IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. X, NO. X, …guxxx035/paper/Liang-TMC-NJN.pdf · IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. X, NO. X, 2012. 3 its practical implementation

IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. X, NO. X, 2012. 1

Evaluating Service Disciplines for On-DemandMobile Data Collection in Sensor Networks

Liang He, Member, IEEE, Zhe Yang, Student Member, IEEE, Jianping Pan, Senior Member, IEEE,Lin Cai, Senior Member, IEEE, Jingdong Xu, Yu Gu, Member, IEEE

Abstract—Mobility-assisted data collection in sensor networks creates a new dimension to reduce and balance the energyconsumption for sensor nodes. However, it also introduces extra latency in the data collection process due to the limited mobilityof mobile elements. Therefore, how to schedule the movement of mobile elements throughout the field is of ultimate importance. In thispaper, the on-demand scenario where data collection requests arrive at the mobile element progressively is investigated, and the datacollection process is modelled as an M/G/1/c-NJN queuing system with an intuitive service discipline of Nearest-Job-Next (NJN).Based on this model, the performance of data collection is evaluated through both theoretical analysis and extensive simulation. NJN isfurther extended by considering the possible requests combination (NJNC). The simulation results validate our models and offer moreinsights when compared with the First-Come-First-Serve (FCFS) discipline. In contrary to the conventional wisdom of the starvationproblem, we reveal that NJN and NJNC have better performance than FCFS, in both the average and more importantly the worst cases,which offers the much needed assurance to adopt NJN and NJNC in the design of more sophisticated data collection schemes, as wellas other similar scheduling scenarios.

Keywords—Wireless ad hoc sensor networks, mobile elements, on-demand data collection

F

1 INTRODUCTION

Many applications in wireless sensor networks are dataoriented [1], [2]. Data collection in sensor networkstypically relies on the wireless communications betweensensor nodes and the sink, which may excessively con-sume the limited energy supply of sensor nodes due tothe super-linear path loss exponents [3]. Furthermore,sensor nodes near the sink also tend to deplete theirenergy much faster than other nodes due to the dataaggregation towards the sink, which imposes them amuch heavier volume of data to forward and leads toa very unbalanced energy consumption in the entirenetwork [4]. In addition, these approaches are basedon a fully connected network, which requires the densedeployment of nodes and thus introduces extra costs.

Another data collection approach utilizes the often-available, controlled mobility of certain devices, referredto as mobile elements in this paper [5]–[7]. By utilizingmobile elements, not only more energy can be con-served and balanced on sensor nodes, but also the com-munications and networking become possible in verysparse networks with the store-carry-forward approach.For example, the seabed crawler deployed in the ob-servatory of the NorthEast Pacific Time-Series Undersea

• L. He and Y. Gu are with the SUTD-MIT International Design Center,Singapore University of Technology and Desing, Singapore, 138682.E-mail: he [email protected]

• Z. Yang, J. Pan and L. Cai are with University of Victoria, Canada.• J. Xu is with Nankai University, China.• An early short version of this work is published at IEEE INFOCOM’12,

which is done when L. He was a Ph.D. candidate at Nankai Universityand a visiting research student at University of Victoria.

Networked Experiments (NEPTUNE) can cruise throughseveral experimentation sites, “talk” to experiment de-vices, and bring the data back to the junction boxes [11].Another mobile element example is the Seaeye Saber-tooth [12], a battery-powered autonomous underwatervehicle, which travels in deep water environments tocollect data from deployed equipments, and uploads thedata to the control center at the docking station. Otherexamples of the mobility-assisted data collection includethe smart buoy equipped with Seatext from WFS [13],the structural health monitoring with radio-controlledhelicopters [14], and so on.

Although mobile elements create a new dimensionfor data collection, they also introduce extra challenges.First, the data collection latency may be large due to therelatively low travel speed of mobile elements [15], [18],especially when compared with that of electromagneticor acoustic waves. This large latency must be addressedfor applications with tight requirements on the timelydata delivery. Second, with a large latency, data lossmight occur due to the buffer overflow of sensor n-odes, which is not desirable for data integrity sensitiveapplications [16]. Finally, mobile elements themselvesare battery-powered as well in most cases, e.g., thetravel distance of Sabertooth is about 20–50 Km witha fully charged battery, so the data collection must beaccomplished before mobile elements deplete their ownenergy.

A lot of efforts have attempted to address these chal-lenges by finding the optimal data collection tour forthe mobile elements, under the offline scenario where thedata collection is carried out in a periodic way [17], [25],[26], [44]. However, in a more practical scenario where


the data collection is carried out in an on-demand manner,we need to determine how the mobile element shouldcarry out the data collection tasks without a priori infor-mation on the data collection demands. There are someexisting efforts aiming to design data collection schemesfor this online scenario [21], [49]. However, a critical andunaddressed issue is how to evaluate the efficiency andoptimality of the proposed schemes, because withouta priori knowledge on the request arrival, no optimalsolution is available as the benchmark in this case.

In this paper, we focus on this on-demand data collec-tion scenario and tackle this limitation by theoreticallyanalyzing the data collection performance when certainclassic disciplines are adopted through a queue-basedmodeling approach, which offers important guidelines indesigning and evaluating more sophisticated on-demanddata collection schemes for the mobile elements.

We first show that data collection requests in theonline scenario can be captured by a Poisson arrivalprocess, and with the travel distance (and time) distribu-tion between any two sensor nodes in the sensing field,the collection process can be modelled as an M/G/1/c-NJN queuing system, which accommodates at most crequests at the same time. The NJN stands for Nearest-Job-Next, a simple and intuitive discipline adopted by themobile element (server) to select the next to-be-servedrequest (client). A challenge with the analysis of the NJNdiscipline is the dynamic and state-dependent service time,as will be explained later. Furthermore, observing thatmultiple requests can be combined and served togetherby the mobile element if a collection site within thecommunication ranges of all the corresponding sensornodes can be found, we extend the NJN discipline toNJN-with-combination (NJNC), i.e., M/G/1/c-NJNC, toexplore how much gain can be achieved by the possiblerequests combination. The resultant data collection effi-ciency is evaluated based on the analytical results on thesystem measures of these models.

To the best of our knowledge, this is the first timethat the distributions of critical performance metrics withNJN and NJNC are obtained through a queue-basedapproach, and the approach can be extended to otherdynamically-prioritized scenarios as well. Our resultsshow that even though NJN may be unfair for farther-away requests temporarily, its average performance out-performs FCFS greatly and more importantly, even itsworst-case performance is better than FCFS, especially inthe case of NJNC. These results offer the much neededassurance to adopt NJN and NJNC in the design of moresophisticated on-demand data collection schemes.

The remainder of this paper is organized as follows.Section 2 briefly reviews the literature on the mobility-assisted data collection. In Section 3, we outline theproblem setting and list the assumptions and definitionsused in this paper. We present the analytical models ofthe NJN and NJNC disciplines in Section 4 and Section 5,respectively. The analytical and simulation results arepresented and compared in Section 6 for model valida-

tion and further insights. In Section 7, we discuss thepossible approaches to further extend and improve thework. Finally, we conclude this paper in Section 8.

2 BACKGROUND AND RELATED WORK

Recently, a lot of efforts have been made on exploringthe mobility-assisted data collection in wireless sensornetworks [25], [27], [29], [44]. Many of them are schemedesign oriented for the offline scenario where the mobileelements (MEs) know all requests in advance and carryout the data collection periodically [17], [19], [28]. Forexample, Ryo et al. focused on the scenario where theME has to accomplish the data collection task by visitingall the nodes within the shortest possible time in [17].They formulated the problem as a label-covering problem,which was proved to be NP-hard. A tour selectionalgorithm for the ME was proposed in [20], which startswith a connected dominating set of the network, obtains aminimum spanning tree based on it, and finally generatesa Hamiltonian circuit for the ME. The case where multipleMEs exist in the network was considered in [28]. Xinget al. tackled this problem by finding the rendezvous lo-cations in [29], and extended the work to jointly optimizethe data routing path and the tour of the ME in [30].

In contrast, the online mobile data collection, whichis carried out in an on-demand manner, is much lessexplored, even though it is more practical in reality [21],[49]. The most intuitive service discipline is First-Come-First-Serve (FCFS), i.e., the order to serve requests isthe same as their arrival order, whose performance isanalytically evaluated in [10]. However, due to the ran-domness of service requests in both of the time and spacedomains, the ME may unnecessarily travel back-and-forth to serve requests, which is clearly undesirable sincethe travel time of the ME dominates the data collectionlatency. Another intuitive service discipline is to servethe geographically nearby requests first, or Nearest-Job-Next (NJN), in order to reduce the distance that theME has to travel and thus the time it has to spendon. However, in the literature and practical systems,NJN and its variants are much less explored, due to theconcerns discussed below.

NJN is similar to the traditional Shortest-Job-Next (SJN)service discipline [22]. However, two extra issues needto be considered. First, for SJN, the service time for eachjob has to be accurately estimated upon its arrival inthe system and remain fixed before its service. But theservice time with NJN for data collection in wirelesssensor networks is jointly determined by the locationof the requesting sensor node and that of the servingME, which cannot be determined in advance until thejob is about to be served. This dynamic service timemakes not only the existing results on SJN not applica-ble to our problem [23], but also the analysis on NJNmuch more challenging due to the dynamic priorityof a particular request. Second, SJN is known to leadto the starvation problem for large jobs, which limits


its practical implementation. Thus whether NJN suffersfrom the similar unfairness problem for mobility-assisteddata collection is another question we need to answer.Another discipline similar to NJN is the Shortest-Seek-Time-First (SSTF) in disk scheduling [24]. The disk tracksare normally treated in a one-dimensional space, whilethe sensing field of NJN is a two-dimensional space.

The design of dynamic scheduling schemes has beenextensively explored in the scenarios such as the dy-namic vehicle routing [45] and dynamic traveling salesmanproblem [46]. Greedy schemes similar to NJN have beenexplored by the research efforts on these problems [47],[48], and our queue-based analysis advances the inves-tigation by obtaining the probability distributions of itscritical performance metrics, such as the queue length,queuing time, and response time. The most relevantwork to ours is [49], in which the communication rangebetween the mobile elements and sensor nodes has beenexploited to facilitate the data collection, as we did inNJNC. A Traveling Salesman Problem with Neighbourhood-s (TSPN)-based schedule has been proposed, and thesystem stability under it has been proved. We treat theTSPN-based schedule as the state-of-the-art in the on-demand data collection problem in sensor networks andadopt it as an important benchmark in our evaluations.

3 ON-DEMAND DATA COLLECTION

3.1 On-Demand Scenario vs. Offline ScenarioNumerous research efforts on the mobile data collectionin sensor networks exist, and most of them focus on theoffline scenario, where the mobile elements carry out thedata collection in a periodic manner. A potential issuewith this periodic data collection is that certain sensornodes may not have data to upload, and visiting thesenodes just to find out no data to collect is clearly notgood for a high-efficient data collection process.

Observing the limitation of the offline scenario, in thispaper we consider the online scenarios, in which the datacollection is carried out by the mobile elements based onthe real-time demands from sensor nodes, i.e., the datacollection is carried out in an on-demand manner.

More specifically, in the on-demand mobile data col-lection problem, we consider the scenario where mobileelements (MEs) travel around in the sensing field tocollect data from sensor nodes with wireless commu-nication techniques. Sensor nodes gather sensory dataaccording to application requirements, and when theyhave gathered enough data to report, or have capturedcertain events that are of particular interests, sensornodes will send data collection requests to the MEs byadopting existing lightweight and efficient protocols thatcan track and communicate with the MEs [3], [31].

Note that usually these tracking protocols rely on themulti-hop forwarding of messages among sensor nodes.Thus adopting these protocols to report the sensory data,which are usually of much larger size, e.g., in camerasensor networks [32], is clearly not a good choice for the

offline scenario on-demand data collection

requesting sensor nodes as input

input is fixed and

known in advance

entire set of sensor nodes as input

input is dynamic and no

a priori information is available

aim to plan the

optimal tour for the ME

aim to design the optimal

service discipline for the ME

need to select the next request

upon each service completion

compute the tour once, and collect

data periodically based on it

Fig. 1: Differences between the on-demand data collec-tion and the offline data collection.

node energy efficiency. This is also the reason that weintroduce the MEs to the network.

The ME maintains a buffer to store the received re-quests, and serve them according to its service discipline.As a baseline, in this paper we consider the case whereonly one ME is available in the network, and discussionon the case of multiple MEs is provided in Section 7.

Different from the offline scenario, the on-demanddata collection shows great dynamics, which reside inboth the temporal domain, i.e., when a new data col-lection request will arrive, and the spatial domain, i.e.,where the new request will come from. This dynamicproperty means our objective will shift from the optimalpath planning for the ME [17], [19], [29], [30], [44], tothe design of efficient real-time service disciplines toselect the request to serve next. The real-time character ofthe on-demand scenario also impose a low computationcomplexity requirement on the service disciplines. Fig-ure 1 summaries the differences between the on-demanddata collection and that under the offline scenario.

3.2 Our ApproachesIn the on-demand data collection, the ME needs toselect request from its buffer as the next one to serve,which shows a clear queuing behavior. Inspired bythis, our approach is to model the on-demand datacollection process as a queuing system (an M/G/1/cqueuing system, as will be explained in Section 4), andtheoretically analyze the performance of data collectionwith different service disciplines. These analytical resultsreveal insights on the data collection performance andguide the design of more sophisticated service schemes.

When talking about service disciplines, the first onecomes to our mind would be the First-Come-First-Serve (FCFS) discipline, whose performance has beenextensively explored by the queuing theory society [39].However, FCFS discipline schedules requests based ontheir temporal features, which may make the ME unnec-essarily move back-and-forth, as shown in Fig. 2(a). Thisis clearly undesirable since usually the travel time of theME dominates the data collection latency.

Noticing the inefficiency of FCFS, we explore twoother disciplines that schedule the requests according totheir spatial features. The first service discipline we ex-plore is the Nearest-Job-Next (NJN) discipline (Fig. 2(b)),


EA

D

B

F

C

EA

D

B

F

C

(a) FCFS (b) NJN

Fig. 2: Demonstration of the FCFS and NJN servicedisciplines. For clarity, only requesting sensor nodes areshown and no new requests are considered. The arrivalorder of requests is shown lexicographically.

i.e., on finishing the service of the current data collectionrequest, the ME selects the spatially nearest requestingsensor node in its service queue according to its currentlocation, and travels to the node to collect data. Fur-thermore, with wireless communications, it is possiblefor the ME to collect data from multiple sensor nodesat a single collection site, provided that the collectionsite is within the communication ranges of all thesenodes. Inspired by this, we extend the NJN discipline toNJN-with-Combination (NJNC), with which other requestsin the queue can be combined with the nearest oneand served together when possible. Analytical resultson the data collection with NJNC are also derived toquantitatively evaluate the performance improvementby the possible requests combination.

