on-demand mobile data collection in cyber-physical...

Research ArticleOn-Demand Mobile Data Collection in Cyber-Physical Systems

Liang He1 Linghe Kong 2 Jun Tao3 Jingdong Xu4 and Jianping Pan5

1University of Colorado Denver CO USA2Shanghai Jiaotong University Shanghai China3Southeast University Nanjing Jiangsu China4Nankai University Tianjin China5University of Victoria Victoria BC Canada

Correspondence should be addressed to Linghe Kong linghekongsjtueducn

Received 10 October 2017 Revised 16 January 2018 Accepted 19 March 2018 Published 30 April 2018

Academic Editor Yin Zhang

Copyright copy 2018 LiangHe et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The collection of sensory data is crucial for cyber-physical systems Employing mobile agents (MAs) to collect data from sensorsoffers a new dimension to reduce and balance their energy consumption but leads to large data collection latency due to MAsrsquolimited velocity Most existing research effort focuses on the offlinemobile data collection (MDC) where the MAs collect data fromsensors based on preoptimized tours However the efficiency of these offline MDC solutions degrades when the data generation ofsensors varies In this paper we investigate the on-demand MDC that is MAs collect data based on the real-time data collectionrequests from sensors Specifically we construct queuing models to describe the First-Come-First-Serve-based MDC with a singleMA and multiple MAs respectively laying a theoretical foundation We also use three examples to show how such analysis guidesonline MDC in practice

1 Introduction

Collecting data from sensors is a core function of large cyber-physical systems such as wind farm and smart grid [1ndash4]Traditional data collection approaches rely on the wirelesscommunications between sensor nodes and the sink exces-sively consuming nodesrsquo limited energy supply and leadingto their unbalanced energy consumption Adopting mobileagents (MAs) for data collection that ismobility-assisted datacollection (MDC) reduces and balances the communicationloads of nodes (and thus their energy consumption) [5ndash7]Also with MAsrsquo controllable mobility the communicationsand networking become possible even in sparse networks viathe store-carry-forward approach A real-lifeMDC example isthe NEPTUNE projectmdasha seabed crawler is deployed to col-lect sensory data from other underwater experiment nodes[8] However MDC leads to large data collection latency dueto MAsrsquo limited velocity degrading the realtimeness of thecollected data and may cause data loss due to the bufferoverflow at sensor nodes

Much research effort on offline MDC exists in the liter-ature where the MAs periodically collect data from nodeswith a preoptimized path [9] On the other end of the spec-trum on-demand MDCmdashsensor nodes send data collectionrequests to the MAs when they have data to report andthe MAs only visit (and collect data from) such requestingnodesmdashis a more efficient approach to exploit MAsrsquo limitedmobility resource especially for event-driven systems withdiverse data generation among sensors [10 11] The challengein the on-demand MDC however is to determine howthe MAs should collect data from nodes without a prioriinformation on future data collection demands

On-demand MDC shows clear queuing behavior In thispaper we formulate two queuing models to capture the on-demand MDC with the First-Come-First-Serve (FCFS) disci-pline (FCFS is a simple and natural choice tomaintain requestfairness and is preferred in certain node-centric scenarios)on the cases where a single MA and multiple MAs aredeployed for data collection respectively and correspondinganalytical results on the data collection performance are

HindawiWireless Communications and Mobile ComputingVolume 2018 Article ID 5913981 13 pageshttpsdoiorg10115520185913981

2 Wireless Communications and Mobile Computing

derived Furthermore we use three examples to show howthe analysis guides the on-demand MDC in practice (i) howto use multiple MAs (ii) When to request data collection(iii) How likely the requests combinationmdashthat is collectdata frommultiple sensor nodes at the same locationmdashwouldhappen via the wireless communication between the MAsand nodes The contributions of this paper include thefollowing

(i) Formulation of an1198721198661 queuing model to captureand analytically evaluate the on-demand MDC whena singleMA is deployed for data collection (Section 4)

(ii) An 119872119866119888 queuing model for the case when mul-tiple MAs are deployed based on which the datacollection performance is explored via approximation(Section 5)

(iii) Three examples to show how the analysis guides theon-demand MDC in practice (Section 7)

The rest of this paper is organized as followsThe literatureon MDC is briefed in Section 2 We formulate the problemin Section 3 The on-demand MDC with a single MA andmultipleMAs is investigated in Sections 4 and 5 and evaluatedin Section 6The practical guidance is presented in Section 7followed by further discussions in Section 8 The paperconcludes in Section 9

2 Related Work

Observing the advantages of MDC over traditional datacollection approaches (eg via direct communication ormultihop forwarding) much effort has been made to explorethe MDC with a single MA [7 11 12] For example Sugiharaand Gupta investigated the MDC with the objective ofminimizing MArsquos travel distance in [9] Zhao et al proposeda three-layer framework for the offline MDC including thesensor layer cluster head layer andMA layerTheMAcollectsdata according to a preoptimized tour with dual antennas[13] A unified framework for analyzing the MArsquos mobilityand data collection latency was presented and solutions tothe involved subproblems were proposed in [14] An MA-tracking protocol was proposed in [15] in which the routingstructure of data collection requests was additively updatedwith MArsquos movement The joint energy replenishment anddata collection was investigated in [16] with the consider-ation of various sources of energy consumption and time-varying nature of energy replenishment The colocation ofMA and wireless charger has been investigated in [17] withthe objective of ensuring sustainable and lossless systemoperation

Scalability is a critical bottleneck when only a single MAis used and a potential mitigation is to employ multiple MAsfor data collection An early investigation on the scenarioof multiple MAs is [18] where the MAs travel along fixedtracks to collect data from nodes with the considerationof load balancing A motion planning algorithm for theMAs was proposed in [19] which minimizes the number ofMAs according to the constraints in distance and time Thiswork was extended for applications with strict distancetime

times104

0

2

4

6

8

of

Rec

eive

d Pa

cket

s

11 21 31 41 51 541Node ID

Figure 1 Diverse data generation among sensor nodes (originaldata from [24])

constraints and a data-gathering algorithm with multipleMAs was proposed in [20] More detailed information onMDC is found in [21] and the references therein

Most of these existing efforts focus on the offline MDCwhile we tackle the on-demandMDC in this paper Althoughsimilar scenarios have been investigated in [12 22 23] ourqueue-based analytical framework provides detailed insightsinto the data collection process such as system size queuingtime and response time and these analytical insights guidethe MDC in practice

3 Preliminaries

31 On-Demand MDC In many offline MDC solutionsthe MAs periodically collect data from sensor nodes basedon preoptimized tours [9] These solutions perform wellwhen nodes generate data at similar paces but degradedramatically when the data generation at nodes variesmdashMAsmay visit nodes with little or no data to report Unfortunatelymany event-driven systems demonstrate such diverse datagenerations [10] For example Figure 1 plots the number ofpackets received from 54 sensor nodes in a trace providedby Intel Berkeley Research Lab [24] showing clear diversityamong nodes Targeting on these scenarios with diverse datagenerations we investigate the on-demand MDC where theMAs collect data based on real-time demands from sensornodes in this paper

32 Network Model We consider the scenario where con-trollable MAs collect data from stationary sensor nodesrandomly deployed in a square sensing field [9] (our modelformulation and analysis are also applicable to sensing fieldsof other shapes as will be explained in Section 4) Sensornodes monitor their surrounding environments store thegathered data in their buffers and send out data collectionrequests toMAswhen their buffers are to be full [10]The datacollection requests can be delivered to the MAs via existingMA-tracking protocols [15] Because the typical data relayspeed is much faster than theMAsrsquo travel we assume that thetime since a request is sent by a sensor node till it is receivedby MAs is short and negligible [12] Note that instead ofusing theseMA-tracking protocols to upload the sensory datadirectly which are normally ofmuch larger volume only data

Wireless Communications and Mobile Computing 3

requestarrival

service pool

mobile agent

FCFS requestdeparture

Figure 2 Queuing model when a single MA is deployed for data collection

collection requests are forwarded to MAs to reduce nodesrsquocommunication loads

MAs maintain a service pool to store the received datacollection requests and serve them according to the First-Come-First-Serve (FCFS) discipline By serving the requestwe mean one of the MAs moves to the correspondingrequesting node to collect its data via short-range wirelesscommunications FCFS albeit not the optimal solution foron-demandMDC is a classic scheduling discipline known tobe fair for clients [25] Moreover the theoretically establisheddata collection performance with FCFS serves as a goodbaseline for the evaluation of more sophisticated MDCsolutions

Also it is not necessary for the MA to travel to theexact location of nodes to collect data because of the wirelesscommunications between theMAand the nodes [12 26]Thisway theMAcan potentially collect data from all nodeswithinits communication range at a single site The impact of thecommunication range on MDC relies on both the field sizeand node density To establish a theoretic foundation we donot directly incorporate the communication range into ourmodeling however we investigate and evaluate its impact inSection 73

33 Problem Statement The on-demand MDC is dynamicboth temporally and spatially that is when a data collectionrequest will be received and where (which sensor node) therequest is from This dynamic property not only shifts ourobjective from MAsrsquo optimal path planning (as in the offlineMDC) to the design of efficient real-time service disciplinesto select the next request (ie the requesting node) to serve(ie collect data from) but also makes the MDC hard tocapture and thus its performance challenging to evaluate Inthis paper we evaluate the on-demand MDC via a queue-based analytical approach

4 MDC with a Single MA

We investigate the on-demand MDC in this section andthe following ones Specifically in this section an 1198721198661queuing model is constructed to capture the MDC when asingle MA is deployed

41 Construction of the MG1 Model The on-demand MDCshows clear queuing behavior inspiring us to capture itwith a queuing modelmdashthe MA serves as the server andthe data collection requests from sensors are treated as the

clients (Figure 2) For any queuing system two fundamentalcomponents to be characterized are the client arrival anddeparture

411 Request Arrival The aggregated request arrival processat the MA is the superposition of 119899 requesting processesof individual sensors where 119899 is the number of sensors inthe system This way the request arrival at the MA can becaptured by a Poisson process according to Palm-Khintchinetheorem [27] This is because (i) for a stable data collectionprocess the number of sensors in the system is large whencompared with the number of to-be-served requests at anygiven time instance indicating low dependency in theirrequesting of data collections (ii) the probability for a sensorto initiate a data collection request at a specific time instanceis small Theoretically if the client population of a queuingsystem is large and the probability by which clients arriveat the queue is low at a specific time the arrival processcan be adequately modeled as Poisson [28] We will furtherstatistically verify this Poisson arrival of requests in Section 6

Assume that a memory buffer of size 119861 is equippedfor each sensor and its asymptotic data generation ratesare 119891119894 (119894 = 1 2 119899) The request arrival rate 120582 can beapproximated as

120582 asymp 1119861119899sum119894=1

119891119894 (1)

Essentially (1) is a lower bound on the aggregated requestsarrival rate because the requesting node would not requestagain before its data has been collected Denoting 120582lowast as thetrue request arrival rate we have

(120582lowast minus 120582) 997888rarr 0+ as 119899 997888rarr infin (2)

412 Request Departure The MA travels to the requestingnode to collect the data therein Because the data propagationspeed is much faster than the MArsquos travel speed we simplifyour investigation by assuming a negligible data transmissionlatency This way the departure process or the service timeof clients can be characterized by the time from the servicecompletion of the current request to the time when the MAmoves to the next requesting node

As the previous data collection site is also the startinglocation when the MA serves the next request the servicetime of consecutively served requests seems not to be inde-pendent However denoting the sequence of service times as1199051 1199052 1199053 if we examine only at every second element of


the original process it is clear that 1199051 1199053 are independentof each other and the distribution-ergodic property of thissubprocess can be observed [29] The same is true forsubprocess 1199052 1199054 The distribution-ergodic property stillholds if we combine these two subprocesses because theirasymptotic behaviors do not change after the combinationA demonstration on this distribution-ergodic property is

shown in Figure 3 This means that if we identify the timedistribution when the MA travels between consecutivelyserved nodes we can use it as the service time distributionfor the queuing model over a long time period

From existing results in geometrical probability [11] thedistance distribution between two random locations in a unitsquare is

119891D (119909) =

2119909 (120587 minus 4119909 + 1199092) 0 le 119909 le 12119909 [2 sinminus1 ( 1119909) minus 2 sinminus1radic1 minus 11199092 + 4radic1199092 minus 1 minus 1199092 minus 2] 1 le 119909 le radic20 otherwise

(3)

Thus with MA travel speed V the service time distribu-tion can be derived as

119865S (119905) = 119875 service time lt 119905 = 119875 travel distance lt V119905

=

intV119905

02119909 (120587 minus 4119909 + 1199092) 119889119909 119905 le 1

V

int102119909 (120587 minus 4119909 + 1199092) 119889119909 + intV119905

12119909 [2 sinminus1 ( 1119909) minus 2 sinminus1radic1 minus 11199092 minus 1199092 minus 2] 119889119909 1

Vlt 119905 le radic2

V

1 119905 gt radic2V

(4)

Its expectation variance and coefficient of variation are

E [S] = E [DV] = 1

VE [D]

V [S] = V [DV] = 1

V2V [D]

Cov [S] = radicV [S]E [S] = radicV [D]

E [D] = Cov [D]

(5)

After characterizing the request arrival and departure wecan model the on-demand MDC with a single MA as an1198721198661 queuing system Note that the distance distributionsbetween the random locations in other field shapes are alsoavailable in the literature [30 31] which can be used in ourmodel accordingly (eg by substituting (3))

42 Analysis Based on the MG1 Queuing Model Withthe 1198721198661 queuing model the data collection latency isequivalently the client response time in the queuing modelWe next derive analytical results on the latter to shed light onthe former

421 System Size Distribution Denote 119883119899 as the number ofrequests in the service pool immediately after the departureof a request at time 119905119899 then

119883119899+1 = 119883119899 minus 1 + 119860119899+1 (119883119899 ge 1)119860119899+1 (119883119899 = 0) (6)

where 119860119899+1 is the number of new arrivals when serving the(119899 + 1)th request It is clear that 119860119899+1 depends only on theservice time of the (119899 + 1)th request rather than any eventsthat occurred earlier (ie the system size at earlier departurepoints 119883119899minus1 119883119899minus2 ) Thus the embedded discrete-timeprocess 1198831 1198832 observed at departure times is aDiscrete-Time Markov Chain (Figure 4) with transition probabilities

119901119894119895 = 119875 119883119899+1 = 119895 | 119883119899 = 119894 (7)

Define the probability that 119894 new requests are receivedwhen serving a request as

119896119894 = 119875 119860 = 119894 = intinfin0119875 119860 = 119894 | 119878 = 119905 times 119891S (119905) 119889119905

= intinfin0

119890minus120582119905 (120582119905)119894119894 119891S (119905) 119889119905(8)


t1

t2

t3

t4

Figure 3 Distribution-ergodic property of the service time

0 1 i minus 1 i i + 1k0

k0 k0 k0

k1

k1

k1 k1 k1

k2k2

k3

Figure 4 Transition diagram of the Markov Chain

With 119896119894 we have the following state transition matrix

P = 119901119894119895 =[[[[[[[[[[[[[

1198960 1198961 1198962 1198963 sdot sdot sdot1198960 1198961 1198962 1198963 sdot sdot sdot0 1198960 1198961 1198962 sdot sdot sdot0 0 1198960 1198961 sdot sdot sdot0 0 0 1198960 sdot sdot sdot

]]]]]]]]]]]]]

(9)

Denote 120587 as the steady-state system size probabilities atthe departure times

120587P = 120587 (10)

from which we know that

120587119894 = 1205870119896119894 + 119894+1sum119895=1

120587119895119896119894minus119895+1 for 119894 = 0 1 2 (11)

Define the following generating functions

Π (119911) = infinsum119894=0

120587119894119911119894 |119911| le 1

119870 (119911) = infinsum119894=0

119896119894119911119894 |119911| le 1(12)

Because 1205870 = 1 minus 120588 we haveΠ (119911) = (1 minus 120588) (1 minus 119911)119870 (119911)119870 (119911) minus 119911 (13)

from which 120587 can be derived Note that 120587 are not the sameas the steady-state system size probabilities 119901119899 in generalHowever for the1198721198661 queue it has been proven that thesetwo quantities are asymptotically identical [25]

422 Response Time Distribution Next we derive theresponse time distribution based on 120587 which consists of twoparts (i) the queuing time since its arrival at the system toits service start and (ii) its service time With FCFS for anew request arriving at the queue with 1198991015840 existing requestsits queuing time is the sum of the service times of these 1198991015840requests By convolution theorem we have

119902 (119905 1198991015840) =

0 1198991015840 = 0119904 (119905) 1198991015840 = 1119902 (119905 1198991015840 minus 1) lowast 119904 (119905) 1198991015840 gt 1

(14)

where lowast is the convolution operatorThis way the probabilitydistribution of the queuing time can be derived as

Q (119905) = infinsum119894=0

Pr 119894 existing requestssdot Pr queuing time le 119905 | 119894 existing requests

= infinsum119894=0

120587119894 int1199050119902 (119909 119894) 119889119909

(15)

and its density function is 119902(119905) = 120597Q(119905)120597119905Similarly the response time distribution for a request

being received by the MA with a system size 1198991015840 is119903 (119905 1198991015840) =

119904 (119905) 1198991015840 = 0119902 (119905 1198991015840) lowast 119904 (119905) 1198991015840 gt 1 (16)

Its probability distribution can be calculated as

R (119905) = infinsum119894=0

Pr 119894 existing requestssdot Pr response time le 119905 | 119894 existing requests

= infinsum119894=0

120587119894 int1199050119903 (119909 119894) 119889119909

(17)

and 119903(119905) = 120597R(119905)1205971199055 MDC with Multiple MAs

Scalability is a critical bottleneck when only a single MA isdeployed for data collection especially with a large sensingfield or with a high node density Employing multiple MAsto collect data collaboratively is a straightforward mitigationwhich we investigate next Specifically we consider thescenario where each MA has the full knowledge on thereceived data collection requests which can be achievedby the communications among the MAs and the sink for


requestarrival

service pool (FCFS)

mobile agents

requestdeparture

Figure 5 Queuingmodel whenmultiple MAs are deployed for datacollection

example via satellite or cellular communications WheneveranMA accomplishes its current data collection task it selectsthe next-to-be-served request with FCFS

51 Construction of the MGc Model Our approach is toextend the previously constructed 1198721198661 queuing modelto 119872119866119888 where 119888 is the number of MAs (Figure 5) Theemployment of multiple MAs does not affect sensorsrsquo datageneration and thus the request arrival process is the same asin the single MA case For the request departure the servicetime for individual data collection requests is still the sameas that with a single MA (ie as in (4)) but the aggregatedsystem departure rate will be 119888E[S]52 Analysis Based on the MGc Queuing Model Althoughextending the queuing model to the multiple MAs caseis straightforward evaluating an 119872119866119888 queue is analyti-cally intractable Even when closed-form solutions can beobtained often they are complicated and require particularprobability distributions [32 33] Thus instead of pursuingthe exact analytical results on system measures we use anapproximation approach The basic idea is to combine theanalytical results on simple queuing systems such as119872119872119888and119872119863119888 to approximate themeasures of the119872119866119888 queue[34]

521 Expected Response Time We first explore the expecteddata collection latency or equivalently the expected responsetime of the 119872119866119888 queue A simple two-moment approxi-mation formula with verified accuracy for the mean queuingtime in an119872119866119888 queue can be derived from [35]

E [Q119872119866119888] asymp 1 + 1205742120574E [Q119872119872119888] + (1 minus 120574) E [Q119872119863119888] (18)

where 120574 = Cov[S]2The above approximation is essentially a weighted com-

bination of E[Q119872119872119888] and E[Q119872119863119888] The former can becalculated by

E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2

sdot [119888minus1sum119894=0

(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)

and the latter can be obtained by Crommelinrsquos formula[36]

E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1

[(119894119888120588)119895minus1(119895 minus 1) minus(119894119888120588)119895120588119895 ] 119890minus119894119888120588 (20)

However the series in (20) converges slowly especiallywith high traffic intensity [37] Again approximations areadopted to speed up its convergence An approximation onE[Q119872119863119888] with simple computation complexity and promis-ing accuracy is presented in [34]

E [Q119872119863119888] asymp [1 + ℎ (120579) 119892 (120588) (1 minus 119890minus120579ℎ(120579)119892(120588))]sdot E [Q119872119872119888] (21)

where

120579 = 1 minus 2119888 + 1 119888 ge 1ℎ (120579) = 120579 [((9 + 120579) (1 minus 120579))12 minus 2]

8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)

Substituting (19) and (21) into (18) we can calculateE[Q119872119866119888] with which the requestsrsquo expected response timein the119872119866119888 queue can be derived as

E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)

The expected response time is crucial because it not onlyoffers us insights into the asymptotic data collection latencybut also helps to obtain themeasures on the size of the119872119866119888queue based on which more insights into the MDC can beobtained Again denote 119883 as the number of requests eitherwaiting or being served at arbitrary time in the119872119866119888 queueLetP119894 = P119883 = 119894 (119894 ge 0) and define

120572 = E [Q119872119866119888]E [Q119872119872119888] (24)

A geometric-form approximation for the system sizeprobability is proposed in [38]

P119894119872119866119888 = [(119888120588)119894119894 ]P0119872119872119888 119894 = 0 119888 minus 1(1 minus 120577) 120577119894minus119888U119872119872119888 119894 ge 119888

(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)


is the probability that a newly arriving request has to waitbefore being served in an 119872119872119888 queue Note that 120577 lt 1if 120588 lt 1 and 120577 = 120588 if the service time is exponentiallydistributed

We calculate the expected system size based on itsapproximated distribution as

119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)

and the probability that a newly arrived request has to waitbefore being served that is the equivalent of U119872119872119888 in119872119866119888 queue can be calculated as

U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)

By distributional Littlersquos law [39] the number of customersin the queue has the same distribution as the number ofarrivals during the waiting time Based on this and theabove approximation results on the system size distributionan approximation for the queuing time distribution in the119872119866119888 queue is proposed in [40]

Q119872119866119888 (119905) asymp 1 minus 119890(minus119888120583(1minus120588)119905)120572U119872119872119888119902119872119866119888 (119905) = 120597119876119872119866119888 (119905)120597119905 (29)

Furthermore because the response time of a request isthe sum of its queuing time and service time which areindependent of each other by convolution theorem we have

119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)

6 Performance Evaluations

We verify the model soundness and the analysis accuracyin this section We consider a system deployed in a squarefield of size 100 times 100m2 A total number of 100 sensorsare randomly deployed unless otherwise specified The MAvelocity is set to 1ms based onPowerBot [41]The simulationis implemented with Matlab A total number of 10000requests are generated and served during each run of thesimulation which is repeated for 50 times

To deal with the inconvenience of the piecewise distanceprobability density function in (3) we approximate it by a 10-order polynomial with least squares fitting

119891 (119889) = 0280211988910 minus 209641198899 + 223491198898+ 2436291198897 minus 10682311198896 + 19449281198895minus 18280931198894 + 9182231198893 minus 2936631198892+ 82843119889 minus 00402

(31)

61 Verifying the Queuing Models To verify the soundnessof the queue-based modeling we examine the request arrival

Exponential KB)4=(bSimulation KB)4=(b


0

02

04

06

08

1

CDF

400 800 12000Request Interarrival Time (s)

Figure 6 Verify the request arrival

with an event-driven simulator where stochastic events occurrandomly in the sensing field (note that when events happenin a clustered manner this actually improves the MDCperformance as theMAsrsquo travel distance is reducedThis wayour models capture the worst cases of on-demand MDCwhich are important to provide performance guarantees)Sensors within a certain distance (ie the sensing range)can detect the event and corresponding sensory data aregenerated The data size for recording each event variesfrom 10 to 100 B Events happen independently in both thespatial and temporal domains and sensor nodes initiate thedata collection requests when their buffers become full Weexplore the cases where the sensor node buffer size is 4 KBand 8KB respectively and record the interarrival time of datacollection requests for comparisonwith an exponential distri-bution with the samemean value (the Poisson arrival processindicates an exponentially distributed request interarrivaltime) Figure 6 indicates that the simulation results matchthe exponential distribution well verifying the assumptionon Poisson arrival Furthermore a larger node buffer resultsin a smaller request arrival rate because the sensor nodes canhold the on-board data longer

We further statistically verify the queuing models byvalidating the Poisson arrival of requests the independenceof request arrivals and the service time independenceKolmogorov-Smirnov (K-S) test with a significance level of5 is used to verify the Poisson arrival We perform thetests with a different number of sensor nodes (20ndash100) eachwith 50 trials We record the number of trials that rejectthe Poisson arrival hypothesis The verification results arelisted in the first row of Table 1 The low rejection ratioindicates that the Poisson arrival in ourmodeling is sound Toevaluate the independence of the request arrival and servicetime we record the request interarrival time and servicetime and calculate their 1-lag autocorrelations Again thesimulation is repeated for 50 times with 20 to 100 sensornodes respectively and the average absolute values of theautocorrelations are shown in the second and third rows ofTable 1 The small correlations of both the request arrival andtheir service time support our queue-based modeling

62 Single MA Next we evaluate our analytical results onthe single MA case Service time distribution is the core


Table 1 Verify the queuing model (with 50 trials) of nodes 20 40 60 80 100 of rejections 0 1 1 3 3Arrival corr 00463 00371 00236 00183 00167Service corr 00942 01023 00985 00967 01000

Analysis ( = 1)Simulation ( = 1)


0

001

002

003

PDF

50 100 1500Service Time (s)

Figure 7 Service time distribution

Analysis (휆 = 0012)Simulation (휆 = 0012)


0

02

04

06

08

1

CDF

500 1000 15000Response Time (s)

Figure 8 Response time distribution

component in the queue-based analysis which is obtainedbased on results from geometrical probability We evaluateour analysis on the service time distribution with an MAvelocity of 1ms and 2ms respectively and the results areshown in Figure 7 We can see that the analytical results andthe simulation match greatlyThe service time is significantlyreduced after increasing theMAvelocity from 1ms to 2msagreeing with (4)

The response time distributions with request arrival ratesof 0012 and 0018 are shown in Figure 8 Besides theaccuracy of the analysis we can see that the response timeof requests or the data collection latency in our focus issignificantly increased when increasing the request arrivalrateThis verifies the potential scalability issue when only oneMA is used for data collection

63 Multiple MAs We evaluate our modeling and analysisresults on the multiple MAs case in the followingWe explore

Analysis (c = 1)Simulation (c = 1)Analysis (c = 2)

Simulation (c = 2)Analysis (c = 3)Simulation (c = 3)

0

100

200

300

400

Aver

age Q

ueui

ng T

ime (

s)

0018 003 0042 00540006Request Arrival Rate (휆)

Figure 9 The average queuing time



1 2 3 4 50System Size

0

02

04

06

08

1

CDF

Figure 10 The system size distribution

the cases with 119888 = 2 and 119888 = 3 respectively and also presentthe results with 119888 = 1 for comparison

The approximation results on the expected queuing timeare verified in Figure 9 The effect of deploying more MAsis obvious especially when the request arrival rate is highNote that no results for 119888 = 1 or 119888 = 2 are shown when 120582is larger than 0018 or 0036 because the further increase of 120582will result in a 120588 greater than 1 and no steady-state measurescan be obtained

Figure 10 shows the evaluation results of the approxima-tion on the system size distribution with 120582 of 0018 Besidesthe accuracy of the approximation we can see that increasing119888 from 1 to 2 can greatly shorten the system size which in turnreduces the data collection latency However the benefit ofincreasing 119888 further from2 to 3 is quite limitedThis is becausethe system utilization factor is already small when 119888 = 2and thus further increasing 119888 cannot significantly improve thedata collection performance anymore

The results on the probability for requests to wait beforebeing served are shown in Figure 11 Intuitively the waitprobability increases when the system becomes more heavilyoccupied which results when (1) fewer MAs are adopted (119888decreases) and (2) the data intensity in the network is higher(120582 increases)The verification of this reasoning can be clearlyobserved from Figure 11




0

02

04

06

08

1

Prob

abili

ty to

Wai

t

0016 0026 0036 0046 00540006Request Arrival Rate (λ)

Figure 11 Waiting probability



0

02

04

06

08

1

CDF

40 80 120 1600Queuing Time (s)

Figure 12 Queuing time distribution



50 100 150 2000Response Time (s)

0

02

04

06

08

1

CDF


The queuing time and response time distributions with120582 of 0018 are shown in Figures 12 and 13 respectivelyBesides the analysis accuracy again we observe that furtherincreasing 119888 from 2 to 3 when 120582 = 0018 cannot significantlyreduce the queuing time (response time) agreeing withFigure 10

7 Practical Guidance

The constructed queuingmodels not only reveal insights intothe on-demand MDC but also guide its practical implemen-tation We use three examples to show how these models canassist the system implementation in this section

71 How to Adopt Multiple MAs In the first example weexplore the problem of how to employ multiple MAs forcollaborative data collection In general two strategies canbe usedmdashthe MAs can collaboratively collect data from theentire system referred to as Strategy-I or the system can bedivided into subareas and each MA is responsible to collectdata from one subarea which is referred to as Strategy-IIThese two strategies can be captured by multiserver systemswith shared (ie all MAs share the knowledge of datacollection requests as with Strategy-I) and separate (ie eachMA is only responsible for a subset of requests that fall in itsservice queue as with Strategy-II) queues respectively

Conventional wisdom says that all things being equal ashared queue outperforms separate queues most times [25]To the best of our knowledge however no results on thecomparison of the two strategies have been reported yetHere we close this gap based on the constructed1198721198661 and119872119866119888 queuing models Our results reveal that Strategy-IIoutperforms Strategy-I in bothMAsrsquo workloads and requestsrsquoresponse time contradicting with the conventional wisdom

Let us consider the case where 119888MAs are deployed in an119871 times 119871 sensing field For the ease of description we assumethat 119888 = 1198942 (119894 = 1 2 ) which can be relaxed as willbe explained later When Strategy-I is adopted the datacollection performance can be evaluated based on the resultsin Section 5 Specifically the utilization factor of individualMAs is

1205881 = 120582119888 E [S] (32)

and the expected data collection latency and its distributioncan be obtained according to (23) and (30) respectively

When Strategy-II is adopted that is the field is dividedinto 119888 subareas of size 119871radic119888 times 119871radic119888 each the data collectionperformance can be evaluated based on the results in Sec-tion 4 but in a smaller sensing field Denoting the requestsarrival rate at individual MA and the service time in this caseas 1205821015840 and S1015840 the utilization factor of individual MAs is

