Architecture of Wireless Sensor Networks with Mobile Sinks

International Journal of Distributed Sensor Networks, 3: 289–310, 2007Copyright © Taylor & Francis Group, LLCISSN: 1550-1329 print/1550-1477 onlineDOI: 10.1080/15501320601062130

289

UDSN1550-13291550-1477International Journal of Distributed Sensor Networks, Vol. 3, No. 3, June 2007: pp. 1–26International Journal of Distributed Sensor Networks

Architecture of Wireless Sensor Networks with Mobile Sinks: Multiple Access Case

Architecture of WSNsL. Song and D. Hatzinakos LIANG SONG, Member, IEEE, and DIMITRIOS HATZINAKOS, Senior Member, IEEE

Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, Canada

We propose to develop Wireless Sensor Networks with Mobile Sinks (MSSN), under high sensornode density, where multiple sensor nodes need to share one single communication channel in thenode-to-sink transmission. By exploiting the tradeoff between the successful information retrievalprobability and the nodes energy consumption, a number of multiple nodes transmission schedulingalgorithms are proposed. Both optimal and suboptimal algorithms, which exhibit exponential andlinear complexity respectively, are discussed under the desired application. Computer simulationsshow that suboptimal algorithms perform nearly as good as the optimal one. The study leads to thecross-layer Wireless Link layer design for MSSN.

Keywords Wireless Sensor Networks; Transmission Scheduling; Multiple Access; Wireless LinkLayer; Energy Consumption

1. Introduction

The unique nature of wireless sensor networks, which are resource-limited and application-specific, requires new approaches in the system design. Recently [1], we proposed todevelop Sensor Networks with Mobile Sinks (MSSN) for two application areas, environ-ment monitoring systems with high latency tolerance [19], and intelligent-space [17]. Thisnew architecture features very high energy-efficiency, because multi-hop transmissions ofhigh volume data over the network is converted to single-hop transmissions. Additionaladvantages of MSSN include: infrastructure free, high security, and ease of implementa-tion. In [1], we discussed the transmission scheduling algorithm TSA-MSSN in a sparselydeploying network setup, where it is assumed that only one sensor node is communicatingwith the mobile sink on a given channel.

However, when the density of sensor networks increases, multiple nodes can bewithin the transmission range to the sink, and the assumption of a sparsely deployingsetup no longer holds. Consider for example the application of micro spies [5]. A microspy is a micro aircraft with the size of less than six inches. For the purpose of battle-field observation and strategic reconnaissance, sensor networks can be deployed in aremote place with limited wireless infrastructures. A micro spy can, on the other hand,

Part of the manuscript was presented in “Dense Sensor Networks with Mobile Sinks,” IEEE InternationalConference on Acoustics. Speech and Signal Processing (ICASSP), vol. 3, pp. 677–680, Philadelphia PA, Mar.2005. Manuscript was submitted to International Journal on Distributed Sensor Networks, in Sept. 2005, revisedin May 2006.

Address correspondence to Dimitrios Hatzinakos, Department of Electrical and Computer Engineering, 10King’s College Road, Toronto, ON, M5S 3G4, Canada. Tel.: 416-978-1613, Fax: 416-978-4425.E-mail: [email protected]

290 L. Song and D. Hatzinakos

act as a mobile sink, which flies over the sensor deployed region and delivers the col-lected information back to the commander. In the communication setup, a number ofsensor nodes need to send packets to the micro spy simultaneously, because of theimportance of the desired information. Also, it may be risky to have the micro spiesstay in the air for too long a time, hence, the communication time slots available arelimited.

The wireless architecture in this scenario is centralized, which is analogous to cellularnetworks, star network setup in IEEE 802.15.4 [6], and Point Coordination Function(PCF) in IEEE 802.11 [7]. Although there is a large number of classical multiple accessschemes [15] available, such as TDMA, FDMA, CDMA, etc., all of them assume thatmultiple wireless channels are available for node-to-sink communication. In our case, thesink can assign one such distinctive channels to any individual node. The TSA-MSSN,proposed in [1], can then operate on different channels respectively. However, when thenumber of nodes exceeds the number of channels, multiple nodes will need to share onechannel. Thus, multiple nodes scheduling algorithms are needed to ensure the Quality ofService (QoS) of the wireless network.

Under different types of wireless networks, there are numerous works investigat-ing the desired scheduling problem. Under cellular networks, from an informationtheoretic approach, [8] showed that, to maximize the total uplink capacity, it is suffi-cient to have only the user with the strongest channel to the base station transmit atany given time. Such a power control scheme is known as multiuser diversity, whichis also extended to downlink channels in [9], and other forms of channel capacity suchas delay-limited capacity [10] and outage capacity [11]. Although strategies in [8] areoptimal in the sense of capacity, in the service layer implementations, more QoS fac-tors should be considered, such as fairness, delay, and connection admission control(CAC). Recent references on this topic can be found in CDMA cellular networks [12]and IEEE 802.11 enabled home networking [13]. In the area of sensor networks,SENMA [20] (Sensor Networks with Mobile Agents) studied the sensor networktopology similar to MSSN, and proposed distributed random access scheduling. How-ever, SENMA assumes a simple reachback sensor network, which differentiates itfrom the proposed MSSN.

Compared to the existing works, the MSSN is distinctive for the following reasons: InMSSN, the objective of scheduling algorithms is to retrieve integrated/compressed infor-mation [14] stored on sensor nodes with the minimum energy consumption. Combinedwith the mobility of the sink, this guideline, [1], makes the system design different fromthe conventional wireless networks. More specifically, in the MSSN transmission sched-uling, we exploit the tradeoff between the successful information retrieval credit and thenodes energy consumption cost, under the latency limitations imposed by the sink mobil-ity. Instead, in conventional wireless networks, “throughput” is usually emphasized,which is the number of retrived packets per second.

Next, we discuss the scenario where multiple sensor nodes are sharing one com-munication channel in sending information to the mobile sink which is travellingthrough the deployed sensor networks. The system description is provided in Section II.The proposed optimal multiple nodes transmission scheduling algorithm (MTSA-MSSN) is described in Section III. The optimal MTSA-MSSN has the complexitywhich grows up exponentially with the number of nodes N. Thus, in Section IV, wepropose suboptimal algorithms with the complexity as low as O(N). A comparison andvalidation of the suboptimal algorithms performance are given in Section V, by meansof computer simulations. Discussions and concluding remarks are given in Section VIand VII, respectively.

Architecture of WSNs 291

2. System Description

2.1. Concepts and Assumptions

The concept of the scheduling design is to reduce the sensor nodes energy consumption,in the process of information retrieval, by exploiting the sink mobility. Utilizing thehigher energy, computing, and storage resources at the sink, the proposed schedulingalgorithms are centralized, and executed solely at the sink. The problem of finding theoptimal scheduling policy is formulated as a Markov Decision Process (MDP), andsolved in the framework of dynamic programming [22]. Essentially, the tradeoffbetween the successful information retrieval credit and the sensor nodes energyconsumption cost is explored.

Let N denote the number of nodes sharing one specific communication channel.Assume that the location and the number of residual packets of every node are known atthe sink, by a registration procedure. More specifically, the registration can be conductedon a separate control channel, where the nodes send the registration, i.e, the location andthe packet number, upon the detection of the sink arrival, by means of carrier sensingmultiple access (CSMA). The sink acknowledges the registration by assigning the node agenerated identification number.

At the current time slot i0, the problem to formulate is how the sink decides the trans-mission strategy, i.e., which one of the N nodes is going to transmit and what is the trans-mission power. The scheduling algorithm is executed at the beginning of everycommunication time slot. One time slot is composed of the transmission of two differentpackets in sequence, which are the acknowledgement (ACK) packet from the sink, and thedata packet from one of the sensor nodes, respectively. The obtained scheduling strategycan be piggybacked on the ACK packet broadcasted by the sink to the N associated sensornodes. The ACK packet, on the other hand, also serves to acknowledge successful orfailed data transmission in the previous time slot. It is assumed that the lost data packetswill be retransmitted.

Assume that the sink has an estimation of its current mobility, which can be obtainedfrom the Global Positioning System (GPS). We do not assume that the future mobility pat-tern is available at the sink. Instead, we make two approximations:

1. No new node is admitted to the channel until all N nodes have finished the trans-mission;

2. The mobility of the sink is maintained constant in future time slots.

Both assumptions are realistic, because the scheduling algorithm is run at the beginning ofevery time slot, where both the nodes and the sink mobility information are updated.

In the scheduling design, we do not consider the possibility of new data arrival at sen-sor nodes, in the procedure of information retrieval. The sensor nodes is dissociated withthe sink, whenever all the data packets have been retrieved.

2.2. System State Vector Ê(i0)

Let Range denote the communication range of the sensor node. It is assumed that Range isthe same for all nodes. Let {L1,….LN} denote the location of the N sensor nodes respec-tively, which are fixed throughout the transmission. At the current time slot i0, let {Kn(i0)|n =1…N} denote the number of residual packets awaiting for transmission at node n. Thelocation, direction, and the speed of the sink are denoted by Ls(i0), qs(i0), vs(i0), respec-tively. Then, the current system state is composed of these 2N + 3 elements,


Let Dn(i0) denote the distance between the sink and the sensor node n, n = 1…N,

The communication channel gain for the node n can be modeled as [16].

where A is a constant decided by the antenna gain; b is the path loss exponent decided bythe propagation environment, e.g. b = 2 for free space; and x is a random variable undernormal distribution , indicating the shadowing effect. denotes the variance of x.

Let Tn(i0) denote the estimated transmission time slots available for node n, n = 1…N.Since the sink is assumed to have constant mobility when passing through all the N circu-lar regions, centered at {Ln|n = 1…N} with the radius Range, there is,

where is the time duration of one slot. Fd and Fb are the size of the data and

ACK packets respectively. R is the transmission data rate. Ls_out,n is illustrated in Fig. 1,and is defined as the point at which the sink goes out of the communication range ofsensor node n. (See Appendix I for a mathematical definition.)

Let,

Then, the vector of estimated states can be defined as,

where Ê(i0) is known as,

and,

E i i i v i i K is s N( ) { ( ), ( ), ( ), , , , ( ), , ( )}.0 0 0 0 1 0 0= L L Ls 1 Nq … …K (1)

D i in ( ) ( ) .0 0= −L Ls n (2)

G i A D i dBn n( ) log( ) log( ( )) ( ),0 010 10= ⋅ − ⋅ + (3)

N( , )0 2sx sx2

T ii

v i tns

( )( )

( ),0

0

0

=−

⋅

⎢

⎣⎢⎢

⎥

⎦⎥⎥

L Ls s_out,n

Δ(4)

ΔtF F

Rd b=+

T i T in N

n( ) max { ( )}.01

0== …

(5)

E E E E^

( ) { ( ), ( ), , ( ( ) )},^ ^ ^

i i i i T i0 0 0 0 01 1= + + +… (6)

( ) ( ),E i E i0 0= (7)

L L^

s s( ) ( ) [cos( ),sin( )];

( ) (^

i j i j v t

i j i

s s s

s s

0 0

0 0

+ = + ⋅ ⋅ ⋅

+ =

Δ q q

q q )) ;

( ) ( ) ;

( ) .

^

=

+ = == +

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

qs

s s sv i j v i v

j T i0 0

01 1…

(8)


Thus, the only variable parameters of each estimated state {Ê(i), i0 < i ≤ i0 + T(i0) + 1} are{Kn(i)}, which ranges from 0 to Kn(i0). The vector Ê(i) (i0 < i ≤ i0 + T(i0) + 1) can then take

discrete values.

2.3. Markov Chain Model of Ê(i0)

The transmission strategy S(i) at time slot i, where i0 ≤ i ≤ i0 + T(i0), is decided by twoparameters,

where η(i) ∈ {1…N} denotes the ID of the transmitting node in time slot i. Pt(i) is, on theother hand, the transmission power at the sensor node h(i). Note that Pt(i) can be 0 if all Nnodes are forced to sleep. We assume the discrete levels transmission power, and thatthe number of optional levels is Sizeof {Pt}. Given Ê(i) and S(i), Ê(i + 1) is not related toany previous states before i. Thus, {Ê(i)} can be modelled as a Markov chain in timedomain, that is,

FIGURE 1 The mobile sink passes through the circular region centered on Ln.

( ( ) )K inn

N01

1+=∏

S( ) { ( ) ( )},i i P it= ⋅h (9)

Prob E i E i E i i i

Prob E i E i

( ( ) | ( ), , ( ), ( ) , ( ))

( ( ) | (

^ ^ ^

^ ^

+= +

1

10 0… …S S

)), ( ), ( )).P i it h(10)

^


The state transferring probability function can be written as,

where PÊR(i) is the packet error rate of transmission in time slot i. Consider a BPSKmodulated uncoded data packet. And assume that the sink is unaware of theshadowing effect ξ, i.e. in the channel model Eq.(3). Then, the PÊR(i) can be writtenas, [15],

where is the noise power. And .

In the definition of the transmission strategy S(i), Eq.(9), we preclude the possi-bility of multiple nodes transmitting simultaneously in one time slot. Although thescenario is not physically infeasible, it will require highly sophisticated codingschemes on sensor nodes, which is generally impractical due to the limited hardwareresource on one single sensor node. Instead, by adopting the definition in Eq.(9), wereduce the size of strategy space from Sizeof {Pt}

N to Sizeof {Pt}·N. Furthermore,from an energy conservation perspective, it is also preferable to not allow multiplenodes transmitting simultaneously on one channel. The proof is provided inAppendix II.

3. Optimal MTSA-MSSN Strategy

At current time i0, the objective of the MTSA-MSSN is to decide the optimal strategy S(i0)so as to maximize the probability of successful transmission while minimizing the energyconsumption. However, these two goals can not be achieved simultaneously. The optimalstrategy, , is searched by maximizing a utility function,J(S(i0), Ê(i0)),

For i0 ≤ i ≤ i0 + T(i0), we define,

Prob E i E i P i i

P E R i K i K i n

t

n n

( ( ) | ( ), ( ), ( ))

( ), ( ) ( ),

^ ^

^ ^ ^

+ =

+ = ∀ =

1

1

h

11

1 1 1

1

…N

P E R i K i K i

K i K i n

i i

n n

;

( ), ( ) ( ) ,

( ) ( ),

^ ^( )

^( )

^ ^

− + = −

+ = ∀ ≠

h h

hh( );

, ;

i

others0

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

(11)

PER i QP i

D i

t

n i

Fd

( )( )

( ),

^( )

= − −⋅

⋅

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

1 12

2

Λ

s hb

(12)

sn2 Q x e dtt

x( ) = ⋅ −∞∫

1 2

2p

S* * *( ( )) { ( ( )), ( ( ))}E i E i P E it0 0 0= h

S SS

*

( )

^( ( )) argmax{ ( ( ), ( ))}.E i J i E i

i0 0 0

0

= (13)


In Eq. (14), Jm (S(i), Ê(i)) denotes the figure of credit, which is the expected achievableutility by adopting S(i) at Ê(i). Jc (S(i)), on the other hand, is the figure of cost, which isdecided by the energy consumption. λ is a coefficient deciding the tradeoff between the two.

Since {Ê(i)} is modeled as a Markov chain, there is:

and,

where max{Pt} is the maximum available RF (Radio Frequency) transmission power andacts as a normalization factor in Jc.

Thus, by combining Eq.(14,15,16), the function J can be written recursively as,

At the final state of the algorithm, i = i0 + T(i0) + 1, the utility function is decided only byÊ(i0 + T(i0) + 1), that is,

The definition of J (Ê(i0 + T(i0) + 1)) is, however, application dependent. For example,assuming that each packet is equally credited, say “1”, then it can be defined as,

If we set λ = 1 in Eq.(14), Eq.(19) suggests that the cost of a maximal power transmissiontime slot equal to the credit of successfully transmitting one packet.

As such, the optimal MTSA-MSSN can be solved by means of dynamic program-ming, and is summarized in Table 1. Compared with the TSA-MSSN [1], the size of strat-egy space in MTSA-MSSN has increased N times, from Sizeof {Pt} to N · Sizeof {Pt}.

J i E i J i E i J im c( ( ), ( )) ( ( ), ( )) ( ( )).^ ^

S S S= − ⋅l (14)

J i E i

J i E i

Prob E

m

E ii

( ( ), ( ))

max { ( ( ), ( ))}.

(

^

( )( )

^

^

S

SS

=

+ ++

+Σ1

1 1 1

^̂ ^( ) | ( ), ( ), ( )),i E i P i it+1 h

(15)

J iP

P ict

t( ( ))max{ }

( ).S = ⋅1

(16)

J i E i J i P i E i

J i

t

E ii

( ( ), ( )) ( ( ), ( ), ( ))

max { ( (

^ ^

( )( )^

S

SS

= =

++

+

h

Σ1

1 11 1

1

), ( ))},

( ( ) | ( ), ( ), ( ))max{ }

(

^

^ ^

E i

Prob E i E i P i iP

P itt

t

+

+ − ⋅hl

)).

(17)

J i T i E i T i J E i T i( ( ( ) ), ( ( ) )) ( ( ( ) ))^ ^

S 0 0 0 0 0 01 1 1+ + + + = + + . (18)

J E i T i K i K i T in n

n

N

( ( ( ) )) { ( ) ( ( ) )}.^ ^

0 0 0 0 01

1 1+ + = − + +=∑ (19)


Moreover, the size of the state space Ê(i) is . Hence, Eq. (17) is executed

times in any one iteration of i. The TSA-MSSN, in

this framework, becomes a special case of MTSA-MSSN, where N = 1. Since there areT(i0) iterations, the complexity of optimal algorithm is,

Due to the fact that (the number of discrete states) increases exponen-tially with the number of nodes N, the complexity of the optimal algorithm also increasesat least exponentially with N.

Moreover, if a continuous power level control is implemented, the concavity of theutility function is only proved under the condition N = 1, [1]. It is unclear whether a com-putational simple way can be found for the optimization in Eq. (17), when N > 1.

4. Suboptimal Algorithms

Since the optimal algorithm has the complexity, C(Optimal) given by Eq.(20), whichincreases exponentially with the number of nodes N, it is infeasible for implementation ona mobile sink, such as a “micro spy.” In developing suboptimal algorithms, the idea is torun the TSA-MSSN (i.e. N = 1) algorithm individually for each node n, and combine the Nindividual strategies. We denote this suboptimal multiple-access transmission strategy asSs(i). As presented in [1], the complexity of TSA-MSSN can be reduced to O(1), when afeasible storage capability is available on the sink. Then, the complexity of the proposedsuboptimal algorithms, requiring the same sized storage capability, equals N times thecomplexity of TSA-MSSN. Thus,

4.1. Preprocessing: TSA-MSSN Runs

We outline the preprocessing procedure of TSA-MSSN for the integrity of this paper.Assume the current time slot is i0. Let Pt,n(i) denote the strategy (transmission power) of

( ( ) )K innN

01 1+=∏O N Sizeof P K it nn

N{ { } ( ( ) )}⋅ ⋅ +=∏ 01 1

C Optimal O N Sizeof P T i K i

O e

t nn

N

N

( ) { { } ( ) ( ( ) )}

( ).

= ⋅ ⋅ ⋅ +

==∏0 01

1(20)

( ( ) )K innN

01 1+=∏

TABLE 1 Optimal MTSA-MSSN Algorithm

Given E(i0);

Get T(i0) by Eq.(5);

Initialize final-state utility functionJ(Ê(i0) + T(i0) + 1))

by Eq.(19);

For i equals i0 + T(i0) down to i0,For all S(i) and Ê(i),

Calculate J (S(i), Ê(i)) by Eq.(17);

Decide the optimal transmission strategy S* (E(i0))by Eq.(13).

C Suboptimal O N( ) ( ).= (21)


TSA-MSSN run on node n, and let Ên(i) denote the estimated system state on node n,which is,

The definition of follows from Eq.(8).The TSA-MSSN searches the maximum of a utility function J (Pt,n(i0), Ên(i0)) in

obtaining the optimal transmission power decision, ,

The utility function for each node n, Jn, is recursively defined as,

where,

And,

assuming that BPSK modulated uncoded signal is transmitted.The final state configuration function is Jn (Ên(i0 + Tn(i0) + 1)). In compliance with

Eq.(19), it is defined as,

E i i i v i K i

i i i T i

n s s n

n

^ ^ ^ ^( ) { ( ), ( ), ( ), , ( )},

, , ( )

== + +

L L^

s nq

0 0 0 1… ..(22)

L^

s ( ), ( ), ( )i i is and q vs

P E it n n,* ( ( ))0

Pt n nP i

t n nE i J P i E it n

,*

( ),

^( ( )) argmax{ ( ( ), ( ))}.

,

0 0 00

= (23)

J P i E i

J P i E i

n t n n

E iP i t n n

nt n

( ( ), ( ))

max { ( ( ), (

,^

( )( ) ,

^

^ ,

=

++

+Σ1

1 1 ++

+ − ⋅

=

1

1

0

))},

( ( ) | ( ), ( ))max{ }

( ),^ ^

, ,Prob E i E i P iP

P i

i i

n n t nt

t nl

,, , ( ).… i T in0 0+

(24)

Prob( ( ) | ( ), ( ))

( ), ( ) ( );

^ ^,

^ ^ ^

E i E i P i

P E R i K i K i

P

n n t n

n n n

+ =

+ =

−

1

1

1 EE R i K i K i

othersn n n

^ ^ ^( ), ( ) ( ) ;

, ;

+ = −

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

1 1

0

(25)

P E R i QP i A

D in

t n

n n

Fd

^ ,

^( )

( )

( ),= − −

⋅

⋅

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

1 12

2s(26)

J E i T i K i K i T in n n n n n( ( ( ) )) ( ) ( ( ) ).^ ^

0 0 0 0 01 1+ + = − + + (27)


Table 2 summarizes the TSA-MSSN preprocessing in suboptimal algorithms. Afterthe preprocessing we obtain the TSA-MSSN power decision on each node,

. Due to the constraint on Eq. (9), only one node has a transmittingpower greater than zero at any time slot. Different suboptimal algorithms will operate indifferent ways in deciding ηs (E(i0)) ∈ {1…N}, and then,

4.2. Suboptimal Algorithm I: Maximal Sensor Power Strategy (MSPS-MSSN)

The strategy here is to choose the node with maximum as the transmittingnode, that is,

and then by Eq.(28),

Thus, in general, the MSPS-MSSN assumes that the sensor node with higher TSA-MSSNpower level has higher “urgency” in transmission.

4.3. Suboptimal Algorithm II: Maximal Sensor Utility Strategy (MSUS-MSSN)

Based on the TSA-MSSN preprocessing results, MSUS-MSSN chooses the node with themaximal achievable TSA-MSSN utility summation, which is conceptually interpreted as

TABLE 2 Suboptimal MTSA-MSSN Algorithm Preprocessing

Given E(i0);

For n equals 1 to N.Get En(i0) from E(i0);Get Tn(i0) by Eq.(4);

Initialize final-state utility functionJn (Ên(i0 + Tn(i0) + 1))

by Eq.(27);

For i equals i0 + Tn(i0) down to i0.For all Ên(i).

Calculate Jn (Pt,n(i), Ên(i))by Eq.(24);

Decide the TSA-MSSN transmission power

by Eq.(23).end

P E it n n,* ( ( ))0

{ ( ( )) | },*P E i n Nt n n 0 1= …

P E i P E its

t E i E is s( ( )) ( ( ))., ( ( ))*

( ( ))0 00 0

= h h (28)

P E it n n,* ( ( ))0

h s

nt n nE i P E i( ( )) argmax{ ( ( ))},,*

0 0= (29)

P E i P E its

nt n n( ( )) max{ ( ( ))}.,*

0 0= (30)


the associated utility functions summation by selecting one node n for transmitting. Thespecific summation is mathematically defined as, for n = 1…N,

We have,

and is obtained through Eq. (28).

4.4. Comparisons of Optimal and Suboptimal Algorithms

To provide an analytical performance comparison between optimal and suboptimal algo-rithms is difficult. However, we discuss two case studies.

1. Case 1: TSA-MSSN Active Case: The definition of “active” is that at least one ofthe N TSA-MSSN power decisions is nonzero. Considera simplified scenario where “ideal coding” is utilized on every sensor node, andthe power control level Pt,n(i0) on the sensor node is continuous. Furthermore, we

assume that there is no shadowing uncertainty about the channel, that is .

These imply that,

holds, instead of Eq.(26) where an uncoded binary information sequence has beenassumed. B in Eq.(33) is a constant that depends on the noise power and thetransmission rate R. Under this scenario, in both optimal and suboptimal algo-rithms, we have the following equations,

Eq. (34) can be interpreted as follows: For the selected transmitting node n, sincePt,n(i0) = B · Gn(i0) is the minimum power which guarantees the successful packettransmission with probability one, it must be the optimal power decision for bothsuboptimal and optimal algorithms.

According to the simplified model of Eq.(34), the optimal and suboptimal algo-rithms differ only in the way of choosing the transmission node, h* (E(i0)) or hs

(E(i0)), respectively. Note that under more realistic settings, this claim holds onlyapproximately. When comparing MSPS-MSSN and MSUS-MSSN, we note that

U E i J E i J P E i E in p p n t n n n

p p

( ( )) { ( , ( ))} ( ( ( )), ( )).^

,* ^

:0 0 0 0

1

0= += ≠nn

N

∑ (31)

h s

nnE i U E i( ( )) argmax{ ( ( ))},0 0= (32)

P E its ( ( ))0

{ ( ( )) | },*P E i n Nt n n 0 1= …

sx2 0=

PER iP i B G i

P i B G int n n

t n n( )

, ( ) ( );

, ( ) ( );,

,0

0 0

0 0

0

1=

≥ ⋅< ⋅

⎧⎨⎩

(33)

sn2

P E i B G i

P E i B G i

t E i

ts

E is

**( ( ))

( ( ))

( ( )) ( );

( ( )) ( );

0 0

0 0

0

0

= ⋅

= ⋅

⎧⎨

h

h

⎪⎪

⎩⎪(34)


the latter takes into considerations both the energy consumption and the number ofresidual packets on every node, while the former takes into considerations only thetransmitting power. A simulation comparison of the optimal and suboptimal algo-rithms is given in Section V.