To simplify the calculation and presentation, we as-sume a unit square sensing field in this paper. Thisassumption can be relaxed without violating the analysisand equivalent results can be obtained accordingly, al-though the computation may be more complicated. Also,we assume that the time from a sensor node sending outits request till the ME receiving the request is negligiblewhen compared with the travel time the ME takes toserve the request. Discussion on how to remove thesetwo simplifications can be found in Section 7.

3.3 Definitions and Symbols

For clarity, we list and briefly explain here the definitionsused in this paper.

• v: the travel speed of the ME (normalized w.r.t. theside length of the field);

• r: the wireless communication range (normalizedw.r.t. the side length of the field) between the MEand sensor nodes;

• c: the maximal number of requests that the ME canaccommodate at the same time;

• S: the service time of requests, or Sl for the to-be-served request selected when there are l datacollection requests in the service queue;

• L: the size of the queuing system, i.e., the numberof requests that are waiting for or under service;

• I: the idle period of the queuing system;• B: the busy period of the queuing system;• π = {π0, π1, ..., πc−1}: the system size probabilities

at the departure time of requests;

• w = {w0, w1, ..., wc}: the system size probabilities atthe arrival time of requests;

• u = {u0, u1, ..., uc}: the steady-state system sizeprobabilities of the queuing system.

4 DATA COLLECTION WITH NJNWe explore the case where the ME serves data collectionrequests with the NJN discipline in this section. Morespecifically, by considering the arrival and departureprocesses of data collection requests, we first model thesystem as a non-preemptive M/G/1/c-NJN queuingsystem, and then obtain the analytical results on thesystem measures, which offer important insights onevaluating the performance of the data collection.

4.1 M/G/1/c-NJN Queuing ModelThe obvious queuing behavior in the data collectionprocess inspires us to construct a queuing model tocapture its performance, in which the ME acts as theserver and data collection requests are treated as clients.For any queuing model, the clients arrival and departureprocesses are the two most fundamental components, sowe characterize them respectively in the following.

4.1.1 Arrival ProcessFor a particular sensor node in a stable network, theprobability for it to send out a data collection requestat a given time instant is small, and the total numberof sensor nodes in the network is usually expected to berelatively large. Theoretically, if the client population of aqueuing system is relatively large and the probability bywhich clients arrive at the queue at a given time instantis relatively small, the arrival process can be adequatelymodeled as a Poisson process [34]. (Similar conclusionsare also obtained in Proposition 1.12 of [35], pp. 11). Basedon this, we adopt a Poisson process to capture the arrivalof data collection requests to the ME, which is furtherverified by an event-driven simulator in Section 6.1.

4.1.2 Departure ProcessThe data propagation speed in sensor networks is aboutseveral hundred meters per second [30], which is muchfaster than the travel speed of the ME. Thus we assumethe communication time between the ME and the sensornode to accomplish the data uploading is negligible(further discussed in Section 7). In this way, the servicetime of each request can be captured by the time fromthe service completion of the current request to the timewhen the ME moves to the next selected requestingsensor node.

Due to the fact that the last collection site is also thestarting point of the travel when the ME serves the nextrequest, the service time of consecutively served requestsseem not to be stochastically independent. However, de-note the sequence of service times as {t1, t2, t3, ...}, and ifwe examine only at every second element of the original


process, it is clear that {t1, t3, t5, ...} are independentof each other, and the distribution-ergodic property ofthis sub-process can be easily observed [36]. The sameis true for sub-process {t2, t4, t6, ...}. The distribution-ergodic property still holds if we combine these twosub-processes since their asymptotic behaviors do notchange after the combination. This means that if we canfind the time distribution when the ME travels betweenconsecutive to-be-served sensor nodes, we can adopt itas the service time distribution for the queuing systemover a long time period, i.e.,

FS(x) = limh→∞

h∑i=0

1

h· Pr{ti ≤ x}. (1)

One challenge in the performance analysis of NJN isits dynamic service time, in the sense that it is determinedby both the location of the requesting sensor node andthat of the ME just before its service. This dynamicproperty of service time makes the existing results onthe traditional Shortest-Job-Next (SJN) discipline not ap-plicable in our problem, which requires the service timesof clients are both known and fixed upon their arrivalto the queue [23]. We tackle this problem by adoptingexisting results on geometrical probability.

By the results in [9], the distribution of the distancebetween two random locations in a unit square is

fD(d) =

2d(π − 4d+ d2) 0 ≤ d ≤ 1

2d[2 sin−1( 1d)− 2 sin−1

√1− 1

d2

+4√d2 − 1− d2 − 2] 1 ≤ d ≤

√2

0 otherwise,(2)

and FD(d) =∫ d

0fD(x)dx.

Based on (2), and with the constant travel speed vof the ME, the travel time distribution between twouniformly and randomly distributed sensor nodes is thus

FT (t) = Pr{D ≤ vt} (0 ≤ t ≤√2/v), (3)

and fT (t) = ∂FT (t)/∂t.The analysis of NJN becomes even more complicated

due to its greedy nature: the service time of the to-be-served request tends to be shorter if more requests arewaiting in the system, which makes the service timestate-dependent. We begin our analysis by conditioningon the system size to tackle the conditional service time.

4.1.3 Finite Capacity SystemNote that for a specific application, certain constraintson the maximal acceptable data collection latency exist,either because of the requirement on the timely deliveryof sensory data or the possible buffer overflow of sensornodes. Thus a finite capacity queuing model is a betterchoice to capture the system behavior when comparedwith the regular M/G/1 queue. However, additionalattention is needed for those requests who arrive andfind a fully occupied system.

4.2 Service Time with a Given System Size

Denote l as the number of requests that are waitingfor service when the ME just accomplishes the serviceof the current request, and is about to select the nextrequest, i.e., the one sent by the nearest requesting sensornode according to the current location of the ME, toserve. Observing the randomness of both the currentlocation of the ME and the requesting sensor nodes,the distances from these l requesting nodes to the MEcan be approximately viewed as l i.i.d. random variableswith distribution fD(d) (further verified in Section 6.3.1).Thus based on the results on the first-order statistic, theprobability distribution of the distance between the MEand the nearest requesting node is

FD,l(d) = 1− (1− FD(d))l, (4)

with probability density function (PDF) fD,l(d) =∂FD,l(d)/∂d. The conditional service time distribution ofthe nearest request with a given system size is thus

fSl(t) = v · fD,l(vt), (5)

where Sl represents the service time with system size l.

4.3 System Size at Departure Time

If we record the system size every time it changes, wecan see the next system size only depends on the currentone, which is exactly the Markov property. Also, the timethe system stays at the current size is a random variablejointly determined by the arrival and the departure pro-cesses of requests. From these, we can see the evolutionprocess of system size is a semi-Markov process. Then if weonly examine the system size at the departure time of therequests, we can observe an embedded discrete-time Markovchain, which is similar to the case with FCFS [10] (Fig. 3).However, since the service time is now dependent on thecurrent system size, the chain becomes heterogeneous inits transition probabilities (Fig. 4).

With the conditional service time distribution obtainedby (5), we can define and calculate the probability of inew arrivals during the time serving a request selectedfrom l available ones as

kli =

∫ √2/v

0

e−λt(λt)i

i!fSl(t)dt, (6)

where λ is the intensity of the arrival process. Note thatfor FCFS shown in Fig. 3,

ki =

∫ √2/v

0

e−λt(λt)i

i!fT (t)dt. (7)

The state transition matrix of the finite capacity queuewith NJN at the request departure time is then


0 1 ii−1 i+1k0

k kk

k1 1k k1kkk0

1

0 0

2 2k3

k1

Fig. 3: State transition at departure time with FCFS.

0 1 ii−1 i+1k0

k k

k1 1k k1kkk0 0 0

2 2k3

k1

k1

11 i+1i−1 i

0

0 i−1i−1

i i+1

i

Fig. 4: State transition at departure time with NJN.

P = {pij} =

k00 k01 · · · k0c−1 1−c−1∑i=0

k0i

k10 k11 · · · k1c−1 1−c−1∑i=0

k1i

0 k20 · · · k2c−2 1−c−2∑i=0

k2i

0 0 · · · k3c−3 1−c−3∑i=0

k3i

· · · · · · · · · · · · · · ·0 0 · · · kc0 1− kc0

, (8)

and πP = π. Thus π can be calculated byc−1∑i=0

πi = 1.

4.4 General Service TimeWe have derived the conditional service time distribu-tion in Section 4.2, and have just obtained the systemsize probabilities at departure times in Section 4.3. Notethat the service time of the current request is determinedat the departure instant of the previous request in mostcases, and the only exception is when the previousdeparture leaves behind an empty system, in which casethe service time simply follows the distribution of fS1(t).Thus, we can obtain the general service time distributionof requests as

FS(t) = π0 ·∫ t

0

fS1(x)dx+

c−1∑l=1

πl ·∫ t

0

fSl(x)dx, (9)

and fS(t) = ∂FS(t)/∂t. The expected service time of thesystem can be calculated by

E[S] =∫ √

2/v

0

t · fS(t)dt. (10)

4.5 Steady-State System SizeIt is proved in [39] that with an infinite system capac-ity, the departure time system size probabilities of thestandard M/G/1 queue are the same as those in thesteady state. However, this conclusion does not hold inthe finite capacity case, since we only have c states forthe departure time system size (0, 1, ..., c−1), while c+1states (0, 1, 2, ..., c) have to be considered for the steadystate distribution. With the level crossing methods, we canobserve the fact that the distribution of system sizes just

prior to the arrival time is identical to the departure timeprobabilities as long as arrivals and departures occurindividually, which clearly holds for the Poisson arrivaland the NJN disciplines. Thus

πi = Pr{new request finds i in queue | request joins}= wi/(1− wc) (0 ≤ i ≤ c− 1),

so

wi = (1− wc)πi (i = 0, 1, ..., c− 1). (11)

To obtain wc, we can equate the arrival rate with thedeparture rate of the system,

λ(1− wc) = (1− w0)/E[S]wc = 1− (1− w0)/(λ · E[S]). (12)

Thus w can be obtained based on (11) and (12).Furthermore, with the PASTA property of the Poissonarrival process, u = w, therefore the steady state systemsize distribution is derived. The expectation of the steadystate system size can be calculated as E[L] =

∑ci=0 i · ui,

and a new request arrives to find a fully occupied systemand thus gets dropped with probability wc. Note thatcertain existing hybrid data collection protocols (i.e.,using multiple homogeneous or heterogeneous MEs)could be good choices to address these dropped requestsif the missing data are indeed needed [40].

4.6 Response TimeThe ultimate metric to evaluate the data collection per-formance is the response time R of requests, i.e., fromthe time the request is sent to the time it is served. Withthe expected system size and by Little’s Law, we know

E[R] = E[L]/ (λ(1− wc)) . (13)

However, similar to the case with the traditional SJNdiscipline, the possible starvation problem needs to beinvestigated when NJN is adopted due to its greedynature.

We argue that although NJN may suffer similar prob-lem, its severity would be much less significant for thefollowing two reasons. First, the service time of requestswith NJN cannot be arbitrarily large, which is boundedby the maximum travel time of the ME when serving twoconsecutive requests, i.e.,

√2/v to cross the sensing field.

Second, the service time of a specific request changes asthe ME travels in the field, and the probability that itkeeps at a large value during a long time period is small.

However, the expected response time shown in (13)is not enough to verify the above reasoning, and moreinsights on the distribution of response time are needed.For a given request, its response time R consists of threeparts, the residual service time Sr of the request underservice upon its arrival, the waiting time from the firstdeparture after its arrival to the time it enters service W ,and its service time S, i.e.,


R = Sr +W + S. (14)

Investigating one step further, we can see that thewaiting time W is actually the sum of the service timeof those requests served before the given request. Basedon the results on the general service time of the queuingsystem (Section 4.4) and by convolution theorem, the dis-tribution of the waiting time when conditioning on thefact that i requests are served ahead can be obtained as

fW(t, i) ∼ f(i)S (t), (15)

where f (x)(·) is the x-fold convolution of f(·).Again, consider the case that l requests are waiting in

the queue for service when the ME selects the next oneto serve. As mentioned above, these l distances from theME to each of the requesting sensor nodes can be viewedas l i.i.d random variables, and thus the probability forany one of them to be selected as the next to serve isapproximately 1/l.

Note that it is clear that given a specific requestingnode, the probability for its request to be selected asthe next one is dependent on its location in the field,and this probability varies for different requesting nodes(consider two requesting nodes located at the cornerand the center of the field, respectively). However, ifwe focus on a random request in the queue and lookat its asymptotic behavior, the selection probability of1/l clearly follows.

Summing over all possible l, the probability for arequest to be selected would be

ps =

c−1∑l=1

πl/ (l(1− π0)) . (16)

Combining (15) and (16) , the distribution of thewaiting time is thus

fW(t) =

N∑i=1

fW(t, i)(1− ps)i−1ps. (17)

where N is the number of sensor nodes in the network.Furthermore, it is well known that the distribution of theresidual service time is [39]

fSr (t) = (1− fS(t))/E[S]. (18)

From (14), (17), and (18), we know the response timeof requests follows a distribution of

fR(t) ∼ fSr (t) ∗ fW(t) ∗ fS(t), (19)

where ∗ represents the convolution operation.With the response time distribution of requests, we can

directly obtain observations on the possible starvationproblem from the tail of the distribution, i.e., the longerthe tail, the more serious is the starvation problem. Wewill examine this again in Section 6.3.5.

The above results on the response time are obtainedbased on the convolution theorem. Although theoreti-cally sound, one potential limitation with convolutiontheorem is that depending on the specific form of thefunctions, sometimes it may not have closed-form an-alytical solutions. To address this limitation, next weinvestigate the busy period of the ME, which can serve asa stochastic upper bound of the response time when theanalytical results on its distribution cannot be obtained.

