Computing Resource Allocation in Three-Tier IoT Fog Networks: a Joint Optimization Approach Combining Stackelberg Game and Matching

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by  Huaqing Zhang

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Computing Resource Allocation in Three-Tier IoT
Fog Networks: a Joint Optimization Approach
Combining Stackelberg Game and Matching
Huaqing Zhang∗, Yong Xiaok, Shengrong Bu†, Dusit Niyato§, Richard Yu‡, and Zhu Han∗
∗Department of Electrical and Computer Engineering, University of Houston, Houston, TX, USA.
kDepartment of Electrical and Computer Engineering at the University of Arizona, Tucson, Arizona, USA.
†School of Engineering, University of Glasgow, UK.
7
§School of Computer Science and Engineering, Nanyang Technological University, Singapore.
1
0 ‡Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada.
2

n
a
J Abstract—Fog computing is a promising architecture to pro- ControlSignal
4 vide economic and low latency data services for future Internet (cid:258)(cid:258) DataStream
1 of things (IoT)-based network systems. It relies on a set of low-
power fog nodes that are close to the end users to offload the DSOd1 DSOd2 DSOdm Virtualized
] services originally targeting at cloud data centers. In this paper, Network
T we consider a specific fog computingnetwork consistingof a set
G of data service operators (DSOs) each of which controls a set of
fog nodes to provide the required data service to a set of data
s. servicesubscribers(DSSs).Howtoallocatethelimitedcomputing (cid:258)(cid:258)
[c resourcesoffognodes(FNs)toalltheDSSstoachieveanoptimal FNf1 FNf2 FNf3 FNfk
and stable performance is an important problem. In this paper,

we propose a joint optimization framework for all FNs, DSOs
1
and DSSs to achieve the optimal resource allocation schemes in
v
2 a distributed fashion. In the framework, we first formulate a (cid:258)(cid:258) Physical
Stackelberg game to analyze the pricing problem for the DSOs Netwotk
2 as well as the resource allocation problem for the DSSs. Under DSSs1 DSSs2 DSSs3 DSSsN
9
3 the scenarios that the DSOs can know the expected amount of Fig. 1: System architecture
resourcepurchasedbytheDSSs,amany-to-manymatchinggame
0
is applied to investigate the pairing problem between DSOs and
.
1 FNs. Finally, within the same DSO, we apply another layer of
0 many-to-many matching between each of the paired FNs and whichcan be intolerablefor theIoT applicationsthatrequires
7 serving DSSs to solve the FN-DSS pairing problem. Simulation
real-time interaction or mobility. In order to overcome these
1 results show that our proposed framework can significantly
: improve the performance of the IoT-based network systems. challenges,fogcomputingisproposedasapromisingsolution.
v
In fog computing, multiple low-power computing devices,
Xi IndexTerms—Fogcomputing,Stackelberggame,match- commonly referred to as the fog nodes (FN), at the edge
ing theory, Internet of things.
r of the networks are deployed to offload the data computing
a
servicesfromthecloud.With the propertyofsmall-scale,low
I. INTRODUCTION construction cost and mobility support, the FNs are generally
With the rapid developmentof Internet of things (IoT), the deployed much closer to the DSSs and therefore can provide
numberofconnecteddeviceshasincreasedataunprecedented low-latency fast-response and location-awareness service [2].
speed[1].Itisknownthatanalyzingthebigdatageneratedby Fog computing networks can consist of a large number
allkindsofIoTdevicesrequiresa largeamountofcomputing of FNs deployed by different DSOs at different locations to
resources.In orderto meetthe demandof the data computing provide various data services and applications to the DSSs.
services, a large number of large-scale data centers has been When DSSs can choose their DSOs as well as the corre-
deployed. In addition, cloud computing has been proposed sponding FNs to further enhance their quality-of-experience,
recently to provide flexible and efficient services to the data how to allocate the limited computingresourcesof all the fog
service subscribers (DSSs). In cloud computing, the data nodes to the DSSs is still an open problem. In this paper,
service operator (DSO) is able to organize a shared pool of we further extend our previous work [3] and focus on the
configurable computing resources (such as servers, storage, resource selection and allocation problem between the FNs,
networks, applications and services), which can be easily DSOs and DSSs. We propose a joint optimization framework
accessed by DSSs on demands. for all FNs, DSOs and DSSs in a distributed fashion. In the
Generally speaking, large-scale data centers or clouds are framework,wefirstformulateaStackelberggametomodelthe
generally built in remote areas which are far from the DSSs. interactionbetweenDSOsandDSSs,wheretheDSOssettheir
This results in high transmission cost and service latency service price first, and the DSSs purchasethe optimalamount
2
TABLE I: List of Notations
of computing resource blocks (CRBs) correspondingly. Once
the prices of DSOs and the purchased resources of DSSs Symbol Definition
havebeenobtained,eachDSScanknowhowmanyCRBs are M TotalnumberofDSOs
N TotalnumberofDSSs
required and can then try to compete for the CRBs owned by
K TotalnumberofFNs
the nearby FNs. Thus, we propose a many-to-manymatching Ψ ThesetofDSOs
game to investigate the interaction between DSOs and FNs Υ ThesetofDSSs
Ω ThesetofFNs
where each DSO has a set of CRBs to offload, and each FN
µ ServicerateofCRBs
hasmanyvacantCRBstosell.AfteralltheDSOsdecidetheir λj Workloadarrival ratefortheDSSsj
DSO-FN pairs, FNs will compete with each other to allocate ri Priceofunitvirtualized CRBsetbytheDCOoi
Ls Thepreference listofDSSsoverDSOs
their CRBs to the DSSs of the DSOs. We also adopt another j
layer of many-to-manymatching framework to solve the FN- tj TotalcostduetothedelayofDSSsj
hj Cost due to network delay from the physical ser-
DSSpairingproblemwithinthesameDSO.Simulationresults viceprovidertoDSSsj
show that our proposed framework can significantly improve oj Costduetoqueuingdelayattheservers
the performance of the fog computing networks. qj TotalamountofCRBspurchasedbytheDSSsj
lkj Distancebetween theFNfk andtheDSSsj
The rest of this paper is organized as follows. We describe Wjs Utility functionofDSSsj
the system modelin Section II and formulatethe problemsin Wid Utility functionofDSOdi
Section III. Based on the formulatedproblem,we analyze the Wkf Utility functionofFNfk
system with the proposed framework in Section IV, where τij ThebooleanvariabledeterminingwhethertheDSO
di servesDSSsj ornot.
the interaction between DSOs and DSSs is considered in αj,βj,γj Weightfactors intheutility functionofDSSsj
Section IV-A, the interaction between FNs and DSOs is tth Themaximumtolerance ofservice delay forDSS
analyzed in Section IV-B, and the interaction between FNs sj
ckj Transmission cost for unit CRB from FN fk to
and DSSs is discussed in Section IV-C. Finally, we evaluate DSSsj
theperformanceofourworkinSectionV,showrelatedworks ei Increment of the energy cost in the massive data
in Section VI and summarize the paper in Section VII. centerforDSOdi
ηkfi NormalizedpreferencefromtheFNfktotheDSO
di
II. SYSTEMMODEL rptkh RUepnpterofbouunnitdCoRfBtotsaeltdbeylatyhecoFsNt fk
Consider a fog computing network where each DSS can qkfjs AmountofCRBsallocated fromtheFNfk tothe
DSSsj
submit its data computing service to a set of neighboring qifkd AmountofCRBsallocated fromtheFNfk tothe
FNs deployed by a set of DSOs as illustrated in Figure 1. DSOdi
Accordingly, we consider a three-tier fog network, where Ldif Preference listofDSOdi onallFNs
the DSOs locate in the middle layer, managing the FNs Lsjf Preference listofDSSsj onallFNs
in the upper layer and serving DSSs in the bottom layer. Lfks Preference listofFNfk onallDSSs
Without loss of generality, we assume there are M DSOs,
labeled as Ψ = {d1,d2,...,dM} and N DSSs, denoted
Assume the DSSs apply real-time interactive applications,
as Υ = {s1,s2,...,sN}. Let λj be the workload arrival
whereQoSismeasuredbytheservicedelay.Inthispaper,for
rate of DSS s , ∀s ∈ Υ. We assume each DSS has a
j j
normalized preference list, denoted as Ls over all DSOs. DSS sj, ∀sj ∈ Υ, we measure the cost of the service delay
j
as
Moreover, K FNs, labeled as Ω = {f1,f2,...,fK}, locate
in the area of consideration. We define the unit amount of
computing resources that can be distributed by each FN as tj =hj +oj, (1)
the “computing resource block (CRB)” [3], each of which
which consists of the cost caused by the queuing delay o at
j
can provide computing service at the rate of µ. The physical
the servers as well as the cost caused by the network delay
data transmission network between FNs and DSSs satisfies
h from the sensors to the physicalservice providerand from
j
the SecondNet topology [4], where the network facilities can
the physical service provider to DSS s .
j
providethe guaranteedquality-of-service(QoS)forthe DSSs.
According to the queuing delay model in [5], [7], which
Accordingly, in order to reduce the risk of potential network
can be easily extended to other models, the cost of queuing
congestion and achieve real-time fast-response interaction,
delay when serving DSS s is
j
each DSO tries to offload the data services submitted by the
DSSstothelarge-scaledatacenterstothelocalFNs.However, λ
o = j , (2)
as the DSSs cannot have the authorization to access the j µ− λj
CRBsdirectly,theDSSsarerequiredtoreceivethevirtualized qj
services from the DSOs, and with the management of DSOs, where q is the amount of CRBs purchased by DSS s .
j j
theCRBs ofthe FNscanfinallybeallocatedtotheDSSs. We Moreover, as the network delay is related to the transmission
assume that different DSOs offer data services with different distance, traffic condition in the network and many other
requirements. Based on the preference list Ls, the DSS s , unpredictedfactors,inrealsituations,wesupposethenetwork
j j
∀s ∈ Υ is required to subscribe to at most one DSO. The delay can be evaluated from training data periodically sent
j
network architecture is illustrated in Fig. 1. from sensors to the physical service provider and from the
3
physical service provider to the DSS. In this paper, we
set the distance between the farthest sensor to the physical
serviceproviderplusthedistancebetweenthephysicalservice
provider (e.g., FN f ) to DSS s is denoted as l . For
k j kj
simplicity, we assume the network resourcefrom the physical FNs
service providerto DSS sj is sufficient, and the cost incurred Many-to-Many
bythenetworkdelayh generallyfollowsalinearfunctionof Matching
j
the distance from the sensor to the physical service provider
DSOs
plusthedistancefromthephysicalserviceprovidertotheDSS
s , i.e., h =θl , where θ is the scalar.
j j kj
As the DSSs in the network pay DSOs for the service,
following the structure of [3], the utility of DSS s , ∀j ∈ Υ,
j
can be denoted as the revenue received from the workload
Stackelberg
data minus both the cost of service delay and payment to the Games
DSOs as follows:
DSS DSS DSS
M
Ws = τ (α λ −β q r −γ t ), (3)
j ij j j j j i j j Many-to-Many
i=1 Matching
X Ajointoptimizationframework
where r is the price set by DSS d for each unit of the
i i
virtualized CRB. αj, βj, and γj are weight factors. τij is Fig. 2: A joint optimization framework
thebooleanvariabledeterminingwhetherDSO d servesDSS
i
s or not. If τ = 1, DSS s is served by the DSO d .
j ij j i
Otherwise, DSS s prefers to be served by other DSOs. The
j
value of τ follows the preference list Ls of DSS s , and K
ij j j Wf = ηf (p −c )qfs, (5)
M k ki k kj kj
sinceeachDSScanatmostchooseoneDSS,i.e., τij =1, k=1
i=1 X
∀j ∈ Υ. For each DSS, assume there is an upPper bound where qkfjs is the amount of CRBs allocated from FN fk to
tth for the service delay. When the service delay is larger DSS sj.
than the threshold, the DSS will regard it as an unsuccessful
connection. Corresponding, we set qth as the lower bound
j III. PROBLEMFORMULATION
of CRBs required for DSS s to guarantee the service delay
j
within the acceptable boundaries. Accordingtothemodeledarchitectureoffognetwork,with
Based on the amount of virtualized CRB purchased by tradings between FNs and DSOs and between DSOs and
servingDSSs,theutilityofeachDSOistherevenuesreceived DSSs, it is impossible to reach the maximum utilities for
from the DSSs minus the payment to the facilities that are all FNs, MDCOs and DSSs simultaneously. Accordingly, we
able to provide the physical CRBs. Each DSO prefers to consider a sequential decision making process. During the
offload its services to the FNs nearby. However, if there are process, the DSOs first predict the total amount of CRBs
not sufficient amount of CRBs from the FNs which can meet purchased from their servings DSSs and set their service
the requirements of all DSSs. Some DSSs will be served by prices to their DSSs based on the prediction. Observing the
the massive data centers, which are located far away from the behaviors of the DSOs, each DSS determines the optimal
DSSs. We suppose the increment of the energy cost in the amount of CRBs to purchase to achieve the maximum utility.
massive data center is e . Therefore, for the DSO d , if qfd Furthermore,knowingthe totalamountof requiredCRBs, the
i i ik
amountofCRBsareoffloadedtotheFNf ,andqo amountof DSOs try to allocate CRBs to the FNs in the neighborhood.
k i
CRBs are still served by the massive data centers, the utility Finally, all FNs selected by the same DSO competes for the
function of DSO d , ∀i∈Ψ, can be denoted as DSSs.
i
We can write the optimization problem can be formulated
N K as follows.
Wd = τ (r q )− p qfd−e qo. (4)
i ij i j k ik i i
Xj=1 kX=1 mqajx Wjs(qj|r), ∀j ∈Υ,
p is the price set by the FN f , which is determined by the
k k M
cost and current traffic of FN fk. τij =1,
ForFNfk inthenetwork,theutilityisthepaymentreceived  i=1 (6)
farsomthethteranDsSmOisssiomnincuosstthfeortreaancshmiusnsiiotnCRcoBs,t.wWhiechsehtasckaj s.t. qτtPjij≤≥αjt0λt,hj,≤τij(βjqjri+γjtj),
linear relationship with the distance l . Moreover, we set j
kj
cηokfinsaisdethriengnotrhmeaplirzeefedrepnrecfeerteondceiffteorethnet DDSSOOsd,it.hAecdcioscrdoiunngtleyd, where r is thepricτiinjg∈p{ro0fi,1le},of a∀lliD∈SΨO,s observed by DSS
utility function for each DSO is s .
j
4
PredictingthebehaviorsoftheDSSs, eachDSO isrequired toleranceandsettingpricesofDSOs,theoptimalamount
tosettheservicepriceto eachservingDSSandcompetewith of CRBs should be determined for high utilities.
other DSOs to choose FNs in the neighborhood. Thus, the 2) PricingproblemfortheDSOs:Inthedataservicewith
formulated problem of the DSO is fog, the DSOs are requiredto providevirtualized CRBs
to the DSSs and try to rent the CRBs from the FNs to
max Wd(r |q∗,p,r∗ ), ∀i∈Ψ, serve DSSs in the physical network. Therefore, how to
i i −i
ri do the pricing for the DSOs is a problem. Considering
N K
τ (r q )≥ p qfd+e qo, (7) the announcedrent from all FNs, the DSOs need to set
s.t. j=1 ij i j k=1 k ik i i apricewhichcanbringprofitsforthemselves.However,
 rPi ≥0, P if the price is set too high, the serving DSSs will not
where q∗ denotes the optimal amount of CRBs purchased by purchasea largeamountofCRBs. Therefore,predicting
allDSSs. p istheprofileofrentforallFNs. r∗ istheprofile the reactions of CRBs and observing the rent of FNs,
−i the DSOs are required to determine its service price so
of optimal service prices set by other DSOs.
as to receive the maximum revenues.
