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Developing User Perceived Value Based PricingModels for Cloud Markets
Peijin Cong, Liying Li, Junlong Zhou, Kun Cao, Tongquan Wei, Mingsong Chen, and Shiyan Hu
Abstract—With the rapid deployment of cloud computing infrastructures, understanding the economics of cloud computing has becomea pressing issue for cloud service providers. However, existing pricing models rarely consider the dynamic interactions between userrequests and the cloud service provider. Thus, the law of supply and demand in marketing is not fully explored in these pricing models.In this paper, we propose a dynamic pricing model based on the concept of user perceived value that accurately captures the real supplyand demand relationship in the cloud service market. Subsequently, a profit maximization scheme is designed based on the dynamicpricing model that optimizes profit of the cloud service provider without violating service-level agreement. Finally, a dynamic closed loopcontrol scheme is developed to adjust the cloud service price and multiserver configurations according to the dynamics of the cloudcomputing environment such as fluctuating electricity and rental fees. Extensive simulations using the data extracted from real-worldapplications validate the effectiveness of the proposed user perceived value-based pricing model and the dynamic profit maximizationscheme. Our algorithm can achieve up to 31.32% profit improvement compared to a state-of-the-art approach.
Index Terms—Cloud Computing, Dynamic Pricing Model, User Perceived Value, Profit Maximization, Augmented Lagrange Function.
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1 INTRODUCTION
CLOUD computing has become a popular commercialcomputing model that distributes user requests to a
set of servers and delivers services over communication net-works. As a business model, it turns resources of computing,storage, and communication into ordinary commodities andutilities in a pay-as-you-go manner [1], [2], [3], [4]. It isnatural for cloud service providers to pursue the goal ofprofit maximization. Thus, the cloud service pricing strategyis of particular importance to cloud service providers.
The pricing model of a cloud service provider consists oftwo parts, namely, the revenue and the cost [5]. The revenueis the income that the cloud service provider gets throughthe sales of cloud services. The cost is the expenditure thatincludes not only the rental and electricity fees to operatemultiserver systems, but also the reward and penalty paidby the cloud service provider to users based on service-level agreement. Profit maximization can be achieved byincreasing revenue or reducing cost. On one hand, cloudservice providers attempt to increase revenue by setting ahigh price for cloud services and attracting a great amountof service purchases. However, service price and purchaseactivity interplay, which cannot be optimized simultane-
This work was partially supported by Shanghai Municipal Natural ScienceFoundation (Grant No. 16ZR1409000) and Natural Science Foundation ofChina (Grant No. 61672230). T. Wei is the corresponding author. Email:[email protected].
• P. Cong, L. Li, K. Cao, and T. Wei are with the Department of Com-puter Science and Technology, East China Normal University, Shanghai200062, China.
• J. Zhou is with the School of Computer Science and Engineering, NanjingUniversity of Science and Technology, Nanjing 210094, China.
• M. Chen is with the Shanghai Key Laboratory of Trustworthy Computing,East China Normal University, Shanghai, 200062, China.
• S. Hu is with the Department of Electrical and Computer Engineering,Michigan Technological University, Houghton, MI 49931, USA.
Manuscript received December 5, 2017; revised February 6, 2018; acceptedMay 16, 2018.
ously [6]. On the other hand, cloud service providers tryto reduce operational cost, such as electricity bill and rentalfees, which are related to multiserver configurations. Thus,aspects including electricity price and multiserver configu-rations need to be considered in cloud pricing modeling.
Numerous investigations have been made into pricingmechanisms for profit maximization in cloud computing.Fixed pricing strategies such as pay-per-use, subscriptionbased pricing, and tiered pricing are the most common pric-ing methods used by major cloud service providers [7], [8],[9]. For example, Li [7] proposes a flat rate pricing strategythat sets a fixed price for all service requests. Kesidis etal. [8] point out that usage-based pricing strategy can usecloud resources more efficiently when compared with flatrate pricing strategy. However, these fixed pricing methodscannot meet the dynamic needs of users and cannot capturethe dynamics of supply and demand in market.
Handling the disadvantages of fixed pricing strategiesnecessitates the dynamic pricing strategies that adjust priceof cloud services according to market situations and userrequirements for service quality. Macias et al. [10] propose agenetic model based dynamic pricing strategy that obtainsoptimal pricing in an iterative way. This strategy offerscompetitive prices in the negotiation of services in cloudcomputing markets. Amazon [11], [12] utilizes a spot pricingstrategy that dynamically adjusts prices for a virtual serviceinstance to accommodate changes in supply and demand.Based on a study of the spot price history of Amazon, Xu etal. [13] propose a dynamic pricing strategy to better under-stand the current market demand. Zhao et al. [14] design anefficient online algorithm for dynamic pricing of virtual ma-chine resources across datacenters in a geo-distributed cloudto pursue long-term profit maximization. Although theseworks investigate dynamic pricing strategies from differentperspectives, service-level agreement is not considered intheir pricing mechanisms which is however important.
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A service-level agreement is defined as an official com-mitment between a service provider and a client [15]. Ituses a price compensation mechanism that provides certaincompensations to users when their service requests areprocessed with low quality of service. Cao et al. [5] presenta pricing model that takes service-level agreement and con-sumer satisfaction into considerations to maximize the profitof cloud service providers. Ghamkhari et al. [16] propose atwo-tier ladder charging method to ensure user satisfaction.Specifically, a cloud service provider will charge users iftheir requests are processed before deadlines. Otherwise, thecloud service provider will not charge users for this execu-tion. Lee and Irwin et al. [17], [18] claim that the price of thecloud service will decrease as the waiting time of servicerequests grows until the cloud service is free. These worksstudy service-level agreement to ensure user satisfactionin the pricing process for profit maximization. However,they ignore the crucial concept of user perceived value intraditional market environment, which reflects the users’willingness to purchase cloud services. User perceived valueis an important concept in pricing process since it ultimatelywill impact the profit of cloud service providers.
In this paper, we propose a user perceived value-baseddynamic pricing mechanism that conforms to the law ofsupply and demand in economics. The contributions of thispaper are summarized as follows.
• A dynamic pricing model that considers the interac-tion between users and the cloud service provideris proposed. The model is built upon the conceptof user perceived value, user reward, and cloudservice provider penalty in the domain of economics,which accurately captures the dynamics of supplyand demand in cloud pricing strategies. In particular,user perceived value is nicely modeled using kerneldensity estimation method in our context.
• A profit maximization scheme is developed basedon the dynamic pricing model to optimize the profitof the cloud service provider by configuring multi-server systems under service-level agreement con-straint. Our scheme also includes a runtime controlloop to adjust service price and muitlserver configu-rations to the dynamics of cloud computing environ-ment such as fluctuating electricity and rental fees.
• Extensive simulations using the data extracted fromreal-world applications validate the effectiveness ofour proposed user perceived value-based pricingmodel and the dynamic profit maximization scheme.Our algorithm achieves 31.32% profit improvementcompared to a state-of-the-art approach.
The remainder of the paper is organized as follows.Section 2 presents the system architecture and models,Section 3 presents the problem definition and overview ofthe proposed scheme. Section 4 describes the proposed userperceived value-based pricing mechanism. The effectivenessof the proposed scheme is validated in Section 5 and con-cluding remarks are given in Section 6.
2 SYSTEM ARCHITECTURE AND MODELS
We consider a common three-tier cloud service provisionstructure that consists of users, cloud service providers,
and infrastructure vendors [5], [9], [19], [20]. Among thethree entities that form a market in cloud computing, theinfrastructure vendor charges the cloud service provider forrenting infrastructures to deploy service capacity, and thecloud service provider charges users for processing theirservice requests. In this paper, users and the cloud serviceprovider are of our particular interest. We introduce our usermodel and cloud service provider model in the followingsubsections.