4.7 Busy Period

Our approach to tackling the busy period of the ME isto approximate its distribution based on the analyticalresults of its statistical moments with verification. Notethat with a Poisson arrival, the idle period distributionof the system is FI(t) = 1 − e−λt. Denote Ti,j as thetime the system takes from entering state i till it enteringstate j, and thus T0,0 is the busy cycle of the system.By conditioning on the number of new arrivals whenserving a request, we know

E[T0,0] = E[I + S1]k10 + E[I + S1 + T1,0]k

11

+E[I + S1 + T2,0]k12 + · · ·

E[T1,0] = E[S1]k10 + E[S1 + T1,0]k

11

+E[S1 + T2,0]k12 + · · ·

E[T2,0] = E[S2 + T1,0]k20 + E[S2 + T2,0]k

21

+E[S2 + T3,0]k22 + · · ·

· · · · · ·

with some simple arrangement, we have

E[Ti,0] =

E[I] + E[S1] +

c∑j=1

E[Tj,0]k1j i = 0

E[Si] +c∑

j=1

E[Tj,0]kij−i+1 i > 0

.(20)

Thus E[T0,0] can be calculated, and the second-ordermoment of T0,0 can be obtained with a similar approach

E[T 20,0] = E[(I + S1)

2]k10 + E[(I + S1 + T1,0)2]k11

+E[(I + S1 + T2,0)2]k12 + · · ·

E[T 21,0] = E[S2

1 ]k10 + E[(S1 + T1,0)

2]k11

+E[(S1 + T2,0)2]k12 + · · ·

E[T 22,0] = E[(S2 + T1,0)

2]k20 + E[(S2 + T2,0)2]k21

+E[(S2 + T3,0)2]k22 + · · ·

· · · · · · ,

and after some arrangement, we have

E[T 2i,0] =

E[(I + S1)2]k10

+c∑

j=1

E[(I + S1 + Tj,0)2]k1j i = 0

E[S21 ]k

10 +

c∑j=1

E[(S1 + Tj,0)2]k1j i = 1

c∑j=1

E[(Si + Tj,0)2]kij−i+1 i > 1

. (21)


Therefore, the first and the second-order moments ofthe ME’s busy period B can be calculated as

E[B] = E[T0,0]− E[I], (22)E[B2] = E[T 2

0,0]− E[I2]− 2E[B]E[I]. (23)

Observing the fact that the busy period of the system isactually the sum of the service times of several consecu-tively served requests, we adopt the Gamma distributionto approximate that of the busy period as

fB(t) = tη−1e−t/θ/ (θηΓ(η)) (t > 0), (24)

where η = 1/(E[B2]/E[B]2 − 1) and θ = E[B]/η. The ac-curacy of this approximation is verified in Section 6.3.6.

5 DATA COLLECTION WITH NJNCWe have theoretically analyzed the system measureswhen the NJN discipline is adopted in the previoussection. In this section, we extend NJN by taking thewireless communication properties into account: withwireless communications, the ME can collect data fromseveral requesting nodes at the same collection site, pro-vided that the site is within the communication rangesof these nodes. We consider this NJNC discipline in thefollowing, with which the ME still selects the spatiallynearest requesting node as the next one to serve, exceptthat if there are other requesting nodes that are withinthe communication range of the nearest one, the ME willcombine these requests and serve them together.

The data collection tends to fail due to the poorcommunication quality when the distance between theME and the corresponding sensor node approaching thecommunication range of adopted devices. However, itis shown through measurement studies that the pack-et reception rates are uniformly high up to a certaintransceiver distance [8]. This means the combination ofrequests is still reasonable since we can find a propervalue for the communication range as the input of theNJNC discipline based on empirical results or deploy-ment measurements.

The combination, if happens, can effectively reducethe system size, and thus shorten the response timeof requests. It can also alleviate the possible starvationproblem, since intuitively, starvation is less likely tohappen with a smaller system size. We mainly deal withtwo questions in this section: how likely the combinationcan happen, and if it happens, how many requests can becombined; to what degree the combination can improvethe data collection performance.

5.1 Combination Probability

For the requests combination to happen, the collectionsite must be covered by the communication ranges of atleast another requesting node, besides the nearest one.Same as before, assume l requests are currently availablewhen the ME selects the next request. The probability

p’i 0

p’i i−1p’i i

p’i i+1

p’i i+2

ii−1 i+10 i+2

Fig. 5: State transition at departure time with NJNC (onlythose for state i are shown for clarity).

that Xl requesting nodes, including the nearest one, canbe combined together and served from one collection sitecan be captured by a binomial distribution

Pr{Xl = x} =

(l − 1

x− 1

)FD(r)x−1(1− FD(r))l−x. (25)

Thus the expected number of combined requests is

E[Xl] =l∑

i=1

i · Pr{Xl = x}, (26)

and the probability for the combination to happen is

Pr{Xl > 0} = 1− (1− FD(r))l−1. (27)

Requests combination improves the system perfor-mance by effectively reducing the system size. Based ona similar M/G/1/c queuing model as that for the NJNdiscipline, we present quantitative analysis on its impacton the system performance in the following.

5.2 M/G/1/c–NJNC Queuing ModelThe ME still selects the nearest request to serve withNJNC, so given the system size, the conditional servicetime distribution is identical to that of NJN. Followinga similar idea of the embedded Markov chain, we canderive its departure time system size probabilities, andthe results shown in (4), (5), and (6) still hold.

The difference between the heterogeneous Markovchains of NJNC and NJN is that, it is now possible tohave multiple departures after one service period, asshown in Fig. 5. With the current system size l, denotean,l as the probability of n arrivals during the serviceof the nearest requests selected from l requests withpossible requests combination. Since the combinationdoes not affect the arrival process, we know

an,l = kln. (28)

Denote dm,l as the probability of m departures afterserving the nearest requests selected from l requestswith possible requests combination, which means m− 1requests have been served together with the nearest one,

dm,l =

(l − 1

m− 1

)FD(r)m−1(1− FD(r))l−m. (29)

Thus the state transition probabilities of the embeddedMarkov chain for NJNC, i.e., the counterparts of (8)under NJNC, are


P ′ = [p′ij ] = [i∑

m=1

dm,i · aj−i+m,i], (30)

and π′P ′ = π′, where π′ is the departure time systemsize distribution with NJNC, and

∑c−1i=0 π

′i = 1. Hence,

the general service time distribution can be derived withthe same approach as in (9) and (10).

Based on (26) and π′, we have the following resulton the system throughput with NJN and NJNC, i.e., thenumber of served requests during a unit time, denotedas ψnjnc and ψnjn, respectively.

ψnjnc

ψnjn= 1 +

(c−1∑i=1

π′i · E[Xi]− 1

)+

(31)

When the requests intensity in the network is heavy,i.e., λ→ ∞, (31) can be simplified as

ψnjnc

ψnjn= E[Xc] as λ→ ∞ (32)

The analysis on the request response time and thebusy period of ME is much more complicated withNJNC, due to the possible multiple departures afterone service, but the idea of obtaining these results stillholds. However, we cannot use (11) and (12) to preciselycalculate the steady state system size in the NJNC caseanymore, because the possible multiple departures afterone service invalidates (11). We can see in this casethe results obtained by (12) are essentially a stochasticupper bound of the arrival time system size probabilities,which makes the results returned by (13) also an upperbound of the expected response time.

6 PERFORMANCE EVALUATIONS

We evaluate our modeling and analytical results on theperformance of data collection with NJN and NJNCdisciplines in this section, and we also show the corre-sponding results with FCFS in certain cases for compar-ison. Note that we have already explored the case withFCFS-with-combination (FCFSC) in [10], whose results arenot shown here due to the space limit. We consider a100 × 100 m2 square sensing field with 100 randomlydistributed sensor nodes, unless otherwise specified.Based on the parameters from real systems [30], thetravel speed of ME is 1 m/s. The communication ranger is 20 m and system capacity c is 8 by default unlessotherwise specified.