1205882 = 1205821015840E [S1015840] (33)

With randomly distributed sensor nodes it is clear that

1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)

Thus from (32) and (33) we have

12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)


Request Arrival Rate (휆)

Strategy-I (c = 4)Strategy-II (c = 4)

004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)

Figure 14 Average response time obtained with the two strategies

This indicates that Strategy-II achieves a lower workloadfor the MAs with reduced MA travel distance which domi-nates the service time in MDC

Because the comparison on the response time achievedwith the two strategies is not so obvious numerical com-parison is performed with a network scale of 100 sensornodes and 119888 = 4 The results are shown in Figure 14 wherethe aggregated request arrival rate at the MAs varies from002 to 005 Again Strategy-II reduces the response time by50 when compared with Strategy-I due to the fact that theservice time is dominated by the MAsrsquo travel time

Although we simplify the above description by assumingthat 119888 = 1198942 (119894 = 1 2 ) the conclusion on the advantageof Strategy-II holds in more general cases However dividingthe field into 119888 identical subareas of square shape may notbe always feasible in which cases the distance distributionsbetween random locations in other field shapes can be used[42]

72 When to Request Data Collection The time for sensorsto request data collection plays a critical role in the on-demand MDC Sending the request too early unnecessarilyincreases the workload of the MAs which also increases thedata collection latency of other requests On the other handa belated request leaves little time for the MAs to completethe data collection before the buffer of the requesting nodeoverflows

The response time distribution derived based on theconstructed queuing models helps identify the proper timeinstant to send out the data collection requests With amemory buffer 119861 for each sensor node and its respective datageneration rate 119891119894 denote 120579 (0 le 120579 le 1) as the remainingmemory buffer ratio when node sends the data collectionrequest This way the aggregated request arrival rate is

120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)

Buffer overflow would occur if the response time is largerthan 120579119861119891119894 With the derived response time distribution (ie(17) and (30)) the probability for buffer overflow to occur can

(1) 120579 = 0 Δ = 001(2) while 119901overflow gt 001 and 120579 lt 1 do(3) 120579 = 120579 + Δ120582 1(1 minus 120579) 119861

119899sum119894=1

119891119894(4) calculate 119877(119905) with 120582(5) 119901overflow = 1 minus int120579119861119891119894

0

119903(119905)119889119905(6) end while

Algorithm 1 Find the optimal remaining buffer ratio 120579

be calculated with given 120579 which in turn allows us to identifythe smallest 120579 (and thus the smallest workloads onMAs) thatguarantees a small enough buffer overflow probability (eg119901overflow lt 001) as illustrated in Algorithm 1

73 Requests Combination and Preemption The MAs canpotentially collect data from multiple sensor nodes at thesame location because of the wireless communication capa-bilities of both theMAs and sensor nodes which correspondsto the scenario of batch service in queuing theory and isreferred to as requests combination here

Clearly the probability for requests combination to occuris jointly determined by (i) the communication range 119877between the MAs and sensors (normalized to the field size)and (ii) the number of requests in the service pool when thenew request arrives that is the queue length Specifically fora new request arriving when 119871 requests are in the queue itcan be combined with at least one of these existing requestswith probability

119875combine (119877 119871) = 1 minus (1 minus int1198770119891D (119909) 119889119909)119871 (37)

where119891D(119909) is the distance distribution between two randomlocations as in (3) This indicates that the effect of requestscombination on improving the on-demand MDC will beprofound when the communication range is large or whenthe service queue is long

Figure 15 shows the effect of requests combination withvarying communication ranges As expected a larger com-munication range has a greater effect in reducing the datacollection latency when compared with the noncombinationcases Also the advantage of requests combination is lesssignificant when the number of MAs increases This isbecause the service pool size is reduced when more MAs areadopted for the data collection tasks and thus the probabilityfor combination to happen is reduced as well

Requests preemption is another potential way to improvethe on-demand MDCmdasha new request arrival may preemptthe service of existing requests if its requesting node is closeto the current locations of the MAs We have analyticallyexplored the possibility and advantage of requests preemp-tion in another work [43]


w combination (c = 1)wo combination (c = 1)


10 15 20 25 30 35 405Communication Range (m)

100

101

102

103

Aver

age R

espo

nse T

ime (

s)

Figure 15 Effect of requests combination with communicationrange

8 Further Discussions

81 System Stability Condition A necessary and sufficientcondition for the data collection process to be stable is120588119888 lt 1[12 22] This way from (1) we know that

120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)

which implies that the minimum requirement on the travelspeed of the MAs is

V gt E [D]119888119861119899sum119894=1

119891119894 (39)

From (39) we can see that to provide a stable data collectionperformance there is a clear trade-off between the numberof required MAs and their capabilities such as travel speed Vand memory size 119861 assisting us in determining the numberof needed MAs in practice

82 Insights for Sophisticated Discipline Design We establisha theoretical foundation on the on-demandMDCwhen FCFSis adopted which reveals insights into the design of moresophisticated service disciplines Through the queue-basedanalysis it is clear that the MAsrsquo travel distance betweentwo consecutively served requests is the dominant factorthat determines the data collection latency which should beminimized to achieve a better performance Inspired by thisin our recent work [11] we have extended these queuingmod-els to investigate the MDC with a greedy service disciplinethat minimizes the travel distance between two consecutivelyserved requests and significant asymptotic improvement canbe observed However the greedy discipline may cause someunfairness among sensor nodes which has to be addressedto guarantee the worst-case performance for every node Alsonote that Petri nets could be another analytical tool to capturesuch data collection process [44] which we will explore morein the future

9 Conclusions

In this paper we have analytically investigated the on-demand MDC in cyber-physical systems Two queuingmodels namely an 1198721198661 and an 119872119866119888 model havebeen constructed to capture the MDC with a single MAand multiple MAs respectively System measures of thequeues for example the expected values and distributionsof queue length queuing time and response time have beenexplored These queuing models shed light on the impactof different parameters on MDC and the correspondinganalytical results serve as guidelines in the design of moresophisticated data collection solutions The soundness of themodels and the accuracy of the analysis have been verified viaextensive simulations

Through the queue-based analysis it is clear that theMAsrsquotravel distance between two consecutively served requestsis the dominant factor that determines the data collectionlatency which should be minimized to achieve a betterperformance Inspired by this in our recentwork [11] we haveextended these queuing models to investigate the MDC witha greedy service discipline that minimizes the travel distancebetween two consecutively served requests and a significantimprovement can be observed asymptotically

Disclosure

A preliminary version of this work was published at IEEEGLOBECOMrsquo11 [45]

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The work reported in this paper was supported in part byCNS-1739577 National Key Research and Development Pro-gram (Grant 2016YFE0100600) NSFC (61672349) the Natu-ral Science Foundation of Jiangsu Province (no BK20151416)and NSERC

References

[1] B Chai J Chen Z Yang and Y Zhang ldquoDemand responsemanagement withmultiple utility companies A two-level gameapproachrdquo IEEE Transactions on Smart Grid vol 5 no 2 pp722ndash731 2014

[2] M Dong K Ota L T Yang S Chang H Zhu and ZZhou ldquoMobile agent-based energy-aware and user-centric datacollection inwireless sensor networksrdquoComputerNetworks vol74 pp 58ndash70 2014

[3] G Li M Dong K Ota J Wu J Li and T Ye ldquoTowards QoEnamed content-centric wireless multimedia sensor networkswith mobile sinksrdquo in Proceedings of the ICC 2017 - 2017 IEEEInternational Conference on Communications pp 1ndash6 ParisFrance May 2017


[4] G Xie K Ota M Dong F Pan and A Liu ldquoEnergy-efficientrouting for mobile data collectors in wireless sensor networkswith obstaclesrdquo Peer-to-Peer Networking and Applications vol10 no 3 pp 472ndash483 2017