2.1 Cloud User ModelTo maximize the profit of a cloud service provider, the cloudservice provider needs to know the aggregate demands ofusers. When a cloud service provider sets up the price of aservice, different users have different responses to this price.Based on the concepts of user perceived value, we give thefirst derivation of such a model in the cloud computingcontext. In the following, we introduce the concepts of userperceived value and then present our derivations.
2.1.1 User Perceived ValueIn conventional markets, the arrival rate of customers to astore is often a response to their regular buying patternsrather than a reaction to individual prices [6]. Thus, it isreasonable to assume that the change of the list price has noeffect on the total number of customers who are visiting thestore. Typically, not all of the customers are willing to buy aspecific commodity. That is, the total number of customerswho buy commodities are no larger than the total numberof people that visit the store.
Customer perceived value is the fundamental basis forall marketing activities [21]. It reflects the worth that aproduct or service has in the mind of a consumer and thisconcept has been widely used in modeling other markets[22]. In general, customers are unaware of the true cost ofproduction for the products they buy, which means thatthey simply have an internal feeling for how much certainproducts are worth to them. In the conventional marketenvironment, only the customer whose perceived value ishigher than the real price of the product is willing to pay forthe product.
In this paper, we use Xi to denote the perceived valuethat user i has for the service. Xi is a continuous randomvariable and 0 ≤ Xi < ∞ holds. As with other pricingmodels [23], X1, X2, · · · , Xn are assumed to be independentand identical random variables. The probability densityfunction of the perceived value X , denoted by f(x), isknown or can be estimated a priori. Perceived value is aprocess of valuing and is much harder to determine. Roig etal. [24] observe that the customer value is perceived by cus-tomers, and cannot be determined objectively by the seller.Factors such as scarcity, marketing efforts, novelty, andbrand associations all play into customer perceived value[25]. Usually, consumers will offer a range of price options.Thus, in the experimental section, a normal distribution isused to describe the initial distribution of user perceivedvalue. Subsequently, the distribution of user perceived valueis fitted using kernel density estimation based on historicalprice data. Kernel density estimation is a non-parametricway to estimate the probability density function of a randomvariable based on a finite data sample [26].
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In the following sections, we adopt the terminology ofcustomer perceived value used in traditional market. Thecloud computing environment is taken as a store and thecloud service is deemed as a special commodity providedin the store. The terminology of customer perceived valueand user perceived value are used interchangeably.
2.1.2 User Demand DistributionUnlike traditional methods that use the expected demand tomodel user behavior [23], [27], the probability distributionof total demands is used in this work to model user servicerequests. We consider a slotted time model that deals withthe pricing decision and constraints for sales periods T ofequal length. Let τ denote the length of each time slot overthe sales period T , and N be the number of time slots τover the sales period T . That is, T = N · τ . A cloud serviceprovider sets list price for the service at the beginning ofregular sales periods. The list price during each sales periodis assumed to be constant, but varies from period to period.
Suppose that the cloud service provider will charge ωper user for a specific cloud service during a sales periodT . Let n denote the total number of users that have interestin the service at the price of ω during the sales period T ,and λu denote the number of users arriving per unit time,respectively. n is assumed to be independent of all otherparameters of the system, and follows a discrete Poissondistribution as [27]
P (n|λu) =(λuT )
ne−λuT
n!, n = 0, 1, 2, · · · ,∞. (1)
The user arrival rate λu may not be constant in many situ-ations. Taking into account the heterogeneity of arrival rate,a Gamma distribution characterized by parameters (α, β)is utilized to represent the arrival rate λu. α is the shapeparameter that determines the shape of the distributioncurve while β is the scale parameter that decides the sizeof the distribution curve. The probability density functionof arrival rate λu is given by
g(λu) =1
Γ(α)βαλ(α−1)u e−λu/β , 0 ≤ λu ≤ ∞, (2)
where the expectation and variance of λu is given byE[λu] = αβ and V ar[λu] = αβ2, respectively, and Γ(α)is a complete gamma function.
Among the n users, any one whose perceived value ofthe service is no less than the list price ω is considered as apotential buyer of the service. Let m denote the numberof potential buyers. It is a non-negative discrete randomvariable taking the value of 0, 1, 2, · · · ,∞ and m ≤ nholds. Based on user perceived value, we use f(x) to denotethe probability density function of the perceived value X ,and F (ω) to represent the cumulative distribution functionof x evaluated at ω. F (ω) describes the probability thatusers do not want to pay ω to buy a cloud service. It isa non-decreasing function of ω, and 0 � F (ω) � 1 andlim
ω→∞F (ω) = 1 hold [28]. Let Pω(m|n) be the probabilitythat m out of n users are inclined to buy in the sales periodwhen the service price is set equal to ω. It follows a binomialdistribution of probability, which is given by
Pω(m|n) = �nm
�[1− F (ω)]m[F (ω)](n−m). (3)
Combining (1)-(3), we can derive the probability of hav-ing m potential buyers during the sales period T when theservice price is set equal to ω. The probability is denoted byPω(m) and given by
Pω(m) =
� ∞
λu=0
∞�n=0
Pω(m|n)P (n|λu)g(λu)dλu
=�m+α−1
m
�[
βT [1− F (ω)]
1 + βT [1− F (ω)]]m[
1
1 + βT [1− F (ω)]]α.
(4)
Clearly, it is a negative binomial distribution. Thus, theexpected number of actual buyers of the service at priceω during period T , denoted by Eω(m), can be computed as
Eω(m) = αβT (1− F (ω)), (5)
where α and β are parameters of the Gamma distribution ofuser arrival rate λu, and F (ω) is the cumulative distributionfunction of x evaluated at ω. The revenue of the cloudservice provider in a sales period T is thus given by
Revenue = ω ·Eω(m) = ωαβT (1− F (ω)). (6)
2.2 Cloud Service Provider Model
The cloud service provider rents a multiserver system thatis constructed and maintained by an infrastructure vendorto serve user service requests. The architecture details of themultiserver system are quite flexible [5]. They can be bladecenters where each server is a server blade [29], clustersof traditional servers where each server is an ordinaryprocessor [30], and multicore server processors where eachserver is a single core [31]. For the ease of presentation, theseblades/processors/cores are simply called servers. Userssubmit their service requests to the cloud service provider,and the cloud service provider serves these service requests(i.e., run these tasks) on the multiserver system.
2.2.1 Multiserver Model
We consider a multiserver system that consists of M ho-mogeneous servers operating at a common speed of s. Themultiserver system can be modeled as an M/M/M queuingsystem where arrivals of user service requests governed bya Poisson process form a single queue and M servers canprocess these service requests in parallel. Let � be the servicerate of user service requests that arrive at the rate of λu. It isclear that � user service requests can be processed by serversif the number of user service requests in the system is notgreater than M . The service time of a user service requeston a server is an exponential random variable denoted byx1 = r/s with mean x1 = r/s, where r is the numberof instructions to be executed for the service request. Afirst-come-first-served (FCFS) queue of infinite capacity ismaintained by the multiserver system for waiting taskswhen all the servers are busy. Let ρ be server utilization,which is defined as the average percentage of time that aserver is busy. It can be expressed as
ρ =λu
M�=
λu
M sr̄
=λur̄
Ms. (7)
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Let Pk be the probability of k service requests beingwaiting or processing in the M/M/M queuing system.Based on queuing theory [5], [32], [33], Pk is given by
Pk =
�P0
(Mρ)k
k! , k ≤ M
P0MMρk
M ! , k ≥ M, (8)
where P0 is the probability that there are no tasks in thequeue, and is formulated into [32]
P0 = (M−1�k=0
(Mρ)k
k!+
(Mρ)M
M !· 1
1− ρ)−1. (9)
The probability that there are exact M service requests in thesystem is thus given by PM = P0
(Mρ)M
M ! . Through Taylorseries expansions of
�M−1k=0 (Mρ)
k/k! ≈ eMρ and M ! ≈√
2πM(Me )M , it can be rewritten as
PM =1− ρ√
2πM(1− ρ)( eρ−1
ρ )M + 1. (10)
This form of PM is necessary for deriving multiserver con-figurations in Section 4.
