Workload Factoring with the Cloud: A Game-Theoretic Perspective

Transcription

1 Workload Factoring with the Cloud: A Game-Theoretic Perspective Amir Nahir Department of Computer Science Technion Israel Institue of Technology Haifa Israel nahira@cstechnionacil Ariel Orda Department of Electrical Engineering Technion Israel Institue of Technology Haifa Israel ariel@eetechnionacil Danny Raz Department of Computer Science Technion Israel Institue of Technology Haifa Israel danny@cstechnionacil Abstract Cloud computing is an emerging paradigm in which tasks are assigned to a combination ( cloud ) of servers and devices accessed over a network Typically the cloud constitutes an additional means of computation and a user can perform workload factoring ie split its load between the cloud and its other resources There is an intrinsic relation between the benefit that a user perceives from the cloud and the usage pattern followed by other users giving rise to a non-cooperative game which we model and investigate We show that the considered game admits a Nash equilibrium Moreover we show that this equilibrium is unique We investigate the price of anarchy of the game and show that while in some cases of interest the Nash equilibrium coincides with a social optimum in other cases it can be arbitrarily large We demonstrate that somewhat counter-intuitively exercising admission control to the cloud may deteriorate its performance Furthermore we demonstrate that certain (heavy) users may scare off other potentially large communities of users Accordingly we propose a resource allocation scheme that addresses this problem and opens the cloud to a wide range of user types I INTRODUCTION Cloud computing is a new and emerging paradigm in which tasks are executed by several services deployed on a set of distributed servers and accessed via a common network This network of servers and devices collectively known as the cloud can offer a significant reduction in computing / storage cost due to economy of scale In fact computing at the scale of the cloud allows users to reach into the cloud for resources as they need them and to attain supercomputer-level power using any standard client with network connectivity For this reason cloud computing has also been described as on-demand computing [6] It is important to note that for many typical users the cloud constitutes an additional means of computation ie in additional to other (typically local) resources A user can thus split its load between the cloud and its other resources that is perform workload factoring [20] Since the cloud offers its services to a multiplicity of potential users there is an intrinsic relation between the benefit that a user perceives from the cloud and the usage pattern followed by other users This relation can come in many different forms For example when the cloud instantaneously shares its resources among current active jobs ie essentially provides a best effort service each user perceives the presence of the other users through the congestion effect of their submitted jobs A second example is a cloud that provides some form of QoS (quality of service) guarantees hence mitigating (or eliminating altogether) such congestion effects In this case the cloud may exercise some admission or rate control in order to meet such guarantees Hence the presence of other users is perceived at a longer time scale namely by occasionally being denied access to the cloud or experiencing rate limitations [15] These two examples are typical of clouds that provide free access eg Google Apps for end users or private clouds for corporate users Yet a third example is a cloud that charges prices for usage possibly offering various levels of QoS for different price ranges eg [2] Here high demand for service would (slowly) translate into higher prices hence relating again between the aggregate behavior of users and the benefit of the cloud as perceived by an individual user 1 While applying to a wide range of time scales all these examples identify a common and fundamental property namely: a dependence between the level of usage of the cloud and the utility perceived by each user The dependence is in fact bi-directional as the perceived utility affects the user s decision on whether to use the cloud and to what extent Since cloud users can be expected to make decisions in a selfoptimizing (selfish) manner we face a non-cooperative game whose investigation is the subject of this study We consider users that can use the cloud as well as local resources in order to execute computation tasks ie jobs More specifically each user needs to decide how to split its job between the cloud and some local private resources such that the the completion time of the job would be minimized Since the local resource is private it provides guaranteed performance ie it does not depend on the decisions nor job patterns of other users; the cloud on the other hand consitutes a more powerful resource yet its performance does depend on the loads presented by other users We establish that the underlying non-cooperative game admits a Nash equilibrium We note that a major complication in establishing this property 1 We note that some price-charging clouds may fall within the framework of the second example rather than the third due to the observed preference of ISP customers of flat fees over usage-based pricing [12] which would call for the employment of rate control schemes [15]

