Analysis of a Yield Management Model for On Demand IT Services Parijat Dube IBM Watson Research Center with Laura Wynter and Yezekael Hayel

On Demand computing services On Demand means offering IT resources to firms when they need it, in the quantity that is required On Demand is a business model – it can be viewed as an alternative to the buy-and-service and lease models for IT hardware. It is also an alternative to purchasing software licenses for use on proprietary hardware. It means paying for use only, of IT hardware, software and networking resources.

On Demand computing services On Demand takes advantage of network speed and sophisticated “middleware”, which allows seamless operation of IT resources, remotely. On Demand is a win-win proposition, for the provider of the service and for the customer: The provider can experience considerable scale economies through resource sharing; The customer saves on outlay expenses, converts purchases to operating costs, and reaps the savings of the scale economies passed on by the provider.

Features of On Demand Temporary (very short term) increases and decreases in resource needs can be satisfied instantaneously, Temporary (very short term) increases and decreases in resource needs can be satisfied instantaneously, Neither space nor human resources need be consumed, or reassigned when no longer needed, Neither space nor human resources need be consumed, or reassigned when no longer needed, There is opportunity to pool resources. There is opportunity to pool resources.

Why Yield Mgmt. for On Demand Marginal cost of providing On Demand services is very low, Marginal cost of providing On Demand services is very low, Market for On Demand services is segmentable, with different job requirements and urgencies, Market for On Demand services is segmentable, with different job requirements and urgencies, While mainly large players (IBM, HP,Sun) are touting On Demand now, field will grow to a large number of mid-size providers -> synchronization of pricing is inevitable. While mainly large players (IBM, HP,Sun) are touting On Demand now, field will grow to a large number of mid-size providers -> synchronization of pricing is inevitable.

Yield management vs. no Y.M. Consider charging a single price. Given a demand curve, one can find the profit- maximizing single price. Revenue = 25; Market share = 50%

Yield management vs. no Y.M. Now consider charging a different price for each segment. Based on the same demand curve,

Yield management vs. no Y.M. Determine the optimal quantities to offer at each price segment. Revenue = 40; Market share = 80%.

Yield management vs. no Y.M. Revenue increases with the number of segments used, under some conditions.

Yield management: Properties We have the following properties of the Yield Management segmented prices: Theorem 1: Let demand be any monotone, decreasing, and nonnegative function, d(p) of price p. Suppose that as the number of price segments increases from i segments to i+1 segments, i=1..N, all price levels are maintained, and a new price level is added as the i+1 st segment. Then, the revenue increases as the number of price segments increases. Sketch of proof: Show that as N increases, R increases. R(N) =  i=1..N  p i ( d(p i ) – d(p i+1 ) ) =  i=1..N  (p i – p i-1 ) d(p i ). R(N+1) =  i=1..N+1  p i (d(p i ) – d(p i+1 )) = R(N) + (p N+! – p N ) d(p N+! ), price difference is positive by assumption as is d(.).

Yield management: Properties Theorem 2: Let demand be any monotone, decreasing, and nonnegative function, d(p) of price p. Then if the price levels are set optimally, the revenue increases as the number of price segments increases, irrespective of whether price levels are maintained or not. Sketch of proof: Let R*(N) be the revenue with N optimally set prices, and R(N) the revenue with N, possibly suboptimal, prices. Then we have that R*(N+1) R(N+1) R*(N), where the first inequality follows from the optimality of the prices, and the second from Thrm.1.

Yield management: Properties Corollary 1: For the case where d(p) = ap+b, with a 0, i.e., the demand is affine and decreasing in price, the optimal price levels for i price segments is: Theorem 3: Let demand be any monotone, decreasing, and nonnegative function, d(p) of price p. Then if the price levels are set optimally, the maximum revenue is obtained when the number of segments goes to infinity, and is given by the integral under the demand curve over the region in which d(p)>0.

Yield management: Properties Theorem 4: Let demand be given by an affine, decreasing function of price, d(p) = ap+b, with a 0. Then, when price levels are optimally set, the larger the market size, the greater the benefit of an increasing number of price segments. That is, let be two different market sizes. That is, let be two different market sizes. Then, Then, where is the revenue with N+1 optimally-set price segments and a market size of and similarly where is the revenue with N+1 optimally-set price segments and a market size of and similarly for and for and

Yield management: Opt. Model The model to determine optimal yield mgmt. quantities on the IT utility takes as input: User (random) discrete choice preference function describing the probability of a user with workload type accepting a YM offering User (random) discrete choice preference function describing the probability of a user with workload type accepting a YM offering Probability that an arriving job is of that type Probability that an arriving job is of that type Random workload, storage req. of jobs Random workload, storage req. of jobs Parallelizability of jobs Parallelizability of jobs Characteristics of the resources (node speeds, storage available, memory and CPU available) Characteristics of the resources (node speeds, storage available, memory and CPU available)

Yield management: Optimization Model T : sojourn time of a job in the system; r and p : unit prices/segments for compute power and storage space; P: multinomial choice probability function; : probability of arrival of a customer of type c c=customer type, i=time, k=fee, q=machine type

Yield management: Model Features nonