THE emergence of cloud computing can already be felt

Transcription

1 158 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 Dynamic Cloud Pricing for Revenue Maximization Hong Xu, Member, IEEE, and Baochun Li, Senior Member, IEEE Abstract In cloud comuting, a rovider leases its comuting resources in the form of virtual machines to users, and a rice is charged for the eriod they are used. Though static ricing is the dominant ricing strategy in today s market, intuitively rice ought to be dynamically udated to imrove revenue. The fundamental challenge is to design an otimal dynamic ricing olicy, with the resence of stochastic demand and erishable resources, so that the exected long-term revenue is maximized. In this aer, we make three contributions in addressing this question. First, we conduct an emirical study of the sot rice history of Amazon, and find that surrisingly, the sot rice is unlikely to be set according to market demand. This has imortant imlications on understanding the current market, and motivates us to develo and analyze market-driven dynamic ricing mechanisms. Second, we adot a revenue management framework from economics, and formulate the revenue maximization roblem with dynamic ricing as a stochastic dynamic rogram. We characterize its otimality conditions, and rove imortant structural results. Finally, we extend to consider a nonhomogeneous demand model. Index Terms Dynamic ricing, revenue maximization, sot market, cloud comuting, ublic cloud, dynamic rogramming Ç 1 INTRODUCTION THE emergence of cloud comuting can already be felt with the burgeoning of cloud service offerings. Beyond technological advances, cloud comuting also shows romises to the economic landscae of comuting. Pricing is a crucial comonent of the cloud economy because it directly affects a rovider s revenue and a customer s budget. Though static ricing is the dominant strategy today, dynamic ricing emerges as an attractive alternative to better coe with unredictable customer demand. The motivation is intuitive and simle: ricing should be leveraged strategically to influence demand to better utilize unused caacity, and generate more revenue. Indeed, Amazon EC2 [2] has introduced a sot ricing feature, where the sot rice for a virtual instance is dynamically udated to match suly and demand as claimed in [4]. Given the flexibility to change the rice on the sot, the fundamental question is, what is the otimal dynamic ricing olicy for a rovider, in terms of maximizing the exected revenue amid random demand? A rovider naturally wishes to set a higher rice to get a higher rofit margin; yet in doing so, it also bears the risk of discouraging demand in the future. An imortant observation is that comuting resources, such as CPU cycles and bandwidth, are inherently erishable: if at some oint in time they are not utilized they are of no value. It is nontrivial to balance this intrinsic tradeoff with erishable caacity and stochastic demand.. H. Xu is with Deartment of Comuter Science, City University of Hong Kong, Kowloon, HKSAR, China. henry.xu@cityu.edu.hk.. B. Li is with the Deartment of Electrical and Comuter Engineering, University of Toronto, 1 King s College Road, Toronto, ON M5S 3G4, Canada. bli@eecg.toronto.edu. Manuscrit received 1 May 213; revised 14 Set. 213; acceted 7 Nov. 213; ublished online 14 Nov Recommended for accetance by A. Liang. For information on obtaining rerints of this article, lease send to: tcc@comuter.org, and reference IEEECS Log Number TCC Digital Object Identifier no /TCC To address this fundamental challenge, we adot a revenue management framework from economics that deals with the roblem of selling erishable resources, such as airline seats and hotel reservations, to maximize the exected revenue from a oulation of rice sensitive customers [43]. Dynamic ricing has become an active field of the revenue management literature, with successful realworld alications in industries such as travel, fashion, and so on [9], [16], [38]. Cloud comuting oses new challenges to solving revenue maximization roblems. First, little is known about how the sot rice is adjusted, and what factors are considered in the ricing algorithm, by a real-world rovider such as Amazon. Also, little is known about demand statistics, and how demand reacts to rice changes. In fact, though Amazon ublishes its sot rice history, very few insights are gained on imortant asects related to modeling of the market. Second, for a cloud rovider, revenue not only deends on the number of customers, but also on the duration of usage. Unlike hotel and car rental reservations where usage durations are known, the exact usage duration of an instance in a cloud is not secified a riori. Thus, not only the arrival but also the dearture of demand is stochastic, and has to be taken into account when collecting revenue. This clearly adds to the modeling comlexity. Our original contributions in this aer are threefold. First, we conduct an emirical study on Amazon s sot rice history. We collect the sot rice trace from both official and unofficial sources, sanning the time eriod from November 3, 29, the incetion of sot instances, to October 27, 211, across all the regions and instance tyes. Surrisingly, we find that, in contrast to the common belief [13], [45], Amazon s sot rice is unlikely to be set according to market suly and demand. Rather, rice oscillates within a very narrow band most of the time, which /13/$31. ß 213 IEEE Published by the IEEE CS, ComSoc, PES, CES, & SEN

2 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 159 Fig. 2. Sot rice of US-East-1 linux m1.small, from November 3, 29 to October 27, 211. Fig. 1. Sot rice of US-east-1.linux.m1.small, from ::, Setember 15, 211 to 23:59:59, Setember 21, 211. is more likely to be an artifact of some ricing algorithm with redetermined reserve rice. Price statistics reveal little about market demand and its relationshi with rice. This suggests that it is questionable to model the market based on the rice statistics of Amazon. Motivated by this observation, we consider the scenario where the cloud rovider with fixed caacity udates the sot rice according to market demand in this aer. Our second contribution is that we formulate the revenue maximization roblem as a finite-horizon stochastic dynamic rogram, with stochastic demand arrivals and deartures. We characterize otimality conditions for the stochastic roblem and rove imortant structural results. Our results show that the otimal ricing olicy exhibits time and utilization monotonicity, and the otimal revenue has a concave structure. They rovide insights on understanding the fundamental tradeoff between ricing to the future to attract more revenue from future demand, and ricing to the resent to extract more revenue from existing customers. We also extend our model to the case with nonhomogeneous demand. We conduct an asymtotic analysis on this more general but difficult roblem. We rove a surrising result that when the demand arrival and dearture rates are linear with system utilization, i.e., number of existing instances, the otimal rice is only a function of time and is indeendent of the system utilization. The remainder of the aer is structured as follows: Section 2 introduces our model after an emirical study of Amazon s sot rice history. In Section 3, we resent our formulation and analysis of the stochastic revenue maximization roblem with a homogeneous demand model. We extend the setting to a nonhomogeneous demand model in Section 4. Numerical results are rovided in Section 5. In Section 6, we discuss issues ertaining to the ractically of dynamic ricing, and in Section 7 we summarize related work. Finally, we conclude the aer in Section 8. 2 MODEL 2.1 Amazon EC2 Sot Price History Before we introduce our model and assumtions, we wish to first do a reality check and examine the sot rice history of Amazon EC2, currently the only rovider that offers services with dynamically changing rices to our knowledge [4]. Our urose is to obtain information on how and when a real-world rovider adjusts its rices, and extract imortant modeling assumtions and/or constraints that we need to consider for our analysis to be meaningful. Amazon sells virtual machines as instances, where different tyes of instances are allocated different amounts of resources. Both Windows and Linux are available. Amazon EC2 oerates in seven geograhical regions with different ricing [2]. Within each region, there are several availability zones with indeendent infrastructures. Amazon starts to offer sot instances and ublish sot rice data in December 29. We collect sot rice traces using the ec2-describe-sot-rice-history API call rovided by the EC2 API tools [3]. Note that this method rovides us 9 days worth of traces because Amazon only makes the most recent 9 days of sot rice history ublicly accessible [4]. To obtain data beyond this limitation, we download the rice data accumulated by interested arties because the incetion of sot instances [27], [41]. This unofficial trace runs from as early as November 3, Notice that this unofficial trace is also collected using the same EC2 API method, and shall be treated with the same credibility. 2 Our combined trace has data until October 27, 211. While we have studied rice data across all regions, availability zones, instance tyes, and oerating systems, the results do not vary across these dimensions. To kee the resentation concise, we only show results for the Linux small standard (API name: m1.small), large standard (m1.large), as well as the extra-large high-memory (m2.xlarge) instance, in the US East (Virginia) (US-East-1) and US West (N. California) (US-West-1) regions. We first consider a relatively short time eriod. Fig. 1 shows a seven-day rice history for US-East-1 linux m1.small instances. Price fluctuates between $.3 and $.86 frequently, demonstrating that Amazon does udate the rice dynamically along the time line. However, if we take a longer ersective, the story becomes drastically different. Fig. 2 shows the comlete sot rice history for the same linux m1.small instances in US-East-1 region. We observe temoral sikes from time to 1. Traces for certain regions such as Asia Pacific Southeast (a-southeast) are only available from when sot instances are made available there. 2. Sot rices also vary across availability zones. However, before the version of EC2 API tools, only the lowest sot rice across the region is returned from the ec2-describe-sot-rice-history method. We, thus, only lot the lowest regional sot rice throughout the study.