2 2 stems from an inherent discontinuity of the users objective functions Informally this discontinuity occurs at the point where a user starts sending part of its job to the cloud making its completion time depend not only on the local resource but on the later between two completion times ie of the portions executed locally and in the cloud Furthermore we establish that the equilibrium point is unique These structural results enable us to investigate the performance of such systems and in particular the inherent inefficiency that stems from the noncooperative behavior Specifically we study the price of anarchy of the game and indicate that it may heavily depend on the specific manner in which users perceive the performance they experience in the cloud as well as on the cloud s behavior under heavy load For example we demonstrate a case where users make worst-case decisions (eg when handling timecritical jobs) in which the unique Nash equilibrium coincides with the social (ie system-wide) optimum; whereas in another case where the cloud performs outsourcing under heavy load (employing techniques such as cloud federation [16]) we demonstrate that the price of anarchy may be arbitrarily large We then turn to investigate practical implications of our formal model and analysis An important question is whether proper design and management of the system can mitigate the defficiencies of noncooperative behavior One defficiency measured by the price of anarchy is the sub-optimality of the social performance at the Nash equilibrium Yet from a practical standpoint there are other defficient aspects that need to be considered for example the potential conquest of the cloud by certain communities of users (eg heavy users ) which scare off all others Our study indicates that the employment of management tools can indeed come at rescue in some cases yet and somewhat counter-intuitively in other cases they may fail to deliver Specifically we demonstrate the latter (negative) finding by showing that the employment of an apparently appealing admission control scheme deteriorates system-wide performance; whereas we demonstrate the former (positive) finding by showing how the partition of the cloud into virtual clouds can result in increasing its attractiveness to various types of users To summarize the main contributions of this paper are as follows Formal modeling of workload factoring with the cloud as a noncooperative game Establishment of the existence and uniqueness of the game s Nash equilibrium Investigation of the game s price of anarchy Investigation of practical implications of the formal model and analysis specifically: demonstrating that admission control may fail to improve system performance; demonstrating how proper resource allocation within the cloud can cope with its potential conquest by certain classes of users The paper is organized as follows After discussing related work in the next section in Section III we formalize the model In Section IV we establish the existence and uniqueness of the Nash equlibrium In Section V we investigate the price of anarchy The next two sections deal with practical implications: admission control is considered in Section VI and resource allocation is considered in Section VII Finally conclusions appear in Section VIII Due to space limits some details are omitted and can be found (online) in [11] II RELATED WORK Game theoretic models have been employed in various related contexts such as networking [7] [9] [13] peer-to-peer networks [10] and distributed systems [3] [5] These studies mainly investigated the structure of the system operating points ie the Nash equilibria of the respective games Such equilibria are inherently inefficient [4] and in general exhibit suboptimal system performance As a result the question of how much worse the quality of a Nash equilibrium is with respect to a centrally enforced optimum has received considerable attention eg [8] [17] [18] A prominent measure proposed to quantify this effect is the price of anarchy [14] which is the worst possible ratio between the social performance at a Nash equilibrium and the corresponding social optimum A study that is particularly relevant to our framework is [1] That study considers a problem where customers arrive to a public mainframe and each needs to decide whether to execute its job there or on a less powerful yet private machine While choosing between executing jobs on a shared infrastructure (namely the cloud) or on a private facility is also the problem investigated in our study there are several major differences between our framework and that of [1] First in [1] a customer appears once in the system and based on the current load of the mainframe makes a single one time and binary decision namely mainframe or private machine ; we on the other hand envision users that consider using the cloud facilities over a long time scale and for multiple jobs hence the decision of each user is a strategy on the portion of its jobs that would be served by the cloud Moreover while [1] focuses solely on a processor-sharing service regime in the mainframe we consider also other and more general classes of service; in particular while [1] assumes that all jobs perceive the service in the mainframe in the same way we allow heterogeneity among users Game theoretic tools have been hardly used in the context of cloud computing One example is [19] which addresses a task scheduling problem in the presence of multiple decision makers taking a game theoretic perspective Another study that is relevant to ours is [20] which also deals with workload factoring; however that study assumed that the resources that a user gets from the cloud are not shared in any way hence the game scenario was not considered III MODEL We consider a cloud workload factoring (CWF) game of N players each of which is a user that has a job that requires processing Player strategies are the choice of the fraction of the job that will be processed by a shared resource (ie