2. Case II: TSA-MSSN Sleeping Case: The definition of TSA-MSSN “sleeping” caseis that the TSA-MSSN power decision by Eq.(23) is zero for all nodes, that is,

Under this condition, all suboptimal algorithms will keep all nodes sleeping in thecurrent time slot i0. The optimal MTSA-MSSM performs strictly better, since itavoids this restriction by jointly deciding the transmission strategy over all Nnodes.

Consider the following illustrative example. We set N = 2, and L1 = L2 = [0,5].Let qs = 0, Ls = [−10,0], and K1(i0) = K2(i0) = 11. The vs is set in such a way that vs.Δt = 1. The TSA-MSSN run on each node will provide the result that

, since there are enough “better” time slots available fortransmission when the sink is moving along the x-axis. However, when jointlyconsidering two nodes, the scheduling algorithm will force one node to transmit,which is equivalent to running TSA-MSSN on one node with K = 20.

Since virtually all the configurations in nodes/sink fall under the two describedcases, one should avoid Case II when employing suboptimal scheduling algo-rithms. This, however, can be achieved by the special design in the Media AccessControl (MAC) layer channel selecting algorithm.

4.5. Implementation of Suboptimal Algorithms

As it has already been mentioned, to avoid the performance degradation in suboptimalalgorithms, one should avoid “Case II.” This can be accomplished by running the channelselection algorithm for one specific node n only when it is active, i.e. . Thiscriterion, on the other hand, also enhances the spectrum efficiency by keeping all theoccupied channels busy.

At the current time i0, assume that there are Nall(i0) nodes associated to the mobilesink. And there are M(i0) occupied channels, where obviously M(i0) < Nall(i0). Define asensor nodes subset A(i0) of {n | n = 1…Nall(i0)} to be,

Then, at time i0, the channel selection algorithm only runs on the newly activated nodesn ∈ Anew(i0), where

Suppose that the maximum number of available channels is Mmax. If M(i0) < Mmax, thechannel selecting algorithm can always randomly pick one vacant channel for the newactive node, n ∈ Anew(i0). However, if M(i0) = Mmax, the channel selection algorithm selectsone of the Mmax occupied channels for the new node, according to some criterion. Although

P E i n Nt n,* ( ( )) , .0 0 1= = … (35)

P E i nt n,* ( ( )) , ,0 0 1 2= =

P E it n n,* ( ( ))0 0>

A( ) { | ( ( )) , ( )}.,*i n P E i n N it n n all0 0 00 1= > = … (36)

A A A Anew ( ) ( ) ( ) ( ).i i i i0 0 0 0 1= − ∩ − (37)


a further discussion on channel selection algorithms is outside the scope of this paper, wegive an illustrative example. When M(i0) = Mmax, assume that the system states associatedwith Mmax channels are Em(i0) (m = 1…Mmax), respectively. Let Am(i0) denote the set ofnodes sharing the channel m. We define the overall packets number of channel m, Km(i0), as,

A simple criterion on deciding the selected channel m* for the new node is that Km*(i0) beminimal. This criterion, however, expresses a rule of thumb, which simply prevents over-crowding any one particular channel. Table 3 describes this algorithm.

5. Simulations

The following simulations scenario is considered. The sink is passing through the sensor net-work region. Some parameters generally comply with IEEE 802.15.4 [6], and are listed inTable 4. The number of transmission power levels is set to 10, and the set of discrete levels is,

λ, which is the parameter to decide the tradeoff between successful transmission andenergy consumption, is set to be ‘1’ throughout all the simulations. The setups of the stim-ulations in Section V-A and V-B, i.e. the nodes/sink locations and the packet numbers, arerandomly chosen, satisfying the “active” and “sleeping” conditions, respectively. All theresults in the figures are the average of 500 Monte-Carlo runs.

5.1. Case 1: TSA-MSSN Active Case

We compare the optimal and suboptimal algorithms under “Case I” specified in SectionIV-D.1. We assume that two sensor nodes are sharing one specific communication channel,that is N = 2. Without loss of generality, let L1 = [−5, 5], L2 = [0, −4], and K1(i0) = K2(i0) = 10.

K i K imn

n i

( ) ( ).( )

0 0

0

= ∑εAm

(38)

TABLE 3 Channel Selecting Algorithm

Given Em(i0), m = 1…M(i0);

Get Km(i0) from Em(i0), m = 1…M(i0) by Eq.(38);

Get Anew by Eq.(37);

For every n ε Anêw

If M(i0) < MmaxGet m* = M(i0) + 1; M(i0) = M(i0) + 1;elseGet m* = arg minm {Km(i0)|m = 1…Mmax};end;

Add node n to Em* (i0);Update Km* (i0) by Eq.(38);

end

P i

dBmt ( ) { , , , , ,

, , , , }( ).

ε −∞ − − − −− − − −

32 28 24 20

16 12 8 4 0(39)


Set θs(i) = 0. With Ls(i0) = [22, 0], the sink passes through the circular communication regionof the nodes in a straight line. That is, the sink is moving away from the nodes.

Simulations are performed to measure the energy consumption Eall and successfultransmission rate Pall of MTSA-MSSN, MSPS-MSSN, and MSUS-MSSN, respectively.In calculating the energy consumption, we omit the RF circuits energy consumption, sinceit is the same for all algorithms. Then, the definitions of Eall and Pall will be,

respectively, where,

Figures 2 and 3 plot Eall and Pall, respectively, when changes.The following observations are made: The energy consumption Eall of MSPS-MSSN

is much higher than the other two algorithms everywhere; however, it also has a higherPall than others when is relatively small. Compared with MSPS-MSSN, the MTSA-MSSN and MSUS-MSSN offer a better tradeoff between energy consumption and a suc-cessful transmission rate. Although MTSA-MSSN is theoretically optimal and has a muchhigher complexity, the suboptimal MSUS-MSSN surprisingly performs at least as good asMTSA-MSSN in the simulation. When is small, MTSA-MSSN outperforms MSUS-MSSN slightly, as shown in Figs. 2 and 3. However, when becomes large, the subopti-mal MSUS-MSSN becomes better than the optimal algorithm in both energy consumptionand the successful transmission rate. The result may seem to be strange at first glance.