When all the servers in the system are busy, a newlysubmitted service request must wait and will be insertedinto the FCFS queue. Let Pq denote the probability ofqueuing a newly arrived task when no servers are idle atthe time of arrival. Pq can be formulated as
Pq =�∞
k=MPk =
PM
1− ρ. (11)
Let N be the average number of requests being waiting orexecuting in the multiserver system. N is calculated as
N =�∞
k=0kPk = Mρ+
ρ
1− ρPq. (12)
The average service response time R is defined as theaverage time elapsed between the time when a servicerequest is submitted and the time when the service requestis finished. In this paper, it is adopted to evaluate the servicequality. It is in fact the sum of task execution time andwaiting time, and can be derived by applying Little’s Law[34] as
R =N
λu= x1(1 +
Pq
M(1− ρ)) = x1(1 +
PM
M(1− ρ)2). (13)
The average service response time R is utilized in this paperas a metric for service-level agreement. If the response timeof a service exceeds the predefined deadline, service-levelagreement is deemed to be violated.
2.2.2 Bill and RentA cloud service provider needs to rent infrastructure andpay electricity to maintain the operation of the computinginfrastructure. Let δ be the fee the cloud service providerpays to rent a server per second during a sales period T , therent the cloud service provider needs to pay for a system ofM servers during the sales period T is
Rent = Mδ ·T. (14)
As a portion of the cloud service cost, electricity feehas become a significant expense for today’s data centers.
It can be derived by multiplying energy consumed by aserver with electricity price. The energy consumed by aserver can be modeled at different levels of abstraction.At the abstraction level of digital CMOS circuit, the powerconsumption, which is denoted by Ptot, can be modeled as
Ptot = Psta + Pdyn, (15)
where Psta is the static power dissipation while Pdyn is thedynamic power dissipation. Psta is independent of switch-ing activity and maintains the basic circuit state, thus can bedeemed as a constant [5]. Pdyn is related to processor switch-ing activity and dominates the total power consumption,which can be formulated as a function of supply voltage vand processing speed s. In addition, the supply voltage isusually linearly proportional to the processing speed, i.e.,v ∝ s. The dynamic power consumption Pdyn is then ex-pressed as ξsγ , where ξ is a processor dependent coefficientand γ is a constant that equals to 2φ+1 (φ > 0). Based on thestatic and dynamic power consumption described above,Equation (16) is used to denote the total power consumptionof a multiserver system, which is,
Ptot = M((Pdyn − Psta)ρ+ Psta), (16)
where M is the server number and ρ is the server utilization.Let ET denote the energy consumed by all M servers in
the system during the sales period T . It is given by
ET = M((Pdyn − Psta)ρ+ Psta) ·T. (17)
Let CT (ET ) be the price of the energy consumed by allservers in the period T , then CT (ET ) can be formulated as
CT (ET ) =
�kT1 , 0 ≤ ET ≤ lTth
kT2 , ET > lTth, (18)
where kT1 , kT2 > 0 are differentiated price and lTth is the
energy consumption threshold in the sales period T . Theelectricity bill of the multiserver system in the sales periodT is hence formulated as
Bill = ET ·CT (ET )
= M((Pdyn − Psta)ρ+ Psta) ·T ·CT (ET ). (19)
2.3 Reward and Penalty Mechanism
Oftentimes, users have different sensitivities to postponingtheir requests. For users whose service requests can bedeferred to a certain extent, the cloud service provider willreward them based on the degree of deferment. However,once the deferment of service requests exceeds a threshold,the cloud service provider will compensate users based onthe degree of the deferment. In the following, we will dis-cuss the reward and penalty mechanism from perspectivesof users and the cloud service provider, respectively.
2.3.1 User Reward ModelWe divide users into I types, each type of users has asensitivity to the service deferment of their service requests.We define a sensitivity factor, denoted by ψi, to quantify thesensitivity of users of type i to the deferment of their servicerequests, which is denoted by Di. For the users of type i,the factor ψi is negatively proportional to the sensitivity of
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the users to the deferment of their service requests. That is, alarger sensitivity factor indicates a more delay-sensitive userservice request. For users running interactive applicationswith no delay tolerance, set ψi = ∞.
The cloud service provider will return more rewardsto those users who are less sensitive to service deferment.Let Li be the monetary loss of the users of type i due totheir degree of sensitivity to service deferment. The userswith a larger degree of sensitivity (ψi) to service deferment(Di) will get less rewards from the cloud service provider,resulting in a larger amount of monetary loss (Li) of theusers of type i. The monetary loss function of the users oftype i is given by Li = ψiDi.
We define a reward function, denoted by �i, to representthe reward the cloud service provider returns to users oftype i. The reward �i is a function of the service defermentDi, and is given by �i = θ · log(1+Di), where θ > 0 and 0 ≤Di ≤ Dmax hold. θ is called the reward factor and Dmax isthe maximum value of service deferment. The log functionis adopted to prevent users from setting excessive servicedeferment, which is unfair to cloud service providers.
Users need to make decisions on their own service defer-ment Di to get the maximum reward from the cloud serviceprovider based on the monetary loss and reward function.Thus, the optimization problem is to maximize (�i − Li)subject to (0 ≤ Di ≤ Dmax), where Di is regarded as acontinuous variable to simplify the optimization problem,and the solution to the problem is given by
Di = max(min(θ
ψi− 1, Dmax), 0). (20)
Substitute Di back into reward function �i, then one has
�i = θ log(1 + max(min(θ
ψi− 1, Dmax), 0)). (21)
Let Reward denote the total monetary reward returnedto users from the cloud service provider over the salesperiod T , then one has
Reward =�N
N ′=1
�I
i=1(�i − Li)λ
iu[N
′]τ, (22)
where λiu[N
′] denotes the arrival rate of the users of type i
in the N′th time slot. In practice, the cloud service provider
can learn the sensitivity factor ψi from experiments orhistorical data. The adopted user reward model is similarto the one presented in [35].
2.3.2 Cloud Service Provider Penalty Model
The high degree of user satisfaction is determined by thefast response of a cloud service provider to service requests.Once the service response time exceeds the threshold valuespecified in service-level agreement, users will be com-pensated by the cloud service provider for low quality ofservice. Given server benchmarking speed (s0), the averageresponse time of service requests (R), and the number ofinstructions for each service request (r), the penalty function
of users of type i, denoted by ui, can be formulated as [5]
ui(r,R) =
���
0, 0 � R � rs0
+Di
d(R− rs0
−Di),rs0
+Di < R � (1 + ωd )
rs0
+Di
ω, R > (1 + ωd )
rs0
+Di
,
(23)
where d is the degree of punishment, Di denotes the servicedeferment of users of type i, and ω is the cloud service pricecharged by the cloud service provider to users. In the futurewe will adopt super linear function to describe the relationbetween compensation and average response time.
The details of Equation (23) are described below. Forusers of type i, if the average response time satisfies 0 � R �rs0+Di, the cloud service provider will regard this execution
of the service request as a successful process with highquality of service and users will not be compensated by thecloud service provider. Otherwise, if the average responsetime satisfies r
s0+ Di < R � (1 + ω
d )rs0
+ Di, the cloudservice provider will regard this execution of the servicerequest as a process with low quality of service. In this case,the compensation provided by the cloud service providerto users will increase linearly as the average response timeR increases. Finally, if the average response time satisfiesR > (1 + ω
d )rs0
+Di, the cloud service provider will regardthis execution of the service request as a failed process andwill not charge users for this execution.
We use Penalty to denote the total compensation pro-vided by the cloud service provider to users, then we have
Penalty =�N
N ′=1
�I
i=1ui(r,R)λi
u[N′]τ, (24)
where λiu[N
′]τ is the average number of user requests of
type i in the time slot τ , and ui(r,R) is the incurred penaltyfor the service requests due to low quality of service.
2.3.3 Gross Profit
The gross profit a cloud service provider earns is the totalrevenue subtracted by the cost of generating that revenue.In other words, gross profit is sales minus cost of thecloud service sold. Assuming the price of cloud service isconstant in a sales period T , the revenue earned is given byω ·Eω(m), where ω denotes the service price per user andEω(m) indicates the expected number of actual buyers atprice ω during the sales period T .
Besides the reward for flexible users and penalty forlow quality of service mentioned above, the cost of cloudservice provider sold also consists of the cost paid to rentcloud computing infrastructures, and the electricity expenseincurred by the cloud service provider to maintain theoperation of the computing infrastructures. We define theprofit of the cloud service provider in a sales period T as therevenue minus the various expenses including the rewardcost, penalty cost, electricity cost, and rental cost, that is,
Profit = Revenue−Reward− Penalty −Bill −Rent,(25)
where Revenue, Reward, Penalty, Bill, and Rent aregiven in Equations (6), (22), (24), (19), and (14), respectively.
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3 PROBLEM DEFINITION AND OVERVIEW OF THEPROPOSED APPROACH
In this section, we formally define the profit maximizationproblem, followed by a brief overview of our proposedapproach to the profit maximization problem.
3.1 Problem Definition
The price of a cloud service interplays with users whopurchase the service, which in turn affects the revenue ofthe cloud service provider. This paper aims to maximize theprofit of the cloud service provider by deriving the optimalnumber of servers, operating speed of servers, and priceof cloud services provided without violating service-levelagreement. Assume that the cloud service provider opti-mizes its decisions at the beginning of each sales period T .Let b1 denote the upper bound on the power consumption ofthe M servers, and b2 be the upper bound on the expectedresponse time of user requests. Our optimization problemcan be formulated as
Maximize: Profit (26)Subject to: θ ≥ 0 (27)
Ptot ≤ b1 (28)
R ≤ b2 (29)
0 ≤ φi[N′] ≤ λi
u[N′]τ, ∀i ∈ I,N ′ ∈ N (30)
N+�Di�
N ′=1
(λiu[N
′]τ − φi[N
′]) ≥
N
N ′=1
λiu[N
′]τ,
∀i ∈ I,N ′ ∈ N (31)
where Profit, Ptot, and R are given in Equations (25), (16),and (13), respectively.
In the above formulation, the reward factor θ is non-negative (Equation 27), the total energy consumption Ptot ofthe multiserver within the sales period T can not exceed b1(Equation 28), and the average service response time R cannot exceed b2 (Equation 29). Equation (30) ensures that theamount of delayed service requests is nonnegative and notlarger than the total number of service requests in each timeslot τ , where φi[N
′] and λi
u[N′]τ are the number of delayed
and total service requests in the N′th time slot, respectively.
Equation (31) ensures that the processing of the arrived userrequests of type i in a sales period T can not be delayedlonger than the allowed service deferment Di of user servicerequests. We will then use the augmented Lagrange multi-plier method to solve the optimization problem, which willbe described in detail in Section 4.
3.2 Overview of the Proposed Approach
The optimization problem given in Equation (26) tries tomaximize the Profit of the cloud service provider un-der the constraints mentioned above. Figure 1 outlines theoverview of our proposed approach to solve the optimiza-tion problem. We first establish user demand distributionbased on the concept of user perceived value. Subsequently,based on user demand distribution, the revenue model andthe expenditure model are developed to construct the profitmaximization problem. This optimization problem is then
solved using the augmented Lagrange multiplier method.Finally, since the parameters of electricity bill and rental feeschange over time, these parameters are monitored and a dy-namic closed loop control scheme is proposed to adapt theservice price and mulitserver configurations to the changesin these parameters. The details of the proposed scheme areprovided in Section 4.
Figure 1: Overview of the proposed approach.
4 USER PERCEIVED VALUE-AWARE PROFIT OPTI-MIZATION SCHEME
In this section, we first leverage the augmented Lagrangemultiplier method to compute the optimal solution, includ-ing the service price, the number of servers, and the speedof servers. Subsequently, a dynamic closed loop controlscheme is proposed to adapt the service price and mul-tiserver configurations to the dynamic cloud computingenvironment.
4.1 Build Augmented Lagrange Function
Unconstrained optimization can be solved in many ways,such as steepest descent method [36], Newton’s method[37], multiplier method [38], etc. However, it is difficultto optimize constrained optimization directly. A commonmethod to solve constrained optimization is to transform theconstrained optimization problem into an unconstrained op-timization problem. Numerous techniques on constrainedoptimization have been investigated in the literature [39],[40], [41]. Of these techniques, the augmented Lagrangemultiplier method is a powerful tool for solving this class ofproblems, which is adopted in this work to solve the profitoptimization problem. Refer to Equation (26), we first buildaugmented Lagrange function to convert the constrainedoptimization problem into an unconstrained optimizationproblem, and then use the multiplier method to solve theunconstrained optimization problem.
The Bill given in Equation (19) is a function of powerconsumption of the multiserver system, the length of thesales period T , and real-time price of electricity. Since real-time price is flat within each sales period T and T itselfis constant, the Bill for T is fixed and can be expressedas Bill = b3Ptot, where Ptot given in Equation (16) is thetotal power consumed by the multiserver system and b3 is a
7
constant coefficient. The profit optimization problem givenin Equation (26) can then be re-written as���������
O(ω,M, s) = ωEω[m]− b3Ptot − δMT
−�NN ′=1
�Ii=1(�i − Li)λ
iu[N
′]τ
−�NN ′=1
�Ii=1 ui(r,R)λi
u[N′]τ
g1(M, s) = b2 − x1(1 +PM
M(1−ρ)2 ) ≥ 0
g2(M, s) = b1 −M((ξsγ − Psta)ρ+ Psta) ≥ 0
,
(32)
where O(ω,M, s) denotes the objective function of Profitgiven in Equation (25), and g1(M, s) and g2(M, s) are con-straint equations of M and s, respectively.
Next, we convert the problem given in Equation (32)with inequality constraints into an augmented Lagrangefunction. Let y be the vector that converts the problemwith inequality constraints to a problem with equalityconstraints, and v be the Lagrange multiplier vector, theaugmented Lagrange function is thus given by
φ(ω,M, s, y, v, σ) = O(ω,M, s)−2�
j=1
vj(gj(M, s)− y2j )
+σ
2
2�j=1
(gj(M, s)− y2j )2, (33)
where the constant parameter σ denotes the penalty factorand σ > 0 holds. The augmented Lagrange function givenin Equation (33) can be converted into the form of
φ(ω,M, s, y, v, σ) = O(ω,M, s)
+2�
j=1
[σ
2[y2j −
1
σ(σgj(M, s)− vj)]
2 − v2j2σ
], (34)
by using the method of completing the square, a techniqueto derive the quadratic formula [42], and the function givenin (34) can be easily maximized when
y2j =1
σmax(0, σgj(M, s)− vj), j = 1, 2. (35)
Plugging y2j given in Equation (35) back into the originalEquation (33), one obtains the augmented Lagrange function
φ(ω,M, s, v, σ) = O(ω,M, s)
+1
2σ
2�j=1
[[max(0, vj − σgj(M, s))]2 − v2j ]. (36)
Through this quadratic relaxation of the original problemgiven in Equation (32), we can derive analytical form ofsolutions to the profit maximization problem. We aim tomaximize the profit of the cloud service provider and obtainthe optimum solutions including service price ω, number ofservers M , and speed of servers s. Specifically, we seek tosolve the augmented Lagrange function given in Equation(36) by first computing partial derivatives of Equation (32)with respect to ω, M , and s. Here, the details are omitted.We only show key steps for solving the partial derivativesof Equation (32) with respect to ω, M , and s.
Calculate the partial derivative of Equation (32) withregard to ω: The partial derivative of Eω[m] with regardto ω is ∂Eω [m]
∂ω = −αβtf(ω), thus, the partial derivative of
O(ω,M, s) with regard to ω is
∂O(ω,M, s)
∂ω=
∂Eω[m]
∂ω·ω + Eω[m]
= αβt(1− ωf(ω)− F (ω)),
where f(ω) and F (ω) are the probability density and thecumulative distribution function at ω, respectively.
Calculate the partial derivative of Equation (32) withregard to M : The partial derivative of g1(M, s) with regardto M can be expressed as
∂g1(M, s)
∂M=
∂
∂M[−x1
M(
1
[√2πM(1− ρ)(eρ/eρ)M + 1](1− ρ)
)]
+x1
M2
PM
(1− ρ)2. (37)
Let D1 =√2πM(1−ρ)(eρ/eρ)M +1 =
√2πM(1−ρ)L+1,
D2 = 1−ρ, and L = (eρ/eρ)M , then Equation (37) becomes
∂g1(M, s)
∂M=
∂
∂M[−x1
M(
1
D1D2)] +
x1
M2
PM
D22
. (38)
The partial derivative of L, D1, and D2 with regard to Mare calculated as follows:
∂L
∂M= L(ρ− ln ρ− 1) + LM(1− 1
ρ)∂ρ
∂M,
∂D1
∂M=
√2π(
1
2√M
(1− ρ)L+√M(− ∂ρ
∂M)L+
√M(1− ρ)
∂L
∂M)
=√2π(
1
2√M
(1 + ρ)L−√M(1− ρ) ln ρL),
∂D2
∂M= − ∂ρ
∂M=
ρ
M.
Substitute ∂L∂M , ∂D1
∂M , and ∂D2
∂M back into the Equation (38),one has
∂g1(M, s)
∂M=
x1
MD1D2[∂D1
∂M D2 +∂D2
∂M D1
D1D2+
1
M].
The partial derivative of g2(M, s) with regard to M can beeasily calculated as
∂g2(M, s)
∂M= (ξsγ − Psta) · ρ+ Psta.
Caculate the partial derivative of Equation (32) withregard to s: The partial derivative of L, D1, and D2 withregard to s are calculated as follows:
∂L
∂s= LM(1− 1
ρ)∂ρ
∂s=
LM
s(1− ρ),
∂D1
∂s=
√2πM [(−∂ρ
∂sL+ (1− ρ)
∂L
∂s)] =
√2πM [ρ+M(1− ρ)2]
L
s
∂D2
∂s= −∂ρ
∂s=
ρ
s.
The partial derivatives of g1(M, s) and g2(M, s) with regardto s are hence computed as
∂g1(M, s)
∂s= −x1
M· ∂
∂s(
PM
(1− ρ)2) = −x1
M[∂D1
∂s D2 +∂D2
∂s D1
(D1D2)2],
8
∂g2(M, s)
∂s= Mρξγsγ−1.
Once we obtain the above partial derivatives of Equation(32) with regard to ω, M and s, we can compute and obtainthe optimal solutions by letting these partial derivatives ofEquation (32) with regard to ω, M , and s equal 0.
4.2 Solve Augmented Lagrange Function
We present in this section an augmented Lagrange multi-plier method based algorithm that solves the profit opti-mization problem given in (33) and derives its optimum so-lutions, including the service price and multiserver configu-rations. The proposed algorithm first computes an optimumLagrange multiplier, which guarantees that the solution oforiginal objective function and the solution of Lagrangefunction are consistent in the case where the optimal mul-tiplier is obtained. Subsequently, the optimal service priceand multiserver configurations are determined.
Let M (k), s(k), and v(k) indicate the kth iteration ofM , s, and v in the algorithm. Let ε, η, and Ψ be threepositive numbers, l be the number of iterations, and L bethe maximum number of iterations. Algorithm 1 describesthe proposed augmented Lagrange algorithm. Inputs to thealgorithm are electricity price Cτ during time slot τ , therent δ, and user requests arrival rate λu. The algorithmiteratively derives the optimal cloud service price ω andmultiserver configurations which includes the optimal num-ber of servers M , the server speed s, and the Profit of thecloud service provider.
The algorithm works as follows. It first formulates theoptimization problem into the form in Equation (26), thensets parameters of ε, η, Ψ, and L, and initializes variablesof M (0), s(0), v1, and l (lines 1-3). In each round of itera-tion, the algorithm calls the augmented Lagrange functionsolver, denoted by ALF-Solver(φ(ω,M (l−1), s(l−1), v(l), σ)),to obtain a local optimum of the ω, M , and s (line5). The ALF-Solver(φ(ω,M (l−1), s(l−1), v(l), σ)) derivesthe local optimum by computing partial derivatives ofφ(ω,M, s, v, σ) with regard to ω, M , and s, and solving asystem of equations of ω, M , and s (lines 18-21).
The algorithm finishes if the Lagrange multiplier vectorv converges and approximates the optimum by an errorof ε. Let Qj(M
(l), s(l)) = gj(M(l), s(l)) − y2j for j = 1, 2
be the penalty item of the augmented Lagrange functiongiven in Equation (33), then the Lagrange multiplier vec-tor v converges if ‖Q(M (l), s(l))‖ < ε holds (lines 6-10).If it does not converge or converges too slowly, that is,‖Q(M (l), s(l))‖/‖Q(M (l−1), s(l−1))‖ ≥ Ψ holds for a pos-itive number Ψ, the penalty factor σ is updated to ησ forη > 1 to speed up the convergence process (lines 11-13).Accordingly, the Lagrange multiplier for the next iterationis updated to vl+1
j = max(0, v(l)j −σgj(M
(l), s(l)))(j = 1, 2)(lines 14-15), and the procedure moves to the next iteration.Once the algorithm converges, the optimum of ω, M , and sare obtained, and the optimal Profit of the cloud serviceprovider can be calculated by using Equation (25) (line7). Line 17 returns the optimal service price, multiserverconfigurations, and Profit of the cloud service provider.
Algorithm 1: Iteratively solve the augmented Lagrangefunction
Input: Electricity price Cτ during sales period τ , rent δ,user requests arrival rate λu;
Output: The optimal service price ω, number of serversM , server speed s, and Profit;
1 Formulate the optimization problem into the form inEquation (26);
2 Set parameters α, β, γ, ε, η, Ψ, and L;3 Initialize M (0), s(0), v(1), and l = 1;4 while l < L do5 [ω(l),M (l), s(l)] = ALF-Solver(φ(ω,M (l−1), s(l−1), v(l), σ));
// exit when {v(l)} converges;6 if ‖Q(M (l), s(l))‖ < ε then7 Calculate the Profit using the Equation (25);
// record the optimal solution;8 Result = [Profit, ω(l),M (l), s(l)];9 break;
10 end// otherwise, increase penalty factor
σ;11 else if ‖Q(M (l), s(l))‖/‖Q(M (l−1), s(l−1))‖ ≥ Ψ then12 σ = ησ;13 end
// update the multiplier vector v;14 vl+1
j = max(0, vlj − σgj(M(l), s(l)))(j = 1, 2);
15 l = l + 1;16 end17 return Result;
// solve the Lagrange function in (36);18 ALF-Solver(φ(ω,M (l−1), s(l−1), v(l), σ))19 Compute partial derivatives of φ w.r.t. ω,M , and s as
∂φ(ω,M (l−1), s(l−1), v(l), σ)/∂(ω,M, s);20 Calculate ω, M , and s based on a system of equations of
∂φ∂ω
, ∂φ∂M
, and ∂φ∂s
;21 return [ω,M, s];
4.3 Design a Dynamic Closed Loop Control Scheme
The solution to the profit maximization problem describedabove focuses on the interaction between users and thecloud service provider. However, the impact of dynamiccloud computing environment such as fluctuating electricitybill and rental fees on profit maximization mechanism is notinvestigated. On one hand, the variation of electricity billor rental fees has a direct impact on the expenditure of thecloud service provider. On the other hand, the variation ofelectricity bill or rental fees has an indirect influence on userperceived value which affects the user demand of the cloudservice, and ultimately impacts the revenue of the cloudservice provider. Thus, it is necessary to design a scheme toadjust service price and multiserver configurations accord-ing to the dynamics of the cloud computing environment.
In this subsection, a closed loop control scheme is de-signed to dynamically update the optimal service priceand multiserver configurations. As illustrated in Figure 2,the runtime control scheme monitors the dynamic cloudcomputing environment. Once the electricity bill or rentalfees changes, the proposed control scheme first fits a newprobability distribution function of user perceived valueusing kernel density estimation based on the historical pricedata set Ω. Subsequently, it reconstructs and resolves theprofit maximization problem based on the new probability
9
Figure 2: Overview of closed loop control scheme
distribution and the variation of electricity bill or rental fees.The kernel density estimation technique adopted in the
proposed control scheme is a stochastic non-parametric wayto estimate the probability density function of a randomvariable [26]. It is a fundamental data smoothing techniquewhere inferences about the population are made based ona finite data sample. Given a univariate independent andidentically distributed sample drawn from some distribu-tion with an unknown density function, the technique canbe used to estimate the shape of the density function.
The details of the proposed runtime control schemeare described in Algorithm 2. Inputs to the algorithm arethe historical price data set Ω and the output of systemmonitor. The closed loop control scheme works as follows.It monitors whether parameters of the cloud computingenvironment change at all times (line 2). If no change,the system will run with the current multiserver config-urations (lines 3-5). Otherwise, it updates and solves theprofit maximization problem (lines 6-15). Lines 7-8 fit theprobability density function (pdf) of user perceived valueusing MATLAB function ksdensity( · ) based on historicalprice data set Ω. Line 9 updates the profit maximizationproblem according to the change of the cloud computingenvironment (i.e., electricity bill or rental fees). Line 10solves the profit maximization problem using algorithm1. The algorithm updates the optimal cloud service priceand multiserver configurations in line 11, and calculatesthe profit of the cloud service provider using Equation (25)in line 12. Finally, it inserts the service price ω into thehistorical price data set Ω in line 13. Line 14 returns theoptimal cloud service price, multiserver configurations, andProfit of the cloud service provider.
The MATLAB function ksdensity( · ) is used to fit theprobability density function of user perceived value basedon historical price data by using kernel smoothing densityestimation. Line 18 first calculates the number of samplesin the historical price data set Ω. The kernel bandwidthh is a free parameter that exhibits a strong influence onthe resulting estimate [26]. Here, Gaussian basis functionsare used to approximate univariate data. Thus, the optimalchoice for kernel bandwidth h is calculated as line 19, whichminimizes the mean integrated squared error used in den-sity estimation [43]. std(Ω) computes the standard deviationof the samples in Ω. Line 20 uses operator @(x) to define thefunction handle g, which represents a normal probability
density function. exp(x) and sqrt(x) represent exponentialfunction and square root function, respectively. Based onnormal probability density function g and bandwidth h, line21 computes the kernel density, that is probability densityfunction by defining the function handle ksden. mean(x) isused to compute the average of the array. Line 22 returnsthe final fitted probability density function.
Algorithm 2: Dynamic closed loop control schemeInput: The historical price data set Ω, the output of
system monitor;Output: The optimal service price ω, number of servers
M , server speed s, and Profit;1 while true do2 Monitor if parameters of cloud computing environ-
ment change;3 if no change then4 continue;5 end6 else7 Ω = historical price data set;
// fit pdf using ksdensity(Ω);8 fX(ω) ← ksdensity(Ω);9 Update profit maximization problem given in (26);
10 Solve profit maximization problem using Algo-rithm 1;
11 Update the optimal service price ω, number ofservers M , and server speed s;
12 Calculate the Profit using Equation (25);13 Insert price ω into historical price data set Ω;14 return [ω,M, s, Profit];15 end16 end
// fit pdf of user perceived value usingMATLAB function ksdensity(Ω);
17 ksdensity(Ω);// get the sample number of Ω;
18 n = length(Ω);// set the optimal bandwidth h;
19 h = std(Ω) · (4/3/n)ˆ(1/5);// obtain the normal pdf;
20 g = @(x)(exp(−.5 ·x.ˆ2)/sqrt(2 · pi));// compute kernel density with g and h;
21 ksden = @(x)mean(g((x− Ω)/h)/h);22 return ksden;
5 SIMULATION-BASED EVALUATION
Extensive simulation experiments have been conducted tovalidate the effectiveness of the proposed scheme. We firstdescribe simulation settings in detail, and then verify theeffectiveness of the proposed user perceived value-baseddynamic pricing model, followed by the validation of theoptimal pricing and multiserver configurations and a com-parison study with benchmarking schemes in terms of theprofit of the cloud service provider.
5.1 Simulation Settings
The simulation experiments are conducted on a machineequipped with 2.56GHz Intel i7 quad-core processor and8GB DDR4 memory, and running a Windows version ofMatlab x64. For the sake of a fair comparison, three typesof users used in [35] are also adopted in our simulation
10
experiments. Users of type 1 are delay-sensitive while usersof type 2 and 3 are delay-insensitive to the deferment ofthe service requests. Data of type 1 are extracted fromYoutube U.S. traffic from January 1, 2014 to January 31, 2014[44]. Data of type 2 and 3 are extracted from GMaps andGMail U.S. traffic from January 1, 2014 to January 31, 2014[44], respectively. The one day ahead real-time pricing datareleased by Ameren Illinois Power Corporation at January2014 are taken as the price input in the experiment [45].We also assume that user perceived value X obeys thefollowing normal distribution, X ∼ N(0, 0.22) [6], [22].In addition, the value of other parameters used in oursimulation experiment are shown in Table 1.
Table 1: Experimental parameters table.
Parameter Definition Value
T sales period 30 dτ time slot 1 hN number of time slot 720Dmax maximum value of service deferment 24ψ1 sensitivity factor of users of type 1 ∞ψ2 sensitivity factor of users of type 2 0.1ψ3 sensitivity factor of users of type 3 0.11
5.2 Verify User Perceived Value-Based Dynamic Pric-ing ModelThis subsection verifies the proposed user perceived value-based dynamic pricing model from the perspective of sup-ply and demand law.
