Core-Selecting Auction Design for Dynamically Allocating Heterogeneous VMs in Cloud Computing Haoming Fu † , Zongpeng Li † , Chuan Wu ‡ , Xiaowen Chu § † University of Calgary ‡ The University of Hong Kong § Hong Kong Baptist University June 24, 2014 Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 1 / 24
Outline Outline Introduction 1 Problem Model 2 Core-Selecting VM Auctions 3 Core-Selecting Payment Rules 4 Simulations 5 Conclusion 6 Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 2 / 24
Introduction Introduction to Cloud Markets Cloud markets Heterogeneity nature of virtual machine (VM) instances; Resource provisioning (resources: CPU, RAM, storage, etc. ) ; Auction for efficient VM allocation ( e.g. Amazon Spot Instances) . Figure: (a) VM configuration from Spot Instances. (b) Illustration of a cloud market. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 3 / 24
Introduction Motivation of Our Work *Shill bidding problem is applicable to combinatorial auctions only and not to simple auctions. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 4 / 24
Introduction Our VM Auction Design We apply the core-selecting auctions, which achieve a revenue at least on par with that of VCG mechanisms and are robust against shill bidding. The core-selecting auction is further tailored to minimize CUs’ incentives to deviate from truthful bidding. A three-dimension auction framework is proposed for dynamic resource provisioning. We propose a combinatorial auction that is expressive enough to sell bundles of heterogeneous VMs. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 5 / 24
Problem Model Problem Model XOR bidding language: a cloud user (CU) can submit multiple bids, but can win a single bid only. B i is the set of all VM bundles CU i bids for. Utility The quasi-linear utility of CU i is: � v i ( S ) − p i if CU i wins a bundle S ∈ B i u i = 0 otherwise The utility of the cloud provider (CP) is: u o = � i ∈N u i . All CUs are individually rational; v i ( S ) is the maximum amount that CU i is willing to pay for S . Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 6 / 24
Problem Model Problem Model winner determination problem (WDP) � � w ( N ) = max b i ( S ) x i ( S ) i ∈N S⊆B i m � n j α k subject to: j ≤ ∀ 1 ≤ k ≤ t ; π k j =1 � x i ( S ) ≤ 1 ∀ i ∈ N ; S⊆B i � � x i ( S ) r j ≤ n j ∀ 1 ≤ j ≤ m , S = ( r 1 , r 2 , . . . , r m ); i ∈N S⊆B i n j ∈ N ∀ 1 ≤ j ≤ m ; x i ( S ) ∈ { 0 , 1 } ∀ i ∈ N , ∀S ⊆ B i . Theorem 1 Relaxing n to take fractional values in the WDP does not change the value of w ( N ). Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 7 / 24
Core-Selecting VM Auctions The Core of Our VM Auction An auction outcome is blocked by coalition C ⊆ N if there is an alternative outcome which generates strictly more revenue for the auctioneer and no less utility for each CU i ∈ C . Core: Set of outcomes that are not blocked. Core � � Core ( N ) = { u ≥ 0 | u i = w ( N ) , u i ≥ w ( C ) , ∀C ⊆ N} i ∈N ∪{ o } i ∈C∪{ o } Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 8 / 24
Core-Selecting VM Auctions The Core of Our VM Auction To keep CU 1 from blocking: p 4 ≥ 4. Example: The cloud provider Similarly: p 5 ≥ 5 , p 6 ≥ 4. (auctioneer) has 25 CPUs and 25 GB To keep CUs 1, 2, 3 from blocking: storage in its resource pool. 7 CUs p 4 + p 5 + p 6 ≥ 33. each submits one bid: Individual rationality. b 1 (6 , 0 , 1) 8 b 2 (2 , 3 , 0) 5 7 = 4 , 11 = 5 , b 3 (0 , 0 , 6) 12 b 4 (7 , 0 , 0) 7 6 = 4 , 7 = 27 , b 5 (0 , 4 , 0) 4 b 6 (0 , 0 , 6) 12 12 = 25 , 6 = 24 , b 7 (5 , 3 , 7) 24 22 = 33 . Table: VM configuration. VM 1 VM 2 VM 3 CPU 1 1 2 storage 1GB 3GB 1GB Figure: Illustration of the core. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 9 / 24
Core-Selecting VM Auctions The Core of Our VM Auction Theorem 2 The payment vector of first price auction is always in the core of our VM auction. (proof by definition) Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 10 / 24
Core-Selecting VM Auctions The Economic problems from VCG Mechanisms The VCG mechanism p VCG = b i ( S i ) − ( w ( N ) − w ( N\{ i } )) i Low revenue: a revenue of 4+5+4=13 is gleaned in view of the fact that the winners are willing to pay a total up to 27+25+24=76. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 11 / 24