3 16 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 Fig. 3. Sot rice of US-East-1 linux m1.large, from November 3, 29 to October 27, 211. time, which we conjecture corresond to the eriods when caacity has to be reclaimed from sot instances to suort other business. Nevertheless, it is evident the rice oscillates within a very narrow band ($.29-$.31) most of the time, rior to Aril 211. After that rice oscillates within a wider band, again with clearly observable lower and uer bounds. Also notice that the temoral rice sikes exhibit a shae of imulse bands, suggesting that rice is instantly adjusted u, oscillates between a certain range, and is instantly adjusted down to the normal level. Figs. 3 and 4 show the rice history for the linux m1.large and m2.xlarge instances in the US-East-1 region. We observe the same trend where rice is almost a straight line most of the time. When rice sikes, it still oscillates within clearly observable uer and lower bound. Notice that in all three figures, we can easily identify a lower bound below which the rice never goes down. These observations imly that the Amazon sot instance market may not be a sot market where rice is set according to suly and demand, as Amazon suggests [4] and many believe in their studies [13], [45]. According to Amazon, users have to submit bids indicating the maximum rice they are willing to ay er instance er hour, and Amazon sets the rice according to suly and user bids. All requests with bids higher than or equal to the sot rice will run. 3 If this were the case, it is highly unlikely that the rice will constantly stay bounded. 4 Our hyothesis is that the sot rice is likely to be artificially controlled with a maximum and minimum rice beyond which it should never go. These maximum and minimum rices may be adjusted by Amazon according to an unknown algorithm. In fact, Ben-Yehuda et al. [8] reach the same conclusion with more thorough examination and modeling, where other ossible exlanations such as collaborative bidding are ruled out. Therefore, we believe it is questionable to study dynamic ricing for a sot cloud market-based closely on the model and mechanism of Amazon, or constraints derived from its sot rice data (e.g., closely bounded rice). As an early theoretical work on this toic, we intentionally choose not to model the secifics of Amazon, but instead to seek to 3. The waiting time for the sot instances to start is not guaranteed though. 4. Consider a real-world sot market, such as a commodity market for oil, cotton, and so on, or a stock exchange, where rice is indeed determined by bids from buyers and sellers. The sot rice evolves continuously and does not have any identifiable bounds, the existence of which would suggest arbitrage oortunities and an inefficient market [36]. Fig. 4. Sot rice of US-East-1 linux m2.xlarge, from February 23, 21 to October 27, 211. develo and analyze market-driven dynamic ricing mechanisms, which may rovide insights on establishing an efficient cloud sot market in the future. As such, it is also exected that our results are different from roerties of Amazon s sot rice. 2.2 Assumtions We now turn to introducing our model and assumtions for the sot market. We focus on an infrastructure cloud rovider that sells virtual machines as instances. Its underutilized caacity is sold through the sot market with a rice dynamically changing over time. Instead of using bids to determine the rice, in our model the sot rice is determined by the rovider according to instantaneous demand and suly. This is simle to imlement and maintain, and avoids the ossible collaborative bidding to game the market. We emhasize that it is necessary for the rovider to have the rice-setting ower, since when it needs to reclaim the caacity it can do so by raising the rice and forcing out customers with low reservation rices. We assume that the sot rice can take any value from an interval ½; max Š. The existence of a maximum rice can be justified by the common tiered service strategy in ractice. For examle, Amazon offers another two tiers of roducts (reserved and on-demand instances) with higher QoS riorities, and the rice of sot instances obviously cannot be higher than those of the higher-tier roducts. Price is charged er instance er time unit. Without loss of generality, we let max ¼ 1 throughout the aer. We consider a finite horizon with continuous time. In this aer, we focus on a monooly setting and do not consider the effect of market cometition on ricing. We also assume that customers are rice takers and do not consider rice anticiating behavior. As the sot market for cloud comuting is still at its early stage, these assumtions hold in general. Analyses of market cometition and rice anticiating behavior are beyond the scoe of this work, and left for our future work. The oerator can influence demand by varying its rice. Demand is determined by two indeendent stochastic rocesses, namely the arrival rocess that corresonds to the births of new instances, and the dearture rocess that models the deaths of existing instances (when customers shut them down). Here, we assume that demand arrivals can be exressed as a Poisson rocess with rate fðþ (number of new instances requested er unit time). Intuitively, as rice

4 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 161 sot instances from emirical studies, the Poisson assumtion rovides a good starting oint for modeling with analytical tractability. Second, it is extensively used in the literature on ricing to model real-world demand rocesses for erishable goods, such as fashion aarel, flight seats, hotels, and so on [14], [15], [43]. Fig. 5. Examles of demand arrival and dearture rate functions. increases, customers have less financial incentives to use the service, therefore a lower arrival rate. The demand dearture rocess is also modeled as a Poisson rocess with rate gðþ, where gðþ is the dearture rate function. When decreases, customers naturally have a lower robability to leave the system, resulting in a lower dearture rate gðþ. fðþ and gðþ can be estimated from ast demand data, and this learning rocess can be refined eriodically. This justifies our use of a finite decision horizon. We imose several mild and ractical assumtions on fðþ and gðþ, which are defined over the interval ½; 1Š. For the arrival rate function fðþ, we assume the following roerties hold: Assumtion 1. fðþ ;f < ;f < ; 8 1, f ðþ ¼, f ð1þ ¼ 1, and fð1þ ¼. The concavity of fðþ is a natural assumtion. It reflects the common sychology that when the sot rice is high, lowering it will be more attractive to customers, comared to when the sot rice is already low. Other assumtions are all intuitive and reasonable requirements commonly used in the literature [9], [14], [15]. An examle of such functions is fðþ ¼kð1 a Þ b, where k>;a>1 and < b<1 as shown in Fig. 5a. The dearture rate function gðþ is clearly increasing in. We further assume the following roerties. Assumtion 2. gðþ ;g > ;g > ; 8 1;g ðþ ¼ 1, g ð1þ ¼, and gðþ ¼. The convexity of gðþ models the henomenon that when rice is high, increasing it further will have a more detrimental effect than when the rice is low. An examle of such functions is gðþ ¼kð1 ð1 a Þ b Þ;k>;a>1, <b<1, as shown in Fig. 5b. Note that the roerty fð1þ ¼ allows us to model the out-of-caacity condition as an imlicit constraint that forces the rovider to set the rice to 1 to shut down the arrival rocess. Similarly, gðþ ¼ allows the rovider to rice at when the system is emty to turn off the dearture rocess. They are often referred to as the null rices in the literature [15]. In reality, we can certainly have demand arrivals and deartures without corresonding sales when the cloud is out of caacity or emty. However, in the context of our model, no generality is lost with this modeling artifact. The Poisson assumtion here is certainly an abstraction. Its use here can be justified for two reasons. First, since there is little knowledge gained on the demand model for 3 A STOCHASTIC REVENUE MAXIMIZATION FORMULATION 3.1 Formulation The ricing roblem can be formulated as follows: At the current time, the oerator has x 2½;CŠ sot instances running in the system with caacity C. It faces a finite decision horizon t> to collect revenue, until it udates the demand functions fðþ and gðþ. Note here t essentially indicates how much time is left for sale, and decreases along the time line. Our rovider uses a nonanticiating ricing olicy ðsþ to maximize the exected revenue over the entire decision horizon. Let XðsÞ denote system utilization, i.e., the number of active instances in the system at any time s 2½;tŠ. A demand is realized at time s if dxðsþ ¼1, and is vanished at time s if dxðsþ ¼ 1. The ricing olicy must be such that the number of active instances does not exceed the caacity C at any time s. We denote by U the set of all such ossible ricing olicies that satisfy dxðmþ 2½ x; C xš; ðsþ 2½; 1Š; 8 s 2½;tŠ: Here, m denotes time in ½;sŠ when s is given. Constraint (1) is the caacity constraint mentioned above. The existence of null rices guarantees that it can always be satisfied. Given a ricing olicy u 2U, we denote the exected revenue collected over the time eriod ½;tŠ by J u ðx; tþ¼ : Z t E u ðsþxðsþds ; 8t >: ð3þ At the very end of the horizon, when t ¼, the exected revenue is clearly zero for any utilization x J u ðx; Þ ¼ : ; 8x 2½;CŠ: The rovider s roblem is to find a ricing olicy u that maximizes the exected revenue generated over ½;tŠ, denoted by J ðx; tþ. Equivalently, J ðx; tþ ¼ : su J u ðx; tþ: u2u 3.2 Otimality Conditions Equation (5) is a stochastic dynamic rogramming roblem. To solve it, we can consider its Hamilton-Jacobi conditions, which are the continuous-time counterart of the Bellman equation. Informally, consider what haens over a small interval of time t. Since both the arrival and dearture rocesses are Poisson, by selecting a rice, the rovider sees one more instance over the next t with ð1þ ð2þ ð4þ ð5þ

5 162 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 robability fðþt þ oðtþ, one fewer instance with robability gðþt þ oðtþ, and no change with the rest of the robability mass. By the Princile of Otimality, J ðx; tþ ¼su xt þ oðtþ þ fðþt J ðx þ 1;t tþ ð6þ þ gðþt J ðx 1;t tþ þð1 ðfðþþgðþþtþj ðx; t tþ : In words, with t time left to the end of the horizon, the otimal exected revenue J ðx; tþ must be equal to the realized revenue during t, which is simly xt, lus the exected value of the otimal exected revenue from the remaining time interval t t, which are the remaining terms of (6). Rearranging the terms and taking the limit as t!, we ðx; tþ ¼ su x þ fðþðj ðx þ 1;tÞ J ðx; tþþ gðþðj ðx; tþ J ðx 1;tÞÞ : Note that (7) holds only for 1 x C 1. When x ¼, the rovider will not see any dearture over t, and is forced to rice at ; when x ¼ C the rovider is forced to set the rice to 1 to shut down the arrival rocess as discussed above. Thus, ð;tþ¼ and ðc; tþ ¼1 in our model. We have the following: J ð;tþ¼fðþt J ð1;t tþ þð1 fðþtþj ð;t tþþoðtþ; J ðc;tþ ¼gð1Þt J ðc 1;t tþ þð1 gð1þtþj ðc;t tþþct þ oðtþ; from which we obtain the following ð;tþ ¼ fðþ J ð1;tþ J ð;tþ ðc;tþ ¼ C gð1þ J ðc;tþ J ðc 1;tÞ : ð9þ We have not yet justified interchanging su and lim t!. This can be done formally using [1, Theorem 2.1]. This is also reminiscent to the technique used in [15]. Thus, a solution to (7) with boundary conditions (4) is indeed the otimal revenue J ðx; tþ, from which we can readily obtain the otimal rices ðx; tþ that together form an otimal ricing olicy u. We now show the existence of a unique solution to (7). Proosition 1. If the demand arrival and dearture rate functions f and g satisfy Assumtions 1 and 2, there exists a unique solution to (7) with boundary conditions (4). Proof. The otimal rice is always within the comact set ½; 1Š. Combining comactness with the fact that f and g are continuous and bounded in establishes the conditions required by [1, Theorem 2.3] for the existence of a unique solution to (7). tu ð7þ ð8þ 3.3 Structural Results Although we have found the otimality conditions, solving them to obtain a closed-form solution is quite difficult for arbitrary demand arrival and dearture functions. Moreover, numerically comuting the otimal solution can also be rohibitive as the state sace grows exonentially with the caacity of the cloud rovider, which is tyically fairly large. However, we are able to characterize several imortant structural roerties of the otimal solution to the dynamic rogram (7). We believe that insights obtained from our analysis are fundamental in understanding the roblem, and instrumental toward designing comutationally efficient heuristics, which is imortant in ractice. Theorem 1 (Monotonicity of otimal revenue). J ðx; tþ is strictly increasing in both x and t. Proof. The fact that J ðx; tþ is strictly increasing in t is intuitive and can be straightforwardly roved, and we omit the details here. The fact that J ðx; tþ is also increasing in x is not trivial because with a smaller number of running instances x to start with, the rovider has an incentive to set a lower rice to attract more customers, and the net effect on revenue can be either ositive or negative. By definition (3), we can write J ðx; tþ Z t Z t ¼ E u u ðsþxds þ E u u ðsþ dx u ðmþds ; where u is the otimal olicy and clearly satisfies dx u ðmþ 2½ x; C xš; 8s 2½;tŠ: ð1þ dx u ðmþ is the otimal birth-death rocess that corresonds to the otimal olicy u. Alternatively, we can also think of u ðmþ as determined by the statistics of the otimal birth-death rocess dx u ðmþ at time m, through E½dX u ðmþ=dmš ¼fð u ðmþþ gð u ðmþþ. Now, we let dx v ðmþ be another birth-death rocess that relates to X u ðmþ by dx v ðmþ ¼ dx u ðmþ 1; 8s 2½;tŠ: dx v ðmþ corresonds to another ricing olicy, denoted by v, which can be obtained from the following relationshi: fð v ðmþþ gð v ðmþþ ¼ E½dX v ðmþ=dmš: Obviously, dx v ðmþ dx u ðmþ holds at all times m 2ð;sŠ. Further, dx v ðmþ < dx u ðmþ must hold for at least m ¼. Thus, E½dX v ðmþš E½dX u ðmþš ) v ðmþ u ðmþ; 8m 2ð;sŠ; E½dX v ðþš <E½dX v ðmþš ) v ðþ > u ðþ: ð11þ We can write out the exected revenue starting with x þ 1 instances under the olicy v as follows:

6 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 163 J v ðx þ 1;tÞ Z t Z t ¼ E v v ðsþðx þ 1Þds þ E v v ðsþ dx v ðmþds Z t Z t ¼ E v v ðsþxds þ E v v ðsþ 1 þ dx v ðmþ ds Z t Z t ¼ E v v ðsþxds þ E v v ðsþ dx u ðmþds ; where v is a feasible olicy because dx v ðmþ 2½ x 1;C x 1Š; 8s 2½;tŠ: ð12þ Comaring to (1), it is readily seen that J v ðx þ 1;tÞ > J ðx; tþ due to (11). Thus, J ðx þ 1;tÞ >J ðx; tþ. tu Theorem 1 asserts that the otimal exected revenue increases with the utilization of the system and/or time. Moreover, we can show that the otimal revenue exhibits diminishing marginal returns with resect to utilization. Theorem 2 (Concavity of otimal revenue). J ðx; tþ is concave in x for any fixed t>. Proof. It suffices to show that 2J ðx; tþ J ðx þ 1;tÞþJ ðx 1;tÞ; 8t > ð13þ holds for all x 2½1;C 1Š, which we rove by constructing a feasible olicy to solve J ðx; tþ with exected revenue equal to ðj ðx þ 1;tÞþJ ðx 1;tÞÞ=2. Suose u and v are otimal olicies that achieve J ðx þ 1;tÞ and J ðx 1;tÞ, resectively. Denote the corresonding otimal birth-death rocesses as dx u ðmþ and dx v ðmþ, resectively. Now, consider a new birthdeath rocess as follows: dx u ðmþ ¼ dx uðmþþdx v ðmþ ; 8m 2½;tÞ: 2 This new rocess corresonds to a olicy u, where u ðmþ ð u ðmþþ v ðmþþ=2 due to the concavity of the function E½dXðmÞ=dmŠ ¼ fððmþþ R gððmþþ by s Assumtions 1 and 2. By construction, dx u ðmþ 2 ½ x; C xš holds for all s 2½;tŠ. Thus, u is a feasible solution for J ðx; tþ. Readily, J ðx; tþ J u ðx; tþ J ðx þ 1;tÞþJ ðx 1;tÞ : 2 tu This concavity roerty of the otimal revenue is not only crucial for further develoment of this aer but also of interest in itself. For examle, it can be useful for determining the otimal number of instances running in the system if it is art of the decisions. When the cost of roviding comuting hardware is linear or strictly convex, the exected rofit becomes a concave function of the number of running instances. In this case, the otimal utilization is the largest quantity for which the marginal exected revenue exceeds the marginal cost. We roceed to consider how the otimal rice changes over time and the utilization x. Our first result is that the otimal rice increases with the system utilization. Theorem 3 (Utilization monotonicity of otimal rice). ðx; tþ < ðx þ 1;tÞ for any fixed t, x 2½;C 1Š. Proof. For convenience, let us denote Mðx; tþ ¼J ðx; tþ J ðx 1;tÞ. From Theorem 2, we know that Mðx þ 1;tÞMðx; tþ. If we take the derivative of the right side of (7) with resect to and set it to zero, we get the necessary and sufficient condition for ðx; tþ which we abbreviate as x : x þ f ð x ÞMðx þ 1;tÞ g ð xþmðx; tþ ¼: ð14þ By Assumtions 1 and 2, we have g ð x ÞMðx; tþ f ð xþmðx; tþ x x ¼)Mðx; tþ g ð x Þ f ð x Þ x þ 1 ¼)Mðx þ 1;tÞ g ð xþ1 Þ f ð xþ1 Þ : Similarly, we have g ð x ÞMðx þ 1;tÞ f ð xþmðx þ 1;tÞx x ¼)Mðx þ 1;tÞ g ð x Þ f ð x Þ : Thus, for the two inequalities to hold, we must have x þ 1 g ð xþ1 Þ f ð xþ1 Þ x g ð x Þ f ð x Þ ¼)g ð xþ1 Þ f ð xþ1 Þ >g ð x Þ f ð x Þ ¼) ðx þ 1;tÞ > ðx; tþ: Theorem 3 has natural economic interretations. When the system is heavily loaded, it is in the interest of the rovider to set a higher rice to obtain a higher revenue from customers, as well as to discourage future demand to revent the system from overloading. On the other hand, when the system is lightly utilized, the rovider can afford to adot a lower rice to attract more customers. We further show that the otimal rice also exhibits time monotonicity. That is, ðx; tþ is decreasing in t. To rove this result, we first need a technical Lemma 1. > for any given t, where Mðx; tþ ¼ J ðx; tþ J ðx 1;tÞ. Proof. Note that at t ¼, J ðx; Þ ¼ for all x 2½;CŠ. Thus, Mðx; Þ ¼. Assume Then, at some t >, Mðx; t Þ, which contradicts with Theorem 1. Thus, the lemma must hold. tu Theorem 4 (Time monotonicity of otimal rice). ðx; tþ is decreasing in t for all x 2½1;C 1Š. Proof. It suffices to rove ðx; tþ < ; 8x 2½1;C 1Š: ð15þ tu

7 164 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 Taking derivative with resect to t in (14) and rearranging the terms, we ðx; tþ ðg ð ðx; tþþmðx; tþ f ð ðx; tþþmðx þ 1;tÞÞ ¼ f ð ðx; þ 1;tÞ g ð ðx; tþ : Alying Lemma 1 and Assumtions 1 and 2, the RHS is seen to be negative. Since g > ;f < ;Mðx; ðx;tþ must be negative in the LHS. tu Therefore, as time runs out, the otimal rice is increasing, and at the end of the horizon when t ¼, ðx; Þ is readily found to be 1, the maximum ossible rice, for all x since Mðx; Þ ¼ in (7). The intuition is that when the rovider has a long eriod of time left (t is large), she should rice to the future and set a lower rice to attract more customers to maximize the exected revenue. As time goes, her focus is shifting to the existing customers, and the otimal strategy is to set a higher rice to extract more revenue. At the end, when t ¼, it simly sets rice to maximize the current revenue and entirely ignores the imact on future revenue. The roerties that we roved in Theorems 1, 2, 3, and 4 are not only intuitively satisfying, but also useful if we wish to comute the otimal olicy numerically because they significantly reduce the search sace of olicies over which one needs to otimize. 4 NONHOMOGENEOUS DEMAND In the receding analysis, we have assumed a time homogeneous demand model, where the demand arrival and dearture rates are time invariant functions of rice. This assumtion is restrictive. In this section, we study the general case where both the demand arrival and dearture functions may change over time. To kee the analysis tractable, we adot a simle nonhomogeneous demand model. We assume that demand arrival at time s can be exressed as a Poisson rocess with a rate XðsÞfððsÞÞ. XðsÞfððsÞÞ denotes the rice-sensitive demand arrival rate with fððsþþ being the endogenous arrival robability function. We use XðsÞ as a linear multilier to model the intuition that when ðsþ decreases, each existing customer has an increased robability fððsþþ to shift more workload to the cloud, resulting in more demand. The demand dearture rocess is similarly modeled as a Poisson rocess with a time-varying rate function XðsÞgððsÞÞ, where gððsþþ is the endogenous dearture robability. When ðsþ decreases, each customer has a lower robability gððsþþ to leave the system, resulting in a lower rate XðsÞgððsÞÞ. The same structural roerties as assumed in Section 3 are carried over here, namely: Assumtion 3. fðþ 1;f < ;f < ; 8 1, f ðþ ¼;f ð1þ ¼ 1;fðÞ ¼1, and fð1þ ¼. Assumtion 4. gðþ 1;g > ;g > ; 8 1; g ðþ ¼1;g ð1þ ¼;gðÞ ¼, and gð1þ ¼ Formulation The revenue maximization roblem under such a nonhomogeneous demand model can then be formulated as follows similar to (5): J ðx; tþ¼ : su u2u s:t: XðsÞ ¼x þ Z t ðsþxðsþds; dxðmþ; ð16þ where R U denotes the set of admissible olicies that satisfy s dxðmþ 2½ xþ1;c xš for all s 2½;tŠ. Notice that in this model, XðsÞ must be larger than or equal to 1. Thus, the rovider is forced to set ð1;tþ¼, and we are only interested in cases where x 2. No generality is lost in making this modeling artifact. Similarly, we obtain the otimality conditions as ðx; tþ ¼ su x þ fðþðj ðx þ 1;tÞ J ðx; tþþ gðþðj ðx; tþ J ðx 1;tÞÞ : The necessary and sufficient condition for ðx; tþ is then ð17þ 1 þ f ð x ÞMðx þ 1;tÞ g ð xþmðx; tþ ¼: ð18þ 4.2 Asymtotic Analysis The nonhomogeneous model creates additional difficulty in obtaining structural roerties regarding the otimal ricing olicy. To overcome the difficulty, while keeing the results relevant, we conduct an asymtotic analysis here. Secifically, we assume C!1, and dro the caacity constraint that limits the set of admissible olicies U. This models a large-scale roblem, where the caacity of a cloud is always enough to accommodate all the virtual instances, and thus, the caacity constraint is usually inactive. This is also useful as an aroximate solution to the original roblem (16), as the asymtotic otimal solution turns out to have a very simle structure. Our first result is that the monotonicity of otimal revenue still holds in the nonhomogeneous demand model. Theorem 5. J ðx; tþ is strictly increasing in both x and t in the nonhomogeneous demand model. Proof. The fact that J ðx; tþ is strictly increasing in t is intuitive. We only rove for the result that J ðx; tþ is also increasing in x. Consider J ðx; tþ and J ðx 1;tÞ. Let u 2 be the otimal ricing olicies that achieve J ðx 1;tÞ. Clearly, u 2 is feasible for J ðx; tþ. From (3), we have R t J u2 ðx; tþ J ðx 1;tÞ ¼ ðsþe u 2 ½X 1 ðsþšds R t ðsþe u 2 ½X 2 ðsþšds : E u2 ½X 1 ðsþš;e u2 ½X 2 ðsþš can be found to grow exonentially over time, i.e., E u2 ½X 1 ðsþš ¼ x e qðsþ ; E u2 ½X 2 ðsþš ¼ ðx 1Þe qðsþ ; where qðsþ ¼ fððsþþ gððsþþ ds: ð19þ

