Propagation control of data forwarding in opportunistic underwater sensor networks

Computer Networks 114 (2017) 80–94

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Propagation control of data forwarding in opportunistic underwater sensor networks Linfeng Liu a,b,∗, Ping Wang a, Ran Wang a,c a

School of Computer Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore School of Computer Science and Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China c School of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China b

a r t i c l e

i n f o

Article history: Received 31 January 2016 Revised 16 December 2016 Accepted 16 January 2017 Available online 18 January 2017 Keywords: Opportunistic underwater sensor networks Opportunistic forwarding Energy consumption Propagation delay Data delivery ratio

a b s t r a c t Opportunistic underwater sensor networks (OUSNs) are developed for a set of underwater applications, including the tracking of underwater creatures and tactical surveillance. The data forwarding objectives of OUSNs differ significantly from those of wireless sensor networks or delay-tolerant networks due to their large energy consumption and large propagation delay underwater. This paper begins with a description of the underwater movement, which consists of the regular movement impelled by external force and the autonomous movement controlled by nodes. Then, a topology determined model is provided to generate a power-law distribution structure, where the communication links that are overlong or are close to the space boundary can be avoided. Finally, a proactive opportunistic forwarding mechanism (POFM) is proposed to minimize the energy consumption from data forwarding. In POFM, at the start of each time slot, each node without data decides whether to request for data or not independently. In particular, the probabilities of nodes requesting for data are calculated according to a dynamic percolation analysis, which suggests the nodes with larger degrees or at earlier time slots should be given larger probabilities. The performance of POFM is analyzed through simulation experiments that produce preferable tradeoff results, indicating that POFM has a minor energy consumption with a guaranteed delivery ratio and a limited propagation delay. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Recently, interest in applying wireless sensor networks (WSNs) [1] into environments to enable/enhance applications such as resource exploration and pollution monitoring has grown. Traditional WSNs consist of large numbers of sensor nodes that are randomly distributed in their detecting fields and vicinities. These nodes are sometimes assumed to be static or to move in a limited range [2]. However, in many practical applications, the movement ranges of nodes are relatively large, such as those in delay-tolerant networks [3], vehicular sensor networks, and pocket switched networks [4]. Accordingly, opportunistic mobile sensor networks (OMSNs) [5–7] are introduced for these types of networks, which are composed of mobile sensor nodes. OMSNs can achieve large-scale sensing at a lower cost compared to a ubiquitous static infrastructure of sensing devices. Nevertheless, due to the node mobility, the available contacts between nodes become scarce and short, which

∗ Corresponding author at: School of Computer Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore. E-mail address: [email protected] (L. Liu).

http://dx.doi.org/10.1016/j.comnet.2017.01.009 1389-1286/© 2017 Elsevier B.V. All rights reserved.

may lead to the unsteadiness of routing paths. Opportunistic underwater sensor networks (OUSNs) are a special case of OMSNs. OUSN technology enables various underwater applications, especially including the tracking of underwater creatures [8] and tactical surveillance [9]. In OUSNs, nodes with autonomous migration ability can move almost arbitrarily. Thus, the future trajectories of nodes are almost unpredictable. As depicted in Fig. 1, the measurements of environmental events are monitored by sensor nodes tied on mobile underwater vehicles (such as whales or submarines) and then transferred to a surface sink through multi-hops. Ultimately, the measurements are aggregated at a LEO satellite or a base station for future processing [10]. Data forwarding in OUSNs is defined as the art of finding and utilizing dynamic paths composed of intermittent contacts for multi-hop data transfer. Data forwarding in OUSNs has several challenges: 1) Energy Consumption. The battery energy of nodes is limited. Batteries usually cannot be replaced easily underwater, and solar energy is rarely exploited. Thus, one aim of OUSNs is to reduce the energy consumption. 2) Propagation Delay. Both electromagnetic waves and laser waves are unsuitable for underwater transmission. Acoustic

L. Liu et al. / Computer Networks 114 (2017) 80–94

81

2. Related work 2.1. Opportunistic forwarding in DTNs

Fig. 1. Architecture of an OUSN.

communication [11] is the typical physical layer technology in OUSNs. Under such case, another distinguishing feature of OUSNs is propagation delay since acoustic waves transmit much slower than electromagnetic waves (the speed of acoustic waves is approximately 1500 m/s). 3) Delivery Ratio. Due to the intermittent links, the data delivery from a source node to a destination node cannot be guaranteed even when the epidemic forwarding is adopted for data routing. It also remains an important issue to improve the delivery ratio. 4) Dynamic Topology. Underwater nodes always move irregularly because of the sophisticated underwater environments (e.g., water current and swimming underwater creature). Given that nodes are mobile, any links between the nodes can generate or disappear over time. Thus, the topology changes constantly, so generated data may not be able to transfer along one single path. Propagation control is vital for the opportunistic forwarding of OUSNs. The number of nodes carrying data will increase over time, and the propagation velocity (the increased number of existing data copies at every time slot) determines whether the data can be delivered under the restrictions of delivery ratio and delay. The appropriate regulation of propagation velocity for different nodes (with different degrees and positions) or at different time slots will be beneficial to the reduction of the energy consumption. Consequently, the propagation control of opportunistic forwarding is an inevitable issue in OUSNs. This research explores the data forwarding problem which aims at minimizing the energy consumption subject to the delivery ratio and propagation delay constraints. To observe the propagation process, a dynamical percolation model is constructed, and through analyzing the model, it is shown that the delivery is more possible to achieve if the nodes with larger degrees are given larger transition probabilities (more prone to request for the data) or if the propagations at earlier slots are given larger velocities. Applying these conclusions, a proactive forwarding algorithm is provided to make a tradeoff among energy consumption, propagation delay and delivery ratio via setting proper transition probabilities. The remainder of this paper is organized: Section 2 discusses related studies; in Section 3, we provide a mathematical network model to describe the data forwarding problem; the proactive opportunistic forwarding mechanism (POFM) is proposed in Section 4; Section 5 gives a POFM analysis from the aspects of complexity, energy consumption, delivery ratio and influence of moving-out probability; Section 6 discusses the performance evaluation of POFM; finally, Section 7 supplies the conclusions.