Core-Selecting VM Auctions The Economic problems from VCG Mechanisms b 1 (6 , 0 , 1) 8 b 2 (2 , 3 , 0) 5 7 = 4 , 11 = 5 , b 3 (0 , 0 , 6) 12 b 4 (7 , 0 , 0) 7 6 = 4 , 7 = 27 , b 5 ( 0 , 4 , 0 ) 4 b 6 (0 , 0 , 6) 12 12 = 25 , 6 = 24 , b 7 (5 , 3 , 7) 24 22 = 33 . available VM 1 VM 2 VM 3 25 CPU 1 1 2 25GB storage 1GB 3GB 1GB Vulnerability to shill bidding: CU 5 impersonates four different CUs, each submitting this bid: b (0 , 1 , 0) 1 3 = 6 . 25. CU 5 still wins 4 instances of VM 2 but reduces its payment from 4 to 0! Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 12 / 24
Core-Selecting VM Auctions The Necessity of Core-Selecting Auctions Theorem 3 (shill bidding proof) In a VM auction formulated with WDP, no CU can earn more than its VCG utility by bidding with shills if and only if the auction is core-selecting. u o + � i ∈N u i is constant since WDP implies efficiency, then Corollary (competitive revenue) The total revenue in a core-selecting VM auction formulated with WDP is at least as high as that in a VCG auction. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 13 / 24
Core-Selecting VM Auctions The Necessity of Core-Selecting Auctions Theorem 3 (shill bidding proof) In a VM auction formulated with WDP, no CU can earn more than its VCG utility by bidding with shills if and only if the auction is core-selecting. Proof sketch of Theorem 3 ∀C ∈ N , their total utility under the VCG mechanism is w ( N ) − w ( N\C ) Our restriction is: � i ∈C u i ≤ w ( N ) − w ( N\C ). Efficiency: w ( N ) = � i ∈N ∪{ o } u i . Then � i ∈ ( N ∪{ o } ) \C u i ≥ w ( N\C ). Note that: Core ( N ) = { u ≥ 0 | � i ∈N ∪{ o } u i = w ( N ) , � i ∈C∪{ o } u i ≥ w ( C ) , ∀C ⊆ N} Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 13 / 24
Core-Selecting Payment Rules Revenue Minimization Rule Revenue minimization rule We pick the payment that minimizes the total revenue over the core. Theorem 4 A core-selecting VM auction formulated with WDP and employing a revenue-minimization payment rule minimizes CUs’ incentives to deviate from truthful bidding. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 14 / 24
Core-Selecting Payment Rules Revenue Minimization Rule The core constraint � i ∈C∪{ o } u i ≥ w ( C ) , ∀C ⊆ N is equivalent to: � p i ≥ w ( ˜ � b i ( S i ) , ∀ ˜ C ∪ ( N\W )) − C ⊆ W i ∈W\ ˜ i ∈ ˜ C C C = w ( ˜ Setting β ˜ C ∪ ( N\W )) − � C b i ( S i ), the core constraint can be i ∈ ˜ compactly written as A p ≥ β . δ = min 1 T · p (revenue minimization) A p ≥ β subject to: (core constraint) p ≤ b (individual rationality) Drawback: revenue minimization rule lacks of precision, since the points minimizing revenue are not unique. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 15 / 24
Core-Selecting Payment Rules Point Reference Rules Point reference rules Among the in-core payment points minimizing the revenue, we pick the one that is closest to some pre-determined point p ′ . min( p − p ′ ) T ( p − p ′ ) (closest to p ′ ) subject to: A p ≥ β (core constraint) p ≤ b (individual rationality) 1 T · p = δ (revenue minimization) VCG-nearest rule and constant p ′ reference rule p ′ can be set to be the VCG payment vector or some constant vector independent of CUs’ bids. Theorem 5 Under the VCG-nearest rule, the set of constraints p ≤ b is unnecessary. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 16 / 24
Core-Selecting Payment Rules Point Reference Rules Origin-nearest rule: p ′ = 0 . available VM 1 VM 2 VM 3 18 CPU 1 1 2 18GB storage 1GB 3GB 1GB Under the constant p ′ reference rule, b 1 (0 , 0 , 6) 12 b 2 (0 , 4 , 0) 4 the winner with high valuation relative 6 = 100 12 = 20 to the auctioneer’s expectation shares b 3 (0 , 4 , 6) 16 b 4 (0 , 0 , 6) 12 18 = 60 6 = 50 less of the burden to conquer a coalitional blocking. Winners are CUs 1 and 2. p VCG = (50 , 0) p VCG − nearest = (55 , 5) p origin − nearest = (60 , 0) Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 17 / 24
Simulations Simulations Figure: Performance of the VM allocation result under core-selecting auctions. Dynamic resource provisioning achieves higher social welfare, stably higher resource utilization but lower user satisfaction. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 18 / 24
Simulations Simulations Figure: The monetary burden shouldered by the winner(s) under (a) the VCG-nearest rule, and (b) the origin-nearest rule. p ∗ i − p VCG ¯ ¯ i = ) . Under the origin-nearest rule, the winner with i µ ¯ i ∈N ( p ∗ i − p VCG � i high valuation relative to the auctioneer’s expectation shares less of the burden to conquer a coalitional blocking. Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 19 / 24
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