8 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 165 Thus, J ðx; tþ J ðx 1;tÞ J u 2 ðx; tþ J ðx 1;tÞ ¼ x x 1 > 1: Moreover, we can rove that the otimal ricing olicy is indeendent of the number of instances in the system. Theorem 6. ðx; tþ ¼ ðtþ; 8x 2. Proof. From the roof of Theorem 5, we know that J ðx;tþ J ðx 1;tÞ x x 1. By the same token, assume that u 1 is the otimal ricing olicy that achieves J ðx; tþ. Since the constraint that R s dxðmþ 2½ xþ1;c xš is droed, u 1 is also feasible for J ðx 1;tÞ. Then, we have J ðx; tþ J ðx 1;tÞ J ðx;tþ J ðx 1;tÞ ¼ x J ðx; tþ J u1 ðx 1;tÞ ¼ x x 1 : tu ð2þ Thus, x 1, and u 1 must be equal to u 2 for the equation to hold, thus the roof. tu Theorem 6 is surrising. It tells us that when the system caacity is not a concern, the otimal ricing olicy with nonhomogeneous demand is only a function of the time remaining to the end of the horizon. This is so because the exected demand is no longer uer bounded by the system caacity. It is always otimal to maximize the exected revenue solely obtained from future demand, which is comletely determined by the time remaining to the end of the horizon and does not deend on the utilization. The otimal exected revenue using the same ricing olicy is, thus, roortional to x as E½XðsÞŠ is roortional to x from (19). This roerty greatly reduces the comlexity of solving the stochastic dynamic rogram by an order of C because time is the only state variable. Further, we can show that for a given number of active instances, the otimal rice still exhibits time monotonicity. Theorem 7. There exists an otimal rice ðtþ that is strictly decreasing in t. J Proof. Since ðx;tþ J ðxþ1;tþ ¼ x xþ1, Mðx; tþ ¼Mðx þ 1;tÞ¼MðtÞ where Mðx; tþ ¼J ðx; tþ J ðx 1;tÞ. Substitute into (18), 1 MðtÞ ¼ g ð ðtþþ f ð ðtþþ : 1 We denote hðþ ¼g ðþ f ðþ. Taking derivative with resect to t, we have M ðtþ ¼h ðþð Þ ðtþ: From the definition of MðtÞ and (17), M ðtþ ðx; ðx 1;tÞ ¼ MðtÞðfð ðtþþ gð ðtþþþ þ ðtþ ðx; tþ > : x Since h ¼ r ðg f Þ ðg f Þr, with Assumtions 3, 4, h <. ðg f Þ 2 ðtþ <. tu 5 NUMERICAL STUDIES In this section, we conduct numerical studies to verify the roerties of the otimal ricing olicy. The system caacity C is set to 1,. This corresonds to a moderate scale data center, such as a single availability zone of Amazon EC2, with several thousand machines caable of running tens of thousands of instances [5]. The decision horizon t is normalized to ½; 1Š, which corresonds to a 1-hour eriod, and rices are charged on a usage time basis as we discussed in Section 2.2. We follow the standard aroach of numerically solving a continuous time dynamic rogram discretizing the time horizon into N intervals of length t and using a difference equation to aroximate the otimality equation. The resulting difference equation J ðx; tþ ¼max xt þ fðþtj ðx þ 1;t tþ 2½;1Š þ gðþtj ðx 1;t tþ þð1 fðþt gðþtþj ðx; t tþ can be solved by backward induction for the discrete time set fnt j n ¼ ; 1;...;Ng with boundary conditions J ðx; Þ ¼. We consider demand functions of the form fðþ ¼k ffiffiffi ð1 2 Þ and gðþ ¼k k ffiffiffi ð1 2 Þ as shown in Fig. 5. In this case, the otimal rice has a closed-form solution ðx; tþ ¼y= ffiffi k 2 þ y 2, where y ¼ x=ðj ðx þ 1; t tþ J ðx 1;t tþþ. To be accurate, the total number of time intervals N ¼ 1=t should be much larger than the maximum number of demand arrivals and deartures k. The time and sace comlexity of the comutation are both OðCNÞ. 5.1 Weak Dynamics Scenarios We first consider a weak dynamics scenario, where the maximum exected demand arrivals and deartures during the decision horizon k is orders of magnitude less than the system caacity C. We assume that a cloud is exected to launch and close several hundreds of instances er hour on average. Thus, we set k to 5, much smaller than the system caacity C ¼ 1;. Time is discretized into N ¼ 1; intervals, each corresonding to 3.6 seconds of time for a 1-hour horizon. The results are shown in Fig. 6. The otimal exected revenue clearly grows with decision horizon t (which decreases with time in the figures) and the utilization x as seen from Fig. 6a. The otimal rice decreases with t (increases with time), and increases with the utilization as seen from Fig. 6b. These observations validate our analysis in Section 3. Note that the otimal rice does not change much when x decreases from 9, to 5,, and is close to 1 for the entire horizon. This is because the effect of demand dynamics is small comared to a moderately loaded system (x ¼ 5; ¼ :5C), and the exected revenue can be maximized without considering much about the future demand, i.e., setting rice close to 1. To facilitate the understanding, Fig. 6d shows a samle ath of the otimal rice, with the corresonding system utilization rocess XðsÞ starting from x ¼ 5;. In the time eriod of ½;:25Š (t decreases from 1 to.75), ðxðsþ; 1 sþ grows only

9 166 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 Fig. 6. Numerical results with C ¼ 1;;fðÞ ¼5 1 2 ;gðþ ¼ , and N ¼ 1; time intervals. marginally from.98 to around.988, while XðsÞ slowly decreases from 5, to about 4,93. Thus, XðtÞ does not deviate much from x, and J ðx; tþ is close to xt, which also exlains its linearity as in Fig. 6c. When the system is lightly loaded (x ¼ 1; ¼ :1C), revenue generated from future demand becomes more imortant, and ðx; tþ is much lower and varies with time as in Fig. 6b. In summary, these results tell us that when the exected dynamics is weak, or equivalently when the decision horizon is short, it is almost otimal to use a static rice close to the maximum rice 1 for a heavily or moderately loaded system. The otimal exected revenue grows linearly with the utilization. When the system is lightly loaded, however, rice has to be dynamically adjusted to obtain maximum revenue. 5.2 Strong Dynamics Scenarios The story becomes quite different when the roblem embraces a significant degree of demand dynamics. Here, we let the maximum exected demand arrivals and deartures k equal to 5,, much larger than the system caacity C. Time is discretized into N ¼ 1 5 intervals. Other arameters remain the same. The results are shown in Fig. 7. The otimal revenue and rice clearly exhibit monotonicity as exected from our analysis. Comared to the weak dynamics case, the first interesting observation is that otimal revenue is insensitive to the utilization. As seen from Fig. 7a, J ð9;;tþ imroves J ð1;;tþ only by a small constant margin, and xt becomes a oor estimate for J ðx; tþ, esecially when t is close to 1. The reason is that when the demand dynamics is strong, revenue from future demand is dominant esecially in the beginning of the horizon (when t is close to 1). Since rice can be adjusted flexibly, the system can always be quickly tuned to a heavy load setting with better revenue, and the end result is that the exected revenue over the horizon is relatively immaterial to the current utilization. This also exlains the stronger concavity of J ðx; tþ in x as seen in Fig. 7c because the marginal benefit of increasing the utilization is diminishing, causing the revenue curve to be bent downwards. The discussion above imlies that dynamic ricing becomes more critical in a strong dynamics setting. It is, thus, exected to see the otimal rice varying significantly with the utilization, which is demonstrated in Fig. 7b. Comared with Fig. 6b, for most of the time ð1;;tþ remains to be much smaller than ð5;;tþ which in turn is much smaller than ð9;;tþ. The reason is that when the effect of dynamics is significant, we should rice to the future even when time left to consider t is relatively small, and adot a low rice when the utilization is low. Only when it is near the end of the horizon, should we raise the rice to harvest more revenue from the existing customers. A critical oint here is that although ðx; tþ remains almost static most of the time for a given x, it does not mean that we can safely use a static rice at a small cost of revenue loss. In fact, the number of instances in the system is exected to fluctuate quickly over time, and whenever it changes we ought to change the rice on the sot. Since ðx; tþ differs widely with x, we ought to use a dynamic ricing olicy to maximize the revenue. This is demonstrated in Fig. 7d with a samle ath of the otimal rice. We can see that the utilization rocess XðsÞ quickly grows from 5, to nearly 1,, and the otimal rice ðxðsþ; 1 sþ rises from about.3 to over.8 in the time eriod of ½; :25Š. Comared with Fig. 6d for the weak dynamics case, the otimal rice is clearly much more dynamic. It is reasonable to conclude that dynamic ricing lays an imortant role in the strong dynamics setting, Fig. 7. Numerical results with C ¼ 1;;fðÞ ¼5; 1 2 ;gðþ ¼5; 5; 1 2, and N ¼ 1 5 time intervals.