Extensive studies has been carried out on the problem of opportunistic forwarding for DTNs (Delay-Tolerant Networks) and mobile sensor networks. The early representative algorithm proposed in [12] was Epidemic Forwarding (EF), where random pair-wise exchanges of messages among mobile nodes ensured the maximum delivery and the minimum delay. However, numerous redundant message copies were generated gradually in the transmission. To reduce the overhead of flooding-based schemes (such as EF), Spyropoulos et al. [13] put forward the Spray-and-Wait algorithm, where a certain number of copies were firstly thrown into the network. Delivery was accomplished when one of the copy-holders arrived at the destination. In [14], a distributed adaptive opportunistic routing scheme that used a reinforcement learning framework was proposed. The scheme can opportunistically route the packets even in the absence of reliable knowledge about channel statistics and network models. Radunovic’ et al. [15] proposed an optimization framework for opportunistic routing, which was eased by network coding. The framework was used to define notions of credits associated with a number of packets in a generation. A primal-dual algorithm was then adopted as the basis for deriving a practical protocol. 2.2. Prediction method in opportunistic forwarding There also exist some studies considering the data forwarding based on mobility prediction or contacting prediction, which is most common method in opportunistic forwarding techniques. For instance, in [16], MaxProp prioritized both the schedule of packets transmitted to other peers and the schedule of packets to be dropped. Each node should keep track of the probabilities of contacting other nodes. Subsequently, contacting history updated the probabilities. LeBrun et al. proposed the Motion Vector (MoVe) [17], which applied movement speed information to make intelligent opportunistic forwarding decisions. MoVe leveraged knowledge regarding the relative speeds of a mobile router and its neighboring nodes to predict the closest distance from trajectories. In MoVe, nodes move in piece-wise linear fashion, following city street structures. The decision of forwarders relies on velocities of nodes, that is, the best result may be achieved if movements of nodes are rarely changed. Niu et al. [18] developed the Predict and Spread (PreS) routing algorithm, in which an adaptive markov chain is adopted to model the node mobility patterns and capture the social characteristics. The simulation results suggested that PreS can improve delivery ratio and reduce delivery latency when proper parameters were set. In [19] the routing algorithm Predict and Relay (PER) for delay-tolerant networks was introduced. PER was based on the assumption that nodes usually move around a set of well-visited landmark points and that node mobility could be predicted if sufficient history information was provided. In spite of that, the regular mobility hypothesis of the above research is unsuitable for OUSNs. 2.3. Data forwarding in OUSNs Relevant research has also been conducted to solve the data forwarding problem in underwater sensor networks. In [20], a network-coding-based protocol named Multiple Paths and Network Coding was proposed, and three disjoint paths were established for different groups of packets. Lee et al. [21] presented the Hydraulic Pressure based Anycast Routing Protocol (HydroCast) for reliable underwater sensor event reporting to one of the surface sinks. HydroCast selected the proper subset of forwarders that maximized

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greedy progress and limited interference. Besides, the follow-up work can be found in [22], where sequence number, hop count, and depth information were used to determine the direction of next-hop and to build a directional trail to the closest sonobuoy. In [23], a generic prediction assisted single-copy routing (PASR) scheme was investigated. The scheme can be configured for multiple mobility models. It differentiated the network mobility patterns according to short-duration trace, and then depicted the features of the best routing paths. The outstanding advantage of PASR is that it was self-adaptive for various node mobilities. However, PASR continued to rely on the historical information to instantiate prediction. The historical information is worthless in OUSNs due to the sophisticated mobility of nodes. Recently, a routing protocol MobiSink (mobile sink) for underwater sensor networks was presented [24] to balance the load on the intermediate nodes through deploying some mobile sink in four horizontal regions. Obviously, the cost of these mobile sinks was extremely expensive, and thereby the availability of MobiSink was restricted. Coutinho et al. [25] proposed the GEographic and opportunistic routing with Depth Adjustment-based topology control (GEDAR) routing protocol. GEDAR was an anycast, geographic and opportunistic routing protocol that routed data packets from sensor nodes to multiple sonobuoys (sinks) at the seas surface. When the node was in a communication void region, GEDAR tried to recover it based on topology control through the depth adjustment of the void nodes. Actually, the depth adjustment of nodes is hard to be realized due to the underwater harsh environment. 2.4. Energy consumption issue in data forwarding The energy consumption issue in opportunistic forwarding also attracts some researchers’ attention. Mao et al. [26] studied how to select and prioritize forwarding lists to minimize the total energy cost of forwarding data to the sink node, and then proposed an energy-efficient opportunistic routing strategy (EEOR). Zuo et al. [27] developed a cross-layer operation aided energy efficient opportunistic routing algorithm for ad hoc networks. The performance of this algorithm was shown to be close to the bound found from an exhaustive search. In [28], a new metric that enabled each sensor to determine a suitable set of forwarders and their relay priorities was exploited. Then, an energy-efficient routing scheme (EFFORT) was presented to maximize the amount of gathered data. Zheng et al. [29] proposed the redundancy based adaptive routing (RBAR) for underwater sensor networks, and it was shown that RBAR achieved a preferable energy efficiency while satisfying different delay requirements of various packets by explicitly controlling the replication process, except that the delivery ratio was not very desirable. A beam width and direction concerned routing protocol (BDCR) [30] was developed for data forwarding problem. Note that BDCR can obtain relatively high delivery ratio and ensure reasonable energy consumption by considering the beam width and three-dimensional direction. 2.5. Motivation of this work Considering the characteristics of OUSNs, energy consumption, delivery ratio and propagation delay should be taken as optimization objectives of data forwarding. Specifically, energy should be conserved as much as possible while proper delivery ratio and propagation delay are guaranteed. Data forwarding in OUSNs is a dynamic process with multifarious unpredictable variations, which affect the final performance of data forwarding significantly. Especially, the propagation velocity is vital for the performance improvement, and a dynamic control of the propagation velocity is advantageous to the expeditious deliveries. However, the propagation control has never been explored

exhaustively in the aforementioned studies. Such gap motivates the propagation control solution advocated in this paper. In this work, the propagation control of data forwarding is treated as an efficient way of achieving these objectives, and a solution is proposed to minimize the energy consumption subject to the delivery ratio and propagation delay constraints. 3. Problem formulation Suppose that a set of mobile nodes (V = {V1 , V2 , · · · , VN }) is uniformly deployed in a convex space D ∈ R3 , |D| = L × W × H. The time is divided into discrete time slots with equal length TS . The explanations of the main notations are presented in Table 1. 3.1. Definitions A three-dimensional node mobility model is put forward in Definition 1. Then, we present Definitions 2–5 to clarify the OUSN model and simplify the problem. Definition 1. Nodes. ∀ Vi ∈ V has a maximum communication range that is denoted as RCmax , and a current radius RC(i)(t) (RC(i)(t) ≤ RCmax ) at the tth slot. The nodes which carry one data copy are called data − holding, and the nodes without data are called data − not − holding. Every node has χ + 1 communication radius levels, i.e., RC0 < RC1 < RC2 < · · · < RCψ < · · · < RCχ = RCmax , and let RCψ = aˆ + bˆ · ψ . The autonomous movement speed of Vi obeys the uniform distribution U (0, MR TS ), where MR is the maximum movement range of autonomous movement during each slot. To simulate the underwater movement of nodes, the coordinate of Vi at the t + 1th slot C (i )(t+1 ) is computed as:

C (i )(t+1) = (x(i )(t+1) +  X (t+1) , y(i )(t+1) +  Y (t+1) , z(i )(t+1) +  Z (t+1) ),

(1)

where ( X (t+1 ) ,  Y (t+1 ) ,  Z (t+1 ) ) denotes the coordinate deviation vector caused by the external force from water current, and (x(i )(t+1) , y(i )(t+1) , z(i )(t+1) ) reflects the autonomous movement of nodes by themselves. The movement range at each slot is confined to:

 |x(i )(t+1) − x(i )(t ) | ≤ |MR · sin (i )(t ) · cos (i )(t ) | |y(i )(t+1) − y(i )(t ) | ≤ |MR · sin (i )(t ) · sin (i )(t ) | |z(i )(t+1) − z(i )(t ) | ≤ |MR · cos (i )(t ) |,

where (i)(t) and (i)(t) are moving azimuth angles. (i)(t) obeys the uniform distribution U(0, π ), and (i)(t) obeys the uniform distribution U(0, 2π ). Besides, the coordinate deviation is expressed as:

⎧   T t1 fx(t ) ⎪ ⎪  X (t+1) = vx(t ) TS + 0 S dt2 dt1 ⎪ ⎪ M ⎪ 0 ⎨  TS  t1 (t ) fy  Y (t+1) = vy(t ) TS + dt2 dt1 ⎪ M 0 ⎪ ⎪  0t1 (t ) ⎪ fz ⎪ ⎩ Z (t+1) = vz(t ) TS +  TS dt2 dt1 , 0 0

(t )

(t )

M

(t )

where (vx , vy , vz ) is the speed vector of water current at the tth slot; M is the water mass of unit volume (the mass of nodes is far smaller than that of water) and F = ( fx(t ) , fy(t ) , fz(t ) ) is the external force, such as gravity, pressure gradient, Coriolis force and shear stress [31], as shown in Fig. 2. The typical underwater force vector is given in the Appendix. Fig. 3 depicts an example of the node mobility during 100 slots. There are five trajectories of different nodes. Definition 2 (Links and Paths). ∀Vi , V j ∈ V , if d(i, j )(t ) ≤ RC (i )(t ) , then (i, j )(t ) ∈ E is a unidirectional link. ∀Vi , V j ∈ V , if links

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83

Table 1 Description of main notations (in alphabetical order). Parameter

Description

Parameter

Description

B Data(i, j) Delay(i, j)(t) Dr(i, j) E(i) Ld Ne(i)(t) Patm (i, j )(t ) RCmax Sk (t) v(i )(t )

Channel capacity Data Packet from Vi to Vj Propagation delay on link Delivery ratio of Data(i, j) Consumption from Vi Size of each data packet Neighboring set of Vi at the tth slot mth path in Pat(i, j )(t ) at the tth slot Maximum communication range Data-not-holding proportion at degree k Transition probability of Vi at the tth slot

C(i )(t ) Deg(i)(t) Dlupp Drlow (i, j )(t ) MR Pat(i, j )(t ) RC(i)(t) Rk (t) Suw ρ k (t)

Coordinate of Vi at the tth slot Degree of Vi at the tth slot Upper bound of forwarding delay Lower bound of delivery ratio Link between Vi and Vj at the tth slot Maximum movement range Paths set from Vi to Vj at the tth slot Communication range of Vi Moving-out proportion at degree k Underwater sound speed Data-holding proportion at degree k

Definition 4 (Delivery Ratio). The delivery ratio of data packet Data(i, j) is Dr(i, j), denoting the probability of Data(i, j) can be delivered from Vi to Vj during the required delay. Definition 5 (Energy Consumption). The energy consumption [33,34] is computed as the consumption summation of all message forwardings, consumption(G ) = V E (k ), where k

t0 +

E (k ) =

Dlupp TS



 

P0 · (RC (k )(t ) )

 t =t0 t0 +

=

Dlupp

TS







P0 · RC (k )(t )

ε

· 10

 RC (k ) (t ) ·α ( f ) 10

.

(3)

 t =t0

Dlupp is the upper bound of forwarding delay. The consumption is calculated based on the characteristics of underwater acoustic communication. Here P0 is the minimum received power level to  guarantee the required quality of reception [33], and (RC (k )(t ) ) is signal attenuation [34]. The energy spreading factor and absorption coefficient are denoted by ε (ε ∈ [1, 2]) and α (f), respectively.

Fig. 2. External force on underwater nodes.

3.2. Problem objectives In summary, in the data forwarding problem of OUSNs we aim at minimizing the energy consumption and simultaneously guarantee the average delivery ratio and the delay of each delivery. Suppose the forwarding starts at the t0 th time slot, and one copy of data (recorded as Data(s, d)) needs to be delivered from the source Vs to the destination Vd , then the objective function can be formally presented as follows:

min consumption(G )

Fig. 3. Example trajectories of the mobility model (during 100 slots).

(i, i1 )(t ) , (i1 , i2 )(t ) , · · · , (j2 , j1 )(t ) , (j1 , j )(t ) exist, then a path from Vi to V j exists. All these paths form a path set Pat(i, j )(t ) , and the mth path is marked as Patm (i, j )(t ) . Definition 3 (Propagation Delay). The delay on link (i, j )(t ) is expressed as [32]:

Delay(i, j )(t ) =

Ld d (i, j )(t ) + , B Suw

(2)

where Ld is the size of each data packet, B is the channel capacity (in bits per second), and Suw is the propagation speed of underwater sound.

⎧ ⎪ ⎪ ⎨

s.t.

⎪ ⎪ ⎩





Delay(k, k )(t ) ≤ Dlupp

(k,k )∈Pat(s,d )(t )



Vs ,Vd ∈V

Dr (s, d )

|s, d|

≥ Drlow , (4)

where  t ∈ [t0 , t0 + and |s, d| is the number of pairs of source destination nodes. Drlow is the lower bound of average delivery ratio. Especially, in this paper the objective regarding the propagation delay is treated as an upper bound Dlupp , and the value of which is set according to the requirements of specific applications. If the cumulative duration of data forwarding exceeds Dlupp , and then all the data-holding nodes will discard the data.

 DlTupp  ), S

4. Algorithm To avoid long-range communications and unnecessary forwarding (the data-holding nodes close to space boundaries are easier to

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L. Liu et al. / Computer Networks 114 (2017) 80–94

move out), we first propose a Topology Determined Model (TDM) to initialize the communication links of nodes at the beginning of every slot. Next, a Proactive Opportunistic Forwarding Mechanism (POFM) is designed to complete the delivery. Detailed descriptions of the TDM model and the POFM algorithm are shown in the following subsections.

1 delta=0.5 delta=1 delta=1.5 delta=2 delta=2.5

0.9 0.8 0.7 0.6

4.1. Topology determined model

0.5

To reduce the massive energy consumption from overlong links, TDM is designed to form an initial topology where the existence of link (i, j) is probabilistic. The probability Pij of adding a link (i, j) is,

⎡

Pi j = C1 · d (i, j )−α · ⎢ ⎢

⎢

⎤−δ

( xi − ) + ( yi −  L 2 2 L2 4

+

W 2

W2 4

0.4 0.3 0.2

)2 ⎥ ⎥ , ⎥

(5)

0.1 0

where α ∈ (0, 1), and C1 is a constant. Pij is irrelevant to the coordinates of Vi and Vj at the Z axis, and (5) indicates that the shorter links closer to space center are more prone to be generated. The determined topology is proven to be a scale-free network by Theorem 1.

0

5

10

15

Fig. 4. Node degree distribution from TDM with different δ .

Theorem 1. After the execution of TDM, the node degree follows the power law distribution. Proof. The mathematical expectation of Vi ’s degree is the total number of adjacent links and is given by

Di =

N

Pil = C1 ·

l=1

N

⎧ ⎪ ⎨

⎡

l=1

⎪ ⎩

⎢

d (i, l )−α · ⎢ ⎢

( xi − ) + ( yi −  L 2 2 L2

+

4

W 2

W2 4

⎤−δ ⎫ ⎪ ) ⎥ ⎬ ⎥ ⎪, ⎥ ⎭ 2

Fig. 5. Transition probability.

