Table Of ContentMulti-Dimensional Payment Plan in Fog
Computing with Moral Hazard
Yanru Zhang, Nguyen H. Tran, Dusit Niyato, and Zhu Han
Abstract—Recently,theconceptoffogcomputingwhichaims
at providing time-sensitive data services has become popular.
In this model, computation is performed at the edge of the
7
network instead of sending vast amounts of data to the cloud.
1
Thus, fog computing provides low latency, location awareness
0
to end users, and improves quality-of-services (QoS). One
2
key feature in this model is the designing of payment plan
n from network operator (NO) to fog nodes (FN) for the rental
a of their computing resources, such as computation capacity,
J spectrum,andtransmissionpower.Inthispaper,weinvestigate
5 the problem of how to design the efficient payment plan to
1 maximize the NO’s revenue while maintaining FN’s incentive Fig. 1: An illustration of fog computing system.
tocooperatethroughthemoralhazardmodelincontracttheory.
] We propose a multi-dimensional contract which considers the
I FNs’ characteristics such as location, computation capacity,
N
storage, transmission bandwidth, and etc. First, a contract cloudcomputinginthesensethatbotharemadeupofvirtual
. which pays the FNs by evaluating the resources they have
s systems providing the flexibility of on demand provisioning
c provided from multiple aspects is proposed. Then, the utility
of compute, storage and network resources. However, fog
[ maximization problem of the NO is formulated. Furthermore,
we use the numerical results to analyze the optimal payment computing has several advantages over cloud computing in
1 plan, and compare the NO’s utility under different payment the sense of a significant reduction in data movementacross
v plans. thenetworkresultinginreducedcongestion,costandlatency,
7
7 IndexTerms—Fogcomputing,contracttheory,adverseselec- eliminationofbottlenecksresultingfromcentralizedcomput-
8 tion, multi-dimension. ing systems, improved security of encrypted data as it stays
7 closer to the end user reducingexposureto hostile elements,
0
I. INTRODUCTION and improved scalability arising from virtualized systems
.
1 [4]. By opening the access to fog computing nodes, service
The rapid developmentsof cloud computinghave brought
0 providers (SPs) can rapidly deploy certain applications and
a centralized solution to application developers and content
7
services to improve the quality of service (QoS) toward end
1 providers. Despite its wide known conveniences and advan-
users. This environment can also create a new value chain
: tages, cloud computing also suffers from certain limitations
v comprising NOs, InPs, SPs, and end users.
i such as high latencyand delay due to long distance between
X Referring to one brief model of fog computing in Fig.
end users and servers [1]. The emerging trends in net-
r workingsuch as large distributed sensor networks, industrial 1, there are variety of end users from the areas such as
a smart grid, industry, vehicular networks, transportation sys-
automation,and high speed transportationthat need location
tem, public safety department,and etc. thatrequirereal-time
dependent fast processing and cannot be satisfied by the
computing services. By subscribing to specific SPs whom
current service form by cloud computing [2].
subscribed to the network operator (NO) to obtain access to
With the motivation of placing the services as close as
physicalresources,endusersareallowedtoaccesscomputing
possible to end users, researchers have proposed a new
resources in both the fog and cloud, with fog close to end
cloudsystem called fogcomputing.Inthis model,fognodes
users while cloud locates far away. Within this network, end
(FNs) such as end-user devices, access points, edge routers
usersdirectlycommunicatewithFNforreal-timecontroland
and switches are deployed at or very close to the edge of
analytic,whiletheFNsonlysendperiodicdatasummariesto
network, and with functionalities like converged computing,
thecloudforfurtheraggregationandprocession.Usually,the
processing, management, networking, storage, physical and
cloud and fog are managed by the NO, who rents the cloud
cyber security [3]. Technically, fog computing is similar to
center and fog nodes from infrastructure providers (InPs).
Y. Zhang ([email protected]) and Z. Han ([email protected]) are with the The NO is aiming at maximize its revenue by efficiently
DepartmentofElectricalandComputerEngineering,UniversityofHouston,
managing and coordinating the computing resources in both
Houston,Texas77004.
N. H. Tran ([email protected]) is with the Department of Computer fog and cloud. To fully utilize the fog computing with
Science andEngineering, KyungHeeUniversity, SouthKorea. minimal rent while ensuring the FNs receive non-negative
D.Niyato([email protected]) iswiththeSchoolofComputerScience
revenue, appropriate payment plan is needed to enable the
and Engineering (SCSE), Nanyang Technological University, Singapore,
639798. NO and FNs to play complementary roles within their
respective business models, and allow all players to benefit
fromgreatercooperation.Inspiredbytheeffort-basedreward
from the labor market where employers pays its employees
based on their work load, we propose a payment plan in
fog computing such that the FNs receive their rental in
accordance with the quantities of computing resources and
the quality of service (QoS) they provide to the NO. The
computing resources include the transmission bandwidth,
power, computation capability (CPU speed), storage size,
as well as a FN’s proximity towards end users. Meanwhile
the QoS can be referred to latency and delay during data
transmission and processing, as well as security [5].
Fig. 2: The multi-dimension resource quantity and QoS
Based on this motivation, we aim at offering a contract
consistent payment contract.
that considers different aspects of the computing resources
providedby FNs to endusers, and assignsdifferentpayment
weightsinordertomaximizetherevenueofNO.Fortunately,
FNandwillpaytherentbasedontheQoSfromthedifferent
the moral hazard model from contract theory provides us a
aspects.
useful tool to design such a payment plan that can solve
the NO’s revenue maximization problems in fog computing
when the FN’s performance in QoS is affected by multiple
A. Operation Cost
aspects[6].FromtheNO’sperspective,it“employs”theFNs
to perform computing tasks and offers them QoS consistent In fog computing, the FN encounters both capital ex-
paymentbymulti-dimensionmeasurements.Insidethisvalue penditure (CapEx) and operational expenditure (OpEx) to
chain,theNOtriestoguaranteethefogcomputingQoSwith provideheterogeneousresources.CapExistheprominentin-
minimal payment, while ensuring the FNs have necessary vestmentwhichincludesthecostofpurchasingandinstalling
incentivestocooperate.Thus,tomaximizetheNO’srevenue, equipmentssuchasrouters,switches,accesspoints,backhaul
the NO needs to find an optimal payment plan that can aggregators, as well as the cost of using licensed spectrum
properly pay the infrastructure rent to FNs [7]. issued by the authorities [8]. Meanwhile the OpEx includes
the energy consumptiondue to signal processing, execution,
The main contributions of this paper are summarized as
and data transmission. We assume that the CapEx is fixed,
follows. First, we are first to propose a QoS consistent
while the OpEx is usage based.
contractwhichconsidersthequantitiesofresourcesprovided
byFNsfrommultipleaspects.Thecontractcharacterizesthe The FN’s heterogeneous resources are often measured
general situation in real world and provides comprehensive in disparate scales or units. It has been proved that those
paymentplantotheFNsforusingofthoseresources.Second, measurements can be mapped into one single unit, such
we formulate NO’s revenue maximization problem, as well as time [9]. Thus by mapping and normalization, we can
as provide the FNs with necessary incentive to cooperate representaFN’scontributionofresourcessuchasbandwidth,
in fog computing. Third, through simulations, we reveal CPU, transmission power by a vector a = (a1,...,an),
different parameter’s impacts on the optimal payment plan, n≥1 for one computing task. After such mapping, each ai
andcomparetheNOutilityundersixdifferentpaymentplans. has the same scale or unit and represents one resource type.
Our results show that by using the proposed payment plan, Suchmappingisbasedonthe knowledgethatthebandwidth
the NO successfully maximizes the utilities and the FNs and CPU speed affect the transmission and processing time,
obtain the continuous incentives to participate in the fog respectively.Duetothecontextawareandlocationdependent
computing. propertiesoffogcomputing,thesizeofdatabeingprocessed
and the geographic distance between FN and end users also
The remainderof this paper is organizedas follows. First,
we will introduce the network model in Section II. Then, havetheimpactsondataprocessingandtransmissionlatency.
TherearemanyotheraspectsthataffecttheQoSoffogcom-
the problem formulation is described in Section III. The
putingthatwe havenotlisted in Fig. 2,such astransmission
performance evaluation is conducted in Section IV. Finally,
power, which can also be mapped to transmission latency.
Section V draws the conclusion.
