Socially Optimal Pricing of Cloud Computing Resources


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1 Socially Optimal Pricing of Cloud Computing Reource Ihai Menache Microoft Reearch New England Cambridge, MA Auman Ozdaglar Laboratory for Information and Deciion Sytem Maachuett Intitute of Technology Cambridge, MA Nahum Shimkin Department of Electrical Engineering Technion Irael Intitute of Technology Haifa 32000, Irael ABSTRACT The cloud computing paradigm offer eaily acceible computing reource of variable ize and capabilitie. We conider a cloudcomputing facility that provide imultaneou ervice to a heterogeneou, timevarying population of uer, each aociated with a ditinct job. Both the completion time, a well a the uer utility, may depend on the amount of computing reource applied to the job. In thi paper, we focu on the objective of maximizing the longterm ocial urplu, which comprie of the aggregate utility of executed job minu loaddependent operating expene. Our problem formulation relie on baic notion of welfare economic, augmented by relevant queueing apect. We firt analyze the centralized etting, where an omnicient controller may regulate admiion and reource allocation to each arriving job baed on it individual type. Under appropriate convexity aumption on the operating cot and individual utilitie, we etablih exitence and uniquene of the ocial optimum. We proceed to how that the ocial optimum may be induced by a ingle perunit price, which charge a fixed amount per unit time and reource from all uer. Keyword Cloud Computing, Pricing, Social Efficiency 1. INTRODUCTION Cloud computing i meant to offer ondemand network acce to hared pool of configurable computing reource [13], uch a virtual erver, application and oftware ervice. Thi paradigm promie to deliver to the uer the economic of cale of a large datacenter, fat and flexible proviioning of reource, and the freedom from longterm invetment in equipment and related technology. The inner working of the datacenter that upport the cloud operation are hidden from the uer, who i preented with virtual erver, computing infratructure, or oftware ervice. The Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. To copy otherwie, to republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. VALUETOOLS 2011, May 1620, Pari, France. Copyright 2011 ACM XXXXXXXXX/XX/XX...$ idea of offering hared computing reource i of coure not new, and ha been extenively tudied and implemented under different computing paradigm, including cluter, grid and utility computing [6]. Recently, however, the notion of cloud computing ha gained prominence, purred by numerou implementation by Amazon, Microoft, Google, IBM and many other, both of public cloud (openly available over the Internet), or private cloud (intended for internal or retricted ue). A recent urvey of the promie and challenge of cloud computing can be for in [1], for example. Shared computing facilitie require effective mechanim for allocating available reource among uer. Thi become a major challenge in view of the diverity of application type and uer need, which are at leat partly hidden from the ytem manager. In public cloud, a major role i taken up by economic mechanim, notably reource pricing, but alo panning more elaborate mechanim uch a bidding and auction. Such economic mechanim are naturally ubject to revenue and profit conideration by commercial ervice provider. In thi paper, however, our main focu i on the ue of pricing a a mean to maximize the ocial welfare aociated with the cloud operation, which conit of the aggregate ervice utility obtained by the cloud uer, le the infratructure and operating cot of the ervice provider. Maximizing the ocial welfare (equivalently, the ocial urplu or ocial efficiency) i epecially relevant for public cloud operated by a public organization or official agencie for the public benefit, a well a for private cloud et up by a commercial company or conortium of firm for internal ue. Socially efficienct operation might a well be aligned with the interet of certain commercial public cloud, a it can help build the company longrun reputation in thi emerging market. Social efficiency, and pricing a a mean to achieve it, are baic notion in economic theory [10, 16]. The preent paper leverage the tandard theory, by conidering explicitly the temporal dynamic of ervice. Our model conider a hared computing facility to which heterogeneou job (or application) arrive equentially. We aume that each job belong to a ditinct uer, and henceforth ue the term job and uer interchangeably. Each arriving uer may acquire a certain amount of the computing reource, while all preent job are erved imultaneouly, each on it allocated reource. The ervice quality experienced by the uer naturally depend on the reource allocated to hi job. Furthermore, depending on application, the job execution time may coniderably cale down with the amount of reource applied to it. Thi i epecially true for batchtype application uch
2 a cientific computing application and buine data analyi, that are computation and data intenive and can be efficiently parallelized. A argued in [1], uch application preent notable opportunitie for the utilization and further advancement of cloud computing. Both thee factor, ervice quality and execution time, are eential part of the uer performance and utility model that we conider. The pricing mechanim we conider i the imple uagebaed pricing with linear tariff, where each uer pay a fixed amount per unit reource and unit time. Such pricing cheme are currently in ue by everal central cloud provider. The deciion on how much reource to ue and for what length of time i thu relegated to the uer. Naturally, potential uer may alo decide to give up the offered ervice altogether, or balk. Such balking deciion effectively hape the arrival rate into the ytem, which together with the reource requirement of erved uer, determine the demand curve for the ytem reource. In the context of ocial welfare, pricing can be viewed a playing a dual role: Firt, regulating the overall ytem load to match available reource (or their operating cot), and econd, inducing a ditribution of available reource among uer which i commenurate with their performance requirement and ervice utilitie. A mentioned, the uggeted model emphaize the equential nature of uer arrival, and the dependence of their ervice time on the allocated reource. Accordingly, we develop the model in ome detail, while clarifying the required aumption. We then examine the ocial welfare under the aumption that ytem reource are ufficient to fully accommodate the ocially optimal demand. Our main reult i twofold: Firt, we etablih the exitence and uniquene of the welfare maximizing olution (in term of arrival rate and allocated reource to each job type). Second, we prove that there exit a unique price that induce thi deired olution. We further elaborate on the relation of our model to the exiting economic model. Some additional topic of interet that have been dealt with within thi model include the iue of load contraint due to limited reource, conideration of profit, and iterative (tatonnementlike) price adjutment cheme that converge to the ocially optimal price. Thee iue have been omitted here for lack of pace and will appear in an extended verion of thi paper. Let u briefly comment on related literature. Marketoriented mechanim have been