On the Throughput, Delay, and Energy Efficiency of Distributed Source Coding in Random Access Sensor Networks

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010 1965

On the Throughput, Delay, andEnergy Efficiency of Distributed Source Coding in

Random Access Sensor NetworksY.-W. Peter Hong, Member, IEEE, Yuh-Ren Tsai, Member, IEEE, Yan-Yu Liao,

Chih-Hsun Lin, and Kai-Jie Yang, Member, IEEE

Abstract—In this work, we analyze the throughput, delay,and energy efficiency of random access sensor networks thatemploy Slepian-Wolf distributed source coding (DSC) and studythe impact of MAC protocol design on these performances.Suppose that 𝑁 sensors observe correlated information fromthe environment and that their local data are sent to a sink nodethrough direct transmission links. To eliminate data redundancy,we allow sensors to encode their local messages using theSlepian-Wolf DSC method. We assume that sensors are orderedsequentially and that each sensor’s message is compressed byexploiting the joint data statistics between itself and the sensorsearlier in the sequence. Due to properties of DSC, a message canbe decoded only if all messages transmitted by sensors earlier inthe sequence are successfully decoded. The loss of one messagemay cause failure in decoding many other messages. Hence,the sensors’ messages are not of equal importance and shouldbe given different transmission priorities by the MAC. Basedon the properties of DSC, we provide analytical tools to studythe throughput, delay, and energy efficiency of slotted ALOHArandom access protocols. Utilizing these tools, we comparebetween the performance of different transmission probabilityassignments and study the impact of MAC protocol design on theperformance of these systems. Furthermore, an adaptive MACprotocol is also proposed to improve upon the throughput anddelay of the original system.

Index Terms—Sensor networks, distributed source coding,medium access control, random access.

I. INTRODUCTION

W IRELESS sensor networks (WSNs) consist of manytiny, low-power and cheap wireless sensor devices that

are deployed to take measurements from the physical envi-ronment, to process the collected data, and to communicateamong each other or to a central processor. Due to the densedeployment of sensors and the spatial correlation betweenthe measured events, the messages to be transmitted by thesensors are often highly correlated and redundant. When theraw data is sent without compression, the energy consumedin transmitting these redundant messages or in retransmitting

Manuscript received March 11, 2009; accepted February 25, 2010. Theassociate editor coordinating the review of this paper and approving it forpublication was D. Zeghlache.

The authors are with the Institute of Communications Engineering,National Tsing Hua University, Hsinchu, Taiwan (e-mail: {ywhong,yrtsai}@ee.nthu.edu.tw, [email protected], {g9664555,d919608}@oz.nthu.edu.tw).

This work was supported in part by the National Science Council, Taiwan,R.O.C., under the grants NSC-95-2221-E-007-043-MY3, NSC-96-2628-E-007-012-MY2 and NSC-96-2752-E-007-003-PAE.

Digital Object Identifier 10.1109/TWC.2010.06.090352

the congested packets may significantly reduce the networklifetime.

Many works in the literature proposed the use of dataaggregation [1]–[3], spatial sampling [4]–[6], and distributedsource coding [7]–[9] to eliminate redundancy in the sensors’messages. Specifically, data aggregation schemes have beenapplied to multihop systems where the messages transmittedby different sensors are jointly processed at the intermediaterelay sensors enroute to its destination. The efficiency of theseschemes relies on the exchange of information among sensorsand the correlation of sensors’ observations in consecutivehops. To compress the data with high efficiency, the messagesmust be routed through a large number of sensors, resultingin large delay and lack of robustness. In spatial samplingschemes, data is gathered only from a subset of sensors inthe network without compression or aggregation. A sensoris chosen to transmit only if its message is sufficiently in-formative, i.e., if its data is less correlated with others. Thisscheme achieves only lossy compression and the compressionefficiency is often unacceptable under strict distortion con-straints.

Due to these reasons, distributed source coding (DSC) [7]–[9] has emerged as a promising technique for data compressionin sensor networks. These works stem from the celebratedSlepian-Wolf DSC theory [7], which shows that the optimalcentralized compression efficiency can be achieved by com-pressing distributively each sensor’s message with only thestatistical knowledge of the other sensors’ data (but not theactual value of the data). Constructive coding schemes havebeen proposed in [10]–[13]. However, when employing DSC,the decoding of a sensor’s message will rely on the successfuldecoding of others. For example, if sensor 𝐴 encodes based onthe statistical knowledge of the data belonging to sensors 𝐵and 𝐶, the messages of sensors 𝐵 and 𝐶 must be successfullydecoded at the destination before sensor 𝐴’s message canbe decoded. Consequently, the loss of a single messagemay cause the decoding failure of multiple other messages.When a message is compressed based on the data statisticsof a large set of sensors, the compression efficiency mayincrease but the reliability decreases since the probability ofsuccessfully decoding this message is reduced. The efficiencyand reliability of DSC with respect to the encoding schemehas been discussed in [14]. Notice that, for a given encodingscheme, certain messages may be more important than others

1536-1276/10$25.00 c⃝ 2010 IEEE

1966 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 6, JUNE 2010

. . . . . . . . . . . . . . .

xxxxxxxxxxxxxxxxxxxxxxxxx Data Sink

Group 1Group k

Group Kk1 k2 kNkk3 . . . . . .

Wk1Wk2

Wk3 WkNk

Fig. 1. Illustration of the system model, where sensor 𝑘𝑖 sends message𝑊𝑘𝑖

which contains 𝐻(𝑋𝑘𝑖∣𝑋𝑘1

, ⋅ ⋅ ⋅ ,𝑋𝑘𝑖−1) bits.

if the successful decoding of these messages is critical todecode a large number of other messages. In this case, thetransmission probability given to each sensor by the mediumaccess control (MAC) protocol may have a significant impacton the throughput, delay, and energy efficiency.

To quantify these effects, we study the basic slottedALOHA [15] random access sensor network where sensorsare connected to the data gathering node through directtransmission links. The data gathering node can be viewedas the access point in a small network or as the cluster-headin a large-scale cluster-based sensor network. Specifically, inthe slotted ALOHA system, time is divided into synchronizedtime slots and each user is allowed to transmit a message ineach time slot with independent probabilities. Conventionally,to maintain fairness, each user is treated as equal and is as-signed the same transmission probability. However, for sensorsemploying DSC, the importance of each sensor’s message isno longer equal and the transmission probability should beassigned accordingly. Although more advanced sensor MACprotocols exist in the literature, slotted ALOHA is fundamentalto many existing schemes and the analysis of its behaviorprovides fruitful insights on future MAC protocol designs.

The main contribution of this work is to analyze theperformance of DSC in terms of the throughput, delay, andenergy efficiency in the slotted ALOHA system. We provideanalytical expressions of the throughput via approximations ofthe traffic load in each time slot. Then, based on the Markovchain analysis, we derive recursive expressions for the averagedelay and energy consumption required to successfully decodea specific message. Three probability assignment schemes arecompared with these analytical tools. Specifically, we showthat the throughput can be significantly improved by assigninghigher transmission probabilities to sensors that are encodedwith less dependency on others when the number of trans-mission time slots in each sensing period is limited. However,when the sensing period is sufficiently large, the delay of sen-sors with lower transmission probabilities may be significantlylarger when compared to the equal probability assignmentscheme. Yet, even in this case, we still save in terms of energyconsumption since assigning higher transmission probabilitiesto less compressed messages will reduce the number of timesthey are retransmitted. Moreover, to reduce the delay of lowpriority sensors, we further propose an adaptive MAC protocolto dynamically increase the sensors’ transmission probabilities(and, thus, the average throughput) as time progresses. This

timeslot 0

timeslot 1

timeslot 2

timeslot M-2

timeslot M-1

timeslot 0

. . . timeslot M-1

. . . . . . . . .

time ofmeasurement

sensing period

time ofmeasurement

Fig. 2. Time of measurement and sensing period.

improves considerably the performance in terms of throughputand delay, but also increases collisions and retransmissions,which in turn leads to larger energy consumption.

The remainder of this paper is organized as follows. In Sec-tion II, we describe the system model considered in this work.Based on this model, we analyze the throughput, delay, andenergy consumption in Sections III, IV, and V, respectively.An adaptive MAC protocol is then given in Section VI. InSection VII, we compare the performance of the proposedschemes and verify our analysis through computer simulations.Finally, we conclude in Section VIII.

II. SYSTEM MODEL

Consider a WSN that consists of one data sink node and 𝑁distributed sensor nodes that communicate to the sink nodethrough direct transmission links as shown in Fig. 1. Thenetwork is divided into 𝐾 encoding groups where the 𝑘-thgroup contains 𝑁𝑘 sensors such that

𝑁 =

𝐾∑𝑘=1

𝑁𝑘.