Moreover, each FN in the network would like to choose
3) DSO-FN pairing problem: As FNs may be private
its preferred DSOs and serve its DSSs with low distance.
computingdevices, which are small-scale and unableto
Thus, competing with other FNs, it is required to determine
communicatewithDSSsdirectly,theFNsareaccessible
the amountof CRBs allocated to DSOs and its serving DSSs,
totheDSOsonly.Inthemulti-DSOscenario,astheFNs
respectively. The optimization problem is denoted as
are accessible to all DSOs, it is a problem for all DSOs
in the network to pair FNs distributedly so as to serve
max Wkf(qifkd,qkfjs|q∗,qfi−d∗k,qf−sk∗j,p−k), ∀k ∈Ω, theirDSSswithlowlatency.Withdifferentrelationsand
qfd,qfs trading history, each FN has different preference on all
ik kj
M DSOs. Observing the rent of all FNs, each DSO also
qifkd ≤qkf−th, hasa preferencesoneach FN. Based on thepreferences
s.t. jiPP=NK=11qqkfifkjsd≤≤qqkfid−−tthh,, ∀i∈Ψ, 4) oDsFwfNSiaO-tlcDl-hFSDNSitSsOppcaasui’irrrirainenngngdtrpeFpsaNruiorslibt’ns,l,geiwmtrehi:sseuArrleetfqtafeuonriryrteahDdehStipoOgahrioereirrancuFghtNiblaietiystsw.taaebbellnee
k=1 DSOs and FNs, each FN has allocated its CRBs to all
whereqfkdPPK=1iqskftjshe≤opqjsti−mthal,amo∀ujn∈toΥfC, RBsrenttoDSOd (f8o)r DfdoiSsrtOathnse.ceHFNobwestewtvoeeear,nllwoeciatahctehintFhoNenieraCDnRdSBOes,acittohiasDllsStDSillSiSassp.vrAaorbsiolteuhmse,
i−k i with a longer transmission distance, the FNs have to
all other FNs, qfs∗ is the optimal amount of CRBs allocated
−kj pay more on the transmission cost. Thus, based on the
to DSS s for all other FNs. During the service, the total
j transmission distance, each FN has a preference over
CRBs distributed to all DSOs or all DSSs cannot exceed its
all DSSs. Moreover, each DSS also have preference
total available CRBs qf−th for FN f . Furthermore, the total
k k over FNs based on the rent. Therefore, following the
CRBs purchased from DSO d or DSS s should not exceed
i j preference of all FNs’ and DSSs’, a stable FN-DSS
its demand qd−th or qs−th, respectively.
i j pairing result should be achieved.
In summary, following the relationships among all FNs,
According to the formulated problems, all FNs, DSOs and
DSOs and DSSs, we focus on the following problems:
DSSs are rational and autonomous individuals, who observe
1) Resource purchasing problem for the DSSs: In the the behaviors of others and make decisions to improve their
network, as the DSSs can only access to the DSOs in own utilities. Therefore, in order to reach the optimal and
a virtualized fashion, they are required to purchase the stable solutions for all FNs, DSOs and DSSs, we model a
optimalamountofCRBs fromtheDSOs. Followingthe three-stage joint optimization framework, as shown in Fig.
systemmodel,asdifferentDSSshavedifferenttolerance 2. In the framework, we first model the Stackelberg games
of servicedelay,whenthe upperboundof servicedelay to solve the pricing problem for the DSOs and resource
is high, the DSSs are able to purchase a small amount purchasing problem for the DSSs. When the DSOs know the
of CRBs to achieve satisfying services. However, when expectedamountofresourcepurchasedbythe DSSs, a many-
the upper bound of service delay is low, the DSSs have to-manymatchingisproposedbetweentheDSOsandtheFNs
to purchase a large amount of CRBs to guarantee the to deal with the DSO-FN pairing problem. Finally, within
service delay is within the tolerated region. Moreover, the same DSO, we apply another many-to-many matching
the service priceset by the DSOs also affectsthe utility between its paired FNs and serving the DSSs to solve the
of DSSs. When the price is in high value, even though FN-DSS pairing problem.
thelargeamountofCRBs isabletoimprovethequality
ofdataservices,theDSSshavetomakealargepayment IV. SYSTEMANALYSIS
to the DSOs for their services. The revenue may not In this section, we analyze the optimal strategies for FNs,
be satisfying. Therefore, considering both the delay DSOs and DSSs. Based on the analysis of the formulated
5
framework, in the following sub-sections, we first investigate
the interactions between the DSOs and DSSs to determine N K
λ β λ β
how many CRBs are required during the service. Given the Wd = τ ( j jr + j jr )− p qfd−e qo.
optimized behaviors of the DSOs and DSSs, we analyze the i j=1 ij µ sγj i µ sγj i k=1 k ik i i
X X
interactions between the FNs and DSOs based on different f (14)
preferences. Finally, with the obtained results, we discuss the In the formulated problem (13), we take the first derivative
interactionsbetween the FNs andDSSs within the same DSO of Wid with respective to ri and discover it is a monotonous
for better services. increasing function with respective to ri. Furthermore, as the
servfice delay cannot surpass tth for the DSSs. The CRB
purchased by DSSs has the following low bound
A. The Interaction between DSOs and DSSs
λ t
In the virtualized network, the DSOs provides CRBs for q ≥ j th . (15)
j
the DSSs. Following the formulated problems for both DSOs µtth−λj
andDSSs,thereisaStackelberggame,wheretheDSOsactas Thus, followingthe relation in (9), the maximumand optimal
leadersandDSSsactasfollowers.Inthegame,whenallDSSs price for the DSO d to DSS s is, ∀i∈Ψ, ∀j ∈Υ,
i j
choose their serving DSOs with their preferences, the DSO
2
first sets the service price. Then, based on the price all DSSs γ µt −λ
r = j th j . (16)
determineoptimalamountof CRBs to purchase.Accordingly, i β λ
j(cid:18) j (cid:19)
considering the optimization problem of DSSs, we have the
following lemma. B. The Interaction between FNs and DSOs
Lemma 1. InthemodeledStackelberggamebetweenDSOd Accordingto the predictedamountofCRBs orderedbythe
i
andDSSs ,whentheDSOannouncesitsservicepricer ,the DSSs, the DSOs try to offload the services from the massive
j i
optimal amount of CRBs q purchased by the DSS is data centers to the DSSs nearby.Observing the service prices
j
set by all FNs, DSO d , ∀i ∈ Ψ, has a preference list Ldf =
i i
λ λ
qj∗ = µ rjiβγjj + µj. (9) phLredif1fe,r.t.o.,cLhodiKfosieotnhealFlNFNs swiinththaelonweigphrbicoer,howoed.seAtstheDSOs
q
Proof: Accordingto the utility functionof DSS sj 3, the Ldf =−p , ‘∀i∈Ψ, ∀k ∈Ω. (17)
second derivative of Ws with respect to q is ik k
j j
Furthermore, each FN also has different preferences over
∂2Wjs =− 2λ2jµ . (10) DSOs. Thus, we set the preference list of FN fk, ∀k ∈ Ω,
∂qj2 (µqj −λj)3 on all DSOs as Lfkd = Lf1kd,...,LdMfk , satisfying,
As ∂2Wjs < 0, Ws is a quasi-concave function with respect Lfd =ηf , h ‘∀i∈Ψi, ∀k ∈Ω. (18)
∂q2 j ik ki
to q . Fjurthermore, the first derivative of Ws with respect to
j j Consideringthepreferencelistsof FNsandDSOs, i.e., Ldf
q is ik
j and Lfd, respectively, we design a many-to-many matching
ik
∂Ws λ 2 β algorithm for the DSO-FN pairing problem. As shown in
j = j −r j. (11) Algorithm 1, after the preference lists are constructed, we set
i
∂q µb −λ γ
j (cid:18) j j(cid:19) j apointerforeachFNinitspreferencelist.Initially,wesetthe
We setthefirstderivativeequaltozeroandobtaintheoptimal pointer at the first DSO in the list. During the each round of
amount of CRBs to purchase so as to achieve the maximum matching, each FN first propose to the DSO in the pointer of
utility, i.e, the preferencelist. Based on behaviorsof the FNs, each DSO
chooses its most preferred FNs in its preference list until the
λ λ
q∗ = j + j. (12) requiredCRBs are all supplied by the FNs. If the FNs supply
j µ r βj µ more CRBs than the DSO requires, the DSO will reject the
iγj CRBs from less favourite FNs. If the FNs supplies less CRBs
q
than the DSO requires, the DSO will choose massive data
Therefore,consideringthereactionsof theDSSs, we adjust centers for the rest of the services. At the end of each round,
the optimization problemfor the DSO di, ∀i∈Ψ, as follows. if all of the CRBs from the FN have been allocated to the
DSOs,thepointersoftheFNwillkeepunchanged.Otherwise,
max Wd(r |q∗,p,r∗ ), ∀i∈Ψ, the pointers of the rejected FNs will move to the next DSO
i i −i
ri in the preference list. In the next round, the FNs which still
N K haveavailableCRBs willproposeto thenewDSOs according
f τ (λj βjr + λj βjr )≥ p qfd+e qo,
s.t. rjP=i1≥i0j, µ qγj i µqγj i kP=1 k ik i i tboyththeeirDpSoiOnteinrs.thSepelcaisfitcraolluyn,di,fbthuet CdiRscBasrdoefdFiNn tahree ccuhrorseennt
(13) round, we suppose the pointer of the FN doesn’t change its
where position, considering the pointed DSO in the current round
6
Algorithm 1 Many-to-Many Matching Algorithm for DSO- whichismorepreferredthanFNf .Inthefuturerounds,when
k
FN pairing problem. there are new FNs proposing to DSO d , if the DSO would
i
like to change its current accepted FNs, the DSO can only
1: for FN fk do choose the FNs that is evenbetter than the FNs in the current
2: ConstructthepreferencelistLfdonallDSOsaccording accepted list. Therefore, for FN f which has been rejected
k k
to (18); once from DSO d , it is impossible for it to be accepted by
i
3: One pointer is set as the indicator pointing at the first the same DSO in the future rounds.
DSO in the preference list.
Lemma 3. Following the Algorithm 2, the DSO-FN pairing
4: end for
problem will ultimately converge and achieve a stable match-
5: for DSO di do
6: Construct the preference list Ldf on all FNs according ing result.
i
to (17); Proof:AsprovedintheLemma2,thepointeroftheFNs
7: end for can only move in one direction. Therefore, in the perspective
8: We set flagk, ∀k ∈ Ω, as the indicator to show if the of the FNs, when the pointer of each FN has moved to the
CRBsofFNfk werechosenbytheDSOinthelastround, end of the preference list, the FN has distributed its available
but discarded in the current round. Initially, flagk =1; CRBstotheDSOsinanoptimizedway.Inotherwords,theFN
9: while the pointers of all FNs have not scanned all the is unable to change its pairing results unilaterally for higher
DSOs in their preference list do utilities. Furthermore, in the perspective of the DSOs, when
10: FNs propose to DSOs with their service price; thepointersofallFNshavemovedtotheendofthepreference
11: forFN fk whostill haveavailableFNsto purchasedo lists, each DSO hasevaluated all proposalsfromthe available
12: if flagk =1 then FNs. Therefore, it also cannot unilaterally change its pairing
13: The pointer keep current position in the list; results, get accepted by the FNs and achieve a higher utility
14: else for itself. Furthermore, according to [6], when every agents
15: The pointer moves to the next position in the list; preference list is substitutable, a pairwise stable matching
16: end if always exists. Based on the above, the DSO-FN pairing will
17: The FN proposes to pointed DSO in its preference ultimately converge and achieve a stable matching result in
list with its available FNs; Algorithm 2.