long been conidered a mean for reource allocation in hared computing ytem, often focuing on ytemoriented performance objective uch a average delay and throughput. A number of paper have conidered the uercentric approach, where the objective i to maximize the aggregate ervice utility, uing variou marketbaed mechanim and concept, including commodity market, bargaining, poted price model, contract baed model, bidbaed proportional reource haring model, bartering, and variou form of auction. Extenive urvey may be found in [17, 3]. The monograph [8] urvey related literature from a queueing theory perpective, while the monograph [12, 4] urvey the ue of pricing in telecommunication ytem and communication network. A related body of work on bidbaed reource allocation ha emerged in the communication network literature, in the context of capacity allocation and congetion control. Recent urvey may be found in [14, 18]. Thee bidding mechanim may be viewed a adaptive, congetiondependent pricing cheme, whereby the available reource are completely divided among the preent uer in proportion to their bid. An application of thee concept to a hared (utility) computing environment ha been conidered in [19], where uer are identifie a peritent flow of job, and bidding i employed to tatically divide the computing reource among them. Our model here i baically different a uer are aociated with ingle job, which arrive equentially and are allocated reource upon arrival. The highly relevant article [7] (oberved after completion of the preent paper) conider a imilar pricing framework to our. That work however doe not conider operating cot, but rather focue on the effect of delay externalitie on the ocial utility. Thi paper i organized a follow. Section 2 lay down the baic ytem and uer model. Section 3 conider the individual optimization problem faced by arriving uer. In Section 4 we conider the ocial optimization problem and how that it admit a unique olution. The next Section 5 etablihe our central reult, namely that that ocial optimality i induced by fixed perunit pricing, and characterize the optimal price. We dicu the economic context of our reult in Section 6, and conclude in Section SYSTEM AND USER MODEL Thi ection introduce our baic model, including the underlying ervice ytem, and the uer characteritic. 2.1 The Service Sytem We conider a hared computing facility to which job (or oftware application) are ubmitted equentially by individual uer. We aociate each job with a ditinct uer, and employ the term job and uer interchangeably. Upon arrival, each job i allocated a certain amount of ervice reource, according to ome reource allocation protocol, and promptly enter ervice which proceed to completion. Alternatively, ome potential arrival may decide to balk (ay, due to high pricing), in which cae they leave the ytem without receiving any ervice. We proceed to decribe quantitatively the parameter of thi model. Uer type: Arriving uer may differ in their ervice requirement and cot parameter. Thee are ummarized by a type identifier, denoted i, which may be conidered a a vector of real or dicrete parameter. Let I denote the et of poible type. Arrival rate: Potential uer arrive according to ome tochatic proce, with pecified rate for each type. with a given rate ditribution. More preciely, let Λ 0 (di) denote a finite poitive meaure on I. Then, for any (meaurable) et I I, the arrival rate of potential uer with type in I i Λ 0 (I). We impoe no further requirement on the arrival procee except that the average number of arrival converge in the long run to to pecified average 1 (almot urely), o that Little law can be applied [15]. An arriving uer may either chooe to enter ervice, or ele might balk with ome probability (to be determined by our deciion model). Thu, the effective arrival rate (an the balking uer), denoted Λ(di), will generally be maller than Λ 0(di). 1 Thi allow to conider fairly general procee, that include, for example, nontationarity due to timeofday variation, Markovmodulated arrival procee, and dependence between the arrival procee of different type.
3 Reource: Let z i 0 denote the quantity of ervice reource allocated to a typei uer (or job) upon arrival. In our context, z i may be thought of a the number of (virtual) computing unit allocated to thi job. We hall aume here that z i i a continuou variable; thi may indeed be the cae in ome ytem, while for other it hould be conidered an approximation to a dicrete variable with fine granularity. It i aumed that z i i fixed throughout the job execution period. Service duration: Let τ i denote the execution (ervice) time of ome typei job. A job are aumed to enter ervice upon arrival, thi coincide with the job ojourn time, namely the total time pent in the ytem. Evidently, the ervice time depend both on the ervice requirement of thi job (a determined by the job type i), a well a the reource z allocated to that job. Thu, τ i i a random variable with a zdependent ditribution, aumed to have finite mean and variance for any z > 0. We let T i (z) = E i,z (τ i ) denote the mean ervice time for type i job uing z reource. It will be naturally aumed that each T i(z) i a decreaing (or at leat nonincreaing) function of z. Thi property, along will ome additional requirement on T i(z), will be formally tated in Subection Steady State We next conider the teady tate load in the ytem for given arrival rate and reource allocation. Recall that type i uer enter the ytem a a Poion proce with (effective) rate Λ(di). Suppoe that each typei uer i allocated a poitive quantity z i of reource. The ervice time i then a random variable with finite mean T i (z i ). A a reult, the ytem may be viewed a a ditribution over an independent collection (indexed by i) of M/G/ queue, each of which i obviouly table. We hall aume that thi ytem i in teady tate. Uing the amplepath verion of Little law [15], the longterm average number of job in ervice for a queue with arrival rate λ i and mean ervice time T i (z i ) i given by N i = λ i T i (z i ). Therefore, the longterm average number of job i ervice (A a function of their type) i ditributed according to N(di) = Λ(di)T i (z i ). A each typei uer occupie z i reource, the longterm average of the total reource utilization i given by Z = z in(di) = z it i(z i)λ(di). (1) i We will refer to Z a the load of the ytem. It may be noted that the decription above preume that the reource pool i unlimited, o that all arriving uer are admitted to ervice. We will relate to the iue of limited reource later in Section??. 2.3 Pricing Uer will be charged ome uage cot for the rendered ervice. We will focu here on fixed peruage pricing, o that the monetary charge M i for a job that occupied z i reource for τ i time unit i i M i = P z iτ i, where P i the perunit price rate (in monetary unit per unit reource and unit time). Therefore, the expected charge for a typei job, given that it wa allocated z i reource, will be E(M i ) = P z i T i (z i ). (2) The latter expreion will form part of the individual uer utility function. We note that the expected ervice time i ued, which reflect the implicit aumption that an arriving uer doe not know beforehand the exact execution time of hi job, but only it ditribution. 