To deal with the inconvenience of the piecewise dis-tance density function in (2), we approximate it by apolynomial function of order 10 using least squares fittingwith a verified accuracy [9].

f̃D(d) = 0.2802d10 − 2.0964d9 + 2.2349d8 + 24.3629d7

−106.8231d6 + 194.4928d5 − 182.8093d4

+91.8223d3 − 29.3663d2 + 8.2843d− 0.04. (33)

6.1 Validation of the M/G/1/c modelingTo validate the M/G/1/c model, we need to examine theassumptions of both the Poisson arrival of requests andthe distribution-ergodic property of their service.

We adopt a hot-spot model [37] to capture the datageneration in our event-driven simulator. Specifically,several hot spots (10 in our simulation) exist in the field,and the probability for an event to occur at a certainlocation is inverse proportional to its distance to theclosest hot spot. When an event occurs, sensor nodeswhose sensing range covers it can detect the event andgenerate sensory data of size αe−αd bits to record it,where d is the distance between the node and the event,and α is set to 0.5 in our simulation. Sensor node sendsout a data collection request when a total volume of1 KB data are accumulated in its buffer. A total numberof 100 data collection requests are generated and servedduring each simulation, which is repeated for 100 times.

For each simulation run, we record the inter-arrivaltime of the requests, and use an exponential distributionwith the same mean value to approximate them. Weadopt the Kolmogorov-Smirnov (K-S) test with a signifi-cance level of 0.05 to verify the goodness-of-fit of the ap-proximation, and record the percentage of runs that passit. We repeat the whole process for 40 times. To verifythe distribution-ergodic property of the service time, wealso record the service time of each request, and calculatetheir one-lag autocorrelation (as in [38]). Figure 6(a) givesthe results of the K-S test and the autocorrelation on theM/G/1/c model, where each point corresponds to one ofthe 100×100 simulation runs. The x-value of the point isthe percentage of simulation runs that pass the K-S test,and the y-value is the one-lag autocorrelation. Thus weexpect that, if these points are clustered around the rightbottom corner, as being observed from the validationresults in Fig. 6(a), the M/G/1/c queuing model adoptedin this paper is confirmed sound and acceptable.

6.2 Combination ProbabilityNext we evaluate our analytical results on the com-bination probability for NJNC. We explore the caseswhere 5, 10, and 15 requests are in the system when theME selects the next request to serve, respectively, andFig. 6(b) shows the results obtained with 10 requests as arepresentative. Three cases with a communication rangeof 10, 20, and 30 m respectively are investigated. Besidesthe accuracy of the analytical results, we can see that thenumber of combined requests increases with a larger r,because it is more likely to collect data from more sensornodes at the same collection site with a longer com-munication range. Furthermore, the probability for thecombination to happen, even with the shortest exploredcommunication range of 10 m, is about 25 percents,which increases to around 90 percents when r is 30 m.The high probability for requests combination indicatesthat NJNC can improve the data collection performancemost of the time, when compared with NJN.


0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

K−S Test

Aut

ocor

rela

tion

FCFS

0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

K−S Test

Aut

ocor

rela

tion

NJN

0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

K−S Test

Aut

ocor

rela

tion

NJNC

(a) Validation of the Queuing Model

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Combined Requests

CD

F

Analytical (r = 10)Simulation (r = 10)Analytical (r = 20)Simulation (r = 20)Analytical (r = 30)Simulation (r = 30)

(b) Requests combination probability

0 10 20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Service Time with a Given System Size

CD

F

Analytical (l=9)Simulation (l=9)Analytical (l=5)Simulation (l=5)

(c) Service time with a given system size

0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022 0.0240

1

2

3

4

5

6

Arrival Rate of Requests (λ)

Ave

rage

Sys

tem

Siz

e

Analytical (FCFS)Simulation (FCFS)Analytical (NJN)Simulation (NJN)Analytical (NJNC)Simulation (NJNC)

(d) Average system size

0 1 2 3 4 5 6 70.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

System Size

CD

F (λ

= 0

.018

)

Analytical (FCFS)

Simulation (FCFS)

Analytical (NJN)Simulation (NJN)

Analytical (NJNC)

Simulation (NJNC)

(e) System size distribution

0 0.005 0.01 0.015 0.020

0.1

0.2

0.3

0.4

0.5

0.6


Ove

rflow

Pro

babi

lity

0 0.005 0.01 0.015 0.02 0.0250

0.05

0.1

0.15

0.2


Ove

rflow

Pro

babi

lity

0 0.005 0.01 0.015 0.02 0.0250

0.05

0.1

0.15

0.2


Ove

rflow

Pro

babi

lity

Analytical (FCFS)Simulation (FCFS)


Analytical (NJNC)Simulation (NJNC)

(f) System overflow probability

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

10

20

30

40

50

Arrival Rate of Requests

Ave

rage

Ser

vice

Tim

e (s

)


0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

10

20

30

40

50

Arrival Rate of Requests

Ave

rage

Ser

vice

Tim

e (s

)

Analytical (NJNC)Simulation (NJNC)

(g) Average service time with NJN and NJNC

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Service Time (s)

CD

F

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Service Time (s)

CD

F

Analytical (FCFS)Simulation (FCFS)Analytical (NJNC)Simulation (NJNC)

Analytical (FCFS)Simulation (FCFS)Analytical (NJN)Simulation (NJN)

(h) General service time distribution

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.02450

100

150

200

250

300

350


Ave

rage

Res

pons

e Ti

me

(s)

Analytical (FCFS)Simulation (FCFS)Analytical (NJN)Simulation (NJN)Analytical (NJNC)Analytical (NJNC)

(i) Average response time

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

Time (s)

CD

F (λ

= 0

.018

)

Analytical(FCFS)Simulation(FCFS)Analytical (NJN)Simulation (NJN)

(j) Response time distribution with NJN

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

300

Requests Ordered by Response Time

Res

pons

e Ti

me

(s)

NJN: Average Response Time = 95.08 sNJNC: Average Response Time = 72.73 s

(k) Response time with NJN and NJNC

0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

CD

F (λ

= 0

.018

)

Analytical(BP of NJN)Simulation (BP of NJN)Analytical (RT of FCFS)Simulation (RT of FCFS)

(l) Gamma approx. of busy period

Fig. 6: Evaluation results on the queue-based modeling and system measures.

6.3 System Measures

We evaluate the analytical results on the system mea-sures with NJN and NJNC in the following.

6.3.1 Service Time with a Given System Size

We first examine the results on the service time distri-bution with a given system size. The conditional servicetime is derived based on the first-order statistic, whichassumes the distances from requesting nodes to thecurrent location of the ME are independent. We start theverification by investigating the independence amongthese distances. In the simulation, after the deployment

of sensor nodes, we randomly select one as the currentlocation of the ME. Then we select two other sensornodes as the requesting sensor nodes, calculate their dis-tance to the ME, and record it. We repeat the calculationfor 10, 000 times, and obtain two distances sequencesof size 10, 000 each. Then we calculate the correlationcoefficient of the two sequences. We repeat the wholeprocess for 20 times, and the mean square error (MSE) ofthe resultant correlation coefficients is 6.8010×10−5, withthe estimation of a strict independence (a correlationcoefficient of 0). This near-zero correlation supports ourapproximation based on their independence.

For the conditional service time distribution, two cases


with a small and a large system size of 5 and 9 are ex-plored respectively, and the results are shown in Fig. 6(c).We can see that the analysis and simulation agree witheach other, and the conditional service time becomesshorter with a larger system size, which is consistentwith their greedy nature: more queued requests in thesystem bring more opportunities to have nearer ones.