When providingthose resources, the FN’s cost incurredis
defined in a quadratic form,
II. SYSTEMMODEL
1
In this section, we consider a monopoly market with one ψ(a)= aTCa, (1)
2
NOtradingwithoneFN.TheNO-FNmutualbenefitmodelis
introducedfirstbyconstructingthemulti-dimensionpayment where C is a symmetric n×n matrix with the form of
plan offered by the NO. Then, we will give the utility
c ··· c
fsuonlucttiioonnsooffthbeothoptthiemaFlNcoanndtraNctO. WbeefoarsesupmroecetehdaitngthteoNthOe C = 1...1 ... 1...n. (2)
considers n aspects of the computing resources provided by cn1 ··· cnn
The diagonal element c of C reflects the FNs resource- with the form of
ii
specific cost coefficient, and the off-diagonal elements cij σ2 ··· σ
rje.present the cost relationship between two resources i and Σ= ...1 ... 1...n, (4)
The sign of cij indicates technologically substitute, com- σn1 ··· σn2
plementary, independent between two resources i and j, whereσ2denotesthevarianceofε ,andσ isthecovariance
if c > 0, < 0, = 0, respectively. If two resources i i ij
ij ofε andε . Thevariancedenotesthedifficultytoguarantee
i j
are technologically substitute, raising the quantity of one
the correctness of measuring the QoS, and also reflects the
resource raises the marginal cost of the effort on the other
resource quantity difference observed at the FN and NO
resources. The example of technologically substitute is the
sides. If the variance is large, the measurability of the QoS
relationship between geographic distance and transmission
is difficult, and there is a high probability that the FN’s
power. To achieve the same data rate at the end user,
performance is poorly measured and far away from the true
longer distance requires a higher transmission power con-
amount of resource the FN has provided. If the QoS is
sumption. In contrast, raising the quantity on one resource
easy to measure, the variance will be small or even zero.
decreases the marginal cost of the other resource if they
For example, the data size is an independent measure with
are technologically complementary. One example is about
variance 0, as well as the geographic distance. While the
the relationship between bandwidth and transmission power.
data processing time not only depends on the data size,
Giventhe same data packagesize and transmissiondistance,
but also the complexity of algorithm, which has a large
the larger bandwidth can achieve the same data rate at
variance.Thecovarianceoftwomeasurementsexistsbecause
receiverwithlowertransmissionpower.Inthisexample,high
themeasurementofoneresourcemayaffectthemeasurement
quality in one resource ease the cost in the other, and thus
of the others; for example the transmission time is affected
called technologically complementary. For technologically
by both bandwidth and power.
independent resources, their operation cost is not dependent
on the quantity of other resources. There are many techno-
C. Payment Plan
logically independent examples in fog computing, such as
the relationships between transmission bandwidth and CPU Inspired by the manager’s reward package in industry,
speed, geographic distance and data size. which comprises a fixed salary, a bonus related to the firm’s
In order to lower the mathematical complexity, we only profits, and stock options related reward based on the firm’s
solve the cases without the technologically complementary share price [10], we define the FN’s payment plan w in fog
in this paper. Thus, the operation cost coefficient matrix is computingasalinearcombinationofa fixedsalaryandQoS
a positive semi-definite matrix with every element in C is relatedpayments.By restrictingthepaymentplanofferedby
non-negative. the NO in the linear form, the payment plan w FN receives
by participating in the fog computing can be written as
w =t+sTq, (5)
B. QoS Measurement
where t denotes the fixed salary, which is a constant in-
TheresourcessuchasCPUspeed,transmissionbandwidth
dependent of QoS and regarded as the subscribing fee to
and power can be easily specified by the FN, the end user
complement the FN’s CapEx. s = (s ,...,s ), n ≥ 1
related parameters such as geographic distance, data size, 1 n
is the payment related to the QoS q. As q is a random
can also be quantifiedeasily, while the measurementof QoS
variable which follows q ∼ N(a,Σ), the payment plan
cannot be that accurate. Though those measurements can be
w is also a random variable with a mean of t + sTa.
mappedintotimescalefollowtheprocedurein[9],errormay
From the scaling property of covariance, we know that
come from the design failure of the measurement system.
Var(sTq) = sTΣs. Thus, the payment plan follows the
Given the actual resources provided by FN is a, which distribution w ∼N(t+sTa,sTΣs).
is hidden from the NO, but the FN’s QoS can be observed
At this point, we can propose the contract that is offered
as a vector of QoS q = (q ,...,q ), n ≥ 1, which can
1 n by the NO as (a,t,s), where a and s are n×1 vectors, and
be regarded as the FN’s performance in latency reduction.
t is a constant value. Under this contract, the NO offers the
Due to aforementioned different measurability on QoS, the
FN a payment plan which includes a fixed salary t, and n
receivedinformationq varies. Therefore,the performanceof
QoSrelatedpayments(s ,...,s ).Fig.2illustrateshowthis
1 n
the FN is a noisy signal of the resources it has provided:
contractworks.TheFN providesthequantitya forresource
i
i in the computing task, which is observed as a QoS q by
i
q =a+ε, (3)
the NO. The NO then offers a payment s related to q .
i i
where the random component ε = (ε ,...,ε ), n ≥ 1,
1 n
is assumed to be normally distributed with mean zero and D. Utility of Fog Node
covariance matrix Σ. Thus, the FN’s performance follows Inthismodel,weassumethattheFNhasconstantabsolute
the distribution of q ∼N(a,Σ). risk averse (CARA) risk preferences, which means the FN
The variance Σ is a symmetric n×n covariance matrix has a constant attitude towards risk as its income increases.
SuchariskpreferencecomesfromtheFN’sconcernaboutits The social welfare is the resource provided by the FN,
security issue when opening access for end users. Thus, FN minus the operation cost and the cost incurred by inaccurate
utility is represented by a negative exponential utility form measurement. Notice that this expression is independent of
[11], the fixed salary t, which serves as an intercept term in the
u(a,t,s)=−e−η[w−ψ(a)], (6) contract.Thus,thefixedsalarytcanonlybeusedtoallocate
the total certainty equivalent between the two parties [13].
where η >0 is the FN’s degree of absolute risk aversion
′′
u
η =− , (7)
u′ III. PROBLEM FORMULATION
whereuistheFN’sutilityfunction.Alargervalueofηmeans
With the system model,we can formulatethe NO’s utility
more incentive for the FN to providemore resources for the
maximization problem while providing the FN necessary
computingtask. The utility and operationcost of the FN are
incentives to cooperate. The NO’s problem can be written
measuredinsuchmonetaryunitsthattheyareconsistentwith
as
the payment from the NO.
From(6),weseethattheFN’sutilityisastrictlyincreasing
∗
max U(a ,t,s), (13)
andconcavefunction.Forlowercomputationcomplexity,we
a,t,s
can make use of the exponentialformof the utility function, s.t. (a) a∗ ∈argmaxu(a,t,s),
and use certainty equivalent as a monotonic transformation a
∗
(b) u(a ,t,s)≥u(w),
of the FN’s expected exponential utility function [12].
Proposition 1. The FN’s utility can be equally represented where u(w) is the reservation utility of the FN when not
by certainty equivalent: providing any resource (a = 0) in the fog computing. The
1 1 NO maximize its own utility under the incentive compatible
CEu =t+sTa− aTCa− ηsTΣs. (8) (IC) constraint (a) that the FN provides the optimal amount
2 2
of resourcea∗ maximizingits own utility,and the individual
The certainty equivalentconsists of the expected payment
rationality (IR) constraint (b) that the utility FN received is
minus the operation cost and measurement cost.
no less than its reservation utility.
Under the assumption of stochastic dependent, the er-
E. Utility of Network Operator
ror terms are stochastically interacted, i.e., σ 6= 0. For
ij
Here,wedefinetheutilityoftheNOastheexpectedgross technologically dependent, we mean that the activities are
benefitsofV(a)minusthepaymentplanw tothe FN.Thus, technologically correlated with each other, i.e., c > 0
ij
the NO’s expected utility is written as and C is a positive definite matrix. We solve this multi-
dimensional problem by using certainty equivalent model
U(a,t,s)=V(a)−w, (9)
withthefollowingsimplereformulationoftheNO’sproblem:
where V(·) is the evaluationfunctionwhich followsV(0)=
0 and V′(·) > 0. Different from the FN who has CARA max βTa−sTa−t, (14)
a,t,s
risk preferences, the NO here is assumed to be risk neutral,
i.e., V′′(·) = 0. Thus, the expected profit of the NO can be s.t. (a) a∗ ∈argmax[t+sTa− 1aTCa− 1ηsTΣs],
simplified to a 2 2
1 1
(b) t+sTa− aTCa− ηsTΣs≥w,
U(a,t,s)=βTa−w, (10) 2 2
where β = (β1,...,βn), n ≥ 1, characterizes the marginal where w also denotes the reservation utility of the FN when
effect of the FN’s contribution a on the NO’s utility V(a). not participating in the fog computing. The IC constraint
Similar tothe definitionofFN’s certaintyequivalent,we can represents the rationality of the FNs choice of contribution.
derive the NO’s certainty equivalent as TheIRconstraintin(b)ensuresthattheNOcannotforcethe
FN into accepting the contract.
CE =E[βTa−w], (11)
p
We first solve the optimal effort by reducing the IC
=βTa−sTa−t.
constraint first. The FN’s certainty equivalent is concave,
since its second-order derivative with respect to a is a
F. Social Welfare negative definite matrix −C. Thus, the optimal effort can
With the definitions of both FN’s and NO’s utility func- bedeterminedbytakingthefirst-orderderivativeoftheFN’s
tionsandcertaintyequivalentpayoffs,wecanhavethesocial certainty equivalent regarding a, and set u′(a,t,s) = 0. In
welfare defined as their joint surplus, i.e., the summation of the matrix differentiation, if we define α = aTCa, as C is
FN’s and NO’s equivalent certainty: a symmetric matrix, we have ∂α/∂a=2aTC [14]. Since C
is symmetric positive definite, its inverse is existent. Thus,
R=CEu+CEp, (12) through numerical derivations, we finally have a=C−1s in
1 1 this multi-dimensioncase. Accordingly,we substitute the IR
=βTa− aTCa− ηsTΣs.
2 2 constraintin (b) with the optimalamountofresourcea∗ and
5 5
General
5 Independent
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1 1 1
0 0 02 3 4 5 6
1 1.2 Cost1 C.4oefficient 1c.6 1.8 1 R2isk Averse D3egree η 4 5 Variance σi2
(a) Cost coefficient cii (b) Risk averse degree η (c) Measurement error variance σi2
Fig. 3: The NO’s utility as different parameters vary
simplify the NO’s problem to that, from the optimal payment plan we have derived, no
matter how those parameters change, the FN’s utility will
max βTC−1s−sTC−1s−t, (15)
a,t,s remain the same. The optimal payment plan will bring FN
1 the utility the same as the reservation utility −e−ηw, which
s.t. (a) t+sTC−1s− (C−1s)TC(C−1s)
2 in our case is −1 as we set w =0.
1 In the previous section, we have solved the optimal
− ηsTΣs=w.
2 payment plan when the measurement errors are stochastic
SubstitutingthevalueoftintheIRconstrainttotheobjective dependent and the resource types are technologically de-
and differentiating the objective function with respect to s, pendent. As this multi-dimension case is the most general
we have the QoS related payment s∗ in the optimal multi- case in reality, we name this mechanism by General. For
dimension payment plan as: comparison,we propose5 more paymentplans. The fist one
is the optimal payment plan when the measurement error
s∗ =(C−1+ηΣ)−1C−1β =(I +ηCΣ)−1β. (16)
and resource type are independent, and thus we name it
With s∗, we havethe optimalamountofcomputingresource by Independent. The second payment plan is called Single
in the multi resource case as Bonusthatisthepaymentplanobtainedintheone-dimension
case. In this one-dimension case, we can regard the NO
a∗ =C−1(I+ηCΣ)−1β. (17) payments FN on the QoS of a single one resource type.
Representingtbyw,s∗ anda∗,weobtainthefixedpayment For example, measuring the data processing latency by only
taking the CPU speed into account. The third and forth
t in the optimal linear payment plan as:
ones are special cases of the General: one is stochastic
1
t∗ =w+ sT(ηΣ−C−1)s, (18) independent but technologically dependent, the other one is
2
technologically independent but stochastic dependent, and
=w+ 1 (I+ηCΣ)−1β T (ηΣ−C−1) (I +ηCΣ)−1β . are named by Stochastic Independent and Technologically
2(cid:2) (cid:3) (cid:2) (cid:3)
Independent, respectively. The last one is called Opening
Using the s∗ in (16), we can indeed determine how Reward, which is the payment plan only contains a fixed
the optimal payment plan varies with the accuracy of QoS salary t. We can regardthis mechanismas the NO offersthe
measuresforeachresourceandtheoperationcostcoefficient FN a one time payment at subscription. But this Opening
of each resource. For example, when two resources are Reward mechanism does not care about FN’s future service
technologically substitution cij > 0, if the measurability of quality.
resourcei worsens, thatis, σ2 increases,then,asis intuitive,
i In Fig. 3a, we compare the NO’s utility from the six
s∗ goesup, but s∗ goesdown. Thus, there is a measurement
j i payment plans as we vary the resource-specific operation
complementarity between the s∗ and s∗ in the presence of
i j cost coefficient cii. From the simulation results we see that,
technologically substitutes problems [6].
as the cost coefficient c increases, the NO’s utility is
ii
decreasingincontrast.Thereasonforthisphenomenonisthat
IV. SIMULATION RESULTSAND ANALYSIS larger cost coefficient c means more operation cost when
ii
In this section, we will first give a detailed analysis of providing such an resource. Therefore, the FN is less likely
how the NO’s utility changes by varying the parameters to contribute in the fog computing. With less computing
suchastheoperationcostcoefficientsandmeasurementerror resources provided by the FN, the QoS will decrease and
covariance. Meanwhile, we will conduct a comparison of the NO’s utility will certainly decrease. In addition, from
the NO’s utility among different payment plans. To set up Fig. 3a, we see that the NO obtains the largest utility in
the simulation, we assume that, the reservation payment of the Opening Reward case. Followed by the Independent,
the FN w = 0 when not cooperating in the fog computing StochasticIndependent,andTechnologicallyIndependent,the
(a = 0). The reason we do not consider the FN’s utility is General case proposed by us brings the fifth highest utility
to the NO, while the Single Bonus gives the least utility. Opening Payment is a “once-for-all” deal which does not
In Fig. 3b, we analyze the impact of FN’s risk averse provide continuous incentives for the FNs, i.e., after the FN
degreeη ontheNO’sutility.AstheNO’sutilityV =a−tin has finished the computing task and receive the payment, it
theOpeningReward isindependentoftheriskaversedegree is likely to stop cooperating in the future.
η, we cannot see any change in the NO’s utility. For the
other five payment plans, we see that the NO’s utility is V. CONCLUSIONS
decreasing as the FN’s risk averse degree η increases. This
In this paper, we have investigated the problem of maxi-
result is intuitive as a larger η means the FN becomes more
mizingNO’srevenuebyefficientlyallocatingFNs’computa-
conservative and sensitive to risk, thus less likely to open
tionresourcesinfogcomputing.Theoptimalpaymentplanis
access to end users. With less resources obtained from the
solvedby payingthe rentof FN’s computingresourcesfrom
FN, the NO’s utility will certainly decrease. From Fig. 3b
a multi-dimension evaluation while ensuring the FN’s coop-
we also obtains the similar ranking of the NO’s utility as
eration.Furthermore,weusethenumericalresultstoanalyze
in the previous figure: the Independent case brings higher
theoptimalpaymentplanbyvaryingdifferentparameters.In
utilitythantheStochasticIndependent,TechnologicallyInde-
addition,wecomparetheNOs’ utilityunderthesix different
pendent, and General one, and the Single Bonus one brings
payment plans, and show that the NO’s utility deteriorates
the smallest utility for the NO.
with large operation cost coefficient, higher risk aversion of
In Fig. 3c, we increase the variance σ2 to see how the
i FNs, and large measurement error variance.
NO’s utility varies. Similar to the previous case, the NO’s
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Independentthanin the TechnologicallyIndependent.There-
fore, the NO’s utility is higher in the Stochastic Independent
than in the Technologically Independentcase, while the Sin-
gle Bonus only payment FN with only one type of resource
evaluation.Asaresult,theFNshavelessincentivetoprovide
computingresources.Inreturn,lessutility isobtainedbythe
NO. For the result of the Opening Reward case, it seems
unreasonableatthefirstsight,asitbringstheNOthehighest
utility than the other payment plans. While we notice that
