the highest priority as this arrangement can maximize the expected packet progress per transmission [16].

Fig 2: Overlapping area that represents the forwarding region

IV. ANALYTICAL FRAMEWORK

In this section, the analytical framework which is based

on a Markov chain-based system with an absorbing state

combined with queueing theory will be proposed in order to

analytically evaluate the performance parameters -

especially the average end-to-end (e2e) delay and reliability

of OR in WSN.

D S

i2

i1

i4

i5

i3

(5, 95%) (15,85%)

(30,35%)

(10, 90%)

(20, 45%)

i6

i7

(10, 90%)

259

A. Absorbing Markov Chains Model

In our work, the analytical framework is based on a

Markov Chain with multiple absorbing states. We model the

routing process for a given flow between nodes i

s and

destination node D (i.e. a source-destination pair) as an

absorbing Markov Chain, in which the wireless sensor

network nodes represent the set of states,

{ }n

sssS ....,,,21

= and a transitional probability matrix,

D where each entry Ddij

∈ represents the transition

probability between states i

s andj

s in the Markov chain.

The absorbing states are associated with the destination

node and some potential dead-end nodes. A dead-end node

is established when the sensor node cannot forward any

packet that it has received due to an empty CRS set. When

this occurs it will drop any packet that is has received. In ad-

hoc WSNs, a dead-end node can cause a degradation of the

overall system performance. In addition, an absorbing state,

1+ns for unsuccessful transmission for each node is also

modeled in our framework. Basically, this state represents

the loss event for the failure of communication between

each node to all of its CRS nodes. The transition probability

between each node and its priority ordered relay in the CRS

is calculated as,

( )∏−

=

−=1

1

1

j

k

ikijijppd { }≠∩∈∈

iiCRSCRSjVji :, . (3)

Whereas, the transition probability betweeni

s and1+n

s is

determined based on the following expression:

∑=

+−=

j

j

ijnidd

1

1,1

iCRSj ∈ (4)

Finally, using the above definitions, the transition matrix )1()1( +×+ nnD is constructed.

According to the theory of Markov Chains with

absorbing states [17], the expected number of times the

sensor node is visited for a communication between i

s and

destination node, D can be determined from the fundamental

matrix of D. The fundamental matrix, F is derived

as 1)( −−= QIF provided that the matrix D with a absorbing

nodes must be in its Canonical form as in (5), where I is the

identity matrix, Q is an ( ) ( )anan −×− matrix for all

transient nodes and R is an ( ) ( )1+×− aan matrix for

absorbing nodes.

=

I

RQD

0 (5)

The entry Ffij

∈ represents the expected number of times

the node j

s is visited for routing flow between i

s and the

absorbing nodes. In addition, the packet success rate (PSR)

for a routing flow between i

s and the destination node D is

given by

∑−

=

=an

j

jDijiDrfpsr

1

. (6)

B. M/D/1/K

To analyze the end-to-end delay of our WSN adopting

OR, we have combined our MC with an M/D/1/K theory of

queuing model [18] since we are dealing with sensor nodes

with a limited memory capacity. The main requirement to

ensure the stability of the model with traffic

intensity 1≤ρ is validated by the assumption that the source

node has a low data rate ( λ ) and each node has high data

service rate ( µ ) due to small data and control packets. In

addition, the average transmission delay due to MAC

operation is determined by,

( )][][][__ BackoffMACWaitMACCSMA

TETEslotsET +×= (7)

where, TMAC_Wait and TMAC_Backoff are the uniform random

time for waiting and back-off during the CSMA channel

access procedure respectively.

The average successful packet service time on each node

in our network can then be determined as:

[ ]txCSMACOORtxsucc

pTTzTT ++= )( , (8)

where )(zTtx

is the average time necessary to transmit a

packet of size z bit and TCOOR is the expected time delay for

a node during candidate coordination procedure.

In the M/D/1/K queuing model, the average number of

packets in the queue can be computed using,

( )ρ

ρ

−=

12

2

qL . (9)

Hence, using Little’s Law the average waiting time can be

computed as:

λ

q

q

LW = . (10)

Based on this definition, the average delay on node i, for

{ }1,...,2,1 −= ni is:

succiqiiTWW += . (11)

Referring to the fundamental matrix once again, the

average end-to-end delay (e2e) between the source nodes

and specified destination node D can be determined using

the following expression:

[ ] ∑−

=

=an

j

jijDiWfeeE

1

,2 . (12)

V. PROTOCOL IMPLEMENTATION

There are many variants of Opportunistic Routing that

have been proposed for wireless networks. In our study, we

focussed our attention on the location-based Opportunistic

Routing protocol. A modified version of energy efficient

260

routing protocol proposed in [8] is adopted in which routing

metric for the selection procedure is modified to incorporate

a quality score parameter, qs. In the situation where the

quality of service factor is not taken into account, we can set

the qs value to 1 because we assume that the CRS nodes can

always support the QoS requirement. This metric calculates

the expected packet advancement (EPA) achieved using

distance, d, quality score, qs, and packet reception rate, p;

given N ordered forwarding candidate set )(Nj

ω .

∑ ∏=

−

=

⋅=r

k

k

n

njkkjkjjpqspdNEPA

1

1

0

))((ω (13)

In terms of prioritizing the selected forwarding nodes, the

criteria chosen in our study is based on distance. Basically,

the closer the node to the destination, the higher the priority

will be to continue to forward the packets towards the

destination. This information is attached to the packet as the

ordered node list where, m

iiiCRS ,..,,21

= and the index

number represents the priority of these

nodes { }mpriority

>>>= ..21ο . A different approach was also

used in [1, 4].

In terms of candidate coordination, we employ a scheme

to schedule the forwarding slot based on priority order and

an implicit acknowledgement for collision and redundancy

avoidance. Basically, when a relay node receives data

packet, it will check whether it is one of the selected

candidate relay nodes. If it is, a delay time to forward the

packet is computed according to the following equation:

)1(min

−×=priorityforward

TT ο (14)

Here, min

T is a constant representing a fixed time interval

for the delay. Otherwise, the packet will be dropped. While

waiting for its turn to forward the packet, if the node

overheard that the same packet has been forwarded towards

the destination by a higher priority node, it will also drop the

packet. This is known as implicit acknowledgement which

is useful to eliminate multiple packets from being forwarded

to the destination as well as to reduce collisions and

interference. The advantage of employing the implicit

acknowledgement mechanism is a dedicated packet

acknowledgement is not necessary and so saving the energy.

TABLE I

SIMULATION PARAMETERS

10 25 50 75 100 110

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Nodes

Avera

ge E

nd-to-e

nd D

ela

y (seconds)

Analytical Results

Simulation Results

Fig. 3: Average End-to-end Delay vs Number of Sensor Nodes

VI. NUMERICAL RESULTS

We now evaluate the performance of the OR protocol with

the proposed Markov chain-based analytical models and

verify them through simulation using Prowler [19], a Matlab-

based probabilistic wireless sensor network simulator.

Prowler provides a radio fading model for packet collisions,

static and dynamic asymmetric links and a CSMA/CA MAC

layer. In addition, to simulate a realistic channel model for a

lossy WSN network, the log-normal shadowing path loss

model derived in [20] is used to estimate the links’ packet

reception rate (PRR) in our study. The adopted model can

simulate highly unreliable links in Mica2 mote. A brief

summary of the various simulation parameters used is

presented in Table 1.

10 25 50 75 100 1100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Nodes

Packet S

uccess R

ate

(%

)

Analytical Results

Simulation Results

Fig. 4: Packet Success Rate vs. Number of Sensor Nodes

The simulated wireless sensor network has stationary

nodes that are randomly distributed within a 100 x 100 m2

square region. The source and the sink node are fixed at

opposite sides of the network and the traffic was generated

at the source node according to an exponential distribution

at a rate of 1 packet per second. All simulations are run for

10 iterations over a period of 500 seconds. The following

performance metrics are evaluated to validate our analytical

framework:

• End-to-end delay – the time delay for a packet from

the source to its destination

• Packet Success Rate – the total number of packets

received at the destinations versus the total number

of packets sent from the source. This is a measure of

reliability.

Parameter Values

Path loss exponent, α 3.5

Log-normal shadowing variance, σ 3.8

Receiver Sensitivity -105 dBm

Transmission Power 6 dBm

Packet Length 400 bits

MAC Minimum Backoff time 2.5ms

MAC Minimum Waiting time 5ms

OR Coordination Delay time, Tmin 0.1s

261

Fig. 3 and Fig. 4 show the comparison results calculated using our analytical framework and the results obtained by means of numerical simulation with 95% confidence interval for both metrics with different network density. Basically, the results confirm our proposed analytical model.

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (second)(a)

Ave

rag

e E

nd-t

o-e

nd

De

lay (

se

co

nd

s)

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (second)(b)

Pa

cke

t S

ucce

ss R

ate

(%

)

Non-QoS OR

QoS OR

Non-QoS OR

QoS OR

Fig. 5: The end-to-end delay (a) and packet success rate (b) performance

analysis between Non-QoS OR and QoS OR. Number of nodes, 50.

VII. ENHANCEMENT STRATEGY FOR BETTER QOS

In WSN applications that require certain level of QoS requirements, OR can be modified to improve overall WSN performance. In this section, we propose a simple mechanism as a strategy to handle WSNs data with specific delay constraint requirement.

A. Adaptive Delay- CRS Score Value

For data with delay constraints, the need to deliver the data on time is critical to ensure that QoS requirement is satisfied. In our OR variant, due to the use of an implicit acknowledgement capability, additional information regarding the quality of communication between a node and its CRS nodes can be inferred - based on previous communications. We refer to this as a score value which measures the ratio of successful packet transmission to nodes

in the CRS. Hence, we can set a threshold,sv

t ( 10 ≤≤sv

t )

that will be used when a node is dealing packet/data with certain time deadlines. The threshold is used to check the score value for a node of higher priority in its CRS. If the score is below the threshold, the waiting time during the coordination stage will be changed adaptively to increase the chance of delivering the packet on time. Fig. 5 shows how the overall end-to-end delay and packet success ratio can be enhanced through this strategy with a tsv threshold value of 0.5.