3 3 the cloud) We denote player i s strategy by σ i The strategy profile of all players is denoted by σ The goal of each player is to minimize the expected completion time of its job Each player i is characterized by the size of its job denoted by ω i (measured in bits) and by its private computational resource whose performance is manifested by a delay (ie computation time) function L i ( ) For example a player may have a private computer that serially processes jobs in which case L i (σ i ) = (1 σi)ωi i for some computing rate 2 i We assume that L i ( ) is monotonically increasing in the amount of work (ie monotonically deceasing in σ i ) In addition we assume that L i (1) = 0 ie if player i submits its entire job to the cloud it local computation time is 0 We denote the estimated completion time of a job of player i in the cloud by Ti cloud noting that in general it depends on the loads submitted to the cloud by the various players We assume that T cloud i is a sum of two components namely d i (σ i ) and D(σ) The first component d i (σ i ) is a delay component that is specific to the player and independent of the load submitted by other players to the cloud ie d i : σ i R + For example d i can correspond to the time required to send a job to the cloud which in turn may depend on the geographical location of the player with respect to the cloud s resources We assume that d i ( ) is continuous monotonically non-decreasing and in addition d i (0) = 0 The second component D(σ) is the cloud s processing delay which depends on the particular loads submitted to it Since the job sizes {ω j } N 1 are assumed to be constant D is a function of the strategy profile σ ie D : σ R + We assume that D( ) is continuous and monotonically increasing (in every one of its parameters) and in addition D(0) = 0 We thus have: T cloud i (σ) = D(σ) + d i (σ i ) A player s completion time is the time in which its job completes processing (both locally and in the cloud) formally: T i (σ) = max { T cloud i (σ) L i (σ i ) } Note that in case the player chooses not to schedule any work to the cloud its completion time is L i (0) We denote the cloud s actual processing completion time by T Cloud (σ) We note that T Cloud (σ) may be different than the expected completion time perceived by the player Ti cloud (σ) For example consider a case in which the cloud processes jobs serially A player may expect to get an average place in the cloud s queue and therefore D(σ) = 1 2 σ i ω i and d i (σ i ) = 1 2 σi ωi However the cloud s actual completion λ i ω i time ie among all jobs would be T Cloud (σ) = A player best-response move is a strategy σ i which given the strategies of all other players σ i yields the lowest value 3 2 Note that (1 σ i ) is the portion of the job that is processed locally 3 We note that if players are concerned with worst-case performance rather than average performance then D(σ) = T Cloud (σ); this point is further discussed in Section V to Ti cloud (< σ i σ i >) A strategy profile σ is said to be at Nash equilibrium if each player considers its chosen strategy to be the best under the given choices of other players We note that for certain values of σ i Ti cloud (< σ i σ i >) is not a continuous function of σ i since the completion time for σ i = 0 (no computation in the cloud) may be much smaller than the completion time where a very small fraction is sent to the cloud This discontinuity prevents us from adopting classic results on the existence of Nash equilibria Moreover we note that each player may have a different d i which further complicates the analysis of Nash equilibria In order to quantify the performance of the CWF game from a social (ie system-wide) perspective we associate with the game a social cost function denoted by C( ) The social cost typically depends on the performance of the users and various forms can be considered eg completion time of the last job (ie makespan) average completion time etc The selfish behavior of the players may lead to system-wide inefficiency We quantify this inefficiency through the worst possible ratio between the social cost at Nash equilibrium and the social cost of the corresponding optimal solution In keeping with common terminology [8] [14] this ratio is termed the price of anarchy and it quantifies the penalty incurred by lack of cooperation (or coordination) among the players in a noncooperative game IV NASH EQUILIBRIUM EXISTENCE AND UNIQUENESS In this section we establish the existence and uniqueness of the Nash equilibrium A Existence First we prove that the CWF game admits a Nash equilibrium As pointed out in the previous section this task is especially challenging due to the discontinuity of the objective function as well as the heterogeneity among users in terms of the d i component We begin by establishing several properties of the CWF game Lemma 1: Let i be some player Let σ denote a profile in which i is at best response Then σ i < 1 Proof: Assume by negation that the claim is wrong Therefore there exists a profile σ and a player i such that σ i = 1 and i is at best response First we choose σ i such that L i (σ i) < D(< σ i σ i >) + d i (σ i) We note that such σ i is guaranteed to exist since L i( ) D( ) and d i ( ) are continuous and L i (1) = 0 Next we observe that both L i (σ i ) < T i cloud (< σ i σ i >) (since σ i was chosen that way) and in addition Ti cloud (< σ i σ i cloud >) < Ti (σ) Therefore it holds that T i (< σ i σ i >) < T i(σ) which is in contradiction to the assumption that i is at best response in σ Corollary 1: Let σ denote a Nash equilibrium profile For every player i σ i < 1 The above corollary implies that at Nash equilibrium every player makes some use of its local resource Following similar

4 4 reasoning it is easy to see that in every Nash equilibrium profile there exists at least one player that makes use of the cloud Next we show that when a player is at best response either it processes its entire job locally or its local completion time is equal to that of the cloud Lemma 2: Let i be some player Let σ denote a profile in which i is at best response Then either L i (σ i ) = T i cloud (σ) or σ i = 0 and D(σ) L i (ω i ) Proof: Assume by negation that the claim is wrong If σ i = 0 and D(σ) < L i (ω i ) then there exists some ɛ > 0 such that D(< σ i ɛ >) + d i (ɛ) L i (ɛ) so player i can unilaterally improve its cost by moving ɛ of its job for processing into the cloud Otherwise (ie σ i 0 ) we consider two cases: in case L i (σ i ) < T i cloud (σ) player i can unilaterally improve its cost by reducing σ i ; and in case L i (σ i ) > T i cloud (σ) player i can unilaterally improve its cost by increasing σ i Corollary 2: Let σ denote a Nash equilibrium profile For every player i either L i (σ i ) = T