C C

⎧ ⎡  −δ ⎤ ⎫ √ ⎪ ⎪ N L 2 W 2 ⎪ ( xi − 2 ) + ( yi − 2 ) ⎪ ⎪ ⎣d (i, l )−α · ⎦ ⎪  ⎪ ⎪ ⎪ ⎪ L2 W2 ⎪ ⎪ ⎨l=1 ⎬ 4 + 4 Di − D j = C1 · ⎡ ⎤  −δ ⎪. ⎪ √ ⎪ ⎪ N ⎪ ⎪ L W 2 2 (x j − 2 ) + (y j − 2 ) ⎪ ⎣d ( j, l )−α · ⎦⎪ ⎪ ⎪  − ⎪ ⎪ ⎩ l=1 ⎭ L2 W2 + 4

degree distribution that satisfies P (k ) ∝ k obtained. 

4

When nodes are uniformly deployed and the signal range is much smaller than L, W and H, the boundary effect can be ignored. Under such case, every node is considered to have the L2 W2 δ −α ·  same neighbors distribution. Then, N l=1 d (i, l ) 4 + 4 ≈ N l=1

d ( j, l )−α · 



L2 4

+

W2 δ 4

can be obtained. Consequently, we

have that Di − D j is equal to

⎧  −δ ⎫ √ ⎪ ⎪ L 2 W 2 ⎪ ⎪ ( x − ) + ( y − ) i i 2 ⎪ ⎪ 2 ⎪ ⎪ ⎨ ⎬ 2 2 N L W

+ 4 4 −α Di − D j = C1 · d (i, l ) ·   −δ . √ ⎪ ⎪ ⎪ ⎪ l=1 (x j − 2L )2 +(y j − W2 )2 ⎪ ⎪ ⎪ ⎪  ⎩− ⎭ L2 W2 4

+

4

According to the definition of (5), the cumulative degree distribution of Vi can be expressed as: Pcd (Di ) = Di = k, hence k = C1 ·

N l=1



d (i, l )−α · 

we have that {(xi − 2L )2 + (yi − Let

another

constant

C2 =

W 2 − 2δ 2 )}

N l=1

2

LW

L2 4

≈

. Let

(xi − 2L )2 +(yi − W )2 2  −δ , then

C1 ·

2

+ W4

k

N

d ( j,l )−α · l=1

d ( j, l )−α · 



L2 4

+



L2 4

W2 δ 4 ,

2

+ W4 δ

.

then

−( 2 +1 ) δ

(Fig. 4) can be

The calculation of TDM is taken as a pre-processing of POFM. At each time slot every node Vi independently computes the probability Pij , and the information about the existence of link (i, j) is recorded by Vi . Thus, the amount of overlong communication links will be restricted by TDM. Moreover, the paracentral nodes will obtain more links (the nodes near boundaries are easy to move out). Accordingly, the energy consumption can be cut down significantly. 4.2. Dynamical percolation analysis To analyze the propagation process, a dynamic percolation model (similar to [35]) for data propagation is constructed, and the proportion variations of data-holding holders (with degree k), datanot-holding nodes (with degree k) and moving-out nodes (with dedρ (t ) dS (t ) gree k) are expressed as the first-order derivatives dkt , dkt and dRk (t ) , dt

π {(xi − 2L )2 +(yi − W2 )}

2

1 2 δ Pcd (Di ) can be rewritten as Pcd (k ) = { LW } . Finally, the node k

which yields

respectively:

⎧ ρ (t ) + Sk (t ) + Rk (t ) = 1 ⎪ ⎪ k ⎪ ⎪ dρ (t ) ⎪ ⎪ = −μρk (t ) + vkSk (t )θ (t ) ⎨ k dt k = 0, · · · , N − 1. dSk (t ) ⎪ = − v kS ( t ) θ ( t ) − μ S ( t ) ⎪ k k ⎪ dt ⎪ ⎪ ⎪ ⎩ dRk (t ) = μρ (t ) + μS (t ). k k

(6)

dt

As shown in Fig. 5, ρ k (t) is the proportion between the number of data-holding nodes (with degree k) and the total number of nodes (with degree k) at the tth time slot. Similarly, Sk (t) and Rk (t) denote the proportion of data-not-holding nodes and

L. Liu et al. / Computer Networks 114 (2017) 80–94

moving-out nodes respectively. v is the average transition probability from status data-not-holding to data-holding (suppose the datanot-holding node at least has a link with one data-holding node), and μ is the average transition probability from one of the status (data-holding or data-not-holding) to moving-out. θ (t) is computed N−1

as

kP (k )ρk (t )

k=0 N−1 k=0

kP (k )

P (k ) = λk−( δ +1 ) , then 2

⎧ ⎨

dSk (t ) = −vkSk (t ) dt N−1

kP (k )Sk (t )

k=0 N−1 k=0

There is

kP (k ) C3 · e − Sk (t ) − μSk (t ), N−1 k=0 kP (k )

k=0

kP (k )

dSk (t ) dt

Sk (t )



−μt = e {μ+C3 ·vke }dt ·

=e

−μt

μt−C3 ·vk e μ





−{μ+C3 ·vke−μt }dt

−vk · e dt + C4    − μ t e · −vk · e−μt+C3 ·vk μ dt + C4

⎩

i=0

and then Ck (t) can be obtained as:



iP (i ) eμt +C ·eμt−C3 ·vi 4 C3 N−1 iP (i ) i=0

e−μt

μ

by (6),

i=0

· C3 · e−μt −

eμt C3

⎫ ⎪ ⎬

iP (i )

⎪ ⎭

−μt μt−C3 ·vi e μ +C4 ·e

N−1

⎪ ⎩

iP (i )

i=0

.

Based on the above derivations, some properties of the proposed model are presented in Theorems 2 and 3, respectively. Theorem 2. If δ is sufficiently large, and then Fk (t) is proportional to k. Proof. The derivative of Ck (t) with respect to k is deduced as:

⎧ ⎪ ⎨

N−1 i=0

C3 · e−μt −

iP (i ) eμt C3

+C4 ·e

−μt μt−C3 ·vi e μ

⎫ ⎪ ⎬

N−1

⎪ ⎪ ⎩ ⎭ i=0 iP (i )   −μt −μt −C ·vk e μ μt−C3 ·vk e μ eμt e 3 v C3 + C4 · e + C3C4 · vk μ ·



eμt C3

+ C4 · eμt−C3 ·vk

e−μt

2

+ C4 · e

−C3 ·vk

dFk (t ) dk

=

dCk (t ) dk

· P (k ) +

dP (k ) dk

μ

e−μt

μ

μ

−

−μt

μt−C3 ·vk e μ

Theorem 3. If t1 < t2 , then

N−1 k=0

2 δ

+1

C3 C4 ·vk e eμt

Fk (t1 ) >

⎫ ⎬ ⎭

−C3 ·vk e

μ

· λk−( δ +1) . 2

−μt

μ

−μt μt−C3 ·vk e μ +C4 ·e

N−1 k=0

> 2δ , and



Fk (t2 ).

Proof. At the tth time slot, the increased proportion of dataholding nodes is calculated as N−1

Fk (t ) ≈

k=0



N−1 0

Ck (t )P (k )dk =

⎧ ⎪ ⎨



N−1

0

N−1

C3 · e−μt −

⎪ ⎩

eμt C3

+C4 ·e

N−1 i=0

vλk− δ

2

−μt μt−C3 ·vi e μ

⎫ ⎪ ⎬ ⎪ ⎭

iP (i )

2

N−1 0

Ck (t ) · λk−( δ +1) dk

iP (i )

i=0

eμt C3

+ C4 · eμt−C3 ·vk

N−1 Evidently, F (t ) will k=0 k N−1 N−1 F ( t ) > F (t ).  1 k k=0 k=0 k 2

e−μt

dk.