2.4 Individual Utilitie An incoming uer ha two deciion to make upon arrival. One i whether to join the ytem or balk. If he decide to join, he further need to determine the amount of reource z i required for hi job. Thee deciion are made individually by each uer, with the goal of maximizing hi own utility. We proceed to define the uer utility function. Firt, the utility of any balking uer i taken to be zero. Note that thi i merely a convenience, a any other baeline value can be ued intead. A for uer who join the ytem, the utility function of a type i uer can be written a: U i(z i) = V i(z i) E(M i) = V i(z i) P z it i(z i), (3) where V i i the (expected) value that the uer aign to executing hi job with reource z i, and E(m i ) i hi expected charge a per (2). The uer value function are aumed to atify to the following propertie. Aumption 1. For each uer type i, (i) V i (z) i continuouly differentiable, and trictly concave increaing in z 0, and bounded above. (ii) V i(0) < 0. The concavity aumption i of coure tandard, implying that the marginal improvement due to additional reource i diminihing. Property (ii) imply guarantee that uer will prefer balking to joining the ytem with z = 0 reource. Dicuion and Elaboration: It i important to note that the value V i may reflect both the execution time of the job, a well a other element related to the quality of ervice (QoS) experienced during the execution time. To make thi pecific, one may conider the eparable form V i (z) = V i (z) E z (c i (τ i )), (4) where the econd component i the cot aociated with the execution time, and the firt repreent the other quality meaure. Here τ i i the actual job execution time, c i ( ) i a delay cot function, and the expected value i taken with repect to the ditribution of τ i given z i = z. We can now ditinguih between two extreme cae: Batch job: Here a certain computation need to be carried out, and the computation time cale with the allocated reource. Such job are common, for example, in cientific and buine computing. Here V i(z) = v i, independently of the allocated reource, and the delay enitivity i the important term. Mild delay enitivity may be captured by linear delay cot, namely, c i(τ i) = γ iτ i for ome γ i > 0. Then V i(z) = V i(z) γ it i(z), where T i(z) = E z(τ i) i the expected ervice time of uer i. Note that the aumed monotonicity and concavity propertie of V i are equivalent here to T i (z)
4 being convex decreaing, which may be reaonably aumed (ee below). Other application may have more critical time contraint, which may be captured by a (convex increaing) function c i ( ) that become teep toward the required completion time. Fixed duration application: In certain application clae, cloud reource may be ecured for fixed period of time. Thi may be the cae, for example, in interactive application that are intended for web cutomer ervice. In that cae the delay term become irrelevant, and V i(z) = V i(z) capture the QoS offered to cutomer during that time period. 2.5 Operating Cot Let C op denote the operating cot of the computing facility per unit time. We aume that thi cot depend on the ytem reource utilization, namely C op = C 0 (Z), where Z denote the average reource utilization, a pecified in (1). We further aume the following. Aumption 2. C 0(Z) i continuouly differentiable and trictly convex increaing in Z 0. Remark 1. Large datacenter normally take advantage of the economy of cale offered by tatitical multiplexing and reource virtualization, o that the overall load on the ytem i maller than the um of individual reource requirement. The cot function C 0 i aumed to take account of thi effect. 2. In addition to the running cot of operation, the cot term may take into account alo alo the required invetment in infratructure, computed for a certain period ahead. Whether thi i included depend of coure on the time cale conidered, and whether invetment in infratructure i conidered a part of the model. 3. Oberve that the operating cot are aumed to depend on the average reource utilization, rather than it temporal ditribution. Thi coarecale approximation i eential for the reult of thi paper. 2.6 Service Time A mentioned, the ervice time, or job execution time, generally depend on the reource z allocated to it. Recall that T i (z) denote the mean ervice time function for typei job. We dicu here ome pecific form for thee function, and then tate our general aumption. With the exception of fixedduration application, we reaonably expect the mean ervice time to be trictly decreaing in z. A common aumption in the proceorharing queueing literature i that of proportional peedup, namely T (z) = D/z. Thi baic model i arguably overly optimitic regarding the benefit of cale in parallel computation, a it ignore factor uch a etup time and parallelization overhead that hould impede further reduction in T beyond a certain point. A lightly modified model that can accommodate uch effect i given by T (z) = a + D z. (5) Here a > 0 preent the noncalable part of the job execution. Obviouly, now T ( ) = a > 0. Thi model ha the ame form a Amdahl law, which i often ued to model poible peedup in parallel computing (e.g., [9]). Our general requirement on T i (z) are given below, and involve the uer value function V i (z) a well. We will ubequently tate more pecific condition on T i that imply thi aumption. Aumption 3. For each uer type i, T i(z) atifie the following propertie: (i) T i(z) i a continuouly differentiable and (weakly) decreaing function of z 0, with lim z T i (z) > 0. (ii) The ratio i (z) (zt i (z)) i trictly decreaing in z. Here V i i the uer utility function, and the prime denote differentiation with repect to z. Oberve that thee condition are atified for the model in (5). In that cae, zt i (z) = az + D and (zt i (z)) = a, a poitive contant, o that (ii) i equivalent to concavity of V i (z) (which i indeed included in our aumption). The cae of a fixed execution time i of coure a pecial cae with D = 0. More generally, property (ii) hold whenever zt i (z) i a convex increaing function of z, a thi implie that the denominator i nondecreaing and poitive. Convexity of zt i (z) i however not a neceary condition for Aumption 3 to hold. A an important example, property (ii) above can be etablihed for certain type of delayenitive utility function, provided that the expected ervice time atifie ome reaonable additional condition on the ervice rate. We ummarize thi obervation in the following lemma. Lemma 1. Let µ i (z) = 1/T i (z) denote the ervice rate function. Suppoe that (i) µ i (z) i a differentiable, trictly concave and trictly increaing function of z 0, with µ i (0) = 0 and µ i ( ) <. (ii) V i (z) = v i c i (T i (z)), where c i i an increaing and convex function of z. Then Aumption 3 i atified. Proof. Item (i) of the aumption i obviou by the tated propertie of µ i. Let f(z) = (z) (zt (z)) (where we omit the index i). Subtituting V (z) = v c(t (z)) and T (z) = µ(z) 1, we obtain f(z) = c (µ(z) 1 )µ (z) µ(z) zµ (z) = N(z) D(z). Item (ii) now follow by howing that N(z) i poitive decreaing and D(z) i poitive increaing (with both monotonicity propertie being trict). The firt claim follow ince µ(z) 1 i trictly decreaing, c ( ) i poitive increaing, and µ (z) i poitive trictly decreaing (by the aumed concavity of µ). Monotonicity of D(z) follow by oberving that D (z) = zµ (z) > 0 for z > 0 (where oneided derivative may be ued if neceary), and poitivity now follow ince D(0) = 0. To illutrate, T (z) = a + D z doe not atify convexity of zt (z), but condition (i) of the lat lemma i eaily een to hold.