cloud i (σ) or σ i = 0 and D(σ) L i (ω i ) The following claims establish some monotonicity among the players in terms of their usage of the cloud according to the power of the local resources Lemma 3: Let i j be two players such that L i (0) L j (0) Let σ be a profile such that j is at best-response and in addition σ j > 0 Let σ be a profile following player i s bestresponse move to σ Then σ i > 0 Proof: First we note that since j is at best-response when σ is applied and σ j > 0 it holds that L j (σ j ) = T cloud j (σ) < L j (0); this is due to the fact that L j (0) is the time required to compute all of player j s work locally Assume by negation that the claim is wrong ie following player i s best-response move σ i = 0 Hence L i(0) D(< σ i 0 >) (ie player i opts not to submit any work to the cloud since the cloud s delay is higher then the time required for player i to process its job locally) Thus we get: L i (0) D(< σ i 0 >) D(σ) T cloud j (σ) = L j (σ j ) < L j (0) which is a contradiction Corollary 3: Let σ denote a Nash equilibrium profile Assume wlog that the players are ordered in ascending order of local computation time ie L 1 (0) L 2 (0) L N (0) Let k be a player such that σk > 0 It holds that for every player i i < k σi > 0 Corollary 4: Let σ denote a Nash equilibrium profile Assume wlog that the players are ordered in ascending order of local computation time ie L 1 (0) L 2 (0) L N (0) Then σ1 > 0 Proof: Follows directly from Lemma 3 and from the observation that at Nash equilibrium at least one player makes use of the cloud We turn to prove the existence of a Nash equilibrium We do so by first establishing the existence of a Nash equilibrium in a 2-player CWF game and then move on to extend it inductively to the general N-player case Theorem 1: The 2-player CWF game admits a Nash equilibrium profile Proof: Assume wlog that L 1 (0) L 2 (0) We show by construction that the game has a Nash equilibrium point We begin with a state in which neither of the players submits any of its workload to the cloud ie σ 0 =< σ1 0 = 0 σ2 0 = 0 > and proceed by having each player in its turn perform a best-response move Let σ1 1 denote the relative amount of workload submitted by player 1 to the cloud following the first best-response move It is easy to see that σ1 1 > 0 Next we consider player 2; if D(< σ1 1 0 >) L 2 (0) (ie the relative workload submitted by player 1 to the cloud generates a delay that is longer than the time required for player 2 to process its entire work locally) it holds that < σ1 1 0 > is a Nash equilibrium profile Otherwise it holds that player 2 will submit some non-zero relative workload denoted by σ2 1 for processing in the cloud Consider then the latter case When player 1 performs a best-response move again it is clear that D(< σ1 1 0 >) < D(< σ1 1 σ2 1 >) (ie the workload submitted by player 2 to the cloud increased the cloud s delay) Thus it holds that L 1 (σ1) 1 < D(< σ1 1 σ2 1 >) + d 1 (σ1) 1 so player 1 s best-response move would be to decrease its relative workload submitted to the cloud ie σ1 2 < σ1 1 In a similar fashion when player 2 performs a best-response move again it holds that D(< σ1 1 σ2 1 >) > D(< σ1 2 σ2 1 >) (ie since player 1 reduced its relative workload in the cloud the cloud s delay has decreased) Therefore player 2 s bestresponse move would be to increase it relative work load ie σ2 2 > σ2 1 Once again since player 2 increased its relative workload in the cloud it holds that player 1 would decrease its as part of his next best-response move ie σ1 3 < σ1; 2 we can apply the same process repeatedly We conclude that the sequence of player 2 s relative workloads following its best-response moves constitues a monotonically increasing sequence {σ 2 } k 1 By definition it holds that this sequence is upper bounded (by 1) and therefore converges to a non-zero value We denote its limit by σ2 Similarly it holds that the sequence of player 1 s relative workloads following its best-response moves constitutes a monotonically decreasing sequence {σ 1 } k 1 By definition it holds that this sequence is lower bounded (by 0) and therefore converges We denote its limit by σ1 Following Corollary 3 since σ2 1 > 0 it holds that σ1 > 0 Finally we conclude that σ =< σ1 σ2 > is a Nash equilibrium profile and the Theorem follows For extending the above result to the N-player case we need the following auxiliary claims which compare between clouds of different performance in terms of best responses and Nash equilibrium profiles

5 5 Lemma 4: Let i j be two players Let D 1 ( ) D 2 ( ) represent two clouds Let σ 1 be a profile such that players i j are at best response with D 1 ( ) Let σ 2 be a profile such that players i j are at best response with D 2 ( ) Assume σi 2 > σ1 i Then σj 2 σ1 j Proof: First we note that since σi 2 > σi 1 it holds that L i (σi 2) < L i(σi 1 ) in addition since player i is at best response both in σ 1 and σ 2 it follows that: D 2 (σ 2 ) + d i (σ 2 i ) = L i (σ 2 i ) < L i (σ 1 i ) D 1 (σ 1 ) Hence it follows that D 1 (σ 1 ) > D 2 (σ 2 ) Assume by negation that σj 2 < σj 1 It follows that L j (σj 2) L j(σj 1 ) In addition since player j is at best response both in σ 1 and σ 2 it follows that: D 2 (σ 2 ) + d j (σ 2 j ) L j (σ 2 j ) L j (σ 1 j ) = D 1 (σ 1 ) + d j (σ 1 j ) which is a contradiction Lemma 5: Let D 1 ( ) D 2 ( ) represent two clouds such that for every σ D 1 (σ) > D 2 (σ) (ie D 2 ( ) represents a better cloud) Let σ 1 σ 2 be Nash equilibrium profiles wrt D 1 ( ) D 2 ( ) respectively Furthermore assume that σ 1 is such that all players submit some work to the cloud Then there exists at least one player i such that σi 2 > σ1 i Proof: Assume by negation that the claim is wrong ie for every player j σj 2 σ1 j Since D1 (σ) > D 2 (σ) and in addition both D 1 ( ) and D 2 ( ) are monotonicaly increasing we get that: D 1 (σ 1 ) D 1 (σ 2 ) > D 2 (σ 2 ) Consider some player k Since σ 1 is a Nash equilibrium profile and σk 1 > 0 it holds that L k(σk 1) = D1 (σ 1 ) + d k (σk 1 ) In addition since σ 2 is a Nash equilibrium profile it holds that