Towards Optimal Capacity Segmentation with Hybrid Cloud Pricing Wei Wang , Baochun Li, Ben Liang Department of Electrical and Computer Engineering University of Toronto
IaaS clouds offer multiple pricing options On-demand (pay-as-you-go) Static hourly rate x run hours = p r t Subscription (reservation) One-time subscription fee Free/discounted usage fee during the reservation period Auction-like pricing (spot market) Users bid for computing instances No service guarantee 2 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
IaaS clouds offer multiple pricing options On-demand GoGrid, � ElasticHosts, BitRefinery, Ninefold Subscription ... On-demand � Subscription Amazon EC2 Auction-like pricing 3 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Why multiple pricing? Compensate the deficiency of individual pricing Static pricing: awkward to market dynamics, easy to understand, risk-free with a static price Spot price: agile to demand/supply changes, hard to understand, risky due to price fluctuations Expand the market demand Long-term users go for subscription Price-sensitive users bid in the spot market X Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
How do cloud providers allocate its capacity to different pricing channels? Wei Wang, Department of Electrical and Computer Engineering, University of Toronto 4
Price Price Subscriptions t t Pay-as-you-go demand Spot market demand Subscription demand Cloud resources How to set the prices? How many instances to offer in each pricing channel? Objective: Revenue maximization 5 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
How many instances to offer in each channel in hour 1? On-demand user $80 $80 $80 Spot user $150 $150 1 2 3 Time(hour) An on-demand user requests 80 instances for 3 hours, starting from hour 1, with on-demand rate $1 A spot user bids for 100 instances each at $1.5 per instance-hour, starting from hour 2 The available capacity of a cloud can only support 100 additional instances 6 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
How many instances to offer in each channel in hour 1? On-demand user $80 $80 $80 Spot user $150 $150 1 2 3 Time(hour) Strategy 1 : Serve the on-demand user in hour 1 (revenue =$240) Strategy 2 : Strategically hold resources in hour 1 and serve the spot user in hour 2 (revenue = $300) 7 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Our focus Price Price t t Periodic auctions demand Pay-as-you-go demand Cloud resources Dynamic capacity segmentation in two channels On-demand channel with a fixed hourly rate Periodic auction channel similar to EC2 spot market 8 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Problem formulation C t − C t C t Price Price a a t t Periodic auctions demand Pay-as-you-go demand Cloud resources : the optimal revenue collected during the prediction Γ τ ( C τ ) window Auction revenue On-demand revenue a ) + γ r ( C t � C t Γ t ( C t ) = E γ a ( C t � max a ) 0 ≤ C t a ≤ C t ⇤ � Γ t +1 ( C t +1 ) + E C t +1 ⇥ , Future revenue 9 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Revenue from the on-demand channel q : the probability that a currently running on-demand instance is terminated by its user in the next time slot 10 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Revenue from the on-demand channel q : the probability that a currently running on-demand instance is terminated by its user in the next time slot Revenue from the on-demand channel, with c instances allocated to it ⇢ p r c/q , if c R t r ; γ r ( c ) = p r R t r /q , otherwise, : # of on-demand requests received at time t R t r A simple model yet gives interesting insights! 11 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Periodic auctions Auctions are carried out periodically Each user i bids for computing instances True demand: instances each with utility v i n i r t b t Bid for instances each at a price i i follows a joint p.d.f. f n,v ( n i , v i ) A uniform clearing price is posted in every time t p t a b t i > p t User i wins if the bid exceeds the clearing price a Upon losing, all running instances are terminated 12 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Auction bidder No partial fulfilment Lose all or win all The same as Amazon EC2 and other clouds Utility function of bidder i Gain Cost ⇢ n i v i � r t i p t if p t a < b t i and r t a , i � n i ; u t i ( r t i , b t i ) = 0 , otherwise. 13 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