5 2.7 The Finite Cla Model The model a decribed above allow a continuum of uer type, with different performance and utility characteritic for each type. While poible to continue the analyi at thi level of generality, we find it ueful to conider here a finite dimenional model, that i amenable to explicit computation and avoid technicalitie aociated with meaurability iue. The firt uch model that come to mind i retricted to a finite number of uer type, each with a poitive ma of arrival. However, uch a model uffer from a couple of hortcoming, both related to the admiion deciion of the uer: (1) Dicontinuou demand: Conider the variation in the arrival rate a the price i increaed. A all uer of a given type hare identical parameter, they will all change their admiion deciion (from join to balk) at the ame price level. Thi will lead to jump in the demand, in repone to ome mall change in price. Such dicontinuitie can hardly be expected in practice. (2) Mixed deciion: Due to the abovementioned dicontinuitie, equilibrium condition will generally require uer of one or more type to chooe probabilitically between join or balk. A uch uer are necearily neutral with repect to thee choice, the precie mechanim through which a given uer come to chooe between them with a given probability remain exogenou to the model. To mitigate thee hortcoming, we will conider a finitecla model that goe beyond the imple finitetype cae. Here uer are grouped into a finite et of uerclae, each haring the ame characteritic except for a continuoulyditributed bia i their ervice utility. A we hall ee, thi addition will indeed induce mooth demand variation, and avoid randomized deciion. The main characteritic of thi model are borrowed from [11], where a imilar utility model wa ued in a queueing context. Let the et I of uer type be divide into a finite et of clae, denoted S = {1,..., S}, with element S. We ue the notation i to indicate that a type i belong to cla. All job of a given cla have imilar ervice time characteritic, namely T i(z) T (z), for all i. Furthermore, the ervice value function V i(z) are taken to have the additive form 2 V i (z) = v i + V (z), i. (6) Thu, the dependence on the reource z i the ame for all uer of a given cla. To that, a typedependent bia v i i added which create intracla variation. We refer to v i a the uer tate parameter. Combining the above with (3) and (2), the utility function of a erved uer i U i (z, P ) = v i + V (z) P zt i (z). (7) The uer type i may now be identified with the pair (, v), namely the uer cla and hi tate parameter. Recall that the potential arrival rate are pecified through a poitive meaure Λ 0 (di) on the et of type I. With i = (, v), we may expre Λ 0 (di) a Λ 0 (, dv); here Λ 0 (, ) i the ditribution of the tate v for cla uer. Let λ max = Λ 0 (, IR) 2 To avoid notational clutter, here and in the following we ditinguih between ome typepecific and clapecific quantitie (uch a V i a V ) through their index only. denote the total arrival rate of potential uer of cla. Some further requirement regarding thee ditribution will be pecified in Aumption 5 below. We oberve that Aumption 1 and 3 regarding V i and T i are in effect, and thee imply imilar propertie for the cla quantitie T and V. We ummarize thee propertie below. Aumption 4. For each uer cla, (i) V (z) i continuouly differentiable, trictly concave increaing for z > 0, and bounded above. (ii) V (0+) =. (iii) T (z) i a continuouly differentiable and decreaing function of z 0, with lim z T (z) > 0. (iv) The ratio (z) (zt (z)) i trictly decreaing in z. Property (ii) enure that Aumption 1(ii) i atified for any tate parameter v i. 2.8 Aggregate Utility Given the utility function in (7), we obtain the following demand function λ (z, P ) = λ max Prob{v + V (z) P zt i (z) 0}, where the probability i taken over v according to it cla ditribution Λ (, dv). Thu, λ (z, P ) i the arrival rate of uer whoe utility i nonnegative when allocated z reource at price P. More important for our purpoe, however, will be the aggregate utility obtained at a given arrival rate. Suppoe that, out of the potential arrival of cla, only thoe uer with higher tate parameter v are admitted up to rate λ [0, λ max ], and each of thoe i allocated z > 0 reource. The aggregate value of ervice for thee admitted uer (per unit time) will be V (λ, z ) = V (λ ) + λ V (z ), (8) where { } V (λ ) = up ve vλ 0(, dv) : e vλ 0(, dv) = λ. {e v [0,1]} (9) Thu, V (λ ) i the um over the higherpercentile tate of uer of cla, up to rate λ. We refer to V a the (tate) aggregate utility. A more explicit expreion i obtained a follow. For each λ, let v 0 and p [0, 1) be o that λ = Λ 0(, dv) + pλ 0(, v 0) (10) v>v 0 (note that p 0 i required only if Λ 0 (, ) ha a point ma at v 0 ). Then V (λ ) = vλ 0 (, dv) + pv 0 Λ 0 (, v 0 ). (11) v>v 0 We next dicu ome propertie of the aggregate utility function V (λ ). It i eaily verified that V (0) = 0, and V i (weakly) concave. With ome additional aumption it atifie tricter propertie (cf. [11]). Lemma 2. For each cla, conider the function V (λ) defined in (9). Suppoe that the meaure Λ 0(, dv) over v i abolutely continuou (relative to the Lebegue meaure), with a denity function g (v). Then