L k (σk 2) D2 (σ 2 ) + d k (σk 2) Since σ2 k σ1 k it holds that L k (σk 2) L k(σk 1 ); it follows that: D 1 (σ 1 ) + d k (σ 1 k) = L k (σ 1 k) L k (σ 2 k) D 2 (σ 2 ) + d k (σ 2 k) which is a contradiction Corollary 5: Let D 1 ( ) D 2 ( ) represent two clouds such that for every σ D 1 (σ) > D 2 (σ) (ie D 2 ( ) represents a better cloud) Let σ 1 σ 2 be Nash equilibrium profiles wrt D 1 ( ) D 2 ( ) respectively Furthermore assume that σ 1 is such that all players submit some work to the cloud For every player i σi 2 σi 1 and in addition for at least one player a strict inequality holds Proof: Follows directly from Lemmas 4 and 5 Theorem 2: The N-player CWF game admits a Nash equilibrium profile Proof: By induction on the number of players For N = 2 the claim was established in Theorem 1 Assume now that any N = k 1 player game has a Nash equilibrium profile We turn to consider a k-player game Assume wlog that players are ordered in ascending order of workload ie L 1 (0) L 2 (0) L k (0) We define a new (k 1)-player s game which includes only players 1 2 k 1 (ie we exclude player k from the game) This game s cloud load function D 1 ( ) is defined based on D( ) (the load function of the original game) in the following manner: D 1 (σ 1 σ 2 σ k 1 ) = D(σ 1 σ 2 σ k 1 0) ie D 1 ( ) is equal to D( ) when player k sets its relative workload to 0 By the inductive hypothesis the k 1-player s game has a Nash equilibrium profile We denote it by σ k 1 Returning to the k-player s game we consider the strategy profile σ 1 =< σ k 1 0 > (ie the first k 1 players play as in the Nash equilibrium profile of the (k 1)-player s game and player k does not submit any work to the cloud) We continue by letting player k perform its best-response move If D(σ 1 ) L k (0) (ie the work submitted by the first k 1 players to the cloud generates a delay which is longer then the time required for player k to process its entire work locally) it holds that σ 1 is a Nash equilibrium profile Assume then that D(σ 1 ) < L k (0) First we note that in this case every player i 1 i k 1 submits some work to the cloud since D(σ 1 ) < L k (0) L k 1 (0) L 1 (0) Next we let player k perform its best-response move and denote the relative workload it submits to the cloud by σ 1 k Again we define a new (k 1)-player s game by excluding player k For this game we define the cloud s delay function D 2 () in the following manner: D 2 (σ 1 σ 2 σ k 1 ) = D(σ 1 σ 2 σ k 1 σ 1 k) By the inductive hypothesis the (k 1)-player s game has a Nash equilibrium profile which we denote by σ k 2 Following Corollary 5 it holds that for every player i i = 1 2 k σ k 2 (i) σ1 k (i) and in addition for at least one player a strict inequality holds Therefore when player k is given another chance to perform a best-response move the cloud delay that it observes is better (lower) than the one that it observed in its previous turn Therefore player k will increase the relative load it submits to the cloud and the process continues repeatedly We conclude that the sequence of player k s relative workloads following its best-response moves {σ k } m 1 is monotonically increasing By definition this sequence is upper bounded (by 1) and therefore converges Denote its limit by σk Similarly it holds that the sequence of Nash equilibrium profiles of the (k 1) player games {σ k } m 1 is monotonically decreasing By definition this sequence is lower bounded (by 0) and therefore converges Denote the limit of this sequence by σ k Following Corollary 3 since σ k > 0 it holds that for every player i σi > 0 We conclude that σ =< σ k σ k > is a Nash equilibrium profile and the Theorem follows

6 6 B Uniqueness We turn to establish the uniqueness of the Nash equilibrium Theorem 3: The CWF game admits a unique Nash equilibrium profile Proof: Assume by negation that the claim is wrong Therefore there exist two different profiles σ 1 σ 2 such that both constitute Nash equilibrium points Hence there exists at least one player i such that σ 1 i σ2 i Assume wlog that σ1 i > σ2 i Since L i ( ) is monotonically increasing in the workload it follows that L i (σ 1 i ) < L i(σ 2 i ) Since both σ1 and σ 2 are Nash equilibrium profiles it holds that: D(σ 1 ) + d i (σ 1 i ) = L i (σ 1 i ) < L i (σ 2 i ) D(σ 2 ) therefore D(σ 1 ) < D(σ 2 ) Next we consider any player j and claim that σj 1 σ2 j Assuming the claim is wrong it holds that there exists at least on player j such that σj 1 < σ2 j Following similar reasoning to that applied above we conclude that: D(σ 2 ) + d j (σ 2 i ) = L i (σ 2 i ) < L i (σ 1 i ) D(σ 1 ) which contradicts our previous conclusion that D(σ 1 ) < D(σ 2 ) We thus conclude that for every player i σj 1 σ2 j and in addition for at least one player a strict inequality applies Since D( ) is monotonically increasing it follows that D(σ 1 ω) > D(σ 2 ω) which contradicts our first conclusion hence the theorem follows V PRICE OF ANARCHY We turn to analyze the Price of Anarchy [8] of the CWF game We shall demonstrate that this value may heavily depends on the specific manner in which users perceive the performance they experience in the cloud as well as on the cloud s behavior under heavy load First we consider the case in which players base their decisions on worst-case cloud performance (ie D( ) T Cloud ( )) In addition we assume players incur no private cost (ie d i ( ) 0) We consider as the game s social cost its makespan ie the time in which the last job completes processing We show that in this case the game s Price of Anarchy is 1 ie it coincides with the social optimum Theorem 4: Considering the makespan as the social cost if d i ( ) 0 and D( ) T Cloud ( ) then the price of anarchy of the CWF game is 1 Proof: Let σ denote the game s Nash equilibrium profile and σ OP T a socially optimal profile We show that σ OP T = σ Assume by negation that the claim is wrong It follows that there exists some player i such that σi OP T σi First we consider the case that for some player i σi > σop i T > 0 In this case it holds that L i (σi ) < L i(σi OP T ) and therefore C(σ ) = D(σ ) = L i (σ i ) < L i (σ OP T i ) C(σ OP T ) where the second equality follows from Corollary 2 The above is in contradiction to σ OP T s optimality Therefore we conclude that for any player i such that σi > 0 it holds that σi OP T σi Next we consider the case that for some player i σi OP T > σi T In this case since for every other player σop i σi it holds that D(σ OP T ) > D(σ ) and therefore C(σ ) = D(σ ) < D(σ OP T ) C(σ OP T ) which is in contradiction to σ OP T s optimality and the theorem follows Consider now the same setting with the following change: now players do not base their decisions on worst-case cloud performance but rather on expected cloud performance Specifically consider a cloud that processes jobs in serial fashion formally: T Cloud (σ) = σ i ω i We assume that all players are identical (in job size and local computation function) and in addition the local computation model is that of linear processing that is L i (σ i ) = (1 σ i) ω i for some arbitrary Players expect the cloud to behave as a simple queue and expect to get an average location in the cloud s job queue that is: D(σ) = N 1 2 σi ω i Note that the above expression relies on the fact that all players are identical and therefore assumes that the Nash equilibrium is symmetric In addition the to waiting time in the queue each player also has to account for the processing of its own job therefore we set d i (σ i ) = σ i ω i As before the game s social cost is its makespan ie { } C(σ) = max T Cloud (σ) max {L i ((1 σ i ) ω i i )} i First we calculate the Nash equilibrium profile σ of this game In view of the above assumptions we have: which yields: (1 σ ) ω σ = = N 1 2 σ ω + σ ω (N + 1)

7 7 Thus the social cost at the Nash equilibrium is C(σ ) = 2Nω 2 + (N + 1) Next we calculate the optimal value of the social cost By Theorem 4 an optimal configuration σ OP T coincides with the Nash equilibrium of a game in which D( ) T Cloud ( ) σ i ω i and d( ) 0 In such a game all players are at Nash equilibrium when for each player which yields: (1 σ) ω σ = = N σ ω + N The optimal value of the social cost is thus: C(σ OP T ) = Therefore the price of anarchy is C(σ ) C(σ OP T ) = 2Nω 2+(N+1) N ω +N N ω + N = 2( + N) 2 + (N + 1) We note that this expression is upper bounded by 2 and the stronger the cloud the lower the value of this expression For example if the cloud has equal power to the aggregate of all users ie = N then the price of anarchy is roughly 4 3 while if = 10N then the price of anarchy is (roughly) Ẇe turn to demonstrate a case where the price of anarchy can be arbitrarily large Consider a cloud that employs cloud federation [16] thus enabling it to outsource some of its load to other clouds More specifically the cloud processes jobs serially but due to limited capacity after reaching a certain load threshold it starts sending (outsourcing) some of its jobs to another cloud That is under such conditions every job that is submitted to the cloud is outsourced with some probability p Furthermore we assume that an outsourced job experiences a (fixed) delay denoted by which is significantly larger than the delays incurred when being processed at the original cloud Specifically we assume: > 1 p ω i As before we assume that all players are identical (in job size and local computation resources) d i (σ i ) 0 and the social cost of the game is its makespan Finally we assume that 2 p < L i (0) < 3 p Since players are identical the load threshold can be expressed in terms of the strategy profile σ namely: the cloud performs outsourcing when N σ i > σ T for some σ T 4 A player considers the expected delay that its submitted work would experience at the cloud including possible outsourcing therefore for N σ i > σ T we have: D(σ) = (1 p) whereas for N σ i < σ T we have: D(σ) = σ i ω i + p σ i ω i Due to the assumption made on it follows that below the threshold D(σ) < p and above the threshold D(σ) < 2 p Since L i (0) > 2 p and L i (1) = 0 the continuity of L i (σ i ) implies that there is a σ i 0 < σ i < 1 such that L i ( σ i ) = p Assume now that the number of users N is large enough so that: N σ i > σ T Let σ be the Nash equilibrium profile Since the players are identical it follows from Corollary 4 that σi > 0 for all i Therefore: L i (σi ) = D(σ ) We proceed to show that at Nash equilibrium outsourcing is in effect By way of contradiction assume that at Nash equilibrium the cloud operates below the threshold ie σi < σ T This implies that D(σ ) < p thus L i (σ i ) < p Since L i( σ i ) = p the monotonicity of L i implies that σ i > σ i thus N σ i > σ T hence establishing that at Nash equilibrium the cloud operates above the threshold Consider now the social cost at Nash equilibrium ie C(σ ) Since at Nash equilibrium all players are subject to outsourcing and the social cost is the makespan C(σ ) is lower-bounded by times the probability of at least one outsourcing occurs ie C(σ ) > (1 (1 p) N ) On the other hand L i (0) < 3 p implies that the optimal social cost is upper bounded by the latter value ie C(σ OP T ) < 3 p ; indeed this can be achieved by executing all jobs locally (since all jobs incur the same delay the makespan is equal to that value) Therefore the price of anarchy is larger than (1 (1 p) N ) 3 p = 1 (1 p)n 3 p which can be made arbitrarily large by considering a large population of users and small values of p eg p = ɛ and N = 1 ɛ for a sufficiently small ɛ 4 For sustaining continuity we could assume that the probability of outsourcing grows linearly from 0 to p in a small neighborhood around the threshold without affecting the discussion and findings that follow For simplicity we ommit these details