The grouping of sensors affects the compression efficiency anddecoding reliability as discussed in [14] and can be determinedin a distributed fashion based on many existing clusteringalgorithms [16], [17]. Here, we assume that the encodinggroups are determined a priori and focus on analyzing theMAC layer performances. In our system, no central control isavailable to determine the encoding groups or to schedule thetransmission of the sensors. Therefore, a random access pro-tocol is adopted in the MAC layer. Specifically, let us considerthe basic slotted ALOHA random access protocol where timeis divided into equal length time slots and the sensors transmitin each time slot with independent probabilities. The time slotshave duration that is greater or equal to the transmission timeof a packet.

Suppose that all sensors take measurements simultaneouslyand periodically at the beginning of every 𝑀 time slots asshown in Fig. 2. The sensors’ data are then encoded into DSCmessages and transmitted in the following 𝑀 time slots. Werefer to the 𝑀 -slot period as a sensing period. Each sensoronly has one message to transmit in each sensing periodand will become inactive once it has successfully deliveredits message. If a sensor does not successfully transmit itsmessage to the sink before a new sensing period begins, theold observation will be discarded.

Let the label 𝑘𝑖 denote the 𝑖-th sensor in the 𝑘-th encodinggroup. Based on the slotted ALOHA protocol, each sensor,e.g. sensor 𝑘𝑖, is assigned a transmission probability 𝑝𝑘𝑖 andattempts a transmission in each time slot with this probability.

HONG et al.: ON THE THROUGHPUT, DELAY, AND ENERGY EFFICIENCY OF DISTRIBUTED SOURCE CODING IN RANDOM ACCESS . . . 1967

The transmission probabilities are independent among sen-sors, and are independent and identically distributed (i.i.d.)over time. We consider the collision channel model where atransmission successfully reaches the destination if and onlyif no other sensor is transmitting in the same time slot. Eachsensor will persist in transmitting its message with the giventransmission probability until it succeeds.

Suppose that, at the beginning of a sensing period, sensorsin the 𝑘-th encoding group observe the data 𝑋𝑘1 , ⋅ ⋅ ⋅ , 𝑋𝑘𝑁𝑘

,where 𝑋𝑘𝑖 is the data observed by sensor 𝑘𝑖, and encode theminto messages 𝑊𝑘1 , ⋅ ⋅ ⋅ ,𝑊𝑘𝑁𝑘

using Slepian-Wolf DSC. Weassume that the encoding is performed in a sequential ordersuch that the message of sensor 𝑘𝑖 is encoded by exploitingthe data statistics of 𝑋𝑘1 , ⋅ ⋅ ⋅ , 𝑋𝑘𝑖−1 . Hence, the data 𝑋𝑘𝑖 ,for 𝑖 > 1, is compressed into the message 𝑊𝑘𝑖 which consistsof 𝐻(𝑋𝑘𝑖 ∣𝑋𝑘1 , ⋅ ⋅ ⋅ , 𝑋𝑘𝑖−1) number of bits, where 𝐻 denotesthe entropy (or conditional entropy of the data). For 𝑖 = 1, themessage 𝑊𝑘1 consists of 𝐻(𝑋𝑘1) bits. Due to the sequentialstatistical dependence, the DSC messages must be decodedin order at the sink, that is, the message 𝑊𝑘𝑖 cannot besuccessfully decoded unless 𝑊𝑘1 , ⋅ ⋅ ⋅ ,𝑊𝑘𝑖−1 are all received.The encoding of messages is performed independently in eachgroup.

With the sequential encoding structure, the messages en-coded earlier in the sequence will have a higher impact onthe successful decoding of others and should be assigned ahigher transmission probability so that it is received with highprobability within the sensing period. Our goal is to analyti-cally evaluate the throughput, delay, and energy consumptionof the system for a given set of transmission probabilities,say {𝑝𝑘𝑖 , ∀𝑘, 𝑖}. The performance of different probabilityassignment schemes will be compared in Sec. VII.

III. THROUGHPUT ANALYSIS

First, let us start by analyzing the system throughput,which is defined as the average number of messages that canbe successfully decoded in each sensing period. The directcomputation of throughput involves probability analysis ona large combination of events, which may rapidly becomeintractable as 𝑁 increases. Therefore, in the following, wefirst derive an approximate expression for the traffic load ineach time slot and use it to derive the average throughput.

In the proposed system, each sensor transmits only onemessage in each sensing period and becomes inactive afterit has transmitted successfully. Therefore, the traffic decreaseswith time as more and more sensors successfully deliver theirmessages to the sink node. The probability that a message issuccessfully received (but not necessarily decoded) by the sinkin slot 0 is

𝑃rx[0] =

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

⎡⎣𝑝𝑘𝑖

𝑁𝑘∏𝑗=1,𝑗 ∕=𝑖

(1− 𝑝𝑘𝑗 )

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

(1 − 𝑝ℓℎ)

⎤⎦=

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

⎡⎣𝑝𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

⎤⎦, (1)

where 𝑞𝑘𝑗 = 1 − 𝑝𝑘𝑗 . Given {𝑝𝑘𝑖 , ∀𝑘, 𝑖}, let us define theaverage total number of transmission attempts at the beginning

of each sensing period (i.e. the 0-th time slot) as

𝐺[0] =

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖 , (2)

which we refer to as the initial traffic load. Then, for large 𝑁and 𝑝𝑘𝑖 ≪ 1, such that 𝑞𝑘𝑖 = 1−𝑝𝑘𝑖 ≈ 𝑒−𝑝𝑘𝑖 , the probabilityin (1) can be approximated as

𝑃rx[0] ≈𝐾∑

𝑘=1

𝑁𝑘∑𝑖=1

[𝑝𝑘𝑖

𝐾∏ℓ=1

𝑁ℓ∏ℎ=1

𝑞ℓℎ

]

≈𝐾∑

𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖 × 𝑒−𝐺[0] = 𝐺[0]× 𝑒−𝐺[0]. (3)

Similarly, let 𝐺[𝑚] be the average number of transmissionattempts (i.e., the average traffic load) in the 𝑚-th time slot. Toevaluate the remaining traffic at time slot 𝑚, one must considerall combinations of messages that could have been transmittedsuccessfully in the previous time slots. The combination ofthese events increases exponentially as the number of timeslots increases. Hence, we compute an approximation of theaverage traffic load by deducting the expected number ofsuccessfully delivered messages in each of the previous timeslots. For example, if 𝑊𝑘𝑖 is successfully delivered to the sinknode in time slot 0, the average traffic that is deducted due tothis event is equal to 𝑝𝑘𝑖 . The probability of this event is

𝑝𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

and the approximated average traffic load in time slot 1 isgiven by

𝐺[1] ≈ 𝐺[0]−𝐾∑

𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

⎡⎣𝑝𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

⎤⎦

= 𝐺[0]−

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

[𝑝2𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

]𝐾∑

𝑘=1

𝑁𝑘∑𝑖=1

[𝑝𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

]

×𝐾∑

𝑘=1

𝑁𝑘∑𝑖=1

⎡⎣𝑝𝑘𝑖


𝑞𝑘𝑗

𝐾∏ℓ=1,ℓ ∕=𝑘

𝑁ℓ∏ℎ=1

𝑞ℓℎ

⎤⎦

≈ 𝐺[0]−

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝2𝑘𝑖𝑒−𝐺[0]

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖𝑒−𝐺[0]

×𝐺[0]𝑒−𝐺[0]

≈ 𝐺[0]−

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝2𝑘𝑖

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

×𝐺[0]𝑒−𝐺[0], (4)

assuming that 𝑁 is large enough and 𝑝𝑘𝑖 ≪ 1 for all 𝑘, 𝑖.Based on the above approximation, we can compute theremaining average traffic load at the beginning of the 𝑚-th


time slot as

𝐺[𝑚] = 𝐺[𝑚− 1]−

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝2𝑘𝑖

𝐾∑𝑘=1

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

×𝐺[𝑚− 1]𝑒−𝐺[𝑚−1], (5)

for 𝑚 ≥ 1.To derive the average throughput, we first compute the

average probability that a particular message is successfullytransmitted within a sensing period. Specifically, given that,in time slots 0, 1, . . ., 𝑚 − 1, the message 𝑊𝑘𝑖 was notsuccessfully transmitted to the sink, it may again fail to reachthe sink in the 𝑚-th time slot if either no transmission attemptis made by the sensor or if a collision occurred, which occurswith probability 𝑞𝑘𝑖 + 𝑝𝑘𝑖

[1 − 𝑒−(𝐺[𝑚]−𝑝𝑘𝑖

)]. Therefore, at

the end of a sensing period that consists of 𝑀 time slots, theprobability that the message 𝑊𝑘𝑖 is successfully received atthe sink is

𝑃𝑆𝑘𝑖(𝑀) ≈ 1−

𝑀−1∏𝑚=0

{𝑞𝑘𝑖 + 𝑝𝑘𝑖

[1− 𝑒−(𝐺[𝑚]−𝑝𝑘𝑖

)]}

. (6)

For sensors in the same encoding group, e.g. group 𝑘, themessage corresponding to the 𝑘𝑖-th sensor can be decoded bythe sink node if the messages 𝑊𝑘1 , ⋅ ⋅ ⋅ ,𝑊𝑘𝑖 have all beensuccessfully received by the sink at the end of the sensingperiod. In this case, if the message 𝑊𝑘𝑖+1 was not successfullyreceived, then only 𝑖 messages will be successfully decodedout of group 𝑘. This occurs with probability

𝑖∏𝑗=1

𝑃𝑆𝑘𝑗(𝑀)× (1− 𝑃𝑆

𝑘𝑖+1(𝑀)).