6 (i) V (λ) i trictly concave in λ [0, λ max ]. (ii) Suppoe g (v) > 0 in ome neighborhood of v 0. Then V (λ) i continuouly differentiable around λ 0 = v v 0 g (v)dv, with the derivative V = d V given by V dλ (λ 0) = v 0. (iii) If g (v) > 0 for all < v <, then V (λ) i continuouly differentiable for all λ [0, λ max ], and lim λ 0 V (λ) =, lim λ λ max (λ) =. Proof. (i) Given the exitence of denity, (10)(11) take the form λ = g (v)dv, (12) v v 0 V (λ) = vg (v)dv. (13) v v 0 In fact, for every λ (0, λ max ) there exit a ome v 0 = v 0(λ) that atifie (12). Thi follow by noting that the righthand ide i continuou in v 0 and (weakly) decreaing from λ max to 0 a v i increaed from to. Now, uing (13), we have that for every λ and ϵ > 0, V (λ + ϵ) V (λ) = v0 (λ) vg (v)dv v 0 (λ+ϵ) v0 (λ) > v 0 (λ) v 0 (λ+ϵ) = v 0(λ)ϵ = v 0(λ) g (v)dv v0 (λ ϵ) v 0 (λ) > V (λ) V (λ ϵ), g (v)dv which implie trict concavity. (ii) From g (v) > 0, it follow that v 0 = v 0 (λ) that atified (12) i continuou and trictly decreaing in λ around λ 0. Oberve now from (12)(13) that dλ = g (v 0 ) and d V dv 0 = v 0 g (v 0 ), o that d V = v dλ 0(λ) around λ 0. Since v 0 (λ) i continuou there, then o i the latter derivative. (iii) The firt part follow from (ii). The limit follow from V (λ) = v 0(λ), a (12) implie that v 0(λ) a λ 0, and that v 0(λ) a λ λ max. dv 0 We will henceforth adopt the following aumption. Aumption 5. The condition of Lemma 2 are atified. That i, for each cla, the tate ditribution Λ 0 (, dv) admit a finite denity g, with g (v) > 0 for all 3 < v <. We finally note that, under the ame condition that lead to (8), the average load of equation (1) can be expreed a Z = λ z T (z ). (14) 3 The infinite upport of g i a modeling convenience, a it enure that at any price level there will be ome uer which chooe to enter ervice, and ome other who chooe to balk. In reality, we need thi property to hold only for price in a reaonable range. 3. INDIVIDUALLY OPTIMAL DECISIONS We proceed to examine the optimization problem faced by an individual uer. Given the advertied perunit price P, an arriving uer need to decide whether to execute hi job at the conidered facility, and if o, the amount of reource to acquire for that purpoe. Conider a uer i of cla and tate v i. Recall that thi uer utility function i given by (7) if he join ervice, and et to 0 if he chooe to balk. In thi ection we conider a fixed price P > 0. We therefore omit P from our notation and write U i (z) for U i (z, P ), etc. The maximal utility for thi uer will be where U max i = max{0, max z 0 U i(z)}, (15) = max{0, v i + U } (16) U = max z 0 {V (z) P zt (z)}. (17) A we how below, the optimization problem in (17) admit a unique maximum. Recall that a calar function the real line i trictly quaiconcave if it i trictly increaing up to a certain point, and trictly decreaing thereafter. Propoition 3. The utility function U (z) = V (z) P zt (z) i trictly quaiconcave, and admit a unique maximizer z > 0, which atifie the following firtorder condition (z ) P (z T (z )) = 0. (18) Proof. We firt oberve that U (z) ha at mot one tationary point z where U (z) = 0. Indeed the latter i equivalent to (z) (zt (z)) = P. But U (0+) = by Aumption 4(ii), and U (z) i eventually decreaing ince lim z zt (z) = by Aumption 3(i), while V i bounded from above. Conequently, there mut exit a maximum point at ome finite z > 0, which clearly mut atify U (z) = 0, namely (18). Since there i no other tationary point, the aertion follow. We may ummarize uer i deciion proce a follow. Firt, he compute the optimal reource allocation z and maximal utility U = U (z ) by olving (17). Next, if v i + U (z ) < 0 he balk, if > 0 he enter ervice uing z reource, and in cae of equality he i neutral between thee two option. For concretene we hall chooe the enter option in thi cae. 4 We can now obtain the effective arrival rate λ of each uer cla. A noted, uer of cla who join ervice are thoe with tate v i U (z ), where U (z) = V (z) P zt (z). Then λ i given by (10) with v = U (z ), and, oberving Lemma 2(ii), V (λ ) = v, or (λ ) + V (z ) P z T (z ) = 0. (19) Since i a trictly increaing function, thi equation uniquely determine λ. 4 A the et of neutral uer will alway have zero meaure under our type aumption, thi choice doe not affect our reult.