5.2.1 Revenue Vs. Service RequirementWe first analyze the relationship between the service re-quirement in terms of the number of instructions, whichis denoted by r, and the revenue of the cloud serviceprovider. In addition to the parameters given in Table 1,we set the average service requirement denoted by r to 1billion instructions. The number of servers M is initializedto 7, the base speed s0 and speed s of servers are bothinitialized to 1 billion instructions per second, and the staticpower consumption Psta is set to 2W. The parameters ofdynamic power consumption are assumed to be γ = 2.0and ξ = 9.4192, and parameters of Gamma distribution areassumed to be α = 2.0 and β = 1.5 [5].
Figure 3(a) shows the relationship between service re-quirements and the revenue of the cloud service providerwhen service request arrival rate λu is 16.15, 16.35, 16.55,16.75, and 16.95 billions instructions per second, respec-tively. It can be seen from Figure 3(a) that the revenueincreases as service requirements increase. This indicatesthat the usage of cloud services and the revenue obtainedare positively correlated under the user perceived value-based pricing model. In addition, as shown in the figure,the revenue decreases as λu increases. This is because withthe increase of λu, servers can not process service requests intime, leading to a higher response time and lower quality ofservice. Low quality of service will result in a smaller num-ber of users to purchase cloud services, thus, the revenue ofthe cloud service provider decreases accordingly.
Figure 3(b) shows the relationship between service re-quirements and the normalized service price when service
(a) Revenue vs. service require-ment.
(b) Normalized price vs. service re-quirement.
Figure 3: Relationship between service requirement andrevenue/normalized price.
request arrival rate λu is 16.15, 16.35, 16.55, 16.75, and16.95 billions instructions per second, respectively. From thefigure, we can see that when service requirement r < 1.4billions, the normalized price fluctuates with the increase ofthe service requirement. When service requirement 1.4 <r < 2.6 billions, the normalized price increases with the in-crease of the service requirement. When service requirementr > 2.6 billions, the normalized price eventually convergesto a stable value with the increase of the service requirement.
5.2.2 Purchase Amount and Revenue Vs. Service PriceFigure 4(a)-4(d) demonstrate how the relationship amongthe cloud service purchase amount, revenue, and the priceof cloud service changes when service request arrival rateλu is 16.15, 16.55, 16.75, and 16.95 billions instructions persecond, respectively. As we can see from these figures,before the service price reaches user perceived value of theservice, the purchase amount of the cloud service increaseswith the increases of the price. Once the price exceedsuser perceived value of the service, the purchase amountdeclines sharply. This observation is consistent with realmarket situation, that is, users are willing to accept a priceand purchase when the price is lower than their perceivedvalue. However, the user’s purchase intention will declinesharply when the price is beyond user perceived value.
It also can be seen from Figure 4(a)-4(d) that the pointwhere purchase amount is maximum is not necessarily thepoint where the revenue is maximum. That is, the revenuefor the scenario of the low price and high purchase amountis not necessarily higher than the revenue for the scenarioof the high price and low purchase amount. When servicerequest arrival rate λu = 16.55, the cloud service providercan get the maximum revenue.
5.2.3 Purchase Amount and Revenue Vs. Service requestarrival rateFigure 5(a) and 5(b) demonstrate that how the cloud ser-vice purchase amount and revenue change when servicerequest arrival rate λu is 16.15, 16.35, 16.55, 16.75, and 16.95billions instructions per second, respectively. From Figure5(a), we can see that the optimal prices for the maximumservice purchase are different under diverse service requestarrival rate λu. For the case where λu = 16.55 billionsinstructions per second, the service purchase amount and
11
(a) λu = 16.15. (b) λu = 16.55. (c) λu = 16.75. (d) λu = 16.95.
Figure 4: Purchase amount and revenue vs. service price.
the service price ω reach the maximum value at the sametime when compared to cases of different service requestarrival rates. Meanwhile, the maximum purchase amountat λu = 16.35 is approximately the same as the maximumpurchase amount at λu = 16.75. This situation holds for thecase where λu = 16.15 and λu = 16.95. This is becausewith the increase of λu, limited number of servers can notprocess arrived service requests in time, leading to a higherresponse time, lower quality of service, and thus a lowermaximum purchase amount of cloud services.
(a) Purchase amount vs. servicerequest arrival rate.
(b) Revenue vs. service request ar-rival rate.
Figure 5: Purchase amount and revenue vs. service requestarrival rate.
The revenue in Figure 5(b) is obtained by multiplyingthe purchase amount and service price in Figure 5(a). Figure5(b) shows that the optimal prices for the maximum revenueare different under various service request arrival rate λu.From this figure, we observe that with the increase of λu, themaximum revenue at different λu increases first and thendecreases. The cloud service provider obtains the maximumrevenue when λu = 16.55 billions instructions per second.Similarly, this is because with the increase of λu, limitednumber of servers can not process arrived service requestsin time, leading to a lower maximum purchase amountof services, and thus a lower revenue. Based on aboveexperimental results, our user perceived value-based pricingmodel conforms to the supply and demand law in market.
5.3 Validate Multiserver Configurations for Profit Maxi-mization
We set the response time constraint for user requests, de-noted by b1, to 0.33 seconds and the power consumption of
the server system, denoted by b2, to 106W. The rental costdenoted by δ is set to 1.5 cents per second [6].
Figure 6(a) shows the relationship between profit andthe number of working servers. It can be seen from thefigure that when service request arrival rate λu = 12.9,13.9, 14.9, 15.9, and 16.9 billions instructions per second,the optimal number of servers denoted by M is 16, 17,19, 18, and 17, respectively. It is clear that when M issmall, the utilization of working servers is approaching 1,leading to a long response time for user requests and lowquality of service accordingly, and in turn a low profit underthe user perceived value-based dynamic pricing model. AsM increases, the number of user requests in the waitingqueue decreases quickly, the user requests do not have towait too long, and thus the profit increases under the userperceived value-based dynamic pricing model. However,as M continues increasing, the profit does not increase.This is because the increase in the number of servers leadsto an increase in the maintenance cost of working serversincluding electricity and rental cost.
Figure 6(b) shows the relationship between profit andthe server speed s. We notice from the figure that in orderto maximize the profit, the optimal speed s is set to 0.7642,0.9435, 1.1044, 1.1293, and 1.2838 billions instructions persecond when the service request arrival rate λu = 12.9,13.9, 14.9, 15.9, and 16.9 billions instructions per second,respectively. It is clear that when the server speed s islow, the utilization of servers is approaching 1, leading toa long response time for user requests and low qualityof service accordingly, and in turn a low profit under theuser perceived value-based pricing model. When the serverspeed s is high, service requests are more likely to beexecuted on time, leading to an increase in the profit underthe user perceived value-based pricing model. However,with the continued increase in s, the profit does not increaseas expected. This is because the increase in s leads to anincrease in the cost of operating a multiserver system.
Figure 6(c) gives the optimal M and s of servers thatmaximize the profit when λu = 16.9 billions instructions persecond. It can be seen that the maximal profit is obtainedwhen s and M is set to 1.4351 billions instructions persecond and 17, respectively. That is, 687.9 cents of profit isobtained when 17 servers are open and each server runs at1.4351 billions instructions per second.
12
(a) Profit vs. number of severs (M ). (b) Profit vs. server speed (s). (c) Optimal server configurations.
Figure 6: Validate server configurations for profit maximization.
(a) λu=16.9, M=17. (b) λu=12.55, M=18.
Figure 7: Compare the maximal profit with two benchmark-ing pricing models.