μ

decrease

as

t

increases,

so

4.3. Proactive opportunistic forwarding mechanism Based on the topology generated by TDM, POFM (Proactive Opportunistic Forwarding Mechanism) is specially proposed to deliver the data within the limited delay. In POFM, every data-notholding node makes a decision on whether to request for the data in its neighborhood actively. Especially, the global topology are not needed for POFM, and each node will independently apply the information about the existences of neighboring links (obtained from TDM) to make decisions, so there are not any centralized computations in POFM. Detailed descriptions of POFM are given as follows: Dl

.

μ

Step 1. At the start of the tth time slot (t ∈ [t0 , t0 +  Tupp ]), S each data-holding node Vj broadcasts (with the maximum radius RCmax ) an update_message (Fig. 6) containing a quadruple (ID, t, C ( j )(t ) , Data_list ( j ))1 after a random backoff time, which is 4 π RC 3

max computed as random(0, 3 |D|max · N · { LBn + RCSuw }], and Ln is the size of notification message of applying for the channels. Besides,

Because the derivative of Fk (t) with respect to k is expressed as

⎭

Theorems 2 and 3 suggest that the number of data-holding nodes will increase faster if the nodes with large degrees are given large transition probabilities or if the propagations at earlier slots are given larger velocities. The propagation control of velocity can shorten the forwarding delay, thus the energy consumption is also reduced, and the similar conclusions can be applied into POFM as well.

e−μt

+ C4 · eμt−C3 ·vk μ N−1

μ

e−μt

C3

=

vk ⎧ ⎪ ⎨

+ C4 · eμt−C3 ·vk

C3C4 · vk e eμt C3

⎫ ⎬

e−μt

> 0, that is, Fk (t) is proportional to k.



which yields that θ (t ) = C3 · e−μt −

eμt C3

dFk (t ) dk

·

N−1

dCk (t ) = dk

· 1+

e−μt 1 · eμt + C4 · eμt−C3 ·vk μ . C3

=

Ck (t ) =

⎧ ⎨

thus

(7)

Formula (7) is clearly a Bernoulli equation. Hence, there is −1

eμt C3

When δ is sufficiently large, there is

can be further sim-

dSk (t ) 2 + μ + C3 · vke−μt · Sk (t ) = vkSk (t ) . dt

eμt +C ·eμt−C3 ·vi 4 C3 N−1 i=0 iP i

()

≈ Sk (t ) approximately when N is large

enough, and hence the formula regarding plified as the following form:

iP (i )

i=0

=

−μt

N−1

v · C3 · e−μt − ⎩

least one end carrying the data. Let Fk (t ) = Ck (t )P (k ) denote the increased proportion of data-holding nodes with k degree at the tth d(ρk (t )+Sk (t )) time slot, where Ck (t ) = vkSk (t )θ (t ). Firstly, there is = dt −μ(ρk (t ) + Sk (t )). We have that ln |ρk (t ) + Sk (t )| = −μ(ρk (t ) + Sk (t )) + C, which yields ρk (t ) + Sk (t ) = C3 · e−μt , so ρk (t ) = C3 · e−μt − Sk (t ). This formula is then substituted into the third subexpression of (6),





dFk (t ) dCk (t ) 2 2 2 = · λk−( δ +1) − λ + 1 k−( δ +2) · Ck (t ) dk dk δ

, which indicates the probability of a link having at

N−1

85

· Ck (t ). According to Theorem 1, let

1

Data_list ( j ) is the set of held data by Vi .

86

L. Liu et al. / Computer Networks 114 (2017) 80–94

Fig. 6. Message structures and sequence diagram.

Table 2 Complexity of POFM.

note that at the start of each time slot, there is a synchronization interaction message, the size of which is denoted as Ls and there are usually Ln , Ls  Lu , Lr , Ld . Step 2. If Vi receives the quadruple from Vj , then Pij will be computed according to (5). The neighboring set of Vi at the tth ! time slot can be determined as Ne(i )(t ) = d (i, j )(t ) ≤RC (i )(t ) {V j }. After

that, the degree of Vi can be obtained and marked as Deg(i)(t) ← |Ne(i)(t) |. Step 3. The transition probability of Vi is calculated based on the following formula,

v(i )(t ) = ⎧ 0, if Deg(i )(t ) = 0 or V j ∈ DH (s, d ), ∀V j ∈ Ne(i )(t ) ⎪ ⎪ ⎪ ⎪ 1, else if Vi ≡ Vd ⎪ ⎪ ⎪ ⎛ ⎞γ ⎨ " #   β ⎜ ⎪ d (i, d ) t −1 ⎟ Deg(i )(t ) ⎪ ⎪ (⎟ , else. 1− √ · · ⎜1− ' ⎪ ⎝ 2 2 2 ⎪ N−1 L +W + H Dlupp ⎠ ⎪ ⎪ ⎩ TS

which indicates that the nodes with closer distance, higher degrees or earlier slots will obtain larger transition probabilities, i. e., those nodes are more inclined to request for the data. Step 4. If Vi is selected to hold Data(s, d), then Vi will request the designated node Vj for the Data(s, d), and Vj is selected

from V j | min d (i, j )(t ) & Data(s, d ) ∈ Data_list ( j ) & V j ∈ Ne(i )(t ) . Vi will send a request _message (Fig. 6) to the designated node with

the radius RC (i )(t ) = min RCψ | RCψ ≥ d (i, j )(t ) . Subsequently, the designated node will reply with a data_message (Fig. 6). Step 5. Once Vi receives Data(s, d), the data-holding set of ! Data(s, d) is updated as DH (s, d ) ← DH (s, d ) Vi . Vi updates ! Data_list (i ) ← Data_list (i ) Data(s, d ), and remaining_delay is modified as remaining_delay ← remaining_delay − {receiving_time − (t − 1 ) · TS }; Step 6. At each slot, the above steps are repeated until Vd receives the first data copy from Vs or Dlupp has been overran. If

Step

Message complexity

Time complexity

1 2 3 4 5 6

O(N2 ) O(1) O(1) O(N2 ) O(1) O(1)

O(1) O(N2 ) O(N) O(N) O(N) O(1)

the data has been delivered to Vd before the required delay, an announcement originated from Vd will be forwarded to all datanot-holding nodes to stop the requestings (The size of announcement message is very small and can be piggybacked with other data messages, so the effect of broadcast is negligible). Given the description of POFM, TS should be set as the minimum period during which each node can accomplish the proceL +L +L max dure Steps 1–5, and TS = u Br d + 3 RCSuw , where Lu , Lr and Ld are the sizes of update_msg, request _msg and data_msg, respectively. TS consists of the maximum time required for three interactions update_msg, request _msg and data_msg, respectively. 5. Algorithm analysis 5.1. Algorithm complexity Table 2 shows the message amount and time consumption of each step in POFM. The messages of POFM are mainly generated in Step 1 and Step 4, respectively. In Step 1, each node broadcasts a notification message, and the message complexity is O(N2 ); in Step 4, the message amount will reach O(N2 ) at the worst case. As a result, the message complexity of POFM is O(N2 ). In Step 2, O(N2 ) probabilities are updated; in Step 3, there will be at most N nodes to calculate their transition probabilities at the Dlupp T th time slot, so the time complexity of Step 3 is O(N). The S

L. Liu et al. / Computer Networks 114 (2017) 80–94

computation quantities of Steps 4 and 5 are associated with N as well. Therefore, the time complexity of POFM is also O(N2 ).