6.3.2 System SizeThe ultimate performance metric of data collection is theresponse time of requests, which is determined by boththe service time and system size, and a shorter servicetime does not necessarily lead to a shorter response time.Thus, we move on to evaluate the system size.

The average system sizes with different request arrivalrates λ for FCFS, NJN, and NJNC are shown in Fig. 6(d).The system sizes under all the three disciplines are smalland comparable to each other when the traffic is light,since the potential for both NJN and NJNC to take effectis quite limited in this case. The system size for FCFSincreases very quickly when λ increases, and cannotkeep stable anymore when λ exceeds a certain threshold(0.018 in Fig. 6(d)). In fact, the case of λ = 0.018 inour simulation roughly corresponds to the case thatρ = λ · E[S] = 1 for FCFS, where ρ is the systemutilization factor, and increasing λ further will resultin an unstable system where the steady state does notexist. When compared with FCFS, NJN can reduce thesystem size greatly because it tries to serve and finishnearby requests in a shorter time, and NJNC can furtherdecrease the system size as a result of possible requestscombination. Note that the capability of NJNC to furtherreduce the system size becomes more obvious when λincreases, since a larger system size leads to a larger po-tential to combine already queued requests. (One thingto mention is that, not surprisingly, the resultant systemsize with FCFSC falls between those with FCFS and NJN,e.g., an average system size of 1.9 is resulted with theFCFSC discipline when λ is 0.018 [10].)

We then explore the system size probabilities for FCFS,NJN and NJNC with λ = 0.018 (Fig. 6(e)). The maximalsystem size resultant by NJN and NJNC is around 5 to 6,but that for FCFS is too large to be shown clearly in thesame figure (given enough system capacity, the maximalsystem size with FCFS observed in our simulation is 33).Furthermore, we can see that the requests combinationshortens the queue consistently.

6.3.3 Overflow ProbabilityData collection requests may be lost due to the limitedsystem capacity. To gain insights on this possible dataloss problem, we explore the overflow probability of thefinite capacity system in Fig. 6(f), in which we considera very small system capacity (c = 3) to amplify theproblem and obtain a clear observation. These threedisciplines result in similar overflow probabilities whenλ is small. However, the overflow probability for FCFSincreases dramatically and becomes much less stable as

λ increases, and is much larger than those achieved byNJN and NJNC. On the other hand, even with a smallcapacity, the overflow probabilities for NJN and NJNCare very small and more stable (less than 0.2 in the worstcase by our simulation settings), and NJNC can furtherreduce the overflow probability by reducing the systemsize through possible requests combination.

6.3.4 General Service TimeWe then examine the general service time for both NJNand NJNC. The average service time of NJN and NJNCwith different arrival rate of requests are shown inFig. 6(g). The decreasing trend of the service time asλ increases agrees with the greedy nature of both NJNand NJNC with more queued requests. When compar-ing these two disciplines, the service time of NJNC isslightly larger than that of NJN because of the possiblecombination, since NJNC reduces the system size moreaggressively and thus also reduces the possibility offinding a closer requesting sensor node in the future.Also note that the difference between the two becomesmore obvious with a larger λ.

To gain more insights on the service time, Fig. 6(h)shows the general service time distribution with λ =0.018, where the service time distribution for FCFS is alsoshown for comparison as well. We can see that whencompared with FCFS, NJN and NJNC can shorten theservice time noticeably because of their nature to servenearby requests with less time, and similar insights canbe observed as in Fig. 6(g).

6.3.5 Response TimeAs mentioned above, both service time and system sizeaffect the response time. Since NJN has a shorter servicetime and NJNC has a smaller system size, next we needto examine their response time.

The average response time for FCFS, NJN and NJNCis shown in Fig. 6(i), from which we can see thatwhen compared with FCFS, NJN and NJNC can greatlyshorten the response time of requests, especially whenλ is large. NJNC can further reduce the response timeas a result of possible requests combination, which alsobecomes more obvious as λ increases. It shows thatbetween a shorter service time for NJN and a smallersystem size for NJNC, the system size reduction is moredominating for the resultant response time. A smallersystem size also indicates a lower overflow probabilityfor a given system capacity limit. Actually, the aboveobservations can be inferred based on Fig. 6(d) andthe linear relationship between the response time andsystem size (by Little’s Law). We still include Fig. 6(i)here for a clear and straightforward comparison.

We have derived the distribution of the responsetime of requests to investigate the potential starvationproblem when the NJN (NJNC) discipline is adopted.One fundamental part to obtain the response time dis-tribution is the probability for a request to be selected forservice, and we verify this before moving to the response


TABLE 1: Verification of requests’ selection probability.

λ 0.03 0.04 0.05 0.06# of Selections 3.7166 5.0175 5.6479 6.1289

# of Waiting Reqs. 3.8036 5.0325 5.6533 6.1312

time. For each request, after its arrival, we record thenumber of selections made by the ME until the requestis chosen, and compare this to the average number ofrequests that are waiting for service in the queue. If (16)holds, then we expect these two results to be close. Weintentionally set higher traffic intensities (and thus largersystem sizes) for this verification to make the observationmore clear. The comparison results are shown in Table 1,which verify the accuracy of (16).

Next we evaluate the results on the response timedistribution, which are shown in Fig. 6(j) (λ = 0.018).The response time by the FCFS discipline (obtainedaccording to [39]) is also shown for comparison. Besidesthe accuracy of the analytical results, we can see even theworst case of NJN, in terms of the longest response timeexperienced by requests, is much smaller than that by theFCFS discipline. This observation is a little unexpectedsince FCFS is known to be able to achieve a good fairnessamong clients, but it also alleviates our concerns on thepossible starvation problem with NJN (NJNC).

To present a clear comparison between the perfor-mance of NJN and NJNC, we presents the response timeobtained by a particular run of simulation (λ = 0.018)in Fig. 6(k), where a total number of 10, 000 requestsare served (or dropped due to system overflow). Wesort the response time in an ascending order, and onlyplot the response time every 200 requests for clarity. Notsurprisingly, NJNC further reduces the response time interms of both the average and worst cases.

6.3.6 Busy Period

Figure 6(l) shows the evaluation results on the distri-bution of the busy period of the ME, where λ is set to0.018. We can see that although the results are obtainedthrough approximation methods, the accuracy is stillgood. For comparison, the response time distributionobtained with the FCFS discipline is shown here again.We can see that with NJN, even its busy period, which isa stochastic upper bound of the response time, is smallerthan the response time with FCFS.

6.4 Comparison with Other Scheduling Schemes

In the following, we compare the performance of NJNand NJNC with two classic scheduling schemes.

• TSP: the offline optimized TSP-based schedulingscheme, with which the mobile element carries outthe data collection according to the optimal solutionof the TSP instance formulated based on the nodedeployment. This is involved in many offline datacollection scheme designs [17], [19], [44].

• Gated-TSPN: the dynamic schedule scheme basedon a TSPN instance according to the available data

0.002 0.006 0.01 0.014 0.018 0.02250

100

150

200


Ave

rage

Res

pons

e T

ime

(s)

Gated−TSPNNJNNJNC

Fig. 7: Response time withrequest arrival rate λ.

50 100 150 200 2500

500

1000

1500

2000

2500

3000

Field Size L (m)

Ave

rage

Res

pons

e T

ime

(s)

TSPGated−TSPNNJNNJNC

Fig. 8: Response time withfield size L.

collection requests [49], which is treated as the state-of-the-art for the on-demand data collection.