18: We set flagk =0;
19: end for
C. The Interaction between FNs and DSSs
20: DSOs determines which FNs to choose;
21: for DSO di do When the CRBs from FNs have been rent to all DSOs,
22: if The total available amount of CRBs proposed by within each DSO, how to allocate the CRBs to all DSSs still
the FNs exceed the requirements then remainsa problem.Accordingto the rent of all FNs, DSS sj,
23: TheDSOdi choosesthemostpreferredamountof ∀j ∈ Υ, has a preference list Lsjf = Lsj1f,...,LsjKf on all
CRBs from FNs, and rejects the rest; FNs in the neighborhood,satisfying,
h i
24: For CRBs of the FN fk which is chosen by the
DSO in the last round, but rejected in the current Lsf =−p , ∀j ∈Υ, ∀k ∈Ω. (19)
kj k
round, we set flag =1;
k
25: end if Furthermore, according to the utility function of the FN,
26: end for the distance between the FN and its serving DSS affect the
27: end while revenues the FN received. With a longer distance, the FN
has to pay more for the data transmission in the network.
Therefore,we set a preferencelist Lfs = Lsf,...,Lsf for
i j1 jK
may need more CRBs from the FN. The matching repeats in FN fk, ∀k ∈Ω, over all DSSs, i.e., h i
circulations until all the pointers of the FNs have moved out
their preference list. According to the algorithm, we have the Lfs =−l , ∀k ∈Ω, ∀j ∈Υ. (20)
kj kj
following lemmas.
Consideringthe preferencelists of DSSs andFNs, i.e., Lsf
Lemma2. ForeachFNinthematchingalgorithm,thepointer kj
and Lfs, respectively, we design a many-to-many matching
oftheFNinitspreferencelistmovesinonedirection.Inother kj
algorithm for the FN-DSS pairing problem within DSO d ,
words, when its pointer has moved to the next DSOs in the i
∀i∈Ψ.AsshowninAlgorithm2,afterthepreferencelistsare
preference list, whatever the matching results of other FNs,
constructed,wesetapointerforeachDSSinitspreferencelist.
the FN cannot achieve a higher utility by moving the pointer
Initially, we set the pointer at the first FN in the list. During
back.
the each round of matching, each DSS first proposes to the
Proof: As shown in Algorithm 2, when the DSOs deter- FN in the pointer of the preference list. Based on behaviors
mines which FNs to choose, they choose the CRBs from the of the DSSs, each FN chooses its most preferred DSSs in its
mostpreferredamount.IfsomeCRBsfromFNf isdiscarded preferencelistuntilthe maximumCRBs availablein theDSO
k
by DSO d , the current accepted CRBs belong to the FNs d are reached. If the DSSs request more CRBs than the FN
i i
7
can supply, the FN will reject the less favourite DSSs. At the Algorithm2Many-to-ManyMatchingAlgorithmforFN-DSS
end of each round, if the DSSs have been allocated all of its pairing problem.
requested CRBs from the FNs, the pointers of the DSS will
keep unchanged. Otherwise, the pointers of the rejected DSS 1: for DSS sj do
will move to the next FN in the preference list. In the next 2: ConstructthepreferencelistLsf onallDSOsaccording
j
round,the DSSs which requireCRBs will proposeto the new to (19);
FNsaccordingtotheirpointers.Specifically,ifsomeCRBs of 3: One pointer is set as the indicator pointing at the first
FN are allocated to the DSS in the last round, but changed FN in the preference list.
to other DSSs in the current round, we suppose the pointer 4: end for
of the DSS in the next round doesn’t change its position, 5: for FN fk do
considering the pointed FN in the current round may be able 6: Construct the preference list Lfs on all FNs according
k
to supply more CRBs to the DSS. The matching repeats in to (20);
circulations until all the pointers of the DSSs have moved 7: end for
out their preference list. Following Lemma 2 and Lemma 3 8: We set flagj, ∀j ∈ Υ, as the indicator to show if the
in a similar way, the FN-DSS pairing problem can ultimately FNs allocate CRBs to the DSS sj in the last round, but
achieve a stable matching result. adjusted in the current round. Initially, flagj =1;
9: while the pointers of all DSSs have not scanned all the
V. SIMULATION RESULTS ANDDISCUSSIONS FNs in their preference list do
In this section, we present simulation results to evaluate 10: DSSs propose to FNs for their services;
ourproposedframeworkwithMATLAB.InthesimulatedIoT 11: forDSSsj whichhasnotbeenallocatedrequiredCRBs
scenario, without specific explanation, there are 120 DSSs, 4 do
DSOs and 20 FNs allocated randomlyin a circle district with 12: if flagj =1 then
adiameterof10kilometers.AswefocusontheIoTscenarios 13: The pointer keep current position in the list;
where DSSs are closely located with its sensors, without loss 14: else
of generality for the methods in this paper, we suppose the 15: The pointer moves to the next position in the list;
sensorsof each DSS are locatedat the same positionwith the 16: end if
DSS. We follow the settings in [7] and set the service rate of 17: The DSS proposes to pointed FN in its preference
each computingresourceblockis 0.1 (ms)−1. For each DSS, list for its data services;
the workload arriving rate is a random number averaged 0.5 18: We set flagj =0;
(ms)−1.Thedatatransmissionspeedis50km/ms.Thedelay 19: end for
tolerance of all DSSs is set to 60 ms. Furthermore, for each 20: Each FN determines which DSSs to choose;
FN, we set its preference to each DSO as a random number 21: for FN fk do
satisfyingtheuniformdistributionbetween[0,1].Basedonthe 22: if The total available CRBs requested by the DSSs
usage of its computing resources and its service cost, we set exceed the available volume then
theannouncedrentasarandomnumbersatisfyingtheuniform 23: TheFNfk allocatetheCRBstothemostpreferred
distributionbetween(0,10),andtheamountofavailableCRB DSSs, and rejects the rest;
as a random number satisfying uniform distribution between 24: ForCRBsallocatedtotheDSSsj inthelastround,
(0,100). The weight factors α, β and γ are set as 50, 0.01 but adjusted in the current round, we set flagj =
and 0.001, respectively. 1;
As shownin Fig.3, we evaluatethe utility ofall FNswhen 25: end if
the number of DSSs increases. When the number of FNs is 26: end for
fixed, we discover with the number of DSSs increasing, the 27: end while
utilityofallFNsgenerallyincreases,andtheincreasingspeed
first increases then gradually decreases to zero. The reason is
that when the number of DSSs increases, but the number of the performanceof theDSSs in ourproposedframeworkwith
FNs is fixed, the FNs will be able to serve more favourable the performance of the DSSs which is served by the massive
DSSs with a low transmission distance. However,when all of data centers only. With the same amount of workload, we
the available CRBs of FNs are allocated to the DSSs nearby discover that when the number of DSSs increases, the utility
and the number of DSS keep increasing, the DSS will be of DSS genrally increases, and the utility with FNs achieve a
allocated with CRBs fromthe massive data centers. Thus, the highervaluethantheonewithoutFNs.Furthermore,whenthe
total utility of the FNs stop increasing. Nevertheless, when number of DSSs is fixed and the average workload for each
we increase the number of FNs, we discover that with more DSS increases, the DSSs are able to receive more revenues
FNs, the utility can converge to a higher bound ultimately. from the services. Thus, the utility of DSSs increases.
