The Constrained Ski-Rental Problem and its Application to Online Cloud Cost Optimization

Transcription

1 The Constrained Ski-Rental Problem and its Application to Online Cloud Cost Optimization Ali Khanafer *, Murali Kodialam **, Krishna P. N. Puttaswamy ** * Coordinated Science Laboratory, UIUC ** Bell Laboratories, Alcatel-Lucent IEEE INFOCOM 2013, Cloud pricing and cost optimization Session Yunhyoung Kim

2 Table of Contents Introduction Problem Definition The Ski-Rental Problem The Constrained Ski-Rental Problem The Designer s Problem Results Performance Comparison Simulation Results Conclusion 2

3 Introduction Problem Definition Cloud Service Provider Request for Block of Data OPTION 1. From Disk OPTION 2. From Cache - Slow I/O - Cheaper Storage Price - Fast I/O - Expensive Storage Price Focused on Cost Optimization by optimally selecting when to use which storage for the request 3

4 Introduction The Ski-Rental Problem On a Fine Day? On a Snowy Day unknown number of days for skiing (= unknown number of snowy days) Every snowy day, you choose to rent/buy skis Renting skis : $1 Buying skis : $B Analogy to Cloud File System Buying Skis = Use a cache for a request Renting Skis = Do not use a cache ( serve from disk) * Skier s Strategy : Competitive Randomized Algorithm for Non-Uniform Problems, SoDA,

5 Introduction The Ski-Rental Problem is Online Problem Online : Input is provided piece-by-piece (Entire input is not avaliable from the start) Example : Online Hiring Problem Boss can choose to 1) decline or 2) hire and stop the interview How to deal with online problem? Performance of offline algo. >> Performance of online algo. Consider Competitiveness Competitive Ratio CR = ( ) 5

6 Introduction Attack on Ski-Rental Problem Deterministic Algorithm (DET) Buy skis at B-th day (Rent for B-1 days) Buying Skis = $B Renting Skis = $1 # of skiing days 1 2 B-1 B B+1 Cost of DET $1 $2 $B-1 $2B-1 $2B-1 Cost of optimal offline algorithm Worst Case Competitive Ratio B-th day : CR = 2 Probabilistic Algorithm (PROB) $1 $2 $B-1 $B $B Buy skis according to the probability distribution between 0, B The skier randomly choose the day to buy skis Worst Case Expected Competitive Ratio : CR =

7 The Constrained Ski-Rental Problem Main Contribution of this paper Add a constraint on the average number of snowy days Improved worst-case expected CR compared to DET and PROB Problem setting $ B Buying price of skis $ 1 Rental price of skis (per day) The time at which the skier buys skis If <, rent for days and buy skis The designer rent and buys skis ( ) Probability distribution over The number of snowy days ( ) Probability distribution over (, ) Expected cost incurred by the skier 0 x y < rent for days The designer rent skis B OPT( ) Cost incurred by optimal offline algorithm 0 y x B 7

8 The Constrained Ski-Rental Problem Matrix Zero-Sum Game Game between the designer and the adversary?? # of skiing days 1 2 B-1 B B+1 Cost of optimal $1 $2 $B-1 $B $B 8

9 The Constrained Ski-Rental Problem To the continuous world! 0, B [0, B), Strategy space of designer and adversary $ B Buying price $ 1 Rental price (per day) The time at which the designer buys skis ( ) Probability distribution over The number of snowy days ( ) Probability distribution over ( = ) where (, ) Expected cost incurred by the designer OPT( ) Cost incurred by optimal offline algorithm and a constraint : 9

10 The Constrained Ski-Rental Problem We can get expected CR and use it as designer s objective function. If <, Then,?, = + ( ) 0 The designer rent and buy skis Expected CR (designer s objective function) B y $ B Buying price $ 1 Rental price (per day) The time at which the designer buys skis ( ) Probability distribution over The number of snowy days ( ) Probability distribution over ( = ) where (, ) Expected cost incurred by the designer OPT( ) Cost incurred by optimal offline algorithm < designer s problem adversary s problem 10

11 The Constrained Ski-Rental Problem Focus on We have an objective function and constraints on ( ) Therefore its Lagrangian form is : And its dual is : Diff. twice Solve ODE Substitute By = 1 11

12 The Constrained Ski-Rental Problem Diff. twice Solve ODE Substitute By = 1 Consider the whole designer s problem : Then it becomes 12

13 The Constrained Ski-Rental Problem If < 1 + ( ) When Otherwise, 1 < ( 2) which is the same as that of PROB. > If the first moment is high, then knowing its value does not provide any information 2 is independent of the first moment, * This paper also tried to use the second moment, but I skip it (similar to the case ) 13 CR becomes

14 Results Competitive Ratio Simulation Results 14

15 Results Cloud File System Based Evaluation Synthetic workload Focus on the I/O cost (B=10) Real workload Traces from data center at MS For 4KB block size, B = 47 (regarding the price of storage) Average Cost : $1509 (DET), $1345 (PROB), $1410 ( -PROB) But the difference between the cheapest and the costliest is reduced 15

16 Summary Conclusion Try to optimize the cost (I/O time, price) for the cloud file system Model the problem into the Ski-Rental problem, and suggest an improved algorithm Constrained Ski-Rental Problem Apply the result to synthetic/real workload Contribution Applied the concept of Ski-Rental problem to the recent system Utilized properties of the distribution (first- and secondmoment) for the constraint to the problem 16

17 Q & A 17