7 We finally etablih ome plauible monotonicity propertie, that will be be needed later on. Let z (P ) and λ (P ) denote the individually optimal reource allocation and effective arrival rate under price P. Lemma 4. z (P ), z (P )T (z (P )) and λ (P ) are all continuou and trictly decreaing function of P. Conequently, o i Z(P ) = λ z T (z ). Proof. Fixing, we remove the cla index from V, z (P ), etc. in the remainder of thi proof. Conidering z(p ), we write (26) a P = (z) (zt (z)). (20) But the righthand ide i continuou and trictly decreae in z (by Aumption 4), o that P = P (z) i trictly decreaing in z. Thi implie that the invere function z = z (P ) i a welldefined continuou function which i trictly decreaing in P. Turning to z(p )T (z(p )), note that (zt (z)) > 0 at any olution z = z(p ) of (20), ince > 0 (Aumption 4) and P > 0. Therefore h(z) = zt (z) i trictly increaing in z at thee point. But we have jut hown that z(p ) i trictly decreaing in P, and therefore o i h(z(p ). Continuity of the latter follow from that of z(p ) and T (z). Finally, conider λ(p ). By (25), (λ) = V (z) + P T (z)z = k(z), o that (λ(p )) = k(z(p )). Recall that (λ) i continuou and trictly decreaing in λ by Lemma 2. Therefore, the required propertie of λ(p ) would follow by howing that k(z(p )) i continuou and trictly increaing in P. But dk(z(p )) dp = k dz z dp + k P = k P = T (z(p ))z(p ) > 0 where we have ued (26) to conclude that k = 0 (which z i of coure related to the Envelope Theorem [16]). Thi conclude the proof. The lat lemma how that, enibly, a the price P i increaed, uer will acquire le reource z, and their arrival rate λ will decreae. Further, The multiple z T (z ) that repreent the total reource uage over time by each uer i decreaing a well (even though the execution time T (z ) increae under our aumption), a doe the average ytem load Z. 4. THE OPTIMAL SOCIAL WELFARE The ocial welfare, or ocial utility, i defined a the um of utilitie of all individual entitie that are conidered part of the ociety. In our model thee are the individual uer together with the ervice provider. We conider here the ocially optimal aignment of arrival and reource allocation, which i intended to maximize the ocial welfare. We allow thi aignment to be managed by an omnicient central controller, that ha full knowledge of the the ytem parameter a well a individual cutomer type and preference. Thi i not a realitic cenario of coure, and i ued only to identify the ocial optimum. Later we will how that thi optimum can be achieved by appropriate pricing. We proceed to preent the ocial welfare function that i to be maximized, and characterize it optimal olution. We tart by preenting a general expreion for the ocial welfare, which we then pecialize to the finitecla model. The lat ubection etablihe exitence and uniquene of the optimal olution. 4.1 The Social Welfare The controller deciion may be expreed in term of the following variable: 1. e i [0, 1], where e i = 1 mean that uer of type i are admitted to ervice, e i = 0 mean rejection, and e i (0, 1) mean randomization between thee two option z i 0, the amount of reource to aign to uer of type i (which i relevant only if e i > 0). Recall that the potential arrival rate ditribution i pecified by Λ 0(di). The ocial welfare i now given by the um of uer utilitie minu the ytem operating expene: W oc = V i (z i )e i Λ 0 (di) C 0 (Z) (21) where i Z = z i T i (z i )e i Λ 0 (di). i Note that thi expree the teadytate expected ocial urplu per unit time, or equivalently it longterm average. Our goal i to maximize W oc over all (meaurable) election of deciion variable. Let W oc denote thi maximal value. 4.2 Aggregate Utility Form Specializing to the finitecla (but infinitetate) model, we proceed to formulate the above optimization problem a a finite dimenional mathematical program. Suppoe within each cla only uer with higher tate are admitted, up to rate λ, and each of thee i allocated reource z. Then, oberving (8) and (14), the ocial welfare rate i given by W (λ, z) = ( V (λ ) + λ V (z ) ) ( ) C 0 λ T (z )z, (22) where λ = (λ 1,..., λ S), z = (z 1,..., z ). Recalling the definition of V, thi can be interpreted a the ocial welfare obtained when admitting cla uer with higher tate up to rate λ, and allocating z reource to each. Conider the following optimization problem: maximize W (λ, z) (23) ubject to λ [0, λ max ], S, z 0, S. Propoition 5. The maximal value Woc of the ocial welfare (21) coincide with the optimal value of the program (23). 5 We aume that all uer of the ame type are ubject the the ame control deciion. Thi can be argued to be optimal; however we will not bother with that here ince under a continuum of type aumption the chance of obtaining two uer of the ame type are null. Furthermore, randomized deciion will not be required in thi cae.
8 Proof. We firt argue that all uer of a given cla can be allocated identical reource. Thi i mot eaily demontrated uing the firt variation of (21). Conider the maximization of (21) over (z i ) i for a ingle cla, with all other deciion variable fixed. Subtituting an ϵvariation zi ϵ = z i + ϵ z i, we obtain after ome calculation W oc (z ϵ ) = W oc (z)+ ϵ [V (z i ) C 0(Z)(z i T (z i )) ] z i e i Λ(di) + o(ϵ). i Note that we ubtituted i =, which follow from (6). In any maximum the variation term mut be nonpoitive. Now, for z i > 0, z i can have arbitrary ign, o that (z i ) C 0(Z)(z i T (z i )) = 0, z i > 0 mut hold for e i Λ(di)almot every i. If z i = 0 then z i i nonnegative (auming ϵ > 0), and we imilarly obtain (z i) C 0(Z)(z it (z i)) 0, z i = 0. Noting that C 0 > 0 by Aumption 2, if follow a in Propoition 3 that the lat two equation have a unique olution z i z, which i valid for all i with e i > 0 (i.e., which are admitted to ervice). We can therefore retrict attention to z i z for all i. Subtituting in (21) give, after noting (6) and (14), ( ) W oc = + V (z )]e i Λ 0 (di)c 0 λ z T (z ) i=(,v)[v = ( ) ve (,v) Λ 0 (, dv) + λ V (z ) v ) C 0 ( λ z T (z ) where λ = eiλ0(di) = e i v (,v)λ 0(, dv) i the effective arrival rate of cla i. Conider now the maximization of the lat expreion for W oc over {e i e (,v) }. For given λ, the only term that i enitive to the choice of e (,v) i the firt one, and it maximum i clearly obtained by V (λ ) in (9). With thi ubtitution, W oc reduce to the expreion in (22). 