8 8 VI ADMISSION CONTROL As demonstrated in the previous section when all players are at Nash equilibrium the system may exhibit degraded performance To account for that a cloud service provider may try to enforce some admission control policy to guarantee that the system s delay remains adequate Indeed this is a rather natural reaction of a provider to high load In this section we explore this possibility focusing on the following scenario: d i ( ) 0 the players perceived cloud delay D( ) depends (only) on the aggregate load all players are identical in terms of job size and local computation and the local computation model is that of linear processing that is L i (σ i ) = (1 σ i) ω i for some arbitrary We show that under this scenario even the application of the most extreme admission control policy namely preventing any number of users from accessing the cloud causes a deterioration in the social cost We first show that the average completion time increases when any number of players are not permitted to access the cloud Lemma 6: For the scenario specified above preventing any number of players from accessing the cloud increases the average job completion time of the CWF game Proof: Assume by negation that the claim is wrong Therefore there exist some K players such that if they are not allowed to access the cloud the average job completion time does not increase We note that since all players are identical the identity of the K players is irrelevant Denote by σ 1 the Nash equilibrium profile when all players may access the cloud since all players are identical it is easy to see that for every two players i j it holds that σi 1 = σ1 j Since D( ) is monotonically increasing in the aggregate load it also holds that L i (σ 1 i ) = D(N σ 1 i ) (1) Denote by σ 2 the Nash equilibrium profile when K players may not access the cloud Therefore (N K) identical players make use of the cloud hence all of them submit the same amount of work for processing in the cloud Therefore L i (σ 2 i ) = D((N K) σ 2 i ) (2) It is easy to see that when K players cannot access the cloud the remaining (N K) players will increase their usage of the cloud ie σi 2 > σ1 i which in turn implies that L i (σ 2 i ) > L i (σ 1 i ) (3) Following our assumption that D( ) is monotonically increasing in the aggregated load expressions (1) (2) and (3) imply that N σ 1 i > (N K) σ 2 i (4) Next we require that the average job completion time remains the same or improves To satisfy this requirement it must hold that K L i (0) + (N K) L i (σi 2) L i (σi 1 ) N Since all local computing resources process jobs in a linear fashion it follows that K ωi + (N K) (1 σ2 i ) ωi N (1 σ1 i ) ω i Simplifying the above expression yields the conclusion that (N K) σ 2 i N σ 1 i which conradicts expression (4) and the theorem follows Next we turn to show that preventing any number of users from accessing the cloud causes a deterioration in the social cost Theorem 5: For the scenario specified above preventing any number of players from accessing the cloud increases the social cost of the CWF game Proof: First we note that since all players are identical it is easy to see that the Nash equilibrium profile (in case all players may access the cloud) is symmetrical This in turn implies that the average completion time is equivalent to the social cost Following Lemma 6 it holds that preventing any number of users from accessing the cloud causes the average completion time to increase and the theorem follows VII RESOURCE ALLOCATION WITHIN THE CLOUD From Corollary 3 it follows that heavy players ie players with large jobs may scare off other players by placing much of their work in the cloud In such a case potentially large communities may not make use of the cloud at all While this might not be detrimental from the point of view of overall performance (eg average or worst-case performance across all users) it may negatively affect other goals sought by the cloud provider For example if the provider charges some fixed fee to open an account driving away many light users ends up in reducing profitability To address this concern we propose a scheme for allocating resources within the cloud The idea is to partition the resources of the cloud effectively making a cluster of virtual clouds such that each would serve a different class of users We demonstrate the idea on the following setting We consider two classes of users: small users each having a job of size ω and large users each having a job of size ω t for some t > 1 We denote by K the number of large players and by N K the number of small players We assume that the local computation model is that of linear processing according to the following expression: L i (σ i ) = (1 σ i) ω i for some > 0 Furthermore we make the following assumptions regarding processing in the cloud:

9 9 d i ( ) 0; the cloud processes user jobs in serial fashion that is: σ i ω i T Cloud (σ) = for some > 0; users base their decisions on worst-case performance hence: D( ) T Cloud ( ) First we identify the conditions under which small players stay out of the cloud Lemma 7: For the scenario specified above in case t + k (5) k only the K large players make use of the cloud Proof: By construction Assume the condition specified in the claim holds and consider the strategy profile σ in which each of the K large players submits σ 0 = c +K while the other (N K) small players do not submit any work to the cloud It holds that for all large players K (1 c ) ω t c+k L i (σ i ) = (1 σ i) ω i = = ( K ) ω t c+k = 1 + K c ω t = K σ 0 ω t = D(σ) That is all large players are at best response with the cloud Next we consider the cloud s completion time It holds that T Cloud (σ) = K σ0 ω t = K ω t + K This equation together with the assumption that t c+k k yield that the cloud s completion time is at least ω ie the delay generated by the work submitted by large players is at least as large as the time required for a small player to process its entire job locally Therefore small players are also at best response with the cloud (staying away from it) and the claim follows We turn to propose concrete resource allocation guidelines for two examplary cases In the first the cloud provider provides large users with enough computational resources such that their performance would not degrade much with respect to the case in which they make use of all of the cloud s resources In the second case that we consider the cloud provider attempts to allocate resources in a way that supports global