For the 𝑘-th group, the average number of messages success-fully decoded by the sink at the end of each sensing period isgiven by

𝑅𝑘(𝑀)=

min{𝑁𝑘,𝑀}∑𝑖=1

𝑖⋅⎡⎣ 𝑖∏𝑗=1

𝑃𝑆𝑘𝑗(𝑀)× (1 − 𝑃𝑆

𝑘𝑖+1(𝑀))

⎤⎦ (7)

where 𝑃𝑆𝑘𝑁𝑘+1

(𝑀) = 0. The average throughput of the entiresystem is

𝑅(𝑀) =

𝐾∑𝑘=1

𝑅𝑘(𝑀).

IV. DELAY ANALYSIS

In the previous section, we provided an approximate anal-ysis of the average throughput in each sensing period. Giventhat the sensing period is sufficiently long, we are also inter-ested in the average delay required to decode each particularmessage. This is derived in the following by modeling theproblem as the average time to absorption in a Markov chain.

Without loss of generality, let us focus specifically onthe behavior of the sensors in the 𝑘-th group. Supposethat the sensors are labeled as 𝑘1, 𝑘2, ⋅ ⋅ ⋅ , 𝑘𝑁𝑘

accord-ing to the sequential encoding order and let B[𝑚] =(𝐵𝑘1 [𝑚], 𝐵𝑘2 [𝑚], ⋅ ⋅ ⋅ , 𝐵𝑘𝑁𝑘

[𝑚]) be the vector indicating thesuccessful delivery of sensor 𝑘𝑖’s message, where 𝐵𝑘𝑖 [𝑚] = 1if the message from sensor 𝑘𝑖 was successfully transmitted

during or before the 𝑚-th time slot and 𝐵𝑘𝑖 [𝑚] = 0, other-wise. Although the success or failure of messages transmittedby sensors in other groups will not affect the decoding ofthe messages in the 𝑘-th group (since they are encodedindependently), their transmissions will cause congestion tothe network and, thus, decrease throughput. From the deriva-tions in Section III, we know that the average traffic loadin the 𝑚-th time slot can be approximated as �̄�[𝑚] andthat, given the state B[𝑚], the conditional probability thatno user outside of group 𝑘 transmits can be approximated

as 𝑒−(�̄�[𝑚]−∑

𝑖:𝐵𝑘𝑖[𝑚]=0 𝑝𝑘𝑖

)+

, where (𝑎)+ = 𝑎, for 𝑎 > 0, and0, otherwise. By this approximation, the state B[𝑚+ 1] willbe independent of B[𝑚− 1] when given B[𝑚] and, therefore,the process {B[𝑚]}∞𝑚=0 forms a Markov Chain.

Let us denote by 𝑆b the state that represents the eventthat B[𝑚] = b = (𝑏1, . . . , 𝑏𝑁𝑘

). In particular, the state 𝑆0

represents the case that no message is successfully received atthe sink, where 0 is the all-zero vector, and 𝑆1(1:𝑁𝑘) representsthe state that all messages are successfully received, where1(𝑖:𝑗) is the vector that consists of 1’s from the 𝑖-th elementto the 𝑗-th element and zero everywhere else. Starting fromthe initial state 𝑆0, there will be a transition to state 𝑆1(𝑖:𝑖)

if sensor 𝑘𝑖 successfully delivers its message in the first timeslot. The transition probability from state 𝑆b to state 𝑆b′ isdenoted by 𝑃bb′ [𝑚]. Notice that the Markov chain is time-varying due to the decrease in average traffic load over time.

Due to the sequential decoding requirement, the message ofsensor 𝑘𝑖 can be decoded successfully at the sink if and only ifthe messages 𝑊𝑘1 , ⋅ ⋅ ⋅ ,𝑊𝑘𝑖 have been successfully received.The number of time slots required in order to successfullydecode the message 𝑊𝑘𝑖 is equal to the number of transitionsfrom state 𝑆0 to one of the states that has 𝑏𝑗 = 1 for all𝑗 ≤ 𝑖. The average delay is then equivalent to the mean timeto absorption [18] of the Markov Chain by setting the set ofabsorbing states as 𝒮 = {𝑆b : 𝑏𝑗 = 1 for all 𝑗 ≤ 𝑖}.

Let 𝑡b[𝑚] be the average number of additional transitionsneeded before the chain arrives at an absorbing state, giventhat it is currently in state 𝑆b at time 𝑚 (i.e., the conditionalmean time to absorption). Moreover, let 𝑃bb′ [𝑚] be theconditional probability that a transition to state 𝑆b′ occurs astime progresses from 𝑚 to 𝑚+1, given that the state at time𝑚 is 𝑆b. When this event occurs, the conditional mean timeto absorption, starting from time 𝑚, will be equal to 1 plus theremaining time to absorption at time 𝑚+ 1, i.e., 𝑡b′ [𝑚+ 1].Then, by taking the expectation over all possible transitions, itfollows that the conditional mean time to absorption satisfiesthe following set of linear recursive equations:

𝑡b[𝑚] =∑b′

𝑃bb′ [𝑚](1 + 𝑡b′ [𝑚+ 1]) for all b /∈ 𝒮. (8)

By utilizing the conditions that 𝑡b[𝑚] = 0 for all b ∈ 𝒮, wecan then compute the values of 𝑡b[𝑚] by solving the aboveset of linear equations. Since the initial state at the beginningof each sensing period (i.e., at time slot 0) is 𝑆0, the averagedelay of interest is 𝑡0[0].

Let 𝑇 (𝑁 ′)𝑛 (𝑝𝑘𝑖1

, 𝑝𝑘𝑖2, . . . , 𝑝𝑘𝑖

𝑁′ )[𝑚] be the average residualdelay required to decode the messages of sensors 𝑘𝑖1 , ⋅ ⋅ ⋅ , 𝑘𝑖𝑛starting from time 𝑚, given that sensors 𝑘𝑖1 , . . . , 𝑘𝑖𝑁′ arecurrently active. This is equal to the value of 𝑡b[𝑚] given


00

10

11

01

pk1qk2e−(G[m]−pk1−pk2)

+ qk1pk2e−(G[m]−pk1−pk2)

+

1 − pk2e−(G[m]−pk2)

+

pk1e−(G[m]−pk1)

+

1 − pk1e−(G[m]−pk1)

+

pk2e−(G[m]−pk2)

+

1 − pk1qk2e−(G[m]−pk1−pk2)

+ − qk1pk2e−(G[m]−pk1−pk2)

+

(a) 𝑆(1,1) absorbing.

00

10

11

01

pk1qk2e−(G[m]−pk1−pk2)

+ qk1pk2e−(G[m]−pk1−pk2)

+

pk1e−(G[m]−pk1)

+

1 − pk1e−(G[m]−pk1)

+

1 − pk1qk2e−(G[m]−pk1−pk2)

+ − qk1pk2e−(G[m]−pk1−pk2)

+

(b) 𝑆(1,0) ,𝑆(1,1) absorbing.

Fig. 3. The state transition diagrams of the average delay analysis for 𝑁𝑘=2.

above for state b = (𝑏1, . . . , 𝑏𝑁𝑘), where 𝑏𝑗 = 0 for

𝑗 ∈ {𝑖1, . . . , 𝑖𝑁 ′} and 1 otherwise, and the absorbing statesbeing those that have 𝑏𝑗 = 1 for all 𝑗 ≤ 𝑖𝑛. Equivalently, thiscan also be obtained by computing 𝑡0[𝑚] for a Markov chainthat is constructed for a group that consists of only sensors𝑘𝑖1 , . . ., 𝑘𝑖𝑁′ and setting the absorbing states to all states 𝑆b

where b = (𝑏1, . . . , 𝑏𝑁 ′) is such that 𝑏𝑗 = 1 for all 𝑗 ≤ 𝑛. Thevalue of 𝑇 (𝑁 ′)

𝑛 (𝑝𝑘𝑖1, 𝑝𝑘𝑖2

, . . . , 𝑝𝑘𝑖𝑁′ )[𝑚] can be computed for

any 𝑛 and for any subset of sensors 𝑘𝑖1 , . . . , 𝑘𝑖𝑁′ as describedin the following. These derivations are necessary to arrive atthe result 𝑇 (𝑁𝑘)

𝑛 (𝑝𝑘1 , 𝑝𝑘2 , . . . , 𝑝𝑘𝑁𝑘)[𝑚], which is desired. To

facilitate the understanding of this approach, let us start outwith the case of 𝑁𝑘 = 2 and extend to cases with 𝑁𝑘 > 2later on. The Markov Chain of the case 𝑁𝑘 = 2 consists offour states 𝑆(0,0), 𝑆(0,1), 𝑆(1,0), 𝑆(1,1), as shown in Fig. 3.