4.3 Exitence and Uniquene We proceed to how exitence and uniquene of the olution to the ocial optimization problem (22)(23), and characterize thi olution in term of the firtorder optimality condition. We note that thi program i not a concave one, even under the convexity propertie impoed in our model aumption. In fact, it i readily een that thi program i a convex one in λ alone (with z held fixed), and the the optimization problem over z can be tranformed into a convex one (a dicued in Section 6). However, the problem i eentially not jointly convex in λ and z, due to the multiplicative term λ V (z ) and λ T (z )z. Hence, we reort to problempecific analyi that relie on monotonicity argument. We tart with the following characterization of the (poibly local) maxima of our optimization problem. Lemma 6. Let λ = (λ 1,..., λ S ), z = (z 1,..., z S ) be a local maximum point of (23), and define η η(λ, z) = C 0 ( λ z T (z ) ). (24) Then λ (0, λ max ), z > 0, and (λ ) + V (z ) η T (z )z = 0, S, (25) (z ) η [ z T (z ) ] = 0, S. (26) Proof. If follow from Lemma 2(iii) that λ i internal, namely λ {0, λ max }. Differentiating (22) with repect to λ yield (25). Now, λ > 0 implie that z > 0 upon noting Aumption 4(ii). Thu, the maximizing z i interior. Equating the derivative of (22) with repect to z to zero and cancelling λ yield (26). We next hown that equation (24) (26) admit a unique olution. Thi will follow by howing that equation (25) (26) imply that {λ, z } are decreaing in η, while the righthand ide of (24) i increaing in thee variable. The required monotonicity propertie of {λ, z } are ummarized in the next lemma. Lemma 7. Conider equation (25) (26), with fixed η > 0. (i) For any η > 0, there exit a unique olution {λ, z } to equation (25) (26). Denote thi olution by {λ (η), z (η)}. (ii) The function z (η), z (η)t (z (η)) and λ (η) are all continuou and trictly decreaing in η. Proof. (i) Fix η > 0 and. Exitence and uniquene of a olution z to equation (26) follow a in Propoition 3. Exitence and uniquene of a correponding olution λ to (25) now follow by the propertie of V in Lemma 2, item (i) and (iii). (ii) The proof i identical to that of Lemma 4. Lemma 8. There exit a unique olution {λ, z } to the ytem of equation (24) (26). Proof. Conider the righthand ide of equation (24) a a function of η > 0, with z = z (η) and λ = λ (η) a pecified in Lemma 7. By the reult of that lemma, the argument of C 0 i trictly decreaing in η, and ince C 0 i a trictly increaing function (by the aumed convexity of C 0 (Z)) it follow that C 0( λ z T (z )) i trictly decreaing in η. Since it i alo poitive, it follow that (24) ha a unique olution η, with correponding z = z (η ) and λ = λ (η ). We finally need to how that the maximum of (23) i obtained in a compact et, namely not for z. Lemma 9. The global maximum of (23) i attained at a finite point. Proof. We how that W (λ, z) i decreaing in z, for z large enough. Oberve that W (λ, z) z =λ ( (z ) C 0(Z)(T (z )z ) ) λ ( (z ) C 0(0)(T (z )z ) ). Since V i bounded and increaing, then V 0 a z, while our aumption that T (+ ) > 0 implie that lim inf z (T (z )z ) > 0. Therefore, there exit ome z o that W z < 0 for z > z, independently of other variable. Thi immediately implie that the upremum of W (λ, z) i attained for {z z }. But thi define a compact region and W (λ, z) i continuou, o that the maximum i attained there.
9 Thi lead u to the main reult of thi ection. Theorem 10. There exit a unique olution {λ, z } to the ocial optimization problem (23). Thi optimal olution i internal (0 < λ < λ max, z > 0) and obey the firt order condition (25)(24). Proof. By the lat lemma, the maximum i attained at a finite point. But Lemma 6 and 8 that there exit at mot one local maximum, which i therefore the global maximum. The econd part follow from Lemma SOCIALLYOPTIMAL PRICING Having identified the ocially optimal olution, we are faced with the tak of implementing thi olution. Ideally, uch an implementation hould not allow the central controller acce to private information of the uer, which in particular include their ervice utility and preference. In thi ection we how that the imple perunit pricing mechanim, with the ame price to all, uffice to induce the ocial optimum. Let {λ, z } be the unique ociallyoptimal olution (23). We et the perunit price to be P = C 0 ( λ z T (z ) ) = P. (27) Recall that each uer maximize hi individual utility given thi price, a decribed in Section 3. The main reult of thi paper i the following one. Theorem 11. Let the perunit price be P, a defined in (27). Then individual optimality lead to the ocially optimal olution {λ, z }. Proof. We will how that the individual optimality condition coincide with the condition for ocial optimum. Let {λ, z } denote the arrival rate and reource allocation that are obtained through individual optimality with price P. Oberve that z i uniquely determined by equation (18), wherea λ i given by (19). Comparing equation (24) (26) with equation (27), (18) and (19), it may be een that both {λ, z } and {λ, z } atify equation (25) (26), with η = P. But by Lemma 7 the olution to thee equation i unique, o that {λ, z } = {λ, z }. We next conider the ocial welfare a a function of the price, and etablih it unimodality. Beide it own interet, thi property will alo be ueful below. Propoition 12. Let W (P ) denote the ocial welfare W (λ, z) obtained under price P. Then W (P ) i trictly increaing in P for P < P, and trictly decreaing for P > P. Proof. Differentiating W (P ) from (22), we obtain dw (P ) = ( ) W (λ, z) dλ W (λ, z) dz + dp λ dp z dp = ( ) V (λ ) + V (z ) T (z )z C 0(Z) dλ dp + ( ) λ V (z ) λ (z T (z )) C 0(Z) dz dp. Oberving (18) and (19), thi give after ome calculation (which we omit here) dw (P ) dp = (P C 0(Z)) dz dp. Now, Z = λ z T (z ) i decreaing in P by Lemma 4. Thu, C 0(Z) i decreaing in P, while the equality P = C 0(Z) hold at P. Therefore P C 0(Z) < 0 for P < P and P C 0(Z) > 0 for P > P. Thi induce oppoite ign for dw dp. 6. ECONOMIC CONTEXT The cloud computing environment examined in thi paper i that of a dynamic ervice ytem with equential arrival of uer, and variable ervice time that depend on the uer choice. The main iue examined, evolving around the notion of ocial welfare and it maximization, are fundamental one in microeconomic theory. It will thu be ueful to elaborate further on the economic context, and compare the tandard model with the one conidered here. The economic etup of thi paper i baically that of a monopoly, namely a ingle firm that can et market price. The textbook verion of thi problem [16], retricted to a ingle continuou product, conider a finite et I of conumer, with v i (x i ) denoting the value of conumer i for conuming quantity x i 0 of the product, and C(x) being the cot of production of quantity x. The ocial welfare i therefore W (x) = i v(x i) C( i x i). With linear pricing, each uer i maximizing V i(x i) P x i, and (under tandard convexity aumption) the ocial optimum i defined by