VIII. CONCLUSIONS AND FUTURE WORK

In this paper, we proposed a Markov Chain-based analytical framework for OR which incorporates the cost of coordination time. Numerical simulation results show that our model is in agreement within the specific confidence limits. From the performance analysis using the analytical model, enhancement strategies for better QoS guarantee of OR can be adopted. Specifically, for delay constraint data, an adaptive Delay-CRS approach was proposed. Our future work will include implementing other strategies to ensure OR can guarantee more QoS requirements.

IX. REFERENCES

[1] S. Biswas and R. Morris, "ExOR: Opportunistic Multi-Hop Routing for Wireless Networks", in SIGCOMM., 2005

[2] H. Dubois-Ferriere, M. Grossglauser, and M. Vetterli, "Least-cost Opportunistic Routing", in 45th Proceedings of the Allerton Conference on Communication, Control and Computing, 2007

[3] J.N. Al-Karaki and A.E. Kamal, "Routing Techniques in Wireless Sensor Networks: A Survey", IEEE Wireless Communications, vol. 11, pp. 6-28, 2004

[4] Z. Zifei and S. Nelakuditi, "On the Efficacy of Opportunistic Routing", in 4th Annual IEEE Communications Society Conference

on Sensor, Mesh and Ad Hoc Communications and Networks, SECON '07, 2007

[5] C. Szymon, et al., "Trading Structure for Randomness in Wireless Opportunistic Routing", in Proceedings of the 2007 Conference on

Applications, Technologies, Architectures, and Protocols for Computer Communications, 2007

[6] D. Junzhao, L. Hui, and C. Ping, "OMR: An Opportunistic Multi-Path Reliable Routing Protocol in Wireless Sensor Networks", in International Conference on Parallel Processing Workshops, 2007

[7] F. Pingyi, et al., "AsOR: An Energy Efficient Multi-Hop Opportunistic Routing Protocol for Wireless Sensor Networks Over Rayleigh Fading Channels", IEEE Transactions on Wireless Communications, vol. 8, pp. 2452-2463, 2009

[8] K. Zheng, et al., "On Geographic Collaborative Forwarding in Wireless Ad Hoc and Sensor Networks", in International Conference on Wireless Algorithms, Systems and Applications, 2007

[9] A.S. Cacciapuoti, M. Caleffi, and L. Paura, "A Theoretical Model for Opportunistic Routing in Ad Hoc Networks", in International

Conference on Ultra Modern Telecommunications & Workshops, 2009

[10] P. Jacquet, et al., "Opportunistic Routing in Wireless Ad Hoc Networks: Upper Bounds for The Packet Propagation Speed", in 5th

IEEE International Conference on Mobile Ad Hoc and Sensor Systems, MASS 08, 2008

[11] R. Zheng and C. Li, "How Good is Opportunistic Routing?: A Reality Check Under Rayleigh Fading Channels", in Proceedings of the 11th

International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems, 2008

[12] Z. Marco, et al., "Efficient Geographic Routing Over Lossy Links in Wireless Sensor Networks", ACM Trans. Sensor Network., vol. 4, pp. 1-33, 2008

[13] A. Cerpa, et al., "Temporal Properties of Low Power Wireless Links: Modeling and Implications on Multi-hop Routing", in Proceedings of

the 6th ACM International Symposium on Mobile Ad Hoc Networking and Computing, 2005

[14] W. Alec, T. Terence, and C. David, "Taming the Underlying Challenges of Reliable Multihop Routing in Sensor Networks", in

Proceedings of the 1st International Conference on Embedded Networked Sensor Systems, 2003

[15] A. Cerpa, et al., "Statistical Model of Lossy Links in Wireless Sensor Networks", in Fourth International Symposium on Information Processing in Sensor Networks, 2005

[16] H. Takagi and L. Kleinrock, "Optimal Transmission Ranges for Randomly Distributed Packet Radio Terminals", IEEE Transactions on Communications, vol. 32, pp. 246-257, 1984

[17] S.M. Ross, Stochastic Process. 2nd ed, John Wiley and Sons, 1996.

[18] U.N. Bhat, An Introduction to Queueing Theory, Birkhauser, 2008.

[19] G. Simon. "Probabilistic wireless network simulator", http://www.isis.vanderbilt.edu/projects/nest/prowler

[20] M. Zuniga and B. Krishnamachari, "Analyzing the Transitional Region in Low Power Wireless Links", in First Annual IEEE

Communications Society Conference on Sensor and Ad Hoc Communications and Networks, IEEE SECON '04, 2004

262