1) The average delay required to decode the message 𝑊𝑘2 :The message 𝑊𝑘2 can be successfully decoded if and only ifboth 𝑊𝑘1 and 𝑊𝑘2 are successfully received. Hence, 𝑆(1,1)

serves as the absorbing state and the mean time to absorptionsatisfies the following set of equations:

𝑡(0,0)[𝑚]=[1− 𝑝𝑘1𝑞𝑘2𝑒

−(�̄�[𝑚]−𝑝𝑘1−𝑝𝑘2

)+

− 𝑞𝑘1𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘1

−𝑝𝑘2)+](1+𝑡(0,0)[𝑚+1])

+ 𝑝𝑘1𝑞𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘1

−𝑝𝑘2)+(1+𝑡(1,0)[𝑚+1])

+𝑞𝑘1𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘1

−𝑝𝑘2)+(1+𝑡(0,1)[𝑚+1]), (9)

𝑡(0,1)[𝑚]=𝑝𝑘1𝑒−(�̄�[𝑚]−𝑝𝑘1

)+(1 + 𝑡(1,1)[𝑚+ 1])

+(1−𝑝𝑘1𝑒−(�̄�[𝑚]−𝑝𝑘1

)+)(1+𝑡(0,1)[𝑚+ 1]), (10)

𝑡(1,0)[𝑚]=𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘2

)+(1 + 𝑡(1,1)[𝑚+ 1])

+(1−𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘2

)+)(1+𝑡(1,0)[𝑚+ 1]), (11)

and 𝑡(1,1)[𝑚] = 0. The first term in (9) expresses the factthat an average delay of 1 + 𝑡(0,0)[𝑚 + 1] will be experi-enced if either a collision occurs or that nobody transmitsin the 𝑚-th time slot. This event occurs with probability(1 − 𝑝𝑘1𝑞𝑘2𝑒

−(�̄�[𝑚]−𝑝𝑘1−𝑝𝑘2

)+ − 𝑞𝑘1𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘1

−𝑝𝑘2)+).

The second term considers the case where sensor 𝑘1 trans-mits while all other sensors remain idle (which occurs withprobability 𝑝𝑘1𝑞𝑘2𝑒

−(�̄�[𝑚]−𝑝𝑘1−𝑝𝑘2

)+ ) and an average delay of1+𝑡(1,0)[𝑚+1] will be experienced. The third term is obtainedsimilarly by considering 𝑘2. Furthermore, the equations (10)and (11) are written by considering the success and failure ofthe transmissions of sensors 𝑘1 and 𝑘2, respectively. Let usfirst solve the above equations for the asymptotic case where𝑚 → ∞. Since the average traffic load �̄�[𝑚] decreases to 0in this case, it follows that

𝑡(0,0)[∞] =(1−𝑝𝑘1𝑞𝑘2−𝑞𝑘1𝑝𝑘2)(1+𝑡(0,0)[∞])

+𝑝𝑘1𝑞𝑘2(1+𝑡(1,0)[∞])+𝑞𝑘1𝑝𝑘2(1+𝑡(0,1)[∞]),

𝑡(1,0)[∞] =𝑝𝑘2(1+𝑡(1,1)[∞])+𝑞𝑘2(1+𝑡(1,0)[∞]),

𝑡(0,1)[∞] =𝑝𝑘1(1+𝑡(1,1)[∞])+𝑞𝑘1(1+𝑡(0,1)[∞]),

and 𝑡(1,1)[∞] = 0, where 𝑡b[∞] = lim𝑚→∞ 𝑡b[𝑚]. Bysolving the above equations, we get

lim𝑚→∞𝑡(0,0)[𝑚]=

1+𝑞𝑘1𝑝𝑘2

𝑝𝑘1+

𝑝𝑘1𝑞𝑘2

𝑝𝑘2

𝑝𝑘1𝑞𝑘2+𝑞𝑘1𝑝𝑘2

,

lim𝑚→∞𝑡(1,0)[𝑚]=1/𝑝𝑘2 , and lim

𝑚→∞𝑡(0,1)[𝑚]=1/𝑝𝑘1 . (12)

Notice that the above asymptotic equations hold for any 𝑚 >𝑚∗ such that

�̄�[𝑚∗] < min𝑖=1,...,𝑁𝑘

𝑝𝑘𝑖 ≜ 𝑝𝑘,min

since this is sufficient to yield (�̄�[𝑚∗] − 𝑝𝑘𝑖)+ = 0 for all

𝑖. By substituting (12) into (9)-(11) for 𝑚∗ + 1, we can thenobtain 𝑡b[𝑚

∗] for all b. By utilizing the equations (9)-(11)recursively from 𝑚 = 𝑚∗ + 1 to 𝑚 = 0, we can then solvefor the average delay of decoding both 𝑊𝑘1 and 𝑊𝑘2 , i.e.,𝑇

(2)2 (𝑝𝑘1 , 𝑝𝑘2)[𝑚] = 𝑡(0,0)[𝑚] for any finite 𝑚. Our eventual

goal is to solve for 𝑡(0,0)[0] which is the average delay startingfrom the initial state (0, 0) at time 0. Notice that, for 𝐾 = 1,the Markov chain is time invariant and the average delay isgiven directly by (12).

2) The average delay required to decode the message 𝑊𝑘1 :The message 𝑊𝑘1 can be successfully decoded whenever itarrives at the destination, regardless of whether or not 𝑊𝑘2 wasreceived. Therefore, 𝑆(1,1) and 𝑆(1,0) serve as the absorbingstates, as shown in Fig. 3(b), and the mean time to absorptionmust satisfy (9), (10), and 𝑡(1,0)[𝑚] = 𝑡(1,1)[𝑚] = 0. Similarly,we can first obtain the asymptotic solutions

lim𝑚→∞𝑡(0,0)[𝑚]=

1+(𝑞𝑘1𝑝𝑘2/𝑝𝑘1)


and lim𝑚→∞𝑡(0,1)[𝑚]=

1

𝑝𝑘1

, (13)

which holds for any 𝑚 > 𝑚∗ such that �̄�[𝑚∗] <𝑝𝑘,min. Recall that 𝑝𝑘,min is the minimum probability amongthose assigned to sensors in group 𝑘. The average delay𝑇

(2)1 (𝑝𝑘1 , 𝑝𝑘2)[𝑚] = 𝑡(0,0)[𝑚] can be computed by solving

(9) and (10) recursively in the decreasing order of 𝑚.


More generally, suppose that the values of𝑇

(𝑁𝑘−1)𝑛 (p

(𝑖)𝑘 )[𝑚] have been computed for all 𝑛, 𝑚,

and 𝑖, where p(𝑖)𝑘 = (𝑝𝑘1 , ⋅ ⋅ ⋅ , 𝑝𝑘𝑖−1 , 𝑝𝑘𝑖+1 , ⋅ ⋅ ⋅ , 𝑝𝑘𝑁𝑘

). Then,given that there are 𝑁𝑘 sensors in the group with transmissionprobabilities p𝑘 = (𝑝𝑘1 , ⋅ ⋅ ⋅ , 𝑝𝑘𝑁𝑘

), the average delay neededto receive the first 𝑛 messages can be expressed as

𝑇 (𝑁𝑘)𝑛 (p𝑘)[𝑚]=

(1−

𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝑒−(�̄�[𝑚]−∑𝑁𝑘

𝑖=1 𝑝𝑘𝑖)+)

×(1+𝑇 (𝑁𝑘)

𝑛 (p𝑘)[𝑚+ 1])

+𝑛∑

𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝑒−(�̄�[𝑚]−∑𝑁𝑘

𝑖=1𝑝𝑘𝑖)+(1+𝑇

(𝑁𝑘−1)𝑛−1 (p

(𝑖)𝑘 )[𝑚+1]

)

+

𝑁𝑘∑𝑖=𝑛+1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗 𝑒−(�̄�[𝑚]−∑𝑁𝑘

𝑖=1𝑝𝑘𝑖)+(1+𝑇 (𝑁𝑘−1)

𝑛 (p(𝑖)𝑘 )[𝑚+1]

).