Furthermore, because of the competition between FNs, when In Fig. 5, we analyze the relation between the utility of
the number of DSS is small, with more FNs, the increasing FNs and average workload arrive rate for DSSs. As shown in
speed is smaller. the figure, when the number of FNs is fixed and the average
In Fig. 4, we consider the utility of all DSSs with the valueofworkloadλincreases,theutilityofFNsfirstincreases
number of DSSs increasing. In the simulation, we compare then gradually converge to a fixed value. The reason is that
8
6000 6000
Numer of FNs = 15 Number of FNs = 30
Numer of FNs = 20 5500 Number of FNs = 25
5000 Numer of FNs = 25 Number of FNs = 20
Numer of FNs = 30 5000
4000
Ns Ns4500
F F
Utility of 3000 Utility of 34500000
2000
3000
1000
2500
0 2000
40 60 80 100 120 140 160 180 200 220 240 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of DSSs Average Value of λ (ms)-1
Fig. 3: The utility of all FNs versus the number of DSSs Fig. 5: The utility of all FNs versus the average value of λ
×106
18000 14
Small duty, without FNs Number of DSSs =240
16000 SMmedailul mdu dtyu, twy,i twh itFhNosut FNs 12 NNuummbbeerr ooff DDSSSSss ==118200
14000 Medium duty, with FNs
Heavy duty, without FNs 10
Heavy duty, with FNs
12000
Utility of DSSs108000000 Utility of DSOs 68
6000
4
4000
2
2000
0 0
40 60 80 100 120 140 160 180 200 220 240 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of DSSs µ (ms)-1
Fig. 4: The utility of all DSSs versus the number of DSSs Fig. 6: The utility of all DCOs versus the value of µ
delay,the utility ofDSS generallydecreaseswith the valueof
when the workload of all DSSs increases, the FNs are able
t increasing.
to allocate more of its CRBs to the DSSs nearby. However, th
whenalltheavailableCRBsofFNsareallocatedtotheDSSs,
the utility of the FNs stops improving and converges to one VI. RELATEDWORKS
specific value. When the number of FNs increases, with the In the literature, fog computing has been advocated to be
same value of λ, as the FNs are able to provide more CRBs the promising future of the cloud. The concept of pulling
to the DSSs, the converged value is higher. the cloud closer to the users has been widely considered in
InFig.6,weobservetheutilityofallDSOswhenthevalue previous work. In [8], the authors put forward the concept
of µ increases. As shown in the figure, when µ increases, of mist computing, aiming to distribute the cloud and its
each DSS is able to be served with a less amount of CRBs. benefits deeply into the network. In [9], the deployment of
Therefore, the DSO is able to set a higher price to the DSSs edge cloud was proposed. From the DSSs’ side, the edge
and receive high revenues. Moreover, when the number of cloud was able to surrogate the requirements and simplify
DSSsincreases,astheDSO isableto servemoreDSSsatthe the management of the network. From the servers’ side, the
same time, the total utility of DSOs also increases. edge cloud can exploit content and support service delivery
In Fig. 7, we evaluate the relationship of utilities and the in an efficient way. Without deploying massive data centers
tolerance service delay t of DSSs. As shown in the Fig. 7a, with high cost and latency, [10] on the other hand strength-
th
whenthevalueoft increases,eachDSSisabletobeserved ened the importance of small distributed data center designs.