they should carry out 3 more sessions as shown in Fig. 6(b). Furthermore, according to Step 3 of POFM, v(t ) can be derived as:



5.2. Parameter settings

v(t )

The parameters ( , β and γ ) should be set so that delivery ratio is kept at a proper level even in the worst case, where data is forwarded randomly. The destination node holding the data is considered as the data delivery, and the delivery ratio of POFM DrPOFM is computed as the proportion summation of data-holding nodes at every possible time slots. Under such case, the following inequality should be established: 

DrPOFM =

Dlupp TS



N−1

Fk (t )

k=1

t=1

⎤ N−1 ⎧⎡ ⎫ iP (i ) i=0 e−μt ⎪ ⎪ eμt +C ·eμt−C3 ·vi μ ⎪ ⎪ 4 C3 ⎪ ⎪ − μ t ⎦ ⎪⎣C3 · e − ⎪ N−1 ⎪ , -⎪ iP ( i ) ⎪ ⎪ i=0 Dlupp ⎪ ⎪ ⎪ ⎪ TS ⎨ ⎬  

γ    β ≈ Deg d t−1 ⎪· 1 − √L2 +W 2 +H2 · N−1 · 1 −  Dlupp  ⎪ ⎪ t=1 ⎪ TS ⎪ ⎪ ⎪ ⎪ ⎪ N−1 ⎪ ⎪ ⎪

 −2 ⎪ ⎪ N−1 λ k δ ⎪ ⎪ ⎩· ⎭ ρ j (t )P ( j ) · 0 −μt dk e μt μt−C ·vk e C3

j=0

+C4 ·e

3

μ

≥ Drlow ,

(8)

where d is the expected distance between nodes in the deployment space:

 3

N

d=

3

≈

N

H 3 | D|

L 3 |D|

W 3 | D|

H 3 |D|

 ·

x1 =0



W 3 |D| N

y2 =0

where  =



L 3 |D| N



W 3 | D| N

y1 =0

H 3 | D| N



H 3 |D| N



z1 =0

The energy consumption from calculation and receiving messages is negligible compared with sending messages. The energy consumption from sending messages is mainly from data-notholding nodes in Steps 1 and 4. Accordingly, the energy consumption E(POFM ) is expressed as the consumption summation from the transmissions of update_msg, request _msg, data_msg and the messages for backoff and synchronization: Dlupp TS



-

 3P0 · (RCmax ) ·

N−1

k=1

t=1

(t )

Sk (t )P (k )

N·

k=0

Sk (t )P (k )

(t )

RC is the expected communication radius (determined in Step 4) and can be expressed as:

⎧ ⎫ Deg· κ −1 i i=1 ⎪ ⎪ 1 1 2  ⎪ ⎪ N−1 Deg 3 ⎪ ⎪ j ⎪ ⎪ L·W ·H k=0 ρk (t )P (k ) 3 j=1 ⎪ ⎪ 1 N−1 κ · · 1 − 4 ⎪ ⎪ ⎬ Deg 3 ⎨ 3 πN k=0 (ρk (t )+Sk (t ))P (k )

(t ) ⎡ ⎤ RC = . κ ·Deg ⎪ 1  N−1 ⎪ ⎪ κ =1 ⎪ Deg 3 ⎪ ⎪ ρ ( t ) P ( k ) j ⎪ ⎪ j=1 ⎦ ·⎣1 − 1 − N−1 k=0 k ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ k=0 (ρk (t )+Sk (t ))P (k ) The above formulas indicate that the energy consumption can be reduced by decreasing Sk (t), v(t ) and RC

(t )

(t )

. Moreover, Sk (t) and

RC drop with the increase of ρ k (t); by contrast, the value of v(t ) can be reduced by decreasing ρ k (t). Note that ρ k (t) is mainly determined by the transition probability, so a proper value for transition probability should be set to minimize the energy consumption while guaranteeing the delivery ratio and propagation delay.

· P0 · (RC

(t )

)·

N−1

4 π ·t 3 ·RC 3

·N

max there will be 3 nodes that have received the data from |D| the source at the end of this interval. Therefore, we have that the 3 |D|− 43 π ·t 3 ·RCmax . In addition, the deliv3 |D|− 43 π ·(t−1 )3 ·RCmax

ery ratio of EF can be expressed as the probability of the delivery being achieved at all possible time slots:

⎛3 ⎜ ⎜ DrEF = P ⎜ ⎜ ⎝

⎞

4 Dlupp Ld RCmax + B Suw

5

t=1

3

3

4 Dlupp

⎟ ⎟ At ⎟ ⎟=1− ⎠

Ld RCmax + B Suw

6

P (At |At−1 )

t=1

4 Dlupp

[1 − Rk (t )]P (k )

Ld RCmax + B Suw

6

≈ 1−

.

Sk (t )P (k ) ,

+ Smax uw

Let At denote the event that the delivery can be achieved durL max max ing the interval [(t − 1 ) · ( Bd + RCSuw ), t · ( LBd + RCSuw )]. Apparently,

probability is P (At |At−1 ) =

5.3. Energy consumption

+ 3v

k=0

d B

− δ +1

E(POFM ) = N ·

N·

EF will obviously achieve the maximum delivery ratio, and duL max ration of each forwarding is Bd + RCSuw , during which the data is forwarded to all the nodes in the range RCmax . Under the delay Dl ·RC constraint Dlupp , the maximum distance of flooding is L upp RC max .

x2 =0

(x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2. Deg is the math N−1 − 2 ematical expectation of node degree, and Deg ≈ 1 λk δ dk = 2 λ · { (N − 1 )− δ +1 − 1} given Theorem 1. 2

,

N−1

5.4. Energy consumption comparison with EF

L 3 |D| N

 dz 2 dy 2 dx 2 dz 1 dy 1 dx 1 ,

z2 =0

0Deg . ρ ( t ) P ( k ) k = 1 − 1 − N−1 k=0 k=0 (ρk (t ) + Sk (t ))P (k ) ⎧  γ ⎫   β ⎪ ⎪ Deg t−1 ⎨ 1− √ d ⎬ , · N−1 · 1 − Dlupp L2 +W 2 +H 2 TS · 1 . 2 ⎪ ⎪ ⎩· 1 − N−1 1 ⎭ + N−1 1 /



x1 =0 y1 =0 z1 =0 x2 =0 y2 =0 z2 =0

|D|

N2  ×

W 3 |D|

N N N N N N









·

N2



L 3 | D|

|D|

87

'

(9) 4 3

k=0

where v(t ) denotes the expected transition probability at the tth slot. For any nodes, there will be a synchronization message, a backoff notification message and an update_msg at each time slot; in addition, with regard to the data holders and the data receivers,

=

π·

t=1 Dlupp Ld B

max + RCSuw

| D|

3 |D| − 43 π · t 3 · RCmax 4 3 3 |D| − 3 π · (t − 1 ) · RCmax (3 3 · RCmax

.

(10)

The broadcasting of data-holding nodes is carried out at most  L Dlupp  times. The energy consumption of EF only includes the RC d B

+ Smax uw

88

L. Liu et al. / Computer Networks 114 (2017) 80–94

transmission consumption of data_msg:

data packet with a probability Pg to a randomly selected destination at each time slot. The delay due to the algorithm calculation and carrier sensing is neglected in the simulations. Note that all statistical results are averaged over all possible random values of the parameters using 500 independent simulation results. Table 3 lists the values of the parameters.