fairness that is each user can access an amount of resources that is proportional to its job size A Preventing Large User Service Degradation Recall that when all cloud resources are shared by all players and condition 5 is met the expected completion time for large players is k t ω + k ; in a similar fashion if only 1 c of the cloud s computational power is allocated for large players their completion time is k t ω 1 c + k Given a permissible degradation ratio ρ we require that k t ω 1 c +k k t ω +k ρ After some algebra the above expression yields: 1 c + (1 ρ) k ρ that is as long as a large player may access at least +(1 ρ) k ρ of the cloud s computational power its performance degradation would be at most ρ We note that since some resources are exclusively allocated to the small players (namely 1 c) they would all make some (nonzero) use of the cloud B Creating System-wide Fairness Here the cloud provider partitions its resources such that at equilibrium the completion time of a large player would be at most t times that of a small player (we recall that t is the proportion between large and small job sizes) We denote by 1 c the amount of resources allocated for large players hence the remainder 1 c is allocated to the small players As shown above when K large players get access to 1 c computational resources their completion time is k t ω 1 c + k In a similar fashion when (N K) small players access ( 1 c) of the cloud s resources their completion time is (N k) ω ( 1 c) + (N k) Requiring that large players completion time is at most t times that of small players yields the following condition: 1 c K N that is large players should get access to resources in an amount that is at least proportional to their relative size Again we note that as long as some resources are exclusively allocated to the small players ie 1 c < they would all make some (nonzero) use of the cloud VIII CONCLUSIONS Paraphrasing on the classic saying to send or not to send is a dilemma confronted by users in the cloud era As opposed to the origin in this case the answer is not binary and a user may decide to split its job between the cloud and local resources ie perform workload factoring Yet in this case too the dilemma is addressed individually We attempted to understand the global behavior of such an environment by investigating the underlying noncooperative game Such

10 10 an understanding enables to identify possible shortcomings that stem from the noncooperative uncoordinated behavior The potentially large price of anarchy is one such finding The possibility that certain populations of users would fully possess the cloud resources scaring off other populations is yet another finding Based on the foundations laid in this study future work can identify other interesting patterns of behavior For example the d i ( ) components could account for the geographical distance between a user and the cloud in which case investigation of the Nash equilibrium could detect to what extent a cloud is efficient in attracting remote users By better understanding possible shortcomings we could explore schemes for better managing the cloud s resources For example we demonstrated how a simple scheme for allocating the cloud s resources can increase its attractiveness to different types of users At the same time we also demonstrated that exercising control even in an apparently right way may end up (quite counter-intuitively) in deteriorated performance This study is a first attempt to understand cloud workload factoring games As such it adopted the traditional assumption of full information ie users have full and instantaneous knowledge regarding the performance of the cloud In reality of course this is not quite the case and a very interesting question for future research is the effect of incomplete information on the game s structure and the implied behavior and performance of the users Another important subject for future work is to consider QoS (quality of service) issues For example some of the users may need to obey strict delay bounds for completing their jobs hence requiring the incorporation of constraints in the determination of their best responses Another example is to consider clouds that offer various (price-dependent) levels of QoS Yet another intriguing subject for future research is to consider a multiplicity of competing clouds giving rise to a two-level game [10] T Moscibroda S Schmid and R Wattenhofer On the topologies formed by selfish peers in PODC 06: Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing New York NY USA: ACM Press 2006 pp [11] A Nahir A Orda and D Raz Workload factoring with the cloud: A game-theoretic perspective Department of Electrical Engineering Technion Haifa Israel Tech Rep 2010 [Online] Available: ArielOrda/Info/Other/NOR10CWFpdf [12] A M Odlyzko Internet pricing and the history of communications Computer Networks vol 36 no 5/6 pp [13] A Orda R Rom and N Shimkin Competitive routing in multiuser communication networks IEEE/ACM Transactions on Networking vol 1 no 5 pp oct 1993 [14] C Papadimitriou Algorithms games and the internet in STOC 01: Proceedings of the thirty-third annual ACM symposium on Theory of computing New York NY USA: ACM 2001 pp [15] B Raghavan K Vishwanath S Ramabhadran K Yocum and A C Snoeren Cloud control with distributed rate limiting in SIGCOMM 07: Proceedings of the 2007 conference on Applications technologies architectures and protocols for computer communications New York NY USA: ACM 2007 pp [16] B Rochwerger D Breitgand E Levy A Galis K Nagin I M Llorente R Montero Y Wolfsthal E Elmroth J Caceres M Ben- Yehuda W Emmerich and F Galan The RESERVOIR model and architecture for open federated cloud computing IBM Journal of Research and Development vol 53 no [17] T Roughgarden Designing networks for selfish users is hard in FOCS 01: Proceedings of the 42nd IEEE symposium on Foundations of Computer Science Washington DC USA: IEEE Computer Society 2001 p 472 [18] T Roughgarden and E Tardos How bad is selfish routing? 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