TABLE 4 Communication Parameters Setup

Parameter Unit Value

Fd bit 128 × 8

Fbbit 20 × 8

R bits/sec 20000b 3A dB −31Range m 50

dBm −92

vsm/sec 20

sn2

E P iF

Rall tr d

i i

i T i

= ⋅−

+

∑ ( ) ,( )

0

0 0

P K i K i T iall n nn

N

= − + +−∑{ ( ) ( ( ) )},0 0 0

1

1

(40)

(41)

P iP E i for optimal

P E i for suboptimaltr t

ts

( )( ( )), ;

( ( )), ;

*

=⎧⎨⎪

⎩⎪(42)

sx2

sx2

sx2

sx2


This is, however, due to the fact that scheduling algorithms are unaware of the shadowingeffect of the channel in Eq.(3) (see Eq.(12)). Suboptimal MTSA-MSSN chooses a highertransmission power level, which consumes more energy on one hand, and enhances theprobability of successful transmission on the other. Generally, in “Case I,” suboptimalalgorithms are efficient alternatives to the optimal one. Especially for the MSUS-MSSN,it performs as good as the optimal algorithm in terms of total energy consumption Eall andsuccessful transmission rate Pall.

5.2. Case II: TSA-MSSN Sleeping Case

We compare Eall and Pall of MTSA-MSSN, MSPS-MSSN, and MSUS-MSSN under “CaseII” specified in Section IV-D.2. Again, we assume two sensor nodes are associated with onespecific communication channel, which is N = 2. let L1 = [15, 5], L2 = [15, −8], and K1(i0) =K2(i0) = 8. Set qs(i) = 0. With Ls(i0) = [0, 0], the sink passes through the circular communica-tion region of the nodes in a straight line. That is, the sink is moving toward the nodes.

Since the number of available time slots are much higher than the number of packetsin this simulation, Pall for all the algorithms is 16, which implies that no packet is lost inthe transmission. Figure 4 shows the variation of Eall when changes. We observe thatthe suboptimal algorithms consume much higher energy than the optimal one for all .The result agrees with our analysis in Section IV-D.2.

FIGURE 2 Case I: total energy consumption Eall vs. .

0 0.5 1.5 2.51 2 30.19

0.21

0.2

0.22

0.23

0.24

0.25

0.26

σξ2

Eal

l (m

J)

MTSA−MSSNMSPS−MSSNMSUS−MSSN

sn2

sx2

sx2


6. Discussions

6.1. Centralized vs. Distributed Scheduling

We hereby compare the multiple access MSSN with the SENMA [20], [21]. Although thetwo have similar network topology, they are nevertheless different types of networks.

In MSSN, we assume certain signal processing capability on individual sensor nodes,which can be viewed as the high level event acquisition [4], instead of raw data acquisi-tion. The sink is dominant in MSSN, since the deterministic scheduling is decided on thesink. SENMA, on the other hand, assumes a simple reachback sensor network, where thesensor nodes transmit raw data to the sink (mobile agent), with no inter-node signal pro-cessing. The sink in SENMA, on the other hand, consecutively retrieves the packets fromthe node with the best channel state. SENMA is a sensor nodes dominant network,because the random access is implemented. SENMA claims low complexity of the sensornodes. MSSN, however, achieves a higher efficiency and a higher application specificflexibility [1], because of the high level event acquisition. The proposed scheduling algo-rithms better utilize the higher processing/power capability of the sink, and keep the sim-plicity of sensor nodes. The choice between MSSN and SENMA should be dependent onthe application requirements.

The proposed scheduling algorithms are centralized, since the mobile sink completelydecides the transmission strategies. Under the setup of SENMA, [21] proposed the distrib-uted opportunistic scheduling algorithm, by assuming channel state information on sensor

FIGURE 3 Case I: successfully transmitted packets Pall vs. . sx2

19.1

19.2

19.3

19.4

19.5

19.6

19.7

19.8

19.9

20P

all

MTSA−MSSNMSPS−MSSNMSUS−MSSN

0 0.5 1.5 2.51 2 3

σξ2


nodes. Compared with centralized algorithms, distributed algorithms can save protocoloverheads. However, in the multiple access MSSN, since the scheduling criterion is morecomplicated, we consider that the development of distributed scheduling algorithms forMSSN is not cost-efficient.

6.2. Algorithmic Extensions and Cross-layer Design

First, the final state configuration function, given by Eq. (19,27), corresponds to the“Application scenario 1” in [1]. As described in [1], the final state configuration is appli-cation specific. Different configurations can be applied with the algorithms proposed inthis paper without further difficulty. In both [1] and this paper, we have assumed that thetransmission rate R is constant. When this assumption does not hold, and multi-rate chan-nel coding is feasible on sensor node, the strategy space of MTSA-MSSN will be the sizeof N·Sizeof {Pt}·Sizeof {R}, which corresponds to an increase by a factor of Sizeof {R},the number of rate control levels. Suboptimal algorithms, such as the MSUS-MSSN, canalso be extended to this multi-rate case.

Second, in the channel mode Eq. (3), the small scale multipath fading was not consid-ered. By assuming a small scale fading model, e.g. the Rayleigh fading, in Eq. (3), weneed to recalculate the PÊR function in Eq. (12), which is a function of the separating dis-tance D(i), the transmission power level Pt(i), and the transmission rate R. The fading maynot be overlooked in some scenarios, especially when the transmission rate is high, and

FIGURE 4 Case II: total energy consumption Eall vs. . sx2

x 10−3E

aII (

mJ)

MTSA−MSSN

MSPS−MSSN

MSUS−MSSN

0 0.52

3

4

5

6

6.5

5.5

4.5

3.5

2.5

1.5 2.51 2 3

σξ2


the time diversity over fading states is not achievable. Although multiple receivingantenna diversity can be a solution, another more cost-efficient alternative is the coopera-tive transmission in a densely deployed sensor network [2]. It utilizes, by the theory ofMIMO structure, nodes space diversity instead of receiving antenna diversity. Diversecoding schemes and channel models can be implemented in this framework with non-sig-nificant modifications on the scheduling algorithms. The details of the implementationsare subject to future research.

Traditionally, packet transmission scheduling is included in the data link layer. How-ever, by exploiting the tradeoff between the successful transmission and the energy con-sumption, the cross layer design of a consolidate physical and data link layer is needed. In[3], [4], Embedded Wireless Interconnect (EWI) was proposed as the potential universalarchitecture of wireless sensor networks. EWI is composed of two layers, which are theWireless Link layer and the System layer, respectively. Under the architecture of EWI, theproposed transmission scheduling algorithms work on the Wireless Link layer of MSSN.It interacts with the upper System layer by the syntax of application information credits,which is then translated into the final state configuration function of the schedulingalgorithms.

