Resource Pricing and Provisioning Strategies in Cloud Systems: A Stackelberg Game Approach

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ACROSS Workshop at ITC 27, Ghent 11/9/2015

Resource Pricing and Provisioning

Strategies in Cloud Systems: A

Stackelberg Game Approach

Valeria Cardellini, Valerio di Valerio and Francesco Lo Presti

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Talk Outline

 Background and Motivation

 Provisioning Model and Assumptions

 Pricing schemes

 Optimal Pricing as IaaS/SaaS Stackelberg Game

 Existence of equilibria  Equilibria computation

 Numerical Examples

 Conclusions

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Backgroung and Motivation

 Software as a Service – SaaS

 Provider’s applications running on a cloud infrastucture

 Infrastucture as a Service – IaaS

 Compute, storage and network resources as on-demand

service  Virtualization techniques - VMs 2 SaaS SaaS IaaS Acquire VMs

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Backgroung and Motivation

 Issues

 Dynamic Resource Provisioning  Pricing Schemes vs Auction

 Reserved, on demand, spot

 SaaS provider strategies, IaaS strategy

3

SaaS

SaaS IaaS

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Backgroung and Motivation

 Issues

 Dynamic Resource Provisioning  Pricing Schemes vs Auction

 Reserved, on demand, spot

 SaaS provider strategies, IaaS strategy

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Related work

 Too many to mention…with different settings and

assumptions  Auction

 Welfare maximization  Profit maximization

 Game Theory/Mechanism Design

 Single round – Equilibria  Multiple rounds – Bidding strategies » MDP, Reinforcement Learning  …

 Closest to our (which inspired this work)

 Generalized Nash Equilibria (Ardagna et al, TSC ’13)

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Contributions

 Formulate the spot VM pricing and provisioning

problem as a Stackelberg game

 Provide proof of existence of game equilibria

under suitable assumptions

 Algorithms for computing equilibria

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Problem Setting and Assumptions

 SaaS providers offer service to end users

 Service are hosted on a IaaS Cloud  Assume al VMs are equal

 Same CPU, memory, etc.

 Revenue/penalty depends on the performance offered to their

customers (SLA)

 Performance depends on workload and number of allocated VMs  Captured by a suitable Utility Function

 Saas Providers want to maximize their utility

 SaaS utility strictly concave function of number of VMs and

twice differentiable

 Diminishing returns

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Problem Setting and Assumptions

 IaaS provider sells VMs to SaaS Providers

 Different VMs pricing schemes (~ Amazon EC2 offer)  Flat instances

 Reserved instances, One time payment + fixed time unit rate f

 On Demand instances

 Fixed time unit rate d

 Spot instances

 Variable rate s

 Typically d > f >> s

 Iaas Provider wants to maximize its revenue

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Provisioning Model

 Periodic Provisioning

 SaaS provision VMs periodically (e.g., every hour) based

on (expected) workload

 Each SaaS has a fixed amount of flat it acquired in the past

 At the beginning of each time interval:

1. Each SaaS provider determines:

 Number of flat instances (he owns) to use (fixed price f)  Number of on demand to buy (fixed price d)

 How many spot instances to buy (variable price s)

2. IaaS provider determines:

 Spot price s

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Provisioning Model

2 step Provisioning

 1° step: SaaS providers decide how many flat VMs to

use and how many on demand VMs to buy  As to optimize their utility

 Assume IaaS provider has enough resources to satisfy flat

and on demand VMs requests

no competition in this step

 2° step: IaaS

 IaaS provider sells its unused capacity as spot VMs

 Cheap

 Finite amount -> competition

 We study the game arising in the 2° provisioning step

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Two Spot instances Pricing schemes

 SaaS announce maximum price they would accept

 ~ budget constraint

 Same Spot Price Model (SSPM): IaaS sets a

unique spot price for all SaaS providers

 Users whose max price is lower than actual price will not

get any spot instances

 Multiple Spost Price (MSPM): IaaS can set

different price for different SaaS providers  Price will not exceed SaaS max price

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Step 2: Different spot prices

SaaS user u

spot price

max spot price s_u number of

spot VMs

Θ_u utility function

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SaaS PROVIDER 1 PROBLEM

Max_s1 Θ₁(s₁,σ₁)

Subject to: residual cap.

u

s

u

å £

SaaS PROVIDER n PROBLEM

Max_sn Θ_n(s_n,σ_n)

Subject to: residual cap. s_u

u å £

. . .