[5] M Zhao and Y Yang ldquoOptimization-based distributed algo-rithms for mobile data gathering in wireless sensor networksrdquoIEEE Transactions on Mobile Computing vol 11 no 10 pp1464ndash1477 2012

[6] MZhao andYYang ldquoBounded relay hopmobile data gatheringin wireless sensor networksrdquo Institute of Electrical and Electron-ics Engineers Transactions on Computers vol 61 no 2 pp 265ndash277 2012

[7] Y Gu Y S Ji J Li F Ren and B Zhao ldquoEMS efficient mobilesink scheduling in wireless sensor networksrdquo Ad Hoc Networksvol 11 no 5 pp 1556ndash1570 2013

[8] ldquoNEPTUNE Canadardquo httpwwwneptunecanadaca[9] R Sugihara and R K Gupta ldquoOptimal speed control of mobile

node for data collection in sensor networksrdquo IEEE Transactionson Mobile Computing vol 9 no 1 pp 127ndash139 2010

[10] X Xu J Luo and Q Zhang ldquoDelay tolerant event collection insensor networks with mobile sinkrdquo in Proceedings of the IEEEINFOCOM March 2010

[11] L He Zhe Yang J Pan L Cai and J Xu ldquoEvaluatingservice disciplines for mobile elements in wireless ad hocsensor networksrdquo in Proceedings of the IEEE INFOCOM 2012- IEEE Conference on Computer Communications pp 576ndash584Orlando FL USA March 2012

[12] G D Celik and E H Modiano ldquoControlled mobility instochastic and dynamic wireless networksrdquo Queueing Systemsvol 72 no 3-4 pp 251ndash277 2012

[13] M Zhao Y Yang and C Wang ldquoMobile data gathering withload balanced clustering and dual data uploading in wirelesssensor networksrdquo IEEE Transactions on Mobile Computing vol14 no 4 pp 770ndash785 2015

[14] Y Gu Y Ji J Li and B Zhao ldquoESWC efficient scheduling forthe mobile sink in wireless sensor networks with delay con-straintrdquo IEEE Transactions on Parallel and Distributed Systemsvol 24 no 7 pp 1310ndash1320 2013

[15] Z Li Y Liu M Li J Wang and Z Cao ldquoUbiquitous datacollection for mobile users in wireless sensor networksrdquo IEEETransactions on Parallel and Distributed Systems vol 24 no 2pp 312ndash326 2013

[16] S Guo C Wang and Y Yang ldquoJoint mobile data gathering andenergy provisioning in wireless rechargeable sensor networksrdquoIEEE Transactions on Mobile Computing vol 13 no 12 pp2836ndash2852 2014

[17] L Xie Y Shi Y T Hou et al ldquoA Mobile Platform for WirelessCharging and Data Collection in Sensor Networksrdquo IEEEJournal on Selected Areas in Communications vol 33 no 8 pp1521ndash1533 2015

[18] D Jea A Somasundara andM Srivastava ldquoMultiple controlledmobile elements (data mules) for data collection in sensor net-worksrdquo in Proceedings of the 1st IEEE International Conferenceon Distributed Computing in Sensor Systems (DCOSS rsquo05) pp244ndash257 July 2005

[19] MMa and Y Yang ldquoData gathering in wireless sensor networkswith mobile collectorsrdquo in Proceedings of the Proceeding ofthe 22nd IEEE International Parallel and Distributed ProcessingSymposium (IPDPS rsquo08) pp 1ndash9 Miami Fla USA April 2008

[20] M Ma Y Yang and M Zhao ldquoTour planning for mobiledata-gathering mechanisms in wireless sensor networksrdquo IEEE

Transactions on Vehicular Technology vol 62 no 4 pp 1472ndash1483 2013

[21] Y Gu F Ren Y Ji and J Li ldquoThe evolution of sink mobilitymanagement in wireless sensor networks A surveyrdquo IEEECommunications Surveys amp Tutorials vol 18 no 1 pp 507ndash5242016

[22] E Altman and H Levy ldquoQueueing in spacerdquo Advances inApplied Probability vol 26 no 4 pp 1095ndash1116 1994

[23] D J Bertsimas and G V Ryzin ldquoA stochastic and dynamicvehicle routing problem in the Euclidean planerdquo OperationsResearch vol 39 no 4 pp 601ndash615 1991

[24] ldquoIntel Lab Datardquo httpwwwselectcscmuedudatalabapp3indexhtml

[25] DGross Fundamentals of QueueingTheory JohnWileyamp SonsNew Jersey 4th edition 2008

[26] LHe J P Pan and J D Xu ldquoA progressive approach to reducingdata collection latency in wireless sensor networks with mobileelementsrdquo IEEE Transactions on Mobile Computing vol 12 no7 pp 1308ndash1320 2013

[27] D R Cox and H D Miller The Theory of Stochastic ProcessesChapman and Hall London UK 1965

[28] GGrimmett andD Stirzaker Probability and randomprocessesOxford Science Publications The Clarendon Press OxfordUniversity Press New York 1982

[29] C Bettstetter H Hartenstein and X Perez-Costa ldquoStochasticproperties of the random waypoint mobility modelrdquo WirelessNetworks vol 10 no 5 pp 555ndash567 2004

[30] D Moltchanov ldquoDistance distributions in random networksrdquoAd Hoc Networks vol 10 no 6 pp 1146ndash1166 2012

[31] L E Miller ldquoDistribution of link distances in a wireless net-workrdquo Journal of research of the National Institute of Standardsand Technology vol 106 no 2 pp 401ndash412 2001

[32] B N Ma and J W Mark ldquoApproximation of the mean queuelength of anMGc queueing systemrdquo Operations Research vol43 no 1 pp 158ndash165 1995

[33] M J Sobel ldquoSimple inequalities for multiserver queuesrdquo Man-agement Science vol 26 no 9 pp 951ndash956 1980

[34] T Kimura ldquoApproximations for multi-server queues systeminterpolationsrdquo Queueing Systems vol 17 no 3-4 pp 347ndash3821994

[35] T Kimura ldquoA two-moment approximation for themeanwaitingtime in the GIGs queuerdquo Management Science vol 32 no 6pp 751ndash763 1986

[36] C Crommelin ldquoDelay probability formulaerdquo Post Office Electri-cal Engineers Journal vol 26 pp 266ndash274 1934

[37] G P Cosmetatos ldquoOn the Implementation of Pagersquos Approx-imation for Waiting Times in General Multi-Server QueuesrdquoJournal of the Operational Research Society vol 33 no 12 pp1158-1159 1982

[38] T Kimura A transform-free approximation for the queue-lengthdistribution in the finite capacity MGs queue vol 18 ofDiscussion Paper Series A Faculty of Economics HokkaidoUniversity 1993

[39] D Bertsimas and D Nakazato ldquoThe distributional Littlersquos lawand its applicationsrdquoOperations Research vol 43 no 2 pp 298ndash310 1995

[40] M H van Hoorn and H C Tijms ldquoApproximations for thewaiting time distribution of the MGc queuerdquo PerformanceEvaluation vol 2 no 1 pp 22ndash28 1982

[41] A Whitbrook httprobotsmobilerobotscom


[42] F Tong M Ahmadi and J Pan Random Distances Associatedwith Arbitrary Triangles A Systematic Approach between TwoRandom Points University of Victoria Victoria Canada 2013

[43] L He L Kong Y Gu J Pan and T Zhu ldquoEvaluating the On-Demand Mobile Charging in Wireless Sensor Networksrdquo IEEETransactions on Mobile Computing vol 14 no 9 pp 1861ndash18752015

[44] G Liu ldquoComplexity of the deadlock problem for Petri netsmodeling resource allocation systemsrdquo Information Sciencesvol 363 pp 190ndash197 2016