What is the optimal auction mechanism? Wei Wang, Department of Electrical and Computer Engineering, University of Toronto 14
Optimal auction design (m+1)-price auction with a seller reservation price Sort all bidders in a descending order of their bid prices, i.e., b t 1 ≥ b t 2 ≥ . . . φ ( v i ) = v i − 1 − F v ( v i | n i ) Reservation price = , φ − 1 (0) f v ( v i | n i ) 15 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Optimal auction design (m+1)-price auction with a seller reservation price Sort all bidders in a descending order of their bid prices, i.e., b t 1 ≥ b t 2 ≥ . . . φ ( v i ) = v i − 1 − F v ( v i | n i ) Reservation price = , φ − 1 (0) f v ( v i | n i ) Keep accommodating top bidders, until (1) there is no available capacity to serve more or (2) no one bids higher than the reservation price. For the former case, winners are charged the highest bid of losers. For the later case, winners are charged the reservation price. 16 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Optimal auction design (Cont.) b t b t 1 Bid 2 b t 3 φ − 1 (0) Case 1: c Capacity 0 r t r t r t 2 3 1 b t b t Bid 1 2 b t 3 Case 2: φ − 1 (0) Capacity 0 r t r t r t c 3 2 1 X Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Optimal auction design (Cont.) Proposition 1 : The design maximizes the revenue among all auctions producing a uniform clearing price Proposition 2 : The design is two-dimensionally truthful A user always reports true demand: i.e., u t i ( n i , v i ) ≥ u t i ( r t i , b t i ) The detailed proof is gi 17 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Optimal auction design (Cont.) Proposition 1 : The design maximizes the revenue among all auctions producing a uniform clearing price Proposition 2 : The design is two-dimensionally truthful A user always reports true demand: i.e., u t i ( n i , v i ) ≥ u t i ( r t i , b t i ) The detailed proof is gi Remarks Generally, (m+1)-price auction suffers from the problem of demand reduction and is neither truthful nor optimal when a bidder bids for multiple items We show that it is truthful and optimal in cloud markets where partial fulfilment is unaccepted 18 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Auction revenue m m m +1 Revenue: ,where X X X r t i φ ( b t r t r t γ a ( c ) = i ) i ≤ c < i i =1 i =1 i =1 φ ( · ) c Capacity 0 r t r t r t 2 3 1 Revenue = shaded area φ ( · ) Capacity 0 r t r t r t c 2 3 1 X Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Optimal capacity segmentation Wei Wang, Department of Electrical and Computer Engineering, University of Toronto 19
Capacity segmentation revisit Find the optimal segmentation at time t C t a Auction On-demand a ) + γ r ( C t � C t Γ t ( C t ) = E γ a ( C t � max a ) 0 ≤ C t a ≤ C t ⇤ � Γ t +1 ( C t +1 ) + E C t +1 ⇥ , Future State transition C t +1 = C t X ∼ B ( C − C t a , k, q ) a + X X : # of instances terminated by on-demand users at time t 20 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Solving the capacity segmentation problem a ) + γ r ( C t � C t Γ t ( C t ) = E γ a ( C t � max a ) 0 ≤ C t a ≤ C t ⇤ � Γ t +1 ( C t +1 ) + E C t +1 ⇥ , C t +1 = C t X ∼ B ( C − C t a , k, q ) a + X Direct solution is via numerical dynamic programming Hight computational complexity: O ( C 3 ) C is the cloud capacity, and is usually huge Capacity segmentation is time sensitive: it has to be made in the beginning of every period 21 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
Approximation: solve the upper-bound problem Wei Wang, Department of Electrical and Computer Engineering, University of Toronto 22
The upper-bound problem a ) + γ r ( C t � C t ¯ Γ t ( C t ) = E γ a ( C t � max ¯ a ) 0 ≤ C t a ≤ C t ⇤ � ⇥ ¯ Γ t +1 ( C t + E X a + X ) . : Revenue upper bound of the auction channel, γ a ( C t ¯ a ) calculated as if partial fulfilment is accepted in periodic auctions 23 Wei Wang, Department of Electrical and Computer Engineering, University of Toronto
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