(14)

This follows from the fact that, if sensor 𝑘𝑖transmits successfully, which occurs with probability𝑝𝑘𝑖

∏𝑗 ∕=𝑖 𝑞𝑘𝑗 𝑒

−(�̄�[𝑚]−∑𝑁𝑘𝑖=1 𝑝𝑘𝑖

)+ , then, in addition to thetime slot that has just elapsed, the residual delay of𝑇

(𝑁𝑘−1)𝑛−1 (p

(𝑖)𝑘 )[𝑚 + 1] will still be needed if 𝑖 ≤ 𝑛 and the

residual delay of 𝑇(𝑁𝑘−1)𝑛 (p

(𝑖)𝑘 )[𝑚 + 1] will be needed if

𝑖 > 𝑛. On the other hand, if no transmission was successful,then 𝑁𝑘 sensors will still be active and a residual delay of𝑇

(𝑁𝑘)𝑛 (p𝑘)[𝑚 + 1] will be needed. Then, asymptotically, as

𝑚 → ∞, we have

𝑇 (𝑁𝑘)𝑛 (p𝑘)[∞] =

[1 +

𝑛∑𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝑇(𝑁𝑘−1)𝑛−1 (p(𝑖)

𝑘 )[∞]

+


𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝑇(𝑁𝑘−1)𝑛 (p(𝑖)

𝑘 )[∞]

]/ 𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗 . (15)

The average delay 𝑇(𝑁𝑘)𝑛 (p𝑘)[𝑚] for any finite 𝑚 can then

be computed recursively with (14) starting from any 𝑚∗ suchthat �̄�[𝑚∗] < 𝑝𝑘,min. Moreover, for 𝑁𝑘 > 2, we start with𝑇

(2)2 (𝑝𝑘1 , 𝑝𝑘2)[𝑚] and 𝑇

(2)1 (𝑝𝑘1 , 𝑝𝑘2)[𝑚] derived previously

and compute 𝑇(3)𝑛 (𝑝𝑘1 , 𝑝𝑘2 , 𝑝𝑘3)[𝑚], for all 𝑛, using the above

procedure. By proceeding recursively in this fashion, we willeventually be able to obtain the average delay 𝑇

(𝑁𝑘)𝑛 (p𝑘)[𝑚]

for any 𝑁𝑘.

V. ENERGY CONSUMPTION

The main purpose of DSC is to remove the redundancy insensors’ messages and, thus, reduce the energy required todeliver them. Here, we also utilize the Markov Chain analysisto derive the average energy required to successfully decodea given message.

Based on the DSC encoding and the dependencies amongthe sensors’ data, the compression efficiency and, thus, theenergy cost may be different for each sensor. Let 𝜀𝑘𝑖 be theenergy consumed when transmitting the message of sensor 𝑘𝑖and assume that it is proportional to the number of bits in 𝑊𝑘𝑖 .Let us denote by 𝐸

(𝑁𝑘)𝑛 (𝑝𝑘1 , 𝑝𝑘2 , . . . , 𝑝𝑘𝑁𝑘

)[𝑚] the averageenergy that must be consumed, starting from time 𝑚, in orderto successfully receive the messages of sensors 𝑘1, . . . , 𝑘𝑛,

given that 𝑘1, . . . , 𝑘𝑁𝑘are currently active. Similarly, let us

start by analyzing the case of 𝑁𝑘 = 2 using also the state-transition diagram in Fig. 3.

1) The average energy cost of decoding the message 𝑊𝑘2 :To compute the average energy cost required to successfullydecode 𝑊𝑘2 , we again set 𝑆(1,1) to be the absorbing state (seeFig. 3(a)) since 𝑊𝑘2 can be successfully decoded only if themessages 𝑊𝑘1 and 𝑊𝑘2 are both successfully received. Let𝑒b[𝑚] be the average energy that will be consumed, countingfrom state 𝑆b at time 𝑚, before the chain enters an absorbingstate. Its values must satisfy the following equations:

𝑒(0,0)[𝑚] =𝑝𝑘1𝑞𝑘2𝑞other[𝑚](𝜀𝑘1 + 𝑒(1,0)[𝑚+ 1])

+ 𝑞𝑘1𝑝𝑘2𝑞other[𝑚](𝜀𝑘2 + 𝑒(0,1)[𝑚+ 1])

+ 𝑝𝑘1𝑞𝑘2(1− 𝑞other[𝑚])(𝜀𝑘1 + 𝑒(0,0)[𝑚+ 1])

+ 𝑞𝑘1𝑝𝑘2(1− 𝑞other[𝑚])(𝜀𝑘2 + 𝑒(0,0)[𝑚+ 1])

+ 𝑝𝑘1𝑝𝑘2(𝜀𝑘1 + 𝜀𝑘2 + 𝑒(0,0)[𝑚+ 1])

+ 𝑞𝑘1𝑞𝑘2𝑒(0,0)[𝑚+ 1], (16)

𝑒(0,1)[𝑚] =𝑝𝑘1𝑒−(�̄�[𝑚]−𝑝𝑘1

)+𝜀𝑘1

+ 𝑝𝑘1(1− 𝑒−(�̄�[𝑚]−𝑝𝑘1)+)(𝜀𝑘1 + 𝑒(0,1)[𝑚+ 1])

+ (1 − 𝑝𝑘1)𝑒(0,1)[𝑚+ 1], (17)

𝑒(1,0)[𝑚] =𝑝𝑘2𝑒−(�̄�[𝑚]−𝑝𝑘2

)+𝜀𝑘2

+ 𝑝𝑘2(1− 𝑒−(�̄�[𝑚]−𝑝𝑘2)+)(𝜀𝑘2 + 𝑒(1,0)[𝑚+ 1])

+ (1 − 𝑝𝑘2)𝑒(1,0)[𝑚+ 1], (18)

and 𝑒(1,1)[𝑚] = 0, where 𝑞other[𝑚] = 𝑒−(�̄�[𝑚]−𝑝𝑘1−𝑝𝑘2

)+

is the probability that no user outside of group 𝑘 transmits.Notice from the above equations that energy 𝜀𝑘𝑖 is consumedevery time user 𝑘𝑖 attempts a transmission, regardless ofwhether or not the transmission was successful. If only user 𝑘1is transmitting, which occurs with probability 𝑝𝑘1𝑞𝑘2𝑞other[𝑚],then, in addition to consuming 𝜀1, an average of 𝑒(1,0)[𝑚+1]is still required since we enter 𝑆(1,0) after 𝑊𝑘1 is successfullydelivered. The remaining terms follow similarly. Using theapproach given in Section IV, we can first compute theasymptotic average energy consumption

lim𝑚→∞ 𝑒(0,0)[𝑚] =

𝑝𝑘1𝑝𝑘2 + 𝑝𝑘1𝑞𝑘2 + 𝑞𝑘1𝑝𝑘2

𝑝𝑘1𝑞𝑘2 + 𝑞𝑘1𝑝𝑘2

(𝜀𝑘1 + 𝜀𝑘2),

lim𝑚→∞ 𝑒(0,1)[𝑚] = 𝜀𝑘1 , and lim

𝑚→∞ 𝑒(1,0)[𝑚] = 𝜀𝑘2 . (19)

This solution holds for any 𝑚 > 𝑚∗ such that �̄�[𝑚∗] <𝑝𝑘,min. Therefore, by solving (16)-(18) recursively startingfrom 𝑚 = 𝑚∗ + 1 and going backwards to 𝑚 = 0, we caneventually obtain the values of 𝐸

(2)2 (𝑝𝑘1 , 𝑝𝑘2)[𝑚] = 𝑒(0,0)[𝑚]

for any finite 𝑚 (or, specifically, for 𝑚 = 0).

2) The average energy cost of decoding the message 𝑊𝑘1 :The message 𝑊𝑘1 can be successfully decoded as long as𝑊𝑘1 is received at the data gathering node, regardless ofwhether or not 𝑊𝑘2 is received. Therefore, both 𝑆(1,1) and𝑆(1,0) serve as the absorbing states. Similarly, the conditionalenergy cost 𝑒b satisfies the equations (16) and (17) with𝑒(1,0)[𝑚] = 𝑒(1,1)[𝑚] = 0. Again, the asymptotic solution


is given by

lim𝑚→∞ 𝑒(0,0)[𝑚] =

𝑝𝑘1𝑞𝑘2𝜀𝑘1+𝑝𝑘2(𝜀𝑘1+𝜀𝑘2)


and

lim𝑚→∞ 𝑒(0,1)[𝑚]=𝜀𝑘1 .