IaaS PROVIDER Max Subject to: u s s u u å min s £su £sumax u s u s umax s

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Step 2: Same spot prices for all SaaS providers

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Max_s1 Θ₁(s₁,σ)

Subject to: residual cap. s₁=0 if s₁>σ₁max

u

s

u

å £

Max_sn Θ_n(s_n,σ)

Subject to: residual cap. s_n=0 if s_n>σ_nmax u s u å £

. . .

IaaS PROVIDER Max Subject to: u s s u å min s £s s

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Step 2 as Stackelberg game

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u

s

u

å £

u å £

. . .

IaaS PROVIDER Max Subject to: u s s u u å min s £su £sumax SaaS subgame

STACKELBERG GAME FOLLOWERS

Play a non-cooperative game to maximize their utility

STACKELBERG GAME LEADER

Maximizes its revenue by adjusting the spot price(s)

IaaS provider sets

the spot price(s) SaaS providers return the _{equilibrium number of spot VMs}

u

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SaaS subgame

 Fixed spot price vector , compute SaaS equilibrium

 Generalized Nash Game

 strategy space depends on other players

 Jointly Convex Gen. Nash Game

 Equilibrium can be computed solving the associated

Variational Inequality (VI)

 Associated VI function is strongly monotone

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u

s

u

å £

Subject to: residual cap. su u

å £

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SaaS subgame

 Theorem: there is exactly one variational equilibrium

of the followers subgame

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u

s

u

å £

Subject to: residual cap. su u

å £

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IaaS Problem

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Max_s1 Θ₁(s₁,s_-1,σ₁)

u

s

u

å £

Max_sn Θ_n(s_n,s_-n,σ_n)

u å £

. . .

IaaS PROVIDER Max Subject to: u s s u u å min s £su £sumax SaaS subgame

IaaS provider sets

the spot price SaaS providers return the _{equilibrium number of} spots VMs

u

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IaaS Problem

 is the solution of the SaaS subgame

 Theorem: there is at least one Stackelberg

equilibrium of the IaaS game  Under both pricing schemes

18 IaaS PROVIDER Max Subject to: u s s u u å min s £su £sumax u s

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IaaS Problem: MSPM

 IaaS Problem is Mathematical Problem with

Equilibrium Constraints  Cannot be directly solved

 Solution computed by solving a sequence of perturbed

(smooth) problem [Facchinei et al ‘99]

19 IaaS PROVIDER Max Subject to: u s s u u å min s £su £sumax u s

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IaaS Problem: SSPM

 Need to resort to a brute force approach

1. Discretize the spot price interval 2. Compute SaaS subgame equilibrium

1. Easy, it turns out to be a potential game

3. Pick up the price that corresponds to the largest IaaS

revenue 20 IaaS PROVIDER Max Subject to: u s s u å min s £s s

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Numerical examples: Homogeneous Scenario

 SaaS provider capacity 160 VMs

 10 SaaS homogenoeus providers, max price 0.5$

 Spot price vs SaaS load

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Number of available and sold spot VMs

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Numerical examples: Heterogenous Scenario

 10 SaaS heterogenous provider, same utility

function but max price=[0:1; 0:14; 0:18; 0:22; 0:26; 0:30; 0:34; 0:38; 0:42; 0:46]$

 Spot price vs SaaS load

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Number of available and sold spot VMs

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Numerical examples: Heterogenous Scenario

 10 SaaS heterogenous provider, same utility

function but max price=[0:1; 0:14; 0:18; 0:22; 0:26; 0:30; 0:34; 0:38; 0:42; 0:46]$

 MSPM vs SSPM optimal spot price

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Conclusions

 Studied Cloud Resource Provisioning problem

 Formulated the spot pricing as a Stackelberg game

 IaaS as leader

 SaaS providers as followers

 Considered two pricing schemes

 Provided proof of existence of equilibria

 Presented algorithms for equilibria computation

Future work

 Study multi IaaS/SaaS problem

 Extend the work to other provisioning/pricing schemes

 Compare the results with auction based schemes