10 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 167 Fig. 8. Numerical results with C ¼ 1;fðÞ ¼ 1 2 ;gðþ ¼1 1 2, and N ¼ 1; time intervals. and is exected to offer substantial revenue imrovement over static ricing. 5.3 Sensitivity Analysis In the revious two sections, we studied the cases where the maximum exected demand arrivals and deartures k is set to 5 and 5,, resectively. These reresent two extreme cases, and naturally one may wonder if our observations above are sensitive to the values of k, in the sense that the results are skewed at the extreme oints. In this section, we rovide evidence to confirm that the observations are robust to k. In this exeriment, we vary k from 5 to 1,, and discretize time into 1, intervals. Other arameters are the same as in the revious two sections. Fig. 1 shows the otimal rice at 5 ercent utilization ð5;;tþ for different values of k. Observe that when k is small comared to C, ð5;;tþ behaves qualitatively similar to the curve in Fig. 6b. As k increases, otimal rice decreases. The curve gradually becomes flatter, meaning that most of the time the otimal rice should be ket low until at the end of the horizon. The shae of the curve also becomes qualitatively similar to Fig. 7b for the strong dynamics case. Thus, our results drawn from two extreme values of k are also valid for intermediate values of k. We also studied the effect of k for otimal revenue J and the observation is the same. For brevity, we omit the figure. 5.4 Imact of Delay We have assumed that the rovider always has erfect information about the system utilization of the cloud at any time. In reality, the infrastructure monitoring software for the cloud may incur delay in rocessing and roagating data, esecially given the large scale of the system. Delayed information inherently limits the rovider s ability to make correct ricing decisions, and causes revenue loss. In this section, we investigate the imact of delay on rovider s revenue. We conduct a set of numerical studies with k ¼ 1;, and N ¼ 1; intervals for T ¼ 1. We consider a moderately loaded system with an initial utilization of x ¼ 5;, with varying information delay ranging from.1 to.1. We consider the time eriod of ½; :25Š. At each time interval s, the rovider sees a delayed utilization Xðs Þ instead of XðsÞ, and makes a ricing decision based on Xðs Þ. For each value of delay, we generate 5 samle aths of the system utilization rocess XðsÞ and ricing decisions ðxðs Þ; 1 sþ, and comute the average revenue for the eriod ½; :25Š across the 5 runs. Fig. 11 lots the ercentage of revenue loss due to delay. As exected, revenue loss increases when delay is more salient, since with a long delay the actual system utilization is significantly different from what the rovider sees. We observe that when delay is small, i.e., less than.4, revenue loss is less than or around 1 ercent. However, when delay is larger than.4, revenue loss is larger than 2 ercent. In reality information, delay is usually small comared to the time horizon T. For examle, T is 1 hour in our numerical study, and a 1-second delay is only :26T. To summarize, we find that information delay has direct imact on revenue with dynamic ricing, and the rovider has financial incentives to develo a resonsive and accurate management system to obtain real-time information about the resource utilization. 5.5 Nonhomogeneous Demand Finally, we consider the nonhomogeneous demand model, where the demand arrival and dearture rate is XðsÞfððsÞÞ and XðsÞgððsÞÞ, resectively. We use fðþ ¼ 1 2 and gðþ ¼ This is consistent with the homogeneous demand functions excet for the constant k, which is equal to 1 for the nonhomogeneous case so that fðþ and gðþ reresent robabilities. The same discretization technique is used to solve the dynamic rogram using backward induction, and the entire decision horizon is discretized into N ¼ 1; intervals. The otimal rice in this case can be readily obtained to be ðx; tþ ¼1= 1 þ y 2, where y ¼ J ðx þ 1;t tþ J ðx 1;t tþ. We first evaluate a small system with C ¼ 1. Fig. 8 shows the results. From Fig. 8a, we can see that the otimal revenue J ðx; tþ is increasing in x and t, which shows that Theorem 5 for the asymtotic case also holds in general. The interesting story is in Fig. 8b. Comared to Figs. 6b and 7b, otimal rice ðx; tþ is largely indifferent for different values of x when t is smaller than around.5. For t>:5 and t<:3, ðx; tþ is still distinct for different x. This demonstrates the intuition revealed by Theorem 6 that the otimal rice becomes much less deendent on the utilization x, when demand is non-homogeneous. We then consider a large system with C ¼ 1;, and Fig. 9 shows the results. The revenue result in Fig. 9a does

11 168 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 Fig. 9. Numerical results with C ¼ 1;;fðÞ ¼ 1 2 ;gðþ ¼1 1 2, and N ¼ 1; time intervals. not show much difference from the small caacity case. However, the otimal rice becomes identical for different values of x when t<:6 as shown in Fig. 9b, and the difference is smaller when t>:6 comared to the small caacity case in Fig. 8b. This is consistent with Theorem 6 that says when the caacity is large, the otimal rice becomes indeendent of the utilization x. Otimal revenue is linear in x as shown in Figs. 8c and 9c and exected from the fact that Mðx; tþ ¼MðtÞ in the asymtotic case. 6 DISCUSSIONS In this section, we discuss several issues ertaining to the racticality and usefulness of this work, which also lead to ossible directions of future work. 6.1 Benefits of Dynamic Pricing for Users In this aer, we mainly focus on the rovider. Nevertheless, dynamic ricing is also beneficial for users of the cloud. With static ricing, the rovider has limited ways to control the user demand. When demand for resources (e.g., virtual machines, bandwidth, etc.) increases, the erformance of the virtual machines degrades and the robability of failures increases, leading to inferior user exerience. With dynamic ricing, the rovider has an effective means to dynamically control the demand, and ensure the overall erformance of the cloud is satisfactory for customers. Thus, we believe that dynamic ricing is also beneficial to users, from the erformance oint of view. We did not exlicitly model the resource contention (e.g., bandwidth, CPU, memory) and its effect on user exerience of using the cloud, which is an interesting future direction of extending the work. Such an effect is in fact an imortant toic in our community. Recently, there has been active research on roviding bandwidth guarantees and different notions of fairness to users of the cloud (for examle, [17], [22], [23], [35]). We have also reached out to ublic cloud roviders such as Microsoft Azure to comment on the racticality of dynamic ricing [7]. They view dynamic ricing as an viable otion that will be increasingly adoted in addition to static ricing. The motivation is that roviders would like to accommodate various customers to increase our revenue: some customers care more on guaranteed SLA with fixed costs so they may still choose static rice, while some have the flexibility to trade resonse time/quality for their cost with dynamic ricing, by using less resources when rices are high [7]. 6.2 Potential Concerns Here, we also discuss some limitations of dynamic ricing and otential concerns a rovider may have. First, as just mentioned, dynamic ricing is more suitable for flexible workloads that can be stoed or adjusted during the execution. For examle, if the workload is batch data rocessing, it can be ostoned to times when the rice is cheaer, or the user can urchase less virtual machines to run it when rice is high. However, if the workload is a web service that has to resonse to user requests, then the user has very little flexibility to adjust the consumtion according to the rice. This may revent the adotion esecially for smaller roviders with few flexible workloads. Second, to imlement dynamic ricing, the rovider needs to collect demand information and emirically derive the demand arrival and dearture rate functions. It may introduce a trial eriod, before the actual adotion, to accumulate enough demand data to calculate the exected demand change corresonding to a rice change. Demand information also needs to be continuously monitored and analyzed to refine the functions and ricing strategy. This may add managerial overhead to the rovider. Finally, for users, dynamic ricing may seem comlicated to understand sychologically, and they may refer fixed ricing for simlicity. Fig. 1. Otimal rice ð5;;tþ for different values of k. C ¼ 1;, and N ¼ 1; time intervals. Fig. 11. Revenue loss due to delay for a eriod of ½;:25Š. C ¼ 1;, k ¼ 1;, x ¼ 5;, N ¼ 1; intervals.

12 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION Alternative Pricing Strategies We studied dynamic ricing from a revenue management ersective, which is widely adoted in traditional industries such as airlines, car rental, and hotels. Public cloud roviders share more similarity with Internet comanies such as Google and Facebook. Here, we briefly survey the ricing strategies used by Internet comanies and comment on their alicability to cloud comuting. Internet comanies usually rovide their services free of charge, and generate revenues from selling ads online. Examles include Google AdWords [6] and Facebook Ads [1]. Since the values of ads for different goods and services could vary significantly and are difficult to determine, bidding/auctions are usually used as the ricing strategy [6]. Users submit bids as the maximum rice they are willing to ay for each click of their ads, and the comany runs an auction to determine which ads are dislayed on their age, and their rank on the age [6]. Comared to bidding, dynamic ricing has the following advantages that are briefly discussed in Section 2.2. First, with dynamic ricing, the rovider has more control over the rice, while with bidding the rovider cannot control the final sale rice, which can be well below the break-even oint for the rovider to recover its costs. Second, dynamic ricing eliminates the otential user collusion, which is a common and difficult roblem in ractice for auctions [3]. With bidding, a grou of users can collaborate to game the mechanism for their own benefits in various ways. A ublic cloud has a large number of users, making collusion detection and revention even more difficult. Some may argue that users are unfairly taken advantage of with dynamic ricing comared to bidding, as the rice is solely decided by the rovider. We emhasize that first rice is determined taking into account users reactions according to the demand arrival rate and dearture rate functions as in Section 2.2. Moreover, cometition in the market rovides choices for users, and forces the rice to remain at a reasonable level. This is an imortant toic by itself and can be exlored further in future work. 6.4 Costs and Profit Maximization We have mainly discussed revenue maximization without considering costs. Oerating costs of a cloud, mostly ower draw of servers and cooling systems, are fixed and do not vary with the infrastructure utilization. The reason is that oerators tyically leave all the servers and switches on at all times [18]. Thus, we did not consider costs here. Recently, some studies roose to dynamically turn off/ on servers and/or switches according to the workload to save energy [26]. In these cases, costs are roortional to demand, and a rofit maximization formulation is more aroriate for studying ricing which is beyond the scoe of this aer. Readers are referred to our work in [44] for more details. 7 RELATED WORK Our work has roots in revenue management. Since the seminal work of [15], dynamic ricing has become an active toic of revenue management, with many successful realworld alications (see [14] and references therein). As discussed in Section 1, with cloud comuting, the unique challenge is that we need to model stochastic demand deartures in addition to demand arrivals. This is because in our roblem, rice is charged er unit of time and sale, while revious works only consider the simle case of charging er unit of sale. An extensive literature exists on ricing in communication networks and Internet. Many works study offline ricing, where ricing is comuted offline to otimize revenue, social welfare, and so on. Odlyzko [32] argues that the redominant flat-rate ricing for selling retail Internet access encourages waste and is incomatible with service differentiation. The benefits of usage-based ricing are studied in [25], [37], where it is shown that with rice differentiation one can use resources more efficiently. Paris Metro Pricing, in which service differentiation and congestion control are autonomously achieved by charging different rices for different service tiers that share the same infrastructure, is thoroughly studied in [11], [12], [31]. In [4], tiered ricing for Internet transit is further studied. Time is another dimension to unbundle connectivity. Hande et al. [2] characterize the economic loss due to the ISP s inability or unwillingness to rice broadband access based on time of day. Ha et al. [19] study the timedeendent ricing for mobile data. Our work is more related to the online ricing literature that deals with instantaneous demand dynamics and adjusts rice on the sot. Comared with offline ricing it is less exlored in the community. Paschalidis and Liu [33], Paschalidis and Tsitsiklis [34] study online ricing based on congestion in networks as a dynamic rogram, and show that static ricing achieves good erformance in large networks. Our main concern in this aer is revenue maximization using dynamic ricing, and we consider a finite horizon formulation, which is different from the infinite horizon setting in [33], [34]. There have been some recent studies on the market and ricing of cloud resources. Javadi et al. [21] roose a stochastic model for the sot rices of EC2. The focus of [21] is to better model and redict sot rices, while our focus is to develo a new dynamic ricing scheme that imroves the revenue of the oerator. Wang et al. [42] argue for the imortance of ricing in cloud comuting for distributed systems design. From a user ersective, Teng and Magoulès [39] study the equilibrium ricing and allocation olicy between multile users using game theory. From a rovider ersective, Mihailescu and Teo [29] roose a comutationally efficient ricing scheme based on mechanism design, and Macías and Guitart [28] adot a genetic algorithm to iteratively otimize the ricing olicy. The most related work to ours are [24] and our own work [44], where ricing strategies are develoed by solving some otimization roblems. These aroaches are rimarily of a one-shot nature without considering the effect of ricing on future demand and revenue. 8 CONCLUDING REMARKS We resented a revenue maximization framework to tackle the dynamic ricing roblem in an IaaS cloud. The unique challenge is that rices are charged er instance er time