This solution holds for any 𝑚 > 𝑚∗ such that �̄�[𝑚∗] <𝑝𝑘,min. Then, with (16), (17), and the fact that 𝑒(1,0)[𝑚] =

𝑒(1,1)[𝑚]=0, we can compute recursively 𝐸(2)1 (𝑝𝑘1 , 𝑝𝑘2)[𝑚]=

𝑒(0,0)[𝑚] for all 𝑚.Similar to the delay analysis, we can generalize the deriva-

tions to the case with arbitrary group sizes, i.e., 𝑁𝑘. Letp𝑘 = (𝑝𝑘1 , 𝑝𝑘2 , ⋅ ⋅ ⋅ , 𝑝𝑘𝑁𝑘

) be the transmission probabilities

of the 𝑁𝑘 sensors and let 𝐸(𝑁𝑘)𝑛 (p𝑘) be the average energy

cost that is required to successfully deliver the messages𝑊𝑘1 , ⋅ ⋅ ⋅ ,𝑊𝑘𝑛 given that there are 𝑁𝑘 sensors currently activein the group. If more than one sensor is transmitting in thecurrent time slot, no packet will be successfully received at thesink node due to collision, but energy will still be consumed.Given any probability vector p, let us define 𝒜(p) as the indexset of sensors that are still active (i.e. the indices of sensorsassociated with the vector p), e.g., 𝒜(p𝑘) = {𝑘1, ⋅ ⋅ ⋅ , 𝑘𝑁𝑘

}.Let Λ ⊂ 𝒜(p) be the set of simultaneously transmitting usersin group 𝑘. If ∣Λ∣ ≥ 2 or if a sensor outside of group 𝑘 istransmitting, collision will occur and the energy consumedduring this time slot will be

∑𝑟∈Λ 𝜀𝑟. In this case, the set

of active sensors will remain unchanged. If sensor 𝑘𝑗 is theonly sensor transmitting in the time slot, the energy 𝜀𝑘𝑗 willbe consumed and the average energy cost that remains will be𝐸

(𝑁𝑘−1)𝑛−1 (p(𝑗))[𝑚] if 𝑗 ≤ 𝑛 and 𝐸

(𝑁𝑘−1)𝑛 (p(𝑗))[𝑚] if 𝑗 > 𝑛.

Hence, we can write

𝐸(𝑁𝑘)𝑛 (p𝑘)[𝑚]

=∑

Λ⊂𝒜(p𝑘):∣Λ∣≥2

∏𝑟∈Λ

𝑝𝑟∏

𝑠∈𝒜(p𝑘)∖Λ

[𝑞𝑠 ⋅ 𝑞other[𝑚]

×(∑

𝑢∈Λ

𝜀𝑢 + 𝐸(𝑁𝑘)𝑛 (p𝑘)[𝑚+ 1]

)]+

∑Λ⊂𝒜(p𝑘)

∏𝑟∈Λ

𝑝𝑟∏

𝑠∈𝒜(p𝑘)∖Λ

[𝑞𝑠(1− 𝑞other[𝑚])

×(∑

𝑢∈Λ

𝜀𝑢 + 𝐸(𝑁𝑘)𝑛 (p𝑘)[𝑚+ 1]

)]

+

𝑁𝑘∏𝑖=1

𝑞𝑘𝑖𝑞other[𝑚]𝐸(𝑁𝑘)𝑛 (p𝑘)[𝑚+ 1]

+𝑛∑

𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗 𝑞other[𝑚](𝜀𝑘𝑖+𝐸

(𝑁𝑘−1)𝑛−1 (p

(𝑖)𝑘 )[𝑚+1]

)

+

𝑁∑𝑖=𝑛+1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗 𝑞other[𝑚](𝜀𝑘𝑖+𝐸(𝑁𝑘−1)

𝑛 (p(𝑖)𝑘 )[𝑚+1]

).

Asymptotically as 𝑚 → ∞, we have 𝑞other[𝑚] = 1 and, thus,it follows that

𝐸(𝑁𝑘)𝑛 (p𝑘)[∞] = lim

𝑚→0𝐸(𝑁𝑘)

𝑛 (p𝑘)[𝑚]

=

[ ∑Λ⊂𝒜(p𝑘)

∏𝑟∈Λ

𝑝𝑟∏

𝑠∈𝒜(p𝑘)∖Λ𝑞𝑠

∑𝑢∈Λ

𝜀𝑢

+

𝑛∑𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝐸(𝑁𝑘−1)𝑛−1 (p

(𝑖)𝑘 )[∞]

+


𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗𝐸(𝑁𝑘−1)𝑛 (p

(𝑖)𝑘 )[∞]

]/ 𝑁𝑘∑𝑖=1

𝑝𝑘𝑖

∏𝑗 ∕=𝑖

𝑞𝑘𝑗 .

The average energy consumption 𝐸(𝑁𝑘)𝑛 (p𝑘)[𝑚] for any finite

𝑚 can then be computed iteratively by the above equationstarting from any 𝑚 > 𝑚∗ such that �̄�[𝑚∗] < 𝑝𝑘,min.

It is worthwhile to note that, when the probabilities areweighted according to the importance of messages, low pri-ority messages will likely result in larger delay comparedto the case with equal probabilities. However, at the sametime, high priority messages (which are typically of largersize) are transmitted with higher probabilities and, thus, lessretransmissions are required and less energy is consumed. Thisis demonstrated via computer simulations in Sec. VII.

VI. AN ADAPTIVE MEDIUM ACCESS CONTROL PROTOCOL

Since each sensor becomes inactive after it successfullydelivers its message to the sink, the average traffic loaddecreases gradually over time, resulting in many idle timeslots. To improve the bandwidth efficiency in later time slots,we propose an adaptive MAC protocol that increases thetransmission probability of the remaining active sensors ineach time slot such that the total traffic load remains constantover time, i.e., 𝐺[𝑚] = 𝐺[0] and 𝐺𝑘[𝑚] = 𝐺𝑘[0], ∀ 𝑘, 𝑚.

Suppose that we are able to maintain a constant averagetraffic load up to time 𝑚, such that 𝐺[𝑚] = 𝐺[0]. The trafficload in slot 𝑚+1 can be computed with (5) and the averagetraffic reduction, as compared to the traffic in the 𝑚-th timeslot, is equal to

�̃�[𝑚] =

∑𝐾𝑘=1

∑𝑁𝑘

𝑖=1 𝑝2𝑘𝑖[𝑚]∑𝐾

𝑘=1

∑𝑁𝑘

𝑖=1 𝑝𝑘𝑖 [𝑚]×𝐺[0]𝑒−𝐺[0] =

𝐾∑𝑘=1

�̃�𝑘[𝑚],

where 𝑝𝑘𝑖 [𝑚] is the transmission probability of node 𝑖 in the𝑘-th encoding group at time 𝑚 and

�̃�𝑘[𝑚] =

∑𝑁𝑘

𝑖=1 𝑝2𝑘𝑖[𝑚]∑𝐾

𝑘=1

∑𝑁𝑘

𝑖=1 𝑝𝑘𝑖 [𝑚]×𝐺[0]𝑒−𝐺[0]

is the average traffic reduction in group 𝑘. Similar to (3), theprobability that a node in group 𝑘 successfully transmits in the𝑚-th time slot is approximated as 𝑃rx,𝑘[𝑚] ≈ 𝐺𝑘[𝑚]𝑒−𝐺[𝑚] =𝐺𝑘[0]𝑒

−𝐺[0], which also represents the expected number ofmessages that are successfully delivered from group 𝑘 to thesink in the 𝑚-th time slot. Therefore, the remaining numberof active nodes in the 𝑘-th encoding group at the beginningof time slot 𝑚+ 1 can be approximated as

�̂�𝑘[𝑚+ 1] = �̂�𝑘[𝑚]−𝐺𝑘[0]𝑒−𝐺[0]. (20)

To maintain a constant average traffic load in the 𝑘-thgroup, the average traffic reduction in the previous time slot iscompensated for by increasing equally the traffic contributionof the remaining active nodes in the group. Consequently, thetransmission probability is adjusted as

𝑝𝑘𝑖 [𝑚+1] =

{𝑝𝑘𝑖 [𝑚] + �̃�𝑘[𝑚]

�̂�𝑘[𝑚+1], if 𝑝𝑘𝑖 [𝑚+1] ≤ 𝑝𝑡ℎ

𝑝𝑡ℎ, otherwise,(21)


where 𝑝𝑡ℎ is an upper threshold on the transmission probabilitythat is chosen to maintain stability of random access. Pleasenote that the transmission probability adjustment is performeddistributively at each node, and no centralized control orinformation exchange is required. The effectiveness of thisprotocol will be shown via numerical simulations in thefollowing section.