marginal cot pricing, o that P = C ( i xi) hold at the optimal point. Comparing with (3) and (21), it may be een that the the role of the quantity x i i taken up in our model by the quantitytime multiple z i T i. That i, the product being offered here i not meaured in term of the reource quantity z i itelf, but rather in term of quantity multiplied by uage time. And indeed, the propoed pricing cheme (and the ocially optimal one in particular) are linear in the latter meaure. Thi i of coure quite reaonable; however, it i important to realize that thi tructure i not aumed a priori, but rather arie out of our model once we determine that that operation cot C 0 i a function of the average load, and employ Little law to decribe the effect of demand on the ytem load. We will briefly comment on other poibilitie below. Let u conider further the ue of x = zt (z) (with the type index removed for convenience) a the baic deciion variable in place of z. Aume for implicity that x(z) = zt (z) i trictly increaing in z, o that the invere z(x) i well defined (thi i indeed the cae for T (z) = a + D, a in (5)). z The individual utility (3) can now be expreed a a function of x a Ũ(x) = U(z(x)) = V (z(x)) P x. It i now eay to verify that condition (ii) of Aumption 3 i equivalent to trict convexity of V (z(x)) in x. Therefore, under our aumption, the individual utility function U i convex when conidered a a function of x. In the preent paper we have choen to work with z throughout, motivated by cloud application where the uer actually chooe the reource z explicitly, with the computing time T (z) being determined a a reult. We note that working with the reourcetime multiple x directly, rather than z, may be natural i other application when reource are automatically adjuted by the manager according to the application need. We leave further elaboration of thi approach for future tudy.
10 An important point to make i the relation between the tructure of the operating cot term C 0 and the form of the price tariff. In thi paper we have aumed that C 0 = C 0 (Z) i a function of the average load Z = λ z T (z ), which indeed we believe to be the dominant term. Conider, however, the addition of a cot term C 1 which depend only on the average number of uer in the ytem (rather the reource utilized, namely C 1 = C 1(N), with N = λt(z). Thi can repreent, for example, the accounting overhead aociated with each uer. Then, uing imilar reaoning a before, we are led to conider a twopart tariff of the form P zt + QT. We conjecture that a proper choice of the price coefficient P, Q will lead the ytem to ocial optimality; again, thi i left for further tudy. Finally, we comment on our aumption of a continuum of uer type. A noted, the textbook model decribed above conider a finite population I of uer. A variant of the model due to Aumann [2] (and ee [5]) conider a continuum of infiniteimal uer, o that the ocial welfare, for example, take the form W (x) = v(xi)m(di) C( xim(di)). Thi i i i indeed mathematically akin to (21). However, we note that thi imilarity i only mathematical. In our model, the uer are of finite ize, and their number i countable; what i aumed continuou i the pool of poible uer type, from which the type of each uer i drawn. Therefore, what make the effect of each uer negligible i not thi miniature ize, but rather the conideration of the average ytem utility over a long (infinite) time horizon. Thi i again a ditinguihing apect of the dynamic model conidered here, a compared with the tandard economic etup. 7. CONCLUSION Thi paper conidered the reource allocation problem in a cloud computing facility, where the underlying objective i to maximize the ocial utility through a imple pricing cheme. We howed that the ocially optimal operating point i unique, and can be utained by a linear, uagebaed tariff, which charge a fixed price per unit reource and unit time. Beide the analytical reult, a major contribution of the paper i in the modeling apect. The propoed model, which i well uited for economic analyi, incorporate everal novel feature that pertain to the cloud computing environment, including: Incorporating temporal apect into the model. Flexible dependence of the computation time on the applied reource, which can be ued to take account of etup and parallelization overhead. Uer heterogeneity, in term of both utility and job proceing requirement. A flexible finitecla, continuoutype model that allow mooth demand function along with finitedimenional problem formulation. Variable arrival rate, which i haped by uer balking, in addition to their choice of reource. The eential model developed here may provide a bai for additional work on economic apect of cloud computing, conidering further apect of revenue, profit and competition among cloud. Intereting extenion to the model include the allocation of multiple reource type, or reource bundle, rather than the ingle reource type conidered here, a well a the conideration of dicrete reource, and more a detailed analyi of reource allocation that are timevarying according to the application need. Finally, it hould be of interet to tudy poible effect of congetion, which were aumed here to be negligible due to proper management. We hope that the model preented in thi paper will provide a convenient tarting point to tudy thee important problem. 8. REFERENCES [1] M. Armbrut, A. Fox, R. Griffith, A. D. Joeph, R. H. Katz, A. Konwinki, G. Lee, D. A. Patteron, A. Rabkin, I. Stoica, and M. Zaharia. A view of cloud computing. Commun. ACM, 53(4):50 58, [2] R. J. Aumann. Market with a continuum of trader. 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Vogelang. Telecommunication Pricing: Theory and Practice. Cambridge, [13] National Intitute of Standard and Technlogy. NIST Definition of Cloud Computing v15, July [14] S. Shakkottai and R. Srikant. Network optimization and control. In Foundation and Trend in Networking, volume 2, page [15] S. Stidham. A lat word on l = λw. Operation Reearch, 22(2): , [16] H. R. Varian. Microeconomic Analyi. Norton, [17] C. S. Yeo and B. Rajkumar. A taxonomy of marketbaed reource management ytem for utilitydriven cluter computing. Software Practice and Experience, 36(13): , [18] Y. Yi and M. Chiang. Stochatic network utility maximization a tribute to Kelly paper publihed in thi journal a decade ago. European Tranaction on Telecommunication, 19(4): , [19] B. Yolken and N. Bambo. Game baed capacity allocation for utility computing environment. In Proc. GameComm Forthcoming in Telecommunication Sytem.
A note on profit maximization and monotonicity for inbound call centers
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