7. Conclusion

In this paper, we have studied the operation of MSSN when multiple nodes share onecommunication channel during the information retrieval. First, we have developed theoptimal multinode scheduling algorithm MTSA-MSSN. Since the complexity of MTSA-MSSN increases exponentially with the number of nodes, we propose suboptimal algo-rithms, which run the single node scheduling algorithm TSA-MSSN individually on everynode and then combine the results. The two suboptimal algorithms, MSPS-MSSN andMSUS-MSSN, exhibit the complexity as low as O(N). The performances of optimal andsuboptimal algorithms are compared by means of computer simulations. It is shown thatthe suboptimal algorithms achieve nearly the same performance as the optimal MTSA-MSSN when the channel is “active.”

About the Authors

Liang Song (S’03,M’06) received the Bachelor’s degree in Electrical Engineering fromShanghai Jiaotong University, China, in 1999; and the M.S. degree in Electronic Engi-neering from Fudan University, China, in 2002. He obtained the Ph.D. degree from theEdward S. Rogers Sr. Department of Electrical and Computer Engineering, University ofToronto, Canada, in 2005. His research interests and expertise are in the area of signal pro-cessing, communications, networking, and information theory, with the focus on theapplications in wireless sensor and pervasive computing networks. His experience alsoincludes industrial consulting through SENNET Communications and CANAMET Inc.His email address is [email protected]

Dimitrios Hatzinakos (M’90,SM’98) received the Diploma degree from the Univer-sity of Thessaloniki, Greece, in 1983, the M.A.Sc degree from the University of Ottawa,Canada, in 1986 and the Ph.D. degree from Northeastern University, Boston, MA, in1990, all in Electrical Engineering. In September 1990 he joined the Department of Elec-trical and Computer Engineering, University of Toronto, where now he holds the rank ofProfessor with tenure. Also, he served as Chair of the Communications Group of theDepartment during the period July 1999 to June 2004. Since November 2004, he is theholder of the Bell Canada Chair in Multimedia, at the University of Toronto. His research


interests are in the areas of Multimedia Signal Processing and Communications. He isauthor/co-author of more than 150 papers in technical journals and conference proceed-ings and he has contributed to 8 books in his areas of interest. His experience includesconsulting through Electrical Engineering Consociates Ltd. and contracts with United Sig-nals and Systems Inc., Burns and Fry Ltd., Pipetronix Ltd., Defense Research Establish-ment Ottawa (DREO), Nortel Networks. Vivosonic Inc. and CANAMET Inc. He hasserved as an Associated Editor for the IEEE Transactions on Signal Processing from 1998till 2002 and Guest Editor for the special issue of Signal Processing. Elsevier, on SignalProcessing Technologies for Short Burst Wireless Communications which appeared inOctober 2000. He was a member of the IEEE Statistical Signal and Array ProcessingTechnical Committee (SSAP) from 1992 till 1995 and Technical Program co-Chair of the5th Workshop on Higher-Order Statistics in July 1997. He is a senior member of the IEEEand member of EURASIP, the Professional Engineers of Ontario (PEO), and the Techni-cal Chamber of Greece. His email address is [email protected]

References

1. L. Song, and D. Hatzinakos, “Architecture of wireless sensor networks with mobile sinks:sparse deploying case,” IEEE Trans. on Vehicular Technology, vol. 56, no. 4, July 2007. URL:www.comm.utoronto.ca/∼songl/pdf/mssnl.pdf

2. L. Song, and D. Hatzinakos, “Cooperative transmission in poisson distributed wireless sensornetworks: protocol and outage probability.” IEEE Trans. on Wireless Communication, vol. 5,no. 10, pp. 2834–2843, Oct. 2006. URL: www.comm.utoronto.ca/∼songl/pdf/ctp.pdf

3. L. Song, and D. Hatzinakos, “Embedded wireless interconnect for sensor networks: concept andexample,” In Proc. IEEE 4th Consumer Communications and Networking Conference (CCNC),Las Vegas, NV, 2007. URL: www.comm.utoronto.ca/∼songl/pdf/cwi.pdf

4. L. Song, and D. Hatzinakos, “A cross-layer architecture of wireless sensor networks for targettracking,” IEEE/ACM Trans. on Networking, vol. 15, no. 1, pp. 145–158, Feb. 2007. URL.:www.comm.utoronto.ca/∼songl/pdf/lesop.pdf

5. Peter Garrison, “Microspics,” Air & Space Smithsonian, pp. 54. April/May 2000.6. Standard IEEE 802.15.4. MAC and PHY Layer specification for low-rate wireless personal area

networks (PR-WPANs).7. Standard IEEE 802.11. Wireless LAN medium access control(MAC) and physical layer(PHY)

specifications.8. D. Tse and S. V. Hanly, “Multiaccess fading channels - Part I: polymatroid structure, optimal

resource allocation and throughput capacities,” IEEE Trans. Inform. Theory, vol. 44, no. 7, pp.2796–2815. Nov. 1998.

9. L. Li and A. J. Goldsmith, “Optimal resource allocation for pading broadcast channels - Part II:outage capacity,” IEEE Trans. on Information Theory, vol. 47, no. 3, pp. 1103–1127, Mar. 2001.

10. S. V. Hanly and D. Tsc, “Multiaccess fading channels - Part II: Delay-limited capacities,” IEEETrans. on Information Theory, vol. 44, no. 7, pp. 2816–2831. Nov. 1998.

11. L. Li, N. Jindal and A. Goldsmith, “Outage capacities and optimal power allocation for fadingmultiple-access channels,” IEEE Trans. on Information Theory, vol. 51, no. 4, Apr. 2005.

12. D. Zhao, X. Shen, and J. W. Mark, “Radio resource management for cellular CDMA systemssupporting heterogeneous services,” IEEE Trans. on Mobile Computing, vol. 2, no. 2, pp.147–169, Apr.–Jun. 2003.

13. C. Oottamakorn and D. Bushmitch, “Resource management and scheduling for QoS-Capablehome network wireless access point,” in Proc. IEEE CCNC’2004, pp. 7–12. 5–8 Jan. 2004.

14. S. S. Pradhan, J. Kusuma, and K. Ramchandran, “Distributed compression in a dense micro-sensor network,” IEEE Signal Processing Magazine, vol. 19, no. 2, pp. 51–60. Mar. 2002.

15. J. G. Proakis, Digital Communications, NJ: McGraw-Hill Inc., 1995.16. H. L. Bertoni, Radio Propogation for Modern Wireless Systems, NJ: Prentice Hall PTR, 2000.17. A. Pentland, “Smart Rooms,” Scientific American, April 1996.