[45] Liang He Jianping Pan and Jingdong Xu ldquoAnalysis on DataCollection with Multiple Mobile Elements in Wireless SensorNetworksrdquo in Proceedings of the 2011 IEEE Global Communi-cations Conference (GLOBECOM 2011) pp 1ndash5 Houston TXUSA December 2011

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derived Furthermore we use three examples to show howthe analysis guides the on-demand MDC in practice (i) howto use multiple MAs (ii) When to request data collection(iii) How likely the requests combinationmdashthat is collectdata frommultiple sensor nodes at the same locationmdashwouldhappen via the wireless communication between the MAsand nodes The contributions of this paper include thefollowing

(i) Formulation of an1198721198661 queuing model to captureand analytically evaluate the on-demand MDC whena singleMA is deployed for data collection (Section 4)

(ii) An 119872119866119888 queuing model for the case when mul-tiple MAs are deployed based on which the datacollection performance is explored via approximation(Section 5)

(iii) Three examples to show how the analysis guides theon-demand MDC in practice (Section 7)

The rest of this paper is organized as followsThe literatureon MDC is briefed in Section 2 We formulate the problemin Section 3 The on-demand MDC with a single MA andmultipleMAs is investigated in Sections 4 and 5 and evaluatedin Section 6The practical guidance is presented in Section 7followed by further discussions in Section 8 The paperconcludes in Section 9

2 Related Work

Observing the advantages of MDC over traditional datacollection approaches (eg via direct communication ormultihop forwarding) much effort has been made to explorethe MDC with a single MA [7 11 12] For example Sugiharaand Gupta investigated the MDC with the objective ofminimizing MArsquos travel distance in [9] Zhao et al proposeda three-layer framework for the offline MDC including thesensor layer cluster head layer andMA layerTheMAcollectsdata according to a preoptimized tour with dual antennas[13] A unified framework for analyzing the MArsquos mobilityand data collection latency was presented and solutions tothe involved subproblems were proposed in [14] An MA-tracking protocol was proposed in [15] in which the routingstructure of data collection requests was additively updatedwith MArsquos movement The joint energy replenishment anddata collection was investigated in [16] with the consider-ation of various sources of energy consumption and time-varying nature of energy replenishment The colocation ofMA and wireless charger has been investigated in [17] withthe objective of ensuring sustainable and lossless systemoperation

Scalability is a critical bottleneck when only a single MAis used and a potential mitigation is to employ multiple MAsfor data collection An early investigation on the scenarioof multiple MAs is [18] where the MAs travel along fixedtracks to collect data from nodes with the considerationof load balancing A motion planning algorithm for theMAs was proposed in [19] which minimizes the number ofMAs according to the constraints in distance and time Thiswork was extended for applications with strict distancetime

times104

0

2

4

6

8

of

Rec

eive

d Pa

cket

s

11 21 31 41 51 541Node ID

Figure 1 Diverse data generation among sensor nodes (originaldata from [24])

constraints and a data-gathering algorithm with multipleMAs was proposed in [20] More detailed information onMDC is found in [21] and the references therein

Most of these existing efforts focus on the offline MDCwhile we tackle the on-demandMDC in this paper Althoughsimilar scenarios have been investigated in [12 22 23] ourqueue-based analytical framework provides detailed insightsinto the data collection process such as system size queuingtime and response time and these analytical insights guidethe MDC in practice

3 Preliminaries

31 On-Demand MDC In many offline MDC solutionsthe MAs periodically collect data from sensor nodes basedon preoptimized tours [9] These solutions perform wellwhen nodes generate data at similar paces but degradedramatically when the data generation at nodes variesmdashMAsmay visit nodes with little or no data to report Unfortunatelymany event-driven systems demonstrate such diverse datagenerations [10] For example Figure 1 plots the number ofpackets received from 54 sensor nodes in a trace providedby Intel Berkeley Research Lab [24] showing clear diversityamong nodes Targeting on these scenarios with diverse datagenerations we investigate the on-demand MDC where theMAs collect data based on real-time demands from sensornodes in this paper

32 Network Model We consider the scenario where con-trollable MAs collect data from stationary sensor nodesrandomly deployed in a square sensing field [9] (our modelformulation and analysis are also applicable to sensing fieldsof other shapes as will be explained in Section 4) Sensornodes monitor their surrounding environments store thegathered data in their buffers and send out data collectionrequests toMAswhen their buffers are to be full [10]The datacollection requests can be delivered to the MAs via existingMA-tracking protocols [15] Because the typical data relayspeed is much faster than theMAsrsquo travel we assume that thetime since a request is sent by a sensor node till it is receivedby MAs is short and negligible [12] Note that instead ofusing theseMA-tracking protocols to upload the sensory datadirectly which are normally ofmuch larger volume only data


requestarrival

service pool

mobile agent













120582 asymp 1119861119899sum119894=1

119891119894 (1)









119891D (119909) =


(3)



=

intV119905

02119909 (120587 minus 4119909 + 1199092) 119889119909 119905 le 1

V

int102119909 (120587 minus 4119909 + 1199092) 119889119909 + intV119905



V

1 119905 gt radic2V

(4)


E [S] = E [DV] = 1

VE [D]

V [S] = V [DV] = 1

V2V [D]


E [D] = Cov [D]

(5)




119883119899+1 = 119883119899 minus 1 + 119860119899+1 (119883119899 ge 1)119860119899+1 (119883119899 = 0) (6)


119901119894119895 = 119875 119883119899+1 = 119895 | 119883119899 = 119894 (7)


119896119894 = 119875 119860 = 119894 = intinfin0119875 119860 = 119894 | 119878 = 119905 times 119891S (119905) 119889119905

= intinfin0

119890minus120582119905 (120582119905)119894119894 119891S (119905) 119889119905(8)


t1

t2

t3

t4


0 1 i minus 1 i i + 1k0

k0 k0 k0

k1

k1

k1 k1 k1

k2k2

k3



P = 119901119894119895 =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(9)


120587P = 120587 (10)


120587119894 = 1205870119896119894 + 119894+1sum119895=1

120587119895119896119894minus119895+1 for 119894 = 0 1 2 (11)


Π (119911) = infinsum119894=0

120587119894119911119894 |119911| le 1

119870 (119911) = infinsum119894=0

119896119894119911119894 |119911| le 1(12)




119902 (119905 1198991015840) =

0 1198991015840 = 0119904 (119905) 1198991015840 = 1119902 (119905 1198991015840 minus 1) lowast 119904 (119905) 1198991015840 gt 1

(14)


Q (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119902 (119909 119894) 119889119909

(15)



119904 (119905) 1198991015840 = 0119902 (119905 1198991015840) lowast 119904 (119905) 1198991015840 gt 1 (16)


R (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119903 (119909 119894) 119889119909

(17)




requestarrival

service pool (FCFS)

mobile agents

requestdeparture








E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2


(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)


E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1




where


8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)


E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)


120572 = E [Q119872119866119888]E [Q119872119872119888] (24)



(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)




119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




0

02

04

06

08

1

CDF










0

001

002

003

PDF





0

02

04

06

08

1

CDF








0

100

200

300

400

Aver

age Q

ueui

ng T

ime (

s)






0

02

04

06

08

1

CDF









0

02

04

06

08

1

Prob

abili

ty to

Wai

t





0

02

04

06

08

1

CDF






0

02

04

06

08

1

CDF








1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



requestarrival

service pool

mobile agent













120582 asymp 1119861119899sum119894=1

119891119894 (1)









119891D (119909) =


(3)



=

intV119905

02119909 (120587 minus 4119909 + 1199092) 119889119909 119905 le 1

V

int102119909 (120587 minus 4119909 + 1199092) 119889119909 + intV119905



V

1 119905 gt radic2V

(4)