VII. COMPUTER SIMULATIONS AND PERFORMANCE

ANALYSIS

In this section, we verify the theoretical results givenin previous sections through Monte Carlo simulations andcompare the performance of different probability assignmentschemes in terms of throughput, delay, and energy effi-ciency. Specifically, we examine three probability assignmentschemes: (I) Equal Scheme, where all sensors in the samegroup are assigned the same transmission probability, i.e., thetransmission probability of sensor 𝑖 in the 𝑘-th encoding groupis 𝑝𝑘𝑖 =

𝐺𝑘[0]𝑁𝑘

= 𝐺[0]𝑁 for all 𝑘, 𝑖; (II) Linear Scheme, where

the transmission probability of sensors in each group decreaseslinearly with respect to their encoding order. That is, we set𝑝𝑘𝑖 = 𝛼𝑘 + 𝛽𝑘 × (𝑁𝑘 − 𝑖), for 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑁𝑘, where 𝛼 is thetransmission probability of 𝑊𝑘𝑁𝑘

and 𝛽𝑘 = 2(𝐺𝑘[0]−𝑁𝑘𝛼𝑘)𝑁𝑘(𝑁𝑘−1)

is the difference of the transmission probabilities betweenneighboring sensors in the encoding sequence; (III) Two-StepScheme, where we set 𝑝𝑘𝑖 = 𝑝ℎ𝑘

, for 𝑖 = 1, ⋅ ⋅ ⋅ , 𝛾𝑘 and set𝑝𝑘𝑖 = 𝑝𝑙𝑘 , for 𝑖 = 𝛾𝑘+1, ⋅ ⋅ ⋅ , 𝑁𝑘, such that 𝑝ℎ𝑘

> 𝑝𝑙𝑘 . Noticethat Scheme I is identical to the probability assignment in con-ventional slotted ALOHA systems to maintain fairness amongusers. Scheme II takes into consideration the importance of thesensors’ messages and gives a higher transmission probabilityto messages encoded earlier in the sequence. Scheme III isapplicable when the information of the first 𝛾𝑘 sensors is themost desirable.

Consider a network with 𝑁 = 100 nodes divided equallyinto 𝐾 = 10 groups such that the number of sensors in eachgroup is 𝑁𝑘 = 𝑁/𝐾 = 10. We set the initial traffic loadas 𝐺[0] = 1 for the fixed probability assignment schemesand as 𝐺[0] = 0.8 for the adaptive probability assignmentschemes. The choice of 𝐺[0] = 1 is given since this isthe maximum number of transmissions that the collisionchannel can accommodate in each time slot and the choiceof 𝐺[0] = 0.8 was numerical validated as the one that yieldsbest performance for the adaptive scheme. The initial trafficload of each group is 𝐺𝑘[0] = 𝐺[0]/𝐾 for all 𝑘. In Figs.4 to 8 and Figs. 10 to 11, we compare the average trafficload, throughput, delay, and energy consumption of the threeprobability assignment schemes described above and for boththe fixed and adaptive MAC protocols. In the Linear scheme,we set 𝛼 = 0.002 and compute 𝛽𝑘 from 𝑁𝑘 and 𝐺𝑘[0]. Inthe Two-Step scheme, we set 𝑝ℎ𝑘

= 0.016, 𝑝𝑙𝑘 = 0.004 and𝛾𝑘 = 5 for the fixed MAC and set 𝑝ℎ𝑘

= 0.013, 𝑝𝑙𝑘 = 0.003and 𝛾𝑘 = 5 for the adaptive MAC, so that the initial trafficload is consistent for all three probability assignment schemes.

In Fig. 4, we plot the average traffic load of the fixed MACversus the time slot index. The analytic results refer to thederivations given in Section III while the simulation resultsare obtained through Monte Carlo experiments. We can see

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Slot Number

Ave

rage

Tra

ffic

Load

Simulation,Equal,K=10Analysis,Equal,K=10Simulation,Linear,K=10Analysis,Linear,K=10Simulation,Two−Step,K=10Analysis,Two−Step,K=10

Fig. 4. Analytic and simulation results of the average traffic load versus thetime slot number.

that the analytic results closely approximate the simulatedtraffic load for the Equal scheme but is less accurate for theLinear and Two-Step schemes since the assumption that alltransmission probabilities are much less than 1 may not beapplicable in these cases. However, the approximation erroris notable only in later time slots where the average trafficload is small and contention among users are less significant.Therefore, when using the average traffic load to estimatethe collision probability in the throughput, delay, and energyanalysis, the approximation error will not have a significantimpact on the accuracy of the analytic results. Yet, since ourapproximation over-estimates the average traffic load in latertime slots, the analytical results on the throughput will beslightly lower than that of the simulation while the delay andenergy consumption will be slightly higher. It is worthwhileto note that the approximation will be more accurate for largervalues of 𝑁𝑘. Moreover, we observe that the linear and two-step schemes have traffic load that reduces much rapidly overtime, compared to the equal scheme. This is because themessages are given different transmission probabilities and theones with higher priority are delivered rapidly in time whilethe remaining messages have only small contribution to thetotal traffic load.

In Figs. 5 and 6, we plot the throughput per node, whichis defined as the average number of decoded messages nor-malized by the number of nodes 𝑁 , for the fixed and adap-tive protocols, respectively. Both the analytic and simulationcurves are plotted versus the number of time slots. In Fig.5, we observe that Linear and Two-Step schemes achievehigher throughput than the Equal scheme when the sensingperiod is small. This is because, in the previous two schemes,higher transmission probabilities are assigned to messagesearlier in the encoding sequence, allowing them to reach thedestination in a shorter amount of time. However, in theseschemes, messages later in the encoding sequence will beassigned smaller transmission probabilities and, thus, reducingthe throughput in later time slots. This allows the Equalscheme to eventually outperform both Linear and Two-Stepschemes when the sensing period is large. However, we showin Fig. 6 that, when using the adaptive MAC protocol, the


0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Slot Number

Thr

ough

put P

er N

ode

Fixed MAC,Simulation,Equal,K=10Fixed MAC,Analysis,Equal,K=10Fixed MAC,Simulation,Linear,K=10Fixed MAC,Analysis,Linear,K=10Fixed MAC,Simulation,Two−Step,K=10Fixed MAC,Analysis,Two−Step,K=10

Fig. 5. Analytic and simulation results of the throughput per node versusthe time slot number for the fixed (or non-adaptive) MAC protocol.

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Slot Number

Thr

ough

put P

er N

ode

Adaptive,Simulation,Equal,K=10Adaptive,Analysis,Equal,K=10Adaptive,Simulation,Linear,K=10Adaptive,Analysis,Linear,K=10Adaptive,Simulation,Two−Step,K=10Adaptive,Analysis,Two−Step,K=10

Fig. 6. Analytic and simulation results of the throughput per node versusthe time slot number for the adaptive MAC protocol.

throughput per node increases much faster and, in fact, reaches1 around the same time for all schemes due to improvedchannel utilization.

In Fig. 7 and 8, we plot the average delay versus thethroughput per node, i.e., the number of time slots neededbefore the given number of messages are decodable per node,for both the fixed and adaptive schemes. We can see that,with weighted probability assignments, the delay of decodingmessages earlier in the sequence is significantly reduced, but atthe cost of increasing the delay of those later in the sequence.However, with adaptive probability assignment, the cost ofhaving large delay for low priority messages are effectivelyeliminated as shown in 8. Yet, there is a tradeoff with energyefficiency as shown in the following.

To study energy consumption, let us consider the sourcemodel given in [20] where the observations at each node arequantized using an infinite-level uniform scalar quantizer withstep size Δ and the conditional entropy is bounded as

𝐻(𝑋𝑘𝑖 ∣X𝑘𝑖−1

𝑘1) ≤1

2log2 MSE(𝑋𝑘𝑖 ∣X𝑘𝑖−1

𝑘1)

− 1

2log2 𝑃𝑒 + 1− log2 Δ (22)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

Throughput Per Node

Ave

rage

Del

ay


Fig. 7. Analytic and simulation results of the average delay versus thethroughput per node for the fixed (or non-adaptive) MAC protocol.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

100

200

300

400

500

600

700

800

Throughput Per Node

Ave

rage

Del

ay

Adaptive,Simulation,Equal,K=10Adaptive,Simulation,Linear,K=10Adaptive,Simulation,Two−Step,K=10

Fig. 8. Simulation results of the average delay versus the throughput pernode for the adaptive MAC protocol .

where X𝑘𝑖−1

𝑘1= [𝑋𝑘1 , ⋅ ⋅ ⋅ , 𝑋𝑘𝑖−1 ] and

MSE(𝑋𝑘𝑖 ∣X𝑘𝑖−1

𝑘1)

=𝐸[𝑋2𝑘𝑖]−𝐸[𝑋𝑘𝑖(X

𝑘𝑖−1

𝑘1)𝑇 ]𝐸[X

𝑘𝑖−1

𝑘1(X

𝑘𝑖−1

𝑘1)𝑇 ]−1𝐸[X

𝑘𝑖−1

𝑘1𝑋𝑘𝑖 ].