5.4 Compare the Maximal Profit with BenchmarkingPricing Strategies
The proposed user perceived value-based profit maximiza-tion scheme is compared with two benchmarking methodsOMCPM [5] and UPMR [35]. OMCPM [5] is an efficientpricing model that takes such factors into considerationsas service-level agreement and customer satisfaction. It de-rives an optimal server configuration and service price forprofit maximization. UPMR [35] is a usage based pricingmodel used by today’s major cloud service providers. TheUPMR model rewards users proportionally based on thetime length that users set as deadlines for completing theirservice requests. Compared with OMCPM and UPMR, ourpricing method is based on user perceived value that reflectsusers willingness to purchase cloud services.
We compare the maximal profit generated by proposedpricing model with that generated by the two benchmarkingpricing models under the same experimental settings. Twocomparison experiments are conducted. In the first exper-iment, user service request arrival rate λu is set to 16.9billions instructions per second and the number of workingservers M is set to 17. In the second experiment, λu isset to 12.55 billions instructions per second and M is setto 18. It is clear from Figure 7 that our proposed dynamicpricing model is superior to the two benchmarking models.For instance, the proposed pricing model can obtain upto 21.55 cents per second more (31.32%) as compared toOMCPM method, and 15.66 cents per second more (22.76%)as compared to UPMR when λu = 16.9 billions instructionsper second, M = 17, and s = 0.93 billion instructions persecond. Thus, the pricing strategy based on user perceivedvalue can better reflect the market demand and the cloud
service provider can obtain higher profit.We further verify how the expected number of actual
buyers (Eω(m)) and the corresponding revenue changewhen user perceived value obeys normal distributions withdifferent parameters. Figure 8(a)-8(f) show the expectednumber of actual buyers (Eω(m)) under different expec-tations μ and variances σ2 of user perceived value in ourproposed dynamic pricing model. From Figure 8(a)-8(c), wecan see that under different expectations μ, as μ increases,the cloud service provider needs to increase the serviceprice ω to obtain the same amount of purchases. FromFigure 8(d)-8(f), we can see that under different variancesσ2, when service price ω is less than μ, as σ2 increases,the cloud service provider needs to decrease the serviceprice ω to obtain the same amount of purchases. However,when service price ω is greater than μ, as σ2 increases, thecloud service provider needs to increase the service priceω to obtain the same amount of purchases. This is becausethe larger the σ2, the more dispersed the perceived value’sdistribution. Thus, in the case of the same service price ω,the purchase amount changes accordingly.
Figure 9(a)-9(f) show the revenue under different ex-pectations μ and variances σ2 of user perceived value inour proposed dynamic pricing model. From Figure 9(a)-9(c), we can find that under the same purchase amount,the cloud service provider needs to increase expectationμ of normal distribution, that is, users’ perceived valueof services, to achieve higher revenue. From Figure 9(d)-9(f), we can see that under the same purchase amount, thecloud service provider needs to decrease variance σ2 ofnormal distribution to achieve higher revenue. In general,to obtain the higher revenue, the cloud service providerneeds to carry out market strategies to improve perceivedvalue of service in users’ mind. This is because under thesame purchase amount, that is, under the same number ofrequests that the cloud service provider needs to process, thecorresponding expenses are the same. Thus, it is reasonableto grow the profit of the cloud service provider by increasingthe revenue.
6 CONCLUSION
In this paper, we first propose a user perceived value-based dynamic profit maximization mechanism that takesinto account the interaction between users and the cloudservice provider. Subsequently, the augmented Lagrange
13
(a) X ∼ N(0, 1). (b) X ∼ N(10, 1). (c) X ∼ N(30, 1).
(d) X ∼ N(10, 0.01). (e) X ∼ N(10, 0.5). (f) X ∼ N(10, 1.0).
Figure 8: Verify the change in the expected number of actual buyers when user perceived value obeys normal distributionswith different parameters (i.e., μ and σ2).
multiplier method is leveraged to solve the optimizationproblem to derive the optimal solution, including the serviceprice, number of servers, and speed of servers. Finally, adynamic closed loop control scheme is designed to updatethe service price and multiserver configurations using ker-nel density estimation method. Extensive simulation resultsdemonstrate that our proposed profit maximization schemefollows the supply and demand law in market, and are ableto obtain 31.32% and 22.76% more profit compared to thestate of the art benchmarking methods OMCPM [5] andUPMR [35], respectively.
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(a) X ∼ N(0, 1). (b) X ∼ N(10, 1). (c) X ∼ N(30, 1).
(d) X ∼ N(10, 0.01). (e) X ∼ N(10, 0.5). (f) X ∼ N(10, 1.0).
Figure 9: Verify the change in the revenue when user perceived value obeys normal distributions with different parameters(i.e., μ and σ2).
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Peijin Cong received her B.S. degree from theDepartment of Computer Science and Technol-ogy, East China Normal University, Shanghai,China, in 2016. She is currently pursuing themaster degree with the Department of Com-puter Science and Technology, East China Nor-mal University, Shanghai, China. Her current re-search interests include power management inmobile devices and cloud computing.
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Liying Li received her B.S. degree from the De-partment of Computer Science and Technology,East China Normal University, Shanghai, China,in 2017. She is currently pursuing the master de-gree with the Department of Computer Scienceand Technology, East China Normal University,Shanghai, China. Her current research interestsinclude cyber physical systems and IoT resourcemanagement.
Junlong Zhou received his Ph.D. degree inComputer Science from East China Normal Uni-versity in 2017. He is currently an Assistant Pro-fessor with the School of Computer Science andEngineering, Nanjing University of Science andTechnology, Nanjing. His research interests in-clude real-time embedded systems, cyber phys-ical systems, and cloud computing. He has beenan Associate Editor for the Journal of Circuits,Systems, and Computers since 2017. He is amember of the IEEE.
Kun Cao is currently pursuing his Ph.D. de-gree with the Department of Computer Scienceand Technology, East China Normal University,Shanghai, China. His current research interestsinclude high performance computing, multipro-cessor systems-on-chip, and cyber physical sys-tems.
Tongquan Wei received his Ph.D. degree inElectrical Engineering from Michigan Technolog-ical University in 2009. He is currently an Asso-ciate Professor in the Department of ComputerScience and Technology at East China NormalUniversity. His current research interests includeInternet of Things, edge computing, cloud com-puting, and design automation of intelligent andCPS systems. He serves as a Regional Editorfor Journal of Circuits, Systems, and Computerssince 2012. He is a member of the IEEE.
Mingsong Chen received his Ph.D. degree inComputer Engineering from the University ofFlorida, Gainesville, in 2010. He is currently afull Professor with the Department of EmbeddedSoftware and Systems of East China NormalUniversity. His research interests include designautomation of cyber-physical systems, formalverification techniques and mobile cloud com-puting. He is a member of the IEEE.
Shiyan Hu received his Ph.D. in Computer En-gineering from Texas A&M University in 2008.He is an Associate Professor at Michigan Tech,and he was a Visiting Associate Professor atStanford University from 2015 to 2016. His re-search interests include Cyber-Physical Sys-tems (CPS), CPS Security, Data Analytics, andComputer-Aided Design of VLSI Circuits, wherehe has published more than 100 refereed pa-pers. His publications have been awarded sev-eral Best Paper Award and Best Paper Finalists,
which include the 2018 IEEE Systems Journal Best Paper Award. Prof.Hu is the Chair for IEEE Technical Committee on Cyber-Physical Sys-tems. He is an Associate Editor for IEEE Transactions on Computer-Aided Design, IEEE Transactions on Industrial Informatics, IEEE Trans-actions on Circuits and Systems, ACM Transactions on Design Automa-tion for Electronic Systems, and ACM Transactions on Cyber-PhysicalSystems. He is a Fellow of IET.