E(EF ) ≈ P0 · (RCmax ) 3

4

Dlupp Ld RCmax + B Suw



·

4 3

t=1

N−1

3 π · t 3 · RCmax ·N· |D|

k=1

ρk (t )P (k )

-, ⎧, ⎫2 3Dlupp 3Dlupp ⎨ ⎬ + 1 4 3 T T π · RCmax · N S S ≈ P0 · (RCmax ) · 3 · | D| 2 ⎩ ⎭ 3

4 Dlupp Ld RCmax + B Suw

·



N−1

t=1

k=1

ρk (t )P (k ).

Thus,

E(POFM ) E(EF ) ≤

,

Dlupp TS

-

 N−1 3

t=1

k=1 4 3

[1 − Rk (t )]P (k ) + 3v

3 π ·RCmax · |D|

(t )

" , 3Dlupp - , 3Dlupp ·

TS

TS

·

N−1

Sk (t )P (k )

k=0



#2

+1

.

2

Then one easily gets that:

E(POFM ) ≤ E(EF )



,

Dlupp TS

-

t=1 4 3

  v(t ) · N−1 k=0 Sk (t )P (k ) " , 3Dlupp -, 3Dlupp - #2

3 π ·RCmax · | D|



TS

2

,

Dlupp TS

(1 )

v · N−1 N " , 3Dlupp -, 3Dlupp -

t=1

≤ 3 π ·RCmax · |D|

4 3

,

Dlupp TS

TS

·

1− √

=



d

L2 +W 2 +H 2

4 3

1− √

#2

+1

2

- "

≤



TS

3 π ·RCmax · |D|



d

L2 +W 2 +H 2

3 π ·RCmax |D| ·

,

Dlupp TS

·



β

Deg N−1

" , 3Dlupp -, 3Dlupp TS

TS

+1

# ·

N−1 N

+

1 N

#2

2

·



-,

Deg N−1 3Dlupp TS

β

-

·

N−1 N

+1

3 Generally, RCmax  |D|, then we obtain that

Fig. 7 illustrates the impacts of α and δ on the energy consumption, delivery ratio, and node degree. Three observations are made: (a) When δ is fixed, energy consumption decreases as α grows, and accordingly the delivery ratio keeps cutting down rapidly as α increases. This is because only a small amount of links can be generated by TDM when α is large (as shown in Fig. 7(c)), then the delivery becomes difficult to be completed. Thus, both delivery ratio and energy consumption are very low. (b) When α ≤ 0.5, first, both energy consumption and delivery ratio show an upper trend with the increase of δ until a threshold δ = 0.4. Then, the curved surfaces behave with a ruleless fluctuation when δ falls into the interval [0.5,0.9]. This phenomenon is attributed to the increase of δ , which indicates that the links closer to space center are more prone to appear. Hence, lower probabilities of moving out will be assigned to the data-holding nodes and the delivery will be easier to achieve. When δ is large enough (δ > 0.4), the impact of δ will be weaken because the moving-out probability μ is small. (c) Average node degree represents the amount of generated links, and the degree will ascend with the increase of δ in general. The reason for this phenomenon is that a larger δ will give rise to larger probabilities of adding links (as expressed in (5)). 6.2. Impacts of  , β and γ

+1

TS

6.1. Impacts of α and δ

+

1 N

2

E (POFM ) E (EF )

< 1.

6. Performance evaluations In this section, we evaluate the performance of POFM under different model parameters and then compare the results with other algorithms. POFM is realized in ONE (Opportunistic Networks Environment)[36], which is a simulation environment that can generate node mobility using different movement models and forward messages between nodes with various routing algorithms. The IEEE 802.15.4 is adopted for medium access control (MAC), where the carrier sensing and collision avoidance techniques are especially adopted to avoid the collisions of data transmissions, as described in the Appendix. To conduct the testing, N mobile nodes are evenly deployed into the underwater convex space. Then, each node will alter its position according to (1). Each node generates a

The impacts of  , β and γ on energy consumption and delivery ratio are illustrated in Figs. 8 and 9 respectively. When β and γ are fixed, the growth of  will lead to a slight abatement in both energy consumption and delivery ratio. Note that the minimum delivery ratio 89%(> Drlow ) can be obtained when  = 0.5, β = 0.9, and γ = 0.2. When β = 0 or γ = 0, the algorithm performance is extremely poor, which reveals the fact that the considerations of node degree and slot order in propagation velocity are efficient to the improvement of algorithm performance, which confirms the conclusions of Theorems 2 and 3. As shown in Figs. 8 and 9, the curved surface declines slowly with respect to β until it climbs up to 0.5; afterwards, the energy consumption reduces quickly and simultaneously the delivery ratio continues to dip slightly. Moreover, the delivery ratio is relatively stable (which varies between 89% and 100%) with the variation of γ . However, the consumption will slump when γ is set properly. The phenomena indicates that  , β , and γ have mutual effects on the algorithm performance, and appropriate settings of  , β , and γ can effectively reduce the energy consumption while the delivery ratio is guaranteed to be larger than Drlow . 6.3. Maximum communication radius In Fig. 10(a), it appears that the energy consumption curves with larger RCmax are higher than those with smaller RCmax because the nodes will have more neighbors when a larger RCmax is adopted, hence more long links will be generated and the energy consumption augments accordingly. Note that the maximum energy consumption is 45192 J (RCmax = 19, N = 60 0 0). Although setting up longer links will consume more energy, yet it achieves shorter delay (Fig. 10(c)) because paths will have less hops when transferring data. As shown in Fig. 10(b), the delivery ratio continues to increase with the growth of RCmax . This is due to the fact

L. Liu et al. / Computer Networks 114 (2017) 80–94

89

Table 3 Parameter settings in the simulation. Parameter

Description

Value

N |D| RCmax MR Lu Lr Ld Ln Ls B Suw TS P0

Number of sensor nodes Deployment space Maximum communication radius Maximum movement range Size of update_message Size of request _message Size of data_message Size of notification messages for backoff time Size of exchanged messages for synchronization Channel capacity Propagation speed of sound underwater Time slot Minimum signal power Energy spreading factor Absorption coefficient Upper bound of delay Lower bound of average delivery ratio Exponent parameter Exponent parameter Exponent parameter Exponent parameter Exponent parameter Constant coefficient The probability of data generating Maximum level of communication radius Communication radius of the ψ th level Initial velocity of water current Coefficient of turbulent viscosity

30 0 0 40m × 100m × 50m 15 m 10 m 500 B 500 B 20 0 0 B 100 B 100 B 8 kbps 1500 m/s 6.02 s 0.07 V 1.5 0.01 dB/km 50s 80% 0.2 0.7 0.7 0.9 0.8 1.0 0.5 10 5+ψ m (0.15, 0.2, 0.1) m/s ( 0 . 5 , 0 . 5 , 1 . 0 ) m2 / s

ε α (f)

Dlupp Drlow

α δ  β γ

C1 Pg

χ

RCψ

(vx(0) , vy(0) , vz(0) ) (Kx , Ky , Kz )

Fig. 7. Impacts of α and δ .

Fig. 8. Impacts of  , β , and γ on total energy consumption.

90

L. Liu et al. / Computer Networks 114 (2017) 80–94

Fig. 9. Impacts of  , β , and γ on delivery ratio.

Fig. 10. Impacts of RCmax .