th
with less CRBs. Thus, the FNs will supply less CRBs in the The authors took email delivery as an example and showed
network,andtheutilityoftheFNsgenerallydecreases.InFig. the advantages of geo-diversity characteristics of micro data
7a,astheDSOisablesetahighpriceforitsservices,withthe centers. In [11], a novel and distributed computing platform,
valueoft increasing,theutilityofDSOgenerallyincreases. called nano data centers was proposed. The authors evaluated
th
Moreover, in Fig. 7c, even though the DSS is able purchase the energy consumption of nano data centers and showed a
less CRBs with high t , the price of CRBs set by the DSOs significant improvement on energy efficiency, compared with
th
alsoincreases.Furthermore,astheDSSsuffersalotwithhigh the traditional data centers. In [12], considering the existing
9
580 6×105 1900
560
5 1800
540
520 4 1700
Utility of FNs458000 Utility of DSOs3 Utility of DSSs1600
460 2 1500
440
1 1400
420
400 0 1300
40 50 60 70 80 90 100 40 50 60 70 80 90 100 40 50 60 70 80 90 100
t (ms) t (ms) t (ms)
th th th
(a) The utility of all FNs versus t (b) The utility of all DSOs versus t (c) The utility of all DSSsversus t
th th th
Fig. 7: The utilities versus t
th
telecommunicationand Internetservice providers,the authors [26], the authors consider a Stackelberg game between data
showed that it was required and beneficial to leverage the center and buses in the smart city, where each buses collect
existing infrastructure and provide value-added services with data along its route and compete with other buses for the
FNs. In[13], theauthorsoutlinedthevisionoffogcomputing reward forwarding to the data center. In the game, following
and overviewed the important features of fog computing. the proposed heuristic algorithm, the Stackelberg equilibrium
In [14], the network optimization with fog computing was isshowntobeachievedwherethedatacenterandeachbusare
considered.Asthedatacenterswereawareofthelocationsof able to reach maximum utility. In [27], the authors formulate
DSSs with fog computing, dynamic adaptation of computing aStackelberggameforpowerallocationofdatacentersinthe
resources to the DSSs’ conditions was proposed. In [15], the cloud.Inthegame,thepowergridcontrolleractsastheleader
authorscomparedthecloudcomputingwiththefogcomputing and sets prices of the provided energy based on the current
and showed some significant characteristics of fog, which amount of renewable energy and costs. Observing the prices,
was required for current data services. In [16], the authors the cloud controller, i.e., the follower, determines the optimal
elaboratedtheroleoffogcomputinginsiximportantscenarios amount of enery to purchase and do resource allocation for
and surveyed the security issues with fog computing. its data centers. With backward induction, the near-optimal
strategiesofbothplayersinthegamecanbeachieved.In[28],
Moreover, fog computing has been widely considered to
the authors model the interaction between the monopolistic
be beneficial for the IoT. In [17], the authors overview the
data center opertor and the customers as a Stackelberg game.
opportunitiesandchallengesoffog,especiallytheapplications
In the game, the pricing strategies of the monopolistic data
offogcomputinginIoT.In[18],theauthorsdevisethemethod
center operator the corresponding behavior of data service
of MEdia FOg Resource Estimation (MeFoRE), to provide
customers is detailedly analyzed in both homogeneous and
resource estimation based on the service give-up ratio and
heterogeneouscustomerscenarios.In[29],theauthorsadopted
to enhance QoS based on the previous QoE. [19] addresses
the Stackelberg game to solve the problem of minimizing
the utility based pairing problem between the fog nodes and
energy consumption in the data center networks. In [30], the
IoT deviceswith the Irving’smatchingalgorithm.In [20], the
authors proposed a multi-leader multi-follower Stackelberg
authors propose a distributed dataflow programming model
gametoaddressandcooperationproblemsamongWi-Fi,small
for IoT devices to optimize resource allocation on computing
cell and macrocell networks. In [31], the authors combined
infrastructuresacrossthefogandthecloud.In[21],theauthors
the Stackelberg game and the bargainingto design a resource
consider issues of resource prediction, customer type based
allocation problem in a multi-tier LTE unlicensed network.
resource estimation and reservation, advance reservation, and
Furthermore, the auction mechanisms are also powerful tools
pricing in the fog computing for IoTs. [22] considers the
to solve the problem. In [32]–[34], the resource management
requirements of mobility, scalability, reliable control and ac-
problem could be perfectly optimized, but it requires high
tuation in some challenging scenarios of IoT to show the
communicationandcomputationoverhead.In[35],theauthors
benefits and significance of fog computing. Considering the
adopted the generic game theoretic framework to identify
advantages of fog computing, the authors in [23] discuss and
importantedgesinthecontextofk-edgeconnectivitybetween
propose a procedure to be implemented in smart phones for
certain pairs of nodes in a general given network. In [36],
UV measurement.
the graphical game was put forward to analyze the optimized
In order to solve the resource management problems in behaviors of each node in a general graph.
a network system with a distributed fashion, game theory
has been shown as a powerful tool [24]. In the literature, VII. CONCLUSIONS
most of the cases, the network system is normally modeled In this paper, we proposed a joint optimization framework
as a bipartite or a multi-tier graph. Based on this model, in the multi-FN, multi-DSO and multi-DSS scenario for IoT
in [25], a Stackelberg game theoretic model was shown for fogcomputing.Inthe framework,we first modeledthe Stack-
dynamic bandwidth allocation between virtual networks. In elberg games to solve the pricing problem of the DSOs and
10
resource purchasing problem of the DSSs. Then a many-to- [19] S. F. Abedin, M. G. R. Alam, N. H. Tran, and C. S. Hong, “A Fog
manymatchingwas proposedbetweenthe DSOs and the FNs BasedSystemModelforCooperative IoTNodePairingUsingMatching
Theory,”in201517thAsia-PacificNetworkOperationsandManagement
todealwiththeDSO-FNpairingproblem.Finally,weapplied
Symposium (APNOMS),pp.309-314,Busan,Korean,Aug.2015.
another many-to-many matching between its paired FNs and [20] N.K.Giang,M.Blackstock,R.Lea,andV.C.M.Leung,“Developing
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162,Seoul,Korean,Oct.2015.
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29thInternationalConferenceonAdvancedInformationNetworkingand
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Workshop on Computer Aided Modeling and Design of Communication
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