18. T. M. Cover and J. A. Thomas, Elements of Information Theory, NJ: John Wiley & Sons. Inc. 1991.19. R. C. Shah, S. Roy, S. Jain, and W. Brunette. “Data MULES: Modeling a three-tier architecture

for sparse sensor networks,” in Proc. of the First IEEE International Workshop on SensorNetwork Protocals and Applications, Anchorage, Alaska, 11 May, 2003.

20. L. Tong, Q. Zhao, and A. Adireddy, “Sensor networks with mobile Agents,” in Proc. IEEEMilCom’2003.

21. Q. Zhao and L. Tong, “Distributed opportunistic trabsmission for wireless sensor networks,” inProc. IEEE ICASSP’2004.

22. D. Bertsekas, Dynamic Programming and Optimal Control, Belmont, MA, Athena Scientific, 1995.

APPENDIX I

Formula on Ls_out,n

The coordinates vector Ls_out,n is shown geometrically in Fig. 1. and can be computed asfollowing:

where,

APPENDIX II

More on the Strategy Definition of MTSA-MSSN

Eq.(9) precludes the possibility of multiple nodes transmitting simultaneously on a singlecommunication channel. This, however, is possible, when sophisticated coding proceduresare feasible on individual sensor nodes, dealing with the co-channel interference. Whensufficient time slots are available, under a simplified channel coding model, we herebyprove that having multiple nodes transmitting simultaneously is strictly worse than thestrategy prescribed by Eq.(9) under energy efficiency considerations.

We define the following two strategies:

• Strategy 1: At time slot i0, Q out of N nodes associated with the communicationchannel are transmitting with power P1,q (q = 1…Q), simultaneously.

• Strategy 2: During the time slots from i0 to i0 + Q – 1, node q transmits with powerP2,q at time slot i0 + q – 1, where q = 1…Q.

Moreover, we assume a simplified coding model. For 1 ≤ q ≤ Q, there is,

L L Ls_out,n = −⎡⎣⎢

⎤⎦⎥⋅−⎡

⎣⎢

⎤Range y y s s

s s

2 2[ ] , [ ]cos sin

sin cos

q qq q ⎦⎦

⎥ +Ln , (43)

L L Ls n= − ⋅−⎡

⎣⎢

⎤

⎦⎥( ( ) )

cos sin

sin cos.i s s

s s0

q qq q (44)

PER i qR i q C i q

R i q

p all p

p all

( ), ( ) ( );

, ( )

,

,0

0 0

0

11 1 1

0 1+ − =

+ − ≥ ⋅ + −

+ −

g

<< ⋅ + −⎧⎨⎪

⎩⎪=

g C i q

p

p ( );

, .

0 1

1 2

(45)


where γ is a constant satisfying 0 < γ < 1. Under the strategy p, Rp.all(i0 + q – 1) is the sum-mation of the transmission rate over the nodes at time i0 + q – 1. Assume the transmissionrate is a constant R. In “Strategy 1”, R1,all(i0) = Q·R, while it is zero when q > 1. In “Strat-egy 2”, R2,all (i0 + q – 1) = R, for all q. Cp(i0 + q – 1) denotes the channel capacity at time i.For “Strategy I”, we have [18],

where W is the channel bandwidth. For “Strategy II”, on the other hand, we have,

Theorem. Under the coding model specified by Eq.(45), when the condition

is satisfied, “Strategy 2” is strictly more energy-efficient than “Strategy 1,” in the sensethat,

Proof. Due to the coding model in Eq.(45), any power control algorithms will have

and,

for 1 ≤ q ≤ Q.Due to Eq.(46) and Eq.(47), Eq.(50) and Eq.(51) can be written as,

C i WP G iq qq

Q

n1 0

1 012

1( ) log{ ( )}

,,

= ⋅ +⋅⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

=∑s

(46)

C i q WP G i qq q

n2 0

2 0

21 1

1( ) log

( ).

,+ − = ⋅ +⋅ + −⎛

⎝⎜⎞

⎠⎟s(47)

max { ( )}

min { ( )}q q

q q

Q R

r W

R

r w

G i

G i qQ

0

0 1

2 1

2 1+ −

< −

⋅ −⎛

⎝⎜⎜

⎞

⎠⎟⎟

⋅⋅

⋅(48)

P Pqq

Q

qq

Q

21

11

, , .= =∑ ∑< (49)

R i Q R C iall1 0 1 0, ( ) ( ),= ⋅ ≤ ⋅g (50)

R i q R C i qall2 0 2 01 1, ( ) ( ),+ − = = ⋅ + −g (51)

P G iq q

Q R

r wn

q

Q

1 02

1

2 1, ( ) ,⋅ ≥ −⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

⋅⋅

=∑ s (52)


and,

respectively.Therefore,

where (a) and (e) are obvious; (b), (d) are due to Eq.(52) and Eq.(53) respectively; (c) isdue to the condition Eq.(48). The theorem is hence proved.

We further comment that the condition of Eq.(48) is usually satisfied, when Q is rela-tively small. Under this small Q condition, we have

On the other hand, we have

due to the fact that . Thus, the inequality of Eq.(48) holds.

PG i qq

R

r wn

q2

2

0

2 1

1,

.

( ),=

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

+ −

⋅ s(53)

PG i

P G i

G

qq

Q a

q qq qq

Q

b

q q

110

1 01

1

1

,( )

,

( )

max { ( )}{ ( )}

max { (

= =∑ ∑≥ ⋅ ⋅

≥ii

Q R

r W

G i qQ

n

c

q q

R

r W

0

2

0

2 1

1

12 1

)}

min { ( )}( )

⋅⋅⋅

−⎛⎝⎜

⎞⎠⎟⋅

>+ −

⋅ ⋅ −⎛

⎝⋅

s

⎜⎜⎜

⎞

⎠⎟⎟⋅

=+ −

⋅ ⋅ + +=∑

sn

d

q qq qq

Q

e

G i qP G i q

2

02 01

1

11( )

,

( )

min { ( )}{ ( )}

≥≥=∑ P qq

Q21 , ,

(54)

max { ( )}

min { ( )}.q q

q q

G i

G i q0

0 11

+ −≈ (55)

2 1 1 2 1 1

2 1

Q R

W

R

W

R

W

Q

Q

⋅⋅ ⋅

⋅

− > + ⋅ −⎛

⎝⎜⎜

⎞

⎠⎟⎟−

= ⋅ −⎛

⎝⎜⎜

⎞

⎠⎟⎟

g g

g ,

(56)

2 1

R

Wg⋅ >

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Architecture of Wireless Sensor Networks with Mobile Sinks