Fig. 11. Comparisons with other algorithms under underwater mobility model.

that a larger communication range leads to a longer contacting period for nodes, which is propitious to the data forwarding. If RCmax ≥ 15, the delivery ratio will exceed Drlow regardless of N. 6.4. Comparisons with other algorithms In the simulation, we compare the energy consumption, delivery ratio, and actual propagation delay of POFM, EF, MoVe, RBAR and BDCR, respectively, which are illustrated in Fig. 11(a)–(c). It indicates that EF outperforms other algorithms in terms of delivery ratio and average delay by an obvious margin. The reason is that EF adopts the flooding mechanism to forward data, which results in optimal solution. However, energy consumption of EF is too large. Note that the energy consumption of POFM is significantly lower than those of EF and MoVe, and the gap becomes larger with the increase of N, as shown in Fig. 11(a). This is because the num-

ber of long links is strictly controlled by POFM, which attempts to finish the data delivery as early as possible to avoid unnecessary energy consumption. The curve of POFM is slightly lower than that of BDCR, but POFM obviously outdoes in terms of other objectives (delivery ratio and actual delay). Besides, although the curve of RBAR is lower than that of POFM, the delivery ratio of RBAR is much smaller than Drlow , as depicted in Fig. 11(b). The reason is that RBAR allows a node to hold a packet as long as possible until it is necessary to make another copy. Fig. 11(b) also indicates that delivery curves of POFM, MoVe, RBAR and BDCR obviously increase with respect to N, although some fluctuations may appear. When N is larger than 3750, the curves of POFM and EF are very close. Additionally, in Fig. 11(c) the curves of average delay have some fluctuations and the reduction behaves indistinctively with the increase of N. This phenomenon occurs because the number of selectable paths increases when the nodes are deployed more

L. Liu et al. / Computer Networks 114 (2017) 80–94

91

Fig. 12. Performance comparison under random waypoint model.

Fig. 13. Performance comparison under mobility vector model.

densely, shortening the delay accordingly, even though the effect of deployment density is very slight. In summary, through the effective propagation velocity control, POFM consumes less energy while delivery ratio is guaranteed and propagation delay is limited. POFM can achieve a good trade-off between algorithm performance and algorithm cost, and it is especially suitable for the densely deployed OUSNs. 6.5. Simulation comparisons under other mobility models To make more comparisons, we compare POFM with other algorithms under more mobility models, such as Random Waypoint Model [37], Mobility Vector Model [38] and Graph-based Mobility Model [39]. Compared to Random Walk and Random Waypoint, Mobility Vector Model and Graph-based Mobility Model are with more regularities. Mobility vector model uses sub vectors for keeping current mobility information and provides partial changing in motion, so it provides realistic and flexibility for reproducing various models within a single framework. Graph-based mobility model reflects the constraints of movement given by the spatial environment in the real world. In this model, the nodes do not move randomly, but always along the edges of a graph that models the given infrastructure. Specially, graph-based mobility model is extended into three-dimensions. Each node maintains a constant depth and moves on its plane as 2D graph-based mobility model. Under the random waypoint model, the movement speed is randomly selected from interval (0,1.67) m/s, and the pause time is randomly selected from interval (0.5,1.5) s. The curves in Fig. 12 vary similarly to those in Section 6.4, and the value of energy consumption is nearly the same. Due to the pause time setting in random waypoint model, actual delay is slightly larger than that in our mobility model. Conversely, the delivery ratio becomes slightly lower owing to the constraint of Dlupp . Under the mobility vector model, the acceleration factor is set as 3 and the speed varying range falls into the interval (0,1.67) m/s.

Fig. 13 indicates the curves of POFM, MoVe and BDCR are very close, though POFM also shows a perceivable superiority. The reason for this mechanism is that the mobility vector model generates smoother trajectories which are more predictable. Thus, MoVe and BDCR can achieve better results compared to those under other movement model. Despite that, POFM still perform better under the mobility vector model, because the performance relies on the dynamic control of propagation velocities more than the exploitation of historical trajectories. The performance comparison under graph-based mobility model is illustrated in Fig. 14, which indicates that POFM has preferable results compared to other algorithm. Another concern is the phenomenon that more obvious fluctuations occur in these curves, especially of POFM. This mechanism is attributed to the setting of graph-based mobility model, where the size of grid depends on the value of N, that is, a smaller N produces a larger movement grid, which gives a larger movement speed is given, and thus the results are optimized. However, the sparser deployment of nodes worsen the results as well, thereby these fluctuations appear due to the mutual effects of movement speed and deployment density. 7. Conclusions This paper explores the data forwarding problem of energy conservation in OUSNs while the lower bound of delivery ratio and the upper bound of propagation delay are constrained. At the beginning of each slot, a topology determined model is used to construct a power-law distribution structure. The analysis on the model suggests that the quantity of data-holding nodes will increase if the nodes with large degrees are given large transition probabilities or if the propagations at earlier slots are given larger velocities, and POFM is developed according to these features. Simulation results demonstrate that POFM can effectively reduce energy consumption

92

L. Liu et al. / Computer Networks 114 (2017) 80–94

Fig. 14. Performance comparison under graph-based mobility model.

while the delivery ratio is guaranteed and the propagation delay is controlled, as predicted by mathematical analysis. Future research will focus on investigating a self-adaptive solution to achieve better tradeoff between the delivery ratio and the message cost. In addition, a more sophisticated heterogeneous model for OUSN nodes (different maximum communication ranges) should be discussed and analyzed mathematically as well.

Acknowledgments This research is supported by National Natural Science Foundation of China under Grants No. 61373139; Natural Science Foundation of Jiangsu Province under Grant No. BK2012833; Postdoctoral Science Foundation of China under Grant Nos. 2014M560379, 2015T80484.

Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.comnet.2017.01.009.

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L. Liu et al. / Computer Networks 114 (2017) 80–94 Linfeng Liu received the B. S. and Ph. D. degrees in computer science from the Southeast University, Nanjing, China, in 20 03 and 20 08, respectively. Since 2015, he has been a visiting fellow at Nanyang Technological University. At present, he is also an associate professor of the School of Computer, Nanjing University of Posts and Telecommunications. His main research interests lies in the areas of wireless sensor networks and multihop mobile wireless networks protocols. In particular, he is now interested in topology control problems, QoS routing and localization algorithms in mobile UWSNs.

Ping Wang received the Ph.D. degree in electrical engineering from University of Waterloo, Canada, in 2008. Since June 2015, she has been an associate professor in the School of Computer Engineering, Nanyang Technological University, Singapore. Her current research interests include resource allocation in multimedia wireless networks, cloud computing, and smart grid. She was a corecipient of the Best Paper Award from IEEE Wireless Communications and Networking Conference (WCNC) 2012 and IEEE International Conference on Communications (ICC) 2007. She is an Editor of IEEE Transactions on Wireless Communications, EURASIP Journal on Wireless Communications and Networking, and International Journal of Ultra Wideband Communications and Systems.

Ran Wang currently works as a Project Officer at the School of Electrical & Electronic Engineering, Nanyang Technological University (NTU), Singapore. He received his B.E. in Electronic and Information Engineering from Harbin Institute of Technology (HIT), P.R. China in 2011 and Ph.D. in Computer Engineering from Nanyang Technological University (NTU), Singapore in 2016. His current research interests include intelligent management and control in Smart Grid and evolution of complex networks.