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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

1

Introduction

2

Problem Model

3

Core-Selecting VM Auctions

4

Core-Selecting Payment Rules

5

Simulations

6

Conclusion

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. Bi is the set of all VM bundles CU i bids for. Utility The quasi-linear utility of CU i is: ui = vi(S) − pi if CU i wins a bundle S ∈ Bi

• therwise

The utility of the cloud provider (CP) is: uo =

i∈N ui.

All CUs are individually rational; vi(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

• i∈N
• S⊆Bi

bi(S)xi(S) subject to:

m

• j=1

njαk

j ≤

πk ∀1 ≤ k ≤ t;

• S⊆Bi

xi(S) ≤ 1 ∀i ∈ N;

• i∈N
• S⊆Bi

xi(S)rj ≤ nj ∀1 ≤ j ≤ m, S = (r1, r2, . . . , rm); nj ∈ N ∀1 ≤ j ≤ m; xi(S) ∈ {0, 1} ∀i ∈ N, ∀S ⊆ Bi.

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|

• i∈N ∪{o}

ui = w(N),

• i∈C∪{o}

ui ≥ w(C), ∀C ⊆ N}

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 8 / 24

Core-Selecting VM Auctions

The Core of Our VM Auction

Example: The cloud provider (auctioneer) has 25 CPUs and 25 GB storage in its resource pool. 7 CUs each submits one bid:

b1(6, 0, 1)8

7 = 4,

b2(2, 3, 0)5

11 = 5,

b3(0, 0, 6)12

6 = 4,

b4(7, 0, 0)7

7 = 27,

b5(0, 4, 0)4

12 = 25,

b6(0, 0, 6)12

6 = 24,

b7(5, 3, 7)24

22 = 33.

Table: VM configuration.

VM1 VM2 VM3 CPU 1 1 2 storage 1GB 3GB 1GB

To keep CU 1 from blocking: p4 ≥ 4. Similarly: p5 ≥ 5, p6 ≥ 4. To keep CUs 1, 2, 3 from blocking: p4 + p5 + p6 ≥ 33. Individual rationality.

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 pVCG

i

= bi(Si) − (w(N) − w(N\{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

b1(6, 0, 1)8

7 = 4,

b2(2, 3, 0)5

11 = 5,

b3(0, 0, 6)12

6 = 4,

b4(7, 0, 0)7

7 = 27,

b5(0, 4, 0)4

12= 25,

b6(0, 0, 6)12

6 = 24,

b7(5, 3, 7)24

22 = 33.

available VM1 VM2 VM3 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 VM2 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. uo +

i∈N ui 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 ui ≤ w(N) − w(N\C).

Efficiency: w(N) =

i∈N ∪{o} ui.

Then

i∈(N ∪{o})\C ui ≥ w(N\C).

Note that: Core(N) = {u ≥ 0|

i∈N ∪{o} ui = w(N), i∈C∪{o} ui ≥ 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} ui ≥ w(C), ∀C ⊆ N is equivalent to:

• i∈W\ ˜

C

pi ≥ w( ˜ C ∪ (N\W)) −

• i∈ ˜

C

bi(Si), ∀ ˜ C ⊆ W Setting β ˜

C = w( ˜

C ∪ (N\W)) −

i∈ ˜ C bi(Si), the core constraint can be

compactly written as Ap ≥ β. δ = min 1T · p (revenue minimization) subject to: Ap ≥ β (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: Ap ≥ β (core constraint) p ≤ b (individual rationality) 1T · 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

Under the constant p′ reference rule, the winner with high valuation relative to the auctioneer’s expectation shares less of the burden to conquer a coalitional blocking. Origin-nearest rule: p′ = 0.

available VM1 VM2 VM3 18 CPU 1 1 2 18GB storage 1GB 3GB 1GB

b1(0, 0, 6)12

6 = 100

b2(0, 4, 0)4

12 = 20

b3(0, 4, 6)16

18 = 60

b4(0, 0, 6)12

6 = 50

Winners are CUs 1 and 2. pVCG = (50, 0) pVCG−nearest = (55, 5) porigin−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.

µ¯

i = p∗

¯ i −pVCG ¯ i

• i∈N (p∗

i −pVCG i

). Under the origin-nearest rule, the winner with

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

Simulations

Simulations

Figure: The monetary burden shouldered by the winner(s) under (a) the VCG-nearest rule,

and (b) the origin-nearest rule.

Core-selecting auction achieves 8% higher revenue over the VCG mechanism.

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 20 / 24

Simulations

Simulations

Figure: The monetary burden shouldered by the winner(s) under (a) the VCG-nearest rule,

and (b) the origin-nearest rule.

Our core-selecting VM auction has a stable performance; revenues do not deteriorate when bidding instances are from Google Cluster Data

• r are randomly generated in different ways.

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 21 / 24

Conclusion

Conclusion

Core-selecting combinatorial VM auctions are novelly designed for the cloud computing market. They are proved to be proof to shill bidding and generate higher revenues than VCG auctions do. Payment rules that minimize CUs’ incentives to deviate from truthful bidding are extensively discussed. We advance the VM auction design by generalizing static resource provisioning to dynamic resource provisioning, and from homogeneous VM instances to heterogeneous VM instances.

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 22 / 24

Main References

1

• R. Day and P. Milgrom, “Core-selecting package auctions,” International Journal of

Game Theory, vol. 36, no. 3-4, pp. 393C407, March 2008.

2

• R. W. Day and S. Raghavan, “Fair payments for efficient allocations in public sector

combinatorial auctions,” Management Science, vol. 53, no. 9, pp. 1389C1406, September 2007.

3

• R. W. Day and P. Cramton, “Quadratic core-selecting payment rules for combinatorial

auctions,” Operations Research, vol. 60, no. 3, pp. 588C603, May 2012.

4

• A. Gulati, A. Holler, M. Ji, G. Shanmuganathan, C. Waldspurger, and X. Zhu, “Vmware

distributed resource management: Design, implementation, and lessons learned,” VMware Technical Journal, vol. 1, no. 1, pp. 45C64, June 2012.

5

• H. Zhang, B. Li, H. Jiang, F. Liu, A. V. Vasilakos, and J. Liu, “A framework for truthful
• nline auctions in cloud computing with heterogeneous user demands,” in Proc. of IEEE

INFOCOM, April 2013, pp. 1558C1566.

6

• Y. Zhu, B. Li, and Z. Li, “Core-selecting combinatorial auction design for secondary

spectrum markets,” in Proc. of IEEE INFOCOM, April 2013, pp. 2034C2042.

7

Google Cluster Data, https://code.google.com/p/googleclusterdata/.

8

• M. Yokoo, Y. Sakurai, and S. Matsubara, “The effect of false-name bids in combinatorial

auctions: New fraud in internet auctions,” Games and Economic Behavior, vol. 46, no. 1, pp. 174C188, January 2004.

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 23 / 24

Q&A

Thank You!

Haoming Fu, Zongpeng Li and Chuan Wu (UofC, HKU) June 24, 2014 24 / 24