E [S] = E [DV] = 1

VE [D]

V [S] = V [DV] = 1

V2V [D]


E [D] = Cov [D]

(5)




119883119899+1 = 119883119899 minus 1 + 119860119899+1 (119883119899 ge 1)119860119899+1 (119883119899 = 0) (6)


119901119894119895 = 119875 119883119899+1 = 119895 | 119883119899 = 119894 (7)


119896119894 = 119875 119860 = 119894 = intinfin0119875 119860 = 119894 | 119878 = 119905 times 119891S (119905) 119889119905

= intinfin0

119890minus120582119905 (120582119905)119894119894 119891S (119905) 119889119905(8)


t1

t2

t3

t4


0 1 i minus 1 i i + 1k0

k0 k0 k0

k1

k1

k1 k1 k1

k2k2

k3



P = 119901119894119895 =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(9)


120587P = 120587 (10)


120587119894 = 1205870119896119894 + 119894+1sum119895=1

120587119895119896119894minus119895+1 for 119894 = 0 1 2 (11)


Π (119911) = infinsum119894=0

120587119894119911119894 |119911| le 1

119870 (119911) = infinsum119894=0

119896119894119911119894 |119911| le 1(12)




119902 (119905 1198991015840) =

0 1198991015840 = 0119904 (119905) 1198991015840 = 1119902 (119905 1198991015840 minus 1) lowast 119904 (119905) 1198991015840 gt 1

(14)


Q (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119902 (119909 119894) 119889119909

(15)



119904 (119905) 1198991015840 = 0119902 (119905 1198991015840) lowast 119904 (119905) 1198991015840 gt 1 (16)


R (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119903 (119909 119894) 119889119909

(17)




requestarrival

service pool (FCFS)

mobile agents

requestdeparture








E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2


(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)


E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1




where


8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)


E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)


120572 = E [Q119872119866119888]E [Q119872119872119888] (24)



(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)




119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




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1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






119891D (119909) =


(3)



=

intV119905

02119909 (120587 minus 4119909 + 1199092) 119889119909 119905 le 1

V

int102119909 (120587 minus 4119909 + 1199092) 119889119909 + intV119905



V

1 119905 gt radic2V

(4)


E [S] = E [DV] = 1

VE [D]

V [S] = V [DV] = 1

V2V [D]


E [D] = Cov [D]

(5)




119883119899+1 = 119883119899 minus 1 + 119860119899+1 (119883119899 ge 1)119860119899+1 (119883119899 = 0) (6)


119901119894119895 = 119875 119883119899+1 = 119895 | 119883119899 = 119894 (7)


119896119894 = 119875 119860 = 119894 = intinfin0119875 119860 = 119894 | 119878 = 119905 times 119891S (119905) 119889119905

= intinfin0

119890minus120582119905 (120582119905)119894119894 119891S (119905) 119889119905(8)


t1

t2

t3

t4


0 1 i minus 1 i i + 1k0

k0 k0 k0

k1

k1

k1 k1 k1

k2k2

k3



P = 119901119894119895 =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(9)


120587P = 120587 (10)


120587119894 = 1205870119896119894 + 119894+1sum119895=1

120587119895119896119894minus119895+1 for 119894 = 0 1 2 (11)


Π (119911) = infinsum119894=0

120587119894119911119894 |119911| le 1

119870 (119911) = infinsum119894=0

119896119894119911119894 |119911| le 1(12)




119902 (119905 1198991015840) =

0 1198991015840 = 0119904 (119905) 1198991015840 = 1119902 (119905 1198991015840 minus 1) lowast 119904 (119905) 1198991015840 gt 1

(14)


Q (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119902 (119909 119894) 119889119909

(15)



119904 (119905) 1198991015840 = 0119902 (119905 1198991015840) lowast 119904 (119905) 1198991015840 gt 1 (16)


R (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119903 (119909 119894) 119889119909

(17)




requestarrival

service pool (FCFS)

mobile agents

requestdeparture








E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2


(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)


E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1




where


8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)


E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)


120572 = E [Q119872119866119888]E [Q119872119872119888] (24)



(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)




119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




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0

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1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



t1

t2

t3

t4


0 1 i minus 1 i i + 1k0

k0 k0 k0

k1

k1

k1 k1 k1

k2k2

k3



P = 119901119894119895 =[[[[[[[[[[[[[


]]]]]]]]]]]]]

(9)


120587P = 120587 (10)


120587119894 = 1205870119896119894 + 119894+1sum119895=1

120587119895119896119894minus119895+1 for 119894 = 0 1 2 (11)


Π (119911) = infinsum119894=0

120587119894119911119894 |119911| le 1

119870 (119911) = infinsum119894=0

119896119894119911119894 |119911| le 1(12)




119902 (119905 1198991015840) =

0 1198991015840 = 0119904 (119905) 1198991015840 = 1119902 (119905 1198991015840 minus 1) lowast 119904 (119905) 1198991015840 gt 1

(14)


Q (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119902 (119909 119894) 119889119909

(15)



119904 (119905) 1198991015840 = 0119902 (119905 1198991015840) lowast 119904 (119905) 1198991015840 gt 1 (16)


R (119905) = infinsum119894=0


= infinsum119894=0

120587119894 int1199050119903 (119909 119894) 119889119909

(17)




requestarrival

service pool (FCFS)

mobile agents

requestdeparture








E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2


(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)


E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1




where


8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)


E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)


120572 = E [Q119872119866119888]E [Q119872119872119888] (24)



(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)




119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




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1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



requestarrival

service pool (FCFS)

mobile agents

requestdeparture








E [Q119872119872119888] = (119888120588)119888119888119888120583 (1 minus 120588)2


(119888120588)119894119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

(19)


E [Q119872119863119888] = 1120583infinsum119894=1

infinsum119895=119894119888+1




where


8 (1 + 120579) 119892 (120588) = 1120588 minus 1

(22)


E [R119872119866119888] = E [Q119872119866119888] + E [S] (23)


120572 = E [Q119872119866119888]E [Q119872119872119888] (24)



(25)

where

120577 = 1205881205721 minus 120588 + 120588120572

P0119872119872119888 = [119888minus1sum119894=0

(119888120588)119888119894 + (119888120588)119888

119888 (1 minus 120588)]minus1

U119872119872119888 = (119888120588119888)119888 1198750119872119872119888119888 (1 minus 120588119888)

(26)




119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




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1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119864 [L] = infinsum119894=0

119894 sdotP119894119872119866119888 (27)


U119872119866119888 = 1 minus 119888minus1sum119894=0

P119894119872119866119888 (28)




119903119872119866119888 (119905) = 119902119872119866119888 (119905) lowast 119904 (119905) (30)





(31)




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0

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1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






0

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0

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0

02

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1

CDF








1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0

02

04

06

08

1

Prob

abili

ty to

Wai

t





0

02

04

06

08

1

CDF






0

02

04

06

08

1

CDF








1205881 = 120582119888 E [S] (32)



1205882 = 1205821015840E [S1015840] (33)


1205821015840 = 120582119888 E [S1015840] = E [S]radic119888

(34)


12058811205882 =E [S]E [S1015840] = radic119888 gt 1 (35)




004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





004 0050030020

20

40

60

80

Aver

age R

espo

nse T

ime (

s)







120582 asymp 1(1 minus 120579) 119861119899sum119894=1

119891119894 (36)



119899sum119894=1


0

119903(119905)119889119905(6) end while













100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






100

101

102

103

Aver

age R

espo

nse T

ime (

s)




120588119888 = 120582E [S]119888 lt E [D]V119888119861

119899sum119894=1

119891119894 lt 1 (38)


V gt E [D]119888119861119899sum119894=1

119891119894 (39)



9 Conclusions



Disclosure




Acknowledgments


References



















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

















































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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