The average data collection latency resultant with theGated-TSPN, NJN, and NJNC are shown in Fig. 7, withthe request arrival rate varies from 0.002 to 0.022. Wecan see that the NJN (NJNC) outperforms the Gated-TSPN scheme noticeably, especially when the requestarrival rate is large. However, the system stability of theGated-TSPN scheme has been theoretically guaranteed,while our analysis on NJN and NJNC has not provedthis property yet, which is the direction of our futurework. The results with the TSP scheme are not shownin the figure because the latency resultant is significant-ly longer, specifically, around 940 s when λ = 0.006.However, the latency with the TSP scheme is relativelyinsensitive to the request arrival rate, e.g., around 960 swhen λ = 0.022. This infers that the TSP scheme may bea good choice when the intensity of the data collectiontasks in the network is quite heavy.

The response time with various field size are shownin Fig. 8, where the x-axis is the side length of thesquare field. The response time with the TSP schemeincreases almost linearly, which agrees with the natureof the optimal TSP tour. The NJN and NJNC outperformthe Gated-TSPN scheme, and the advantage becomesmore obvious with a larger field. Again, we would like toemphasize that although verified to be efficient throughsimulation, unlike Gated-TSPN, the system stability withNJN and NJNC has not be theoretically proved yet.

7 DISCUSSION AND ONGOING WORK

Our modeling and analysis are shown accurate but someissues need to be further explored. Here we discuss someof these issues and the directions of ongoing work.

Clearly, the assumption of a square sensing field maynot hold in practice. However, our queue-based mod-eling and analysis approaches are still feasible even forgeneral sensing fields, provided that the distance dis-tribution between arbitrary locations in the field can beobtained, e.g., we have also analytically derived the ran-dom distance distributions associated with rhombusesand hexagons [41]. Furthermore, even if the sensing fieldis of irregular shape, which may be true in certain cases,we can adopt the polygon-approximation approach toapproximate the field by the combination of several


regular shapes [42], and derive the random distancedistributions within and between them. Of course, morecomputation efforts will be needed.

Another simplification we made is that the time fortransmitting the data from sensors to the ME is negli-gible, which may not hold if the data volume is large.However, note that given any sensor network deploy-ment, the knowledge on the data transmission rate andcommunication range is available, or at least can beestimated through experimental measurements. Basedon such knowledge, we can estimate the data volumesthat can be collected if the ME travels without any stops,and then the time that the ME has to pause or slow downto collect the remaining data can be calculated as well.The statistics of this additional time can be incorporatedwhen formulating the service time distribution.

The request response time in this work does notinclude the time since the request is sent by the sensornode to its reception at the ME, which we assumed tobe negligible. Observing the fact that these two times areindependent to each other, thus by the convolution the-orem, we can easily take the latter into account providedthat its distribution g(t) can be estimated1, i.e.,

f ′R(t) = g(t) ∗ fR(t). (34)

When multiple MEs are available, a straightforwardapproach is to extend the model to the multi-server case.However, in addition to the queue length and responsetime, we also need to consider the load balance amongthe MEs as another metric to evaluate the system per-formance. Some preliminary results on the on-demanddata collection with multiple MEs can be found in [9].

8 CONCLUSIONS

In this paper, we have analytically evaluated an intuitiveservice discipline, NJN, for data collection with mobileelements in wireless sensor networks, and also exploredthe case where the ME follows NJN and combines re-quests whenever possible, i.e., NJNC. We have modeledthe system as an M/G/1/c queue, and then with differ-ent service disciplines (NJN and NJNC), critical systemmetrics have been derived. We have verified our ana-lytical results through extensive simulation, and gainedmore insights on the starvation problem that NJN andNJNC may suffer from. Our results have showed that notonly the average performance of NJN is much better thanthat of FCFS, but also the worst-case performance of NJNis still better than that of FCFS, even though accordingto the conventional wisdom, NJN may suffer from thestarvation problem. A possible reason is that the servicetime is not arbitrary for data collection applications inwireless sensor networks. Moreover, NJNC can furtherimprove the performance as a result of possible requests

1. We have taken this delivery time into account by proposing apartition-based data collection scheme in a recent work [43].

combination. We have also discussed several possibleextensions as the ongoing and future work.

Acknowledgment: The work is supported in part bySingapore-MIT International Design Center IDG31000101,grant SUTD SRG ISTD 2010 002, Project GREaT IPMD13012,and grant K93-9-2010-01 from Ministry of Educations KeyLab for Computer Network and Information Integration atSoutheast University, China.

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Liang He (S’09-M’12) is currently a postdoc re-search fellow at Singapore University of Technol-ogy and Design. He received his B.Eng. degreein 2006 and Ph.D degree in 2011 from TianjinUniversity, China, and Nankai University, China,respectively. During Oct. 2009 to Oct. 2011, heworked at Panlab at University of Victoria as avisiting research student. He has been a recipi-ent of the the best paper awards of IEEE WCSP2011 and IEEE GLOBECOM 2011.

Zhe Yang (S09) received his B.S. degree in In-formation Engineering in 2005 and M.S. degreein Control Theory and Engineering in 2008, bothfrom Xian Jiaotong University, Xian, China. Heis currently a Ph.D. candidate at the Departmentof Electrical and Computer Engineering, Univer-sity of Victoria, British Columbia, Canada. Hiscurrent research interests include cross-layerdesign for cooperative networks, scheduling andresources allocation for wireless networks andsynchronization.

Jianping Pan (S’96-M’98-SM’08) is currently anassociate professor of computer science at theUniversity of Victoria, Canada. He received hisBachelor’s and PhD degrees in computer sci-ence from Southeast University, China, and hedid his postdoctoral research at the Universityof Waterloo, Canada. He also worked at FujitsuLabs and NTT Labs. He received the IEICE BestPaper Award in 2009, the Telecommunication-s Advancement Foundation’s Telesys Award in2010, the WCSP 2011 Best Paper Award and

the IEEE Globecom 2011 Best Paper Award, and has been serving onthe technical program committees of major computer communicationsand networking conferences including IEEE INFOCOM, ICC, Globecom,WCNC and CCNC.

Lin Cai (S00-M06-SM10) received M.A.Sc. andPh.D. degrees in electrical and computer engi-neering from the University of Waterloo, Cana-da, in 2002 and 2005, respectively. Since 2005,she is currently an Associate Professor with theDepartment of Electrical and Computer Engi-neering, University of Victoria, Canada. She hasbeen an Associate Editor for IEEE Transactionson Wireless Communications, IEEE Transaction-s on Vehicular Technology, EURASIP Journal onWireless Communications and Networking, the

International Journal of Sensor Networks, and the Journal of Communi-cations and Networks.

Jingdong Xu received her B.S., M.S., and Ph.D.degrees in the Department of computer sciencefrom Nankai University, China, in 1988,1991,and 2002, respectively. She is currently a pro-fessor and the department chair of ComputerScience, Nankai University. Prof. Xu’s researchfocuses on wireless ad hoc network,wirelesssensor network and mobile computing.

Yu (Jason) Gu is an assistant professor at theSingapore University of Technology and Design.He received the Ph.D. degree in the Departmentof Computer Science and Engineering at theUniversity of Minnesota, 2010. He is the authorand co-author of over 21 papers in premierjournals and conferences. His publications havebeen selected as graduate course materials byover 10 universities in the United States and oth-er countries. He has received several prestigiousawards from the University of Minnesota.

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