The correlation between 𝑋𝑘𝑖 and 𝑋𝑘𝑗 is given by𝐸[𝑋𝑘𝑖𝑋𝑘𝑗 ] = 𝜎2𝑒−𝜂∣𝑑𝑘𝑖,𝑘𝑗

∣, where 𝑑𝑘𝑖,𝑘𝑗 is their locationdistance. Here, we set 𝜎 = 1, Δ = 𝜎/100, 𝑃𝑒 = 0.001,and 𝜂 = 1/1000. Assume that the sensor topology in eachgroup and their relative entropy are given as in Fig. 9. Wecompare, in Figs. 10 and 11, the energy consumption versusthe throughput per node for the fixed and adaptive MACprotocols. By assuming that each transmitted bit consumesthe same amount of energy, the energy consumption in thefigure is represented by the number of bits transmitted by thesensors. In other words, one unit of energy in the plot is equalto the energy used to transmit a single bit. We can see that alarge part of the energy is consumed in transmitting the firstfew messages since each of these messages require more bitsto represent and also because more sensors are competing forthe channel at this stage (thus, increasing the retransmissions).However, after the first few messages are decoded, less energyis required to retrieve the remaining messages since they are


Fig. 9. Two-dimensional topology of sensors in each group and theirconditional entropy values

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Throughput Per Node

Ene

rgy

Con

sum

ptio

n


Fig. 10. Analytic and simulation results of the total energy consumptionversus the throughput per node for the fixed (or non-adaptive) MAC protocol.

encoded with less bits and experience less channel contention.Hence, one pays little extra energy for doubling the throughputfrom 0.5 to 1. Moreover, as indicated previously, we can seethat, even though the adaptive MAC is effective in reducingthe delay, it consumes more energy due to increased collisionsand retransmissions.

VIII. CONCLUSION

The impact of transmission probability assignments on thethroughput, delay, and energy efficiency of DSC systems inrandom access networks has been analyzed. Specifically, wegave closed-form expressions of the average throughput basedon approximations of the average traffic load in each timeslot and derived the average delay and energy consumptionvia Markov Chain analysis. The analytic results are verifiedthrough simulations and the performance of different prob-ability assignment schemes are compared. We observed atradeoff between the average delay and energy consumptionfor different probability assignment schemes and for fixedand adaptive MAC protocols. Through these discussions, wehighlight the importance of cross-layered transmission controlfor the efficient delivery of DSC messages.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

Throughput Per Node

Ene

rgy

Con

sum

ptio

n

Adaptive,Simulation,Equal,K=10Adaptive,Simulation,Linear,K=10Adaptive,Simulation,Two−Step,K=10

Fig. 11. Simulation results of the total energy consumption versus thethroughput per node for the adaptive MAC protocol.

REFERENCES

[1] C. Intanagonwiwat, R. Govindan, D. Estrin, J. Heidemann, and F. Silva,“Directed diffusion for wireless sensor networking," IEEE/ACM Trans.Networking, vol. 11, no. 1, pp. 2-16, Feb. 2003.

[2] B. Krishanamachari, D. Estrin, and S. Wicker, “The impact of data ag-gregation in wireless sensor networks," in Proc. International WorkshopDistributed Event Based Syst. (DEBS), Vienna, Austria, July 2002.

[3] A. Scaglione and S. Servetto, “On the interdependence of routing anddata compression in multi-hop sensor networks," Wireless Netw., vol. 11,no. 1-2, Jan. 2005.

[4] P. Ishwar, A. Kumar, and K. Ramchandran, “Distributed sampling indense sensor networks: a “bit-conservation” principle," in Proc. Int’lWorkshop Inf. Process. Sensor Netw., Apr. 2003.

[5] M. Dong, L. Tong, and B. M. Sadler, “Impact of data retrieval patternon homogeneous signal field reconstruction in dense sensor networks,"IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4352-4364, Nov. 2006.

[6] M. C. Vuran and I. F. Akyildiz, “Spatial correlation-based collaborativemedium access control in wireless sensor networks," IEEE/ACM Trans.Networking, Apr. 2006.

[7] D. Slepian and J. Wolf, “Noiseless coding of correlated informationsources," IEEE Trans. Inf. Theory, vol. 19, no. 4, pp. 471-480, July1973.

[8] S. S. Pradhan and K. Ramchandran, “Distributed source coding: sym-metric rates and applications to sensor networks," in Proc. IEEE DataCompression Conf., 2000.

[9] Z. Xiong, A. D. Liveris, and S. Cheng, “Distributed source coding forsensor networks," IEEE Signal Process. Mag., vol. 21, no. 5, pp. 80-94,Sep. 2004.

[10] S. S. Pradhan and K. Ramchandran, “Distributed source coding usingsyndromes (DISCUS): design and construction," IEEE Trans. Inf. The-ory, vol. 49, no. 3, Mar. 2003.

[11] J. Garcia-Frias and Y. Zhao, “Compression of correlated binary sourcesusing turbo codes," IEEE Commun. Lett., vol. 5, pp. 417-419, Oct. 2001.

[12] A. Liveris, Z. Xiong, and C. Georghiades, “Compression of binarysources with side information at the decoder using LDPC codes," IEEECommun. Lett., vol. 6, pp. 440-442, Oct. 2002.

[13] N. Gehrig and P. Dragotti, “Symmetric and asymmetric Slepian-Wolfcodes with systematic and non-systematic linear codes," IEEE Commun.Lett., vol. 9, no. 1, pp. 61-63, Jan. 2005.

[14] D. Marco and D. L. Neuhoff, “Reliability vs. efficiency in distributedsource coding for field-gathering sensor networks," in Proc. ThirdInternational Symp. Inf. Process. Sensor Netw. (IPSN), Apr. 2004.

[15] N. Abramson, “The Aloha system—another alternative for computercommunications," in Proc. Fall Joint Comput. Conf., AFIPS Conf.,vol. 44, Montvale, NJ, 1970.

[16] S. Bandyopadhyay and E. J. Coyle, “An energy efficient hierarchicalclustering algorithm for wireless sensor networks," in Proc. INFOCOM,vol. 3, Mar. 2003, pp. 1713-1723.

[17] W. R. Heinzelman, A. Chandrakasan, and H. Balakrishnan, “Energy-efficient communication protocol for wireless microsensor networks,"in Proc. 33rd Annual Hawaii International Conf. Syst. Sciences, Jan.2000.


[18] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochas-tic Processes, 4th edition. McGraw-Hill, 2001.

[19] D. Marco and D. L. Neuhoff, “Entropy of quantized data at highsampling rates," in Proc. ISIT 2005.

[20] C. Fischione, F. Graziosi, F. Santucci, and S. Tennina, “A simple upperbound of the coding rate of a popular distributed source coding tech-nique," UC Berkeley, Technical Report, July 2008. [Online]. Available:http://www.ee.kth.se/ carlofi/publications.html

Y.-W. Peter Hong received his B.S. degree in Elec-trical Engineering from National Taiwan Univer-sity, Taipei, Taiwan, in 1999, and his Ph.D. degreein Electrical Engineering from Cornell University,Ithaca, NY, in 2005. He joined the Institute ofCommunications Engineering and the Departmentof Electrical Engineering at National Tsing HuaUniversity, Hsinchu, Taiwan, in Fall 2005, where heis now an Associate Professor. He was also a visitingscholar at the University of Southern Californiaduring June-August of 2008. His research interests

include cooperative communications, distributed signal processing for sensornetworks, and PHY-MAC cross-layer designs for next generation wirelessnetworks. Dr. Hong received the best paper award for young authors from theIEEE IT/COM Society Taipei/Tainan chapter in 2005, the best paper awardamong unclassified papers in MILCOM 2005, and also the Junior FacultyResearch Award from the College of EECS at National Tsing Hua Universityin 2009.

Yuh-Ren Tsai received the B.S. degree in electricalengineering from National Tsing Hua University,Hsinchu, Taiwan, in 1989, and the Ph.D. degree inelectrical engineering from National Taiwan Uni-versity, Taipei, Taiwan, in 1994. From 1994 to2001, he was a Researcher in TelecommunicationLaboratories of Chunghwa Telecom Co., Ltd., Tai-wan. Since 2001, he has been with the Depart-ment of Electrical Engineering and the Institute ofCommunications Engineering at National Tsing HuaUniversity, Hsinchu, Taiwan, where he is currently

an Associate Professor. His research interests include wireless transmission,sensor networks, mobile cellular systems and cryptography.

Yan-Yu Liao received the B.S. degree in AppliedMathematics from National Chiao Tung University,Hsinchu, Taiwan, in 2005, and the M.S. degree inCommunications Engineering from National TsingHua University, Hsinchu, Taiwan, in 2007. Since2009 he has been with HT mMobile Inc.

Chih-Hsun Lin received his B.S. degree in Depart-ment of Communications Engineering from FengChia University, Taichung, Taiwan, in 2007, andhis M.S. degree in the Institute of Communica-tions Engineering at National Tsing-Hua Univer-sity, Hsinchu, Taiwan. His research interest is themedium access control in wireless sensor networks.

Kai-Jie Yang received the B.S. degree in ElectricalEngineering from National Chung Cheng Univer-sity, Chiayi, Taiwan, in 2000, and the M.S. andPh.D degrees in Communications Engineering fromNational Tsing Hua University, Hsinchu, Taiwan,in 2002 and 2009, respectively. He is a Postdoc-toral Researcher with the Institute of Communica-tions Engineering at National Tsing Hua University,Hsinchu, Taiwan. His research interests include linkstability and mobility estimation in mobile wirelessnetworks.

Documents

On the Throughput, Delay, and Energy Efficiency of Distributed Source Coding in Random Access Sensor Networks