13 17 IEEE TRANSACTIONS ON CLOUD COMPUTING, VOL. 1, NO. 2, JULY-DECEMBER 213 unit, and as a result the demand dearture rocess has to be exlicitly modeled. A stochastic dynamic rogram is formulated, and otimality conditions and imortant structural results of the otimal ricing olicies are resented. We then extended to a general nonhomogeneous demand model. We wish to emhasize that the broad literature of revenue management rovides many meaningful future directions to study cloud resource ricing. For examle, we can model the resuly of comuting resources to the roblem, which corresonds to the inventory control asect of the framework. Since different kinds of resources are involved, how to choose the otimal mix of resources to create a menu of final roducts, such as Amazon s various instance tyes, is also an imortant issue. REFERENCES [1] Advertising on Facebook, htts:// [2] Amazon EC2, htt://aws.amazon.com/ec2/ [3] Amazon EC2 API Tools, htt://aws.amazon.com/develoertools/ 351, 213. [4] Amazon EC2 Sot Instances, htt://aws.amazon.com/ec2/sotinstances, 213. [5] Amazon s EC2 Generating 22M+ Annually, htt://cloudscaling. com/blog/cloud-comuting/amazons-ec2-generating-22mannually, 213. [6] How Costs Are Calculated in AdWords, htts://suort.google. com/adwords/answer/174424, 213. [7] Personal Communication with Yuxiong He, Microsoft Research, Set [8] O.A. Ben-Yehuda, M. Ben-Yehuda, A. Schuster, and D. Tsafrir, Deconstructing Amazon EC2 Sot Instance Pricing, Proc. IEEE Third Int l Conf. Cloud Comuting Technology and Science (CloudCom), 211. [9] G. Bitran and R. Caldentey, An Overview of Pricing Models for Revenue Management, Manufacturing & Service Oerations Management, vol. 5, no. 3, , Set. 23. [1] P. Bremaud, Point Processes and Queues, Margingale Dynamics. Sringer-Verlag, 198. [11] X.R. Cao, H.X Shen, R. Milito, and P. Wirth, Internet Pricing with a Game Theoretical Aroach: Concets and Examles, IEEE/ ACM Trans. Networking, vol. 1, no. 2, , Ar. 22. [12] C.-K. Chau, Q. Wang, and D.-M. Chiu, On the Viability of Paris Metro Pricing for Communication and Service Networks, Proc. IEEE INFOCOM, 21. [13] J. Chen, C. Wang, B.B. Zhou, L. Sun, Y.C. Lee, and A.Y. Zomaya, Tradeoffs between Profit and Customer Satisfaction for Service Provisioning in the Cloud, Proc. 2th Int l Sym. High Performance Distributed Comuting (HPDC), 211. [14] W. Elmaghraby and P. Keskinocak, Dynamic Pricing in the Presence of Inventory Considerations: Research Overview, Current Practices, and Future Directions, Management Science, vol. 49, no. 1, , Oct. 23. [15] G. Gallego and G. van Ryzin, Otimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons, Management Science, vol. 4, no. 8, , Aug [16] M.K Geraghty and E. Johnson, Revenue Management Saves National Car Rental, Interfaces, vol. 27, no. 1, , [17] A. Ghodsi, M. Zaharia, B. Hindman, A. Konwinski, S. Shenker, and I. Stoica, Dominant Resource Fairness: Fair Allocation of Multile Resource Tyes, Proc. USENIX Conf. Networked Systems Design and Imlementation (NSDI), 211. [18] A. Greenberg, J. Hamilton, D.A. Maltz, and P. Patel, The Cost of a Cloud: Research Problems in Data Center Networks, ACM SIGCOMM Comuter Comm. Rev., vol. 39, no. 1, , 29. [19] S. Ha, S. Sen, C. Joe-Wong, Y. Im, and M. Chiang, TUBE: Time Deendent Pricing for Mobile Data, Proc. Proc. ACM SIGCOMM, 212. [2] P. Hande, M. Chiang, R. Calderbank, and J. Zhang, Pricing under Constraints in Access Networks: Revenue Maximization and Congestion Management, Proc. IEEE INFOCOM, 21. [21] B. Javadi, R. Thulasiramy, and R. Buyya, Statistical Modeling of Sot Instance Prices in Public Cloud Environments, Proc. IEEE Int l Conf. Utility and Cloud Comuting (UCC), 211. [22] V. Jeyakumar, M. Alizadeh, D. Mazieres, B. Prabhakar, C. Kim, and A. Greenberg, EyeQ: Practical Network Performance Isolation at the Edge, Proc. USENIX Conf. Networked Systems Design and Imlementation (NSDI), 213. [23] C. Joe-Wang, S. Sen, T. Lan, and M. Chiang, Multi-Resource Allocation: Fairness-Efficiency Tradeoffs in a Unifying Framework, Proc. IEEE INFOCOM, 212. [24] V. Kantere, D. Dash, G. Francois, S. Kyriakooulou, and A. Ailamaki, Otimal Service Pricing for a Cloud Cache, IEEE Trans. Knowledge and Data Eng., vol. 23, no. 9, , Feb [25] G. Kesidis, A. Das, and G. de Veciana, On Flat-Rate and Usage- Based Pricing for Tiered Commodity Internet Services, Proc. 42nd Ann. Conf. Information Sciences and Systems (CISS), 28. [26] M. Lin, A. Wierman, L.L.H. Andrew, and E. Thereska, Dynamic Right-Sizing for Power-Proortional Data Centers, Proc. IEEE INFOCOM, 211. [27] T. Lossen, Cloud Exchange, htt://cloudexchange.org/, 213. [28] M. Macías and J. Guitart, A Genetic Model for Pricing in Cloud Comuting Markets, Proc. Sym. Alied Comuting, 211. [29] M. Mihailescu and Y.M. Teo, On Economic and Comutational- Efficient Resource Pricing in Large Distributed Systems, Proc. 1th IEEE/ACM Int l Sym. Cluster, Cloud and Grid Comuting, 21. [3] P. Milgrom, Putting Auction Theory to Work. Cambridge Univ. Press, 24. [31] A. Odlyzko, Paris Metro Pricing for the Internet, Proc. ACM First ACM Conf. Electronic Commerce (EC), [32] A. Odlyzko, Should Flat-Rate Internet Pricing Continue? IEEE IT Professional, vol. 2, no. 5, , Set. 2. [33] I.C. Paschalidis and Y. Liu, Pricing in Multiservice Loss Networks: Static Pricing, Asymtotic Otimality, and Demand Substitution Effects, IEEE/ACM Trans. Networking, vol. 1, no. 3, , June 22. [34] I.C. Paschalidis and J.N. Tsitsiklis, Congestion-Deendent Pricing of Network Services, IEEE/ACM Trans. Networking, vol. 8, no. 2, , Ar. 2. [35] L. Poa, G. Kumar, M. Chowdhury, A. Krishnamurthy, S. Ratnasamy, and I. Stoica, Faircloud: Sharing the Network in Cloud Comuting, Proc. ACM SIGCOMM, 212. [36] P.V Schaeffer, Commodity Modeling and Pricing: Methods for Analyzing Resource Market Behavior. John Wiley & Sons, 28. [37] S. Shakkottai, R. Srikant, A. Ozdaglar, and D. Acemoglu, The Price of Simlicity, IEEE J. Selected Areas in Comm., vol. 26, no. 7, , Set. 28. [38] B. Smith, R.J Leimkuhler, and J.S Darrow, Yield Management at American Airlines, Interfaces, vol. 22,. 8-31, [39] F. Teng and F. Magoulès, Resource Pricing and Equilibrium Allocation Policy in Cloud Comuting, Proc. 1th IEEE Int l Conf. Comuter and Information Technology (CIT 1), 21. [4] V. Valancius, C. Lumezanu, N. Feamster, R. Johari, and V.V. Vazirani, How Many Tiers? Pricing in the Internet Transit Market, Proc. ACM SIGCOMM, 211. [41] K. Vermeersch, Sot Watch, htt://sotwatch.eu/inut/, 213. [42] H. Wang, Q. Jing, R. Chen, B. He, Z. Qian, and L. Zhou, Distributed Systems Meet Economics: Pricing in the Cloud, Proc. Second USENIX Conf. Hot Toics in Cloud Comuting (HotCloud), 21. [43] L. Weatherford and S. Bodily, A Taxonomy and Research Overview of Perishable-Asset Revenue Management: Yield Management, Overbooking, and Pricing, Oerations Research, vol. 4, no. 5, , Set [44] H. Xu and B. Li, A Study of Pricing for Cloud Resources, ACM SIGMETRICS Performance Evaluation Rev., Secial Issue on Cloud Comuting, vol. 4, no. 4,. 3-12, Mar [45] Q. Zhang, E. Gurses, R. Boutaba, and J. Xiao, Dynamic Resource Allocation for Sot Markets in Clouds, Proc. 11th USENIX Conf. Hot Toics in Management of Internet, Cloud, and Enterrise Networks and Services (Hot-ICE), 211.

14 XU AND LI: DYNAMIC CLOUD PRICING FOR REVENUE MAXIMIZATION 171 Hong Xu received the BEngr degree from the Deartment of Information Engineering, The Chinese University of Hong Kong in 27 and the MASc and PhD degrees from the Deartment of Electrical and Comuter Engineering, University of Toronto, in 21 and 213, resectively. He is currently an assistant rofessor in the Deartment of Comuter Science, City University of Hong Kong. His research interests include the design, analysis, and imlementation of large-scale networked systems in cloud comuting, data centers, and wireless networks. He is a member of the IEEE, IEEE Comuter Society, and ACM. Baochun Li received the BEngr degree from the Deartment of Comuter Science and Technology, Tsinghua University, China, in 1995 and the MS and PhD degrees from the Deartment of Comuter Science, University of Illinois at Urbana-Chamaign, Urbana, in 1997 and 2, resectively. Since 2, he has been with the Deartment of Electrical and Comuter Engineering at the University of Toronto, where he is currently a rofessor. He holds the Nortel Networks junior chair in Network Architecture and Services from October 23 to June 25, and the Bell Canada Endowed chair in Comuter Engineering since August 25. His research interests include large-scale distributed systems, cloud comuting, eer-to-eer networks, alications of network coding, and wireless networks. He received the IEEE Communications Society Leonard G. Abraham Award in the field of communications systems in 2. In 29, he received the Multimedia Communications Best Paer Award from the IEEE Communications Society, and the University of Toronto McLean Award. He is a senior member of the IEEE and IEEE Comuter Society and a member of ACM.. For more information on this or any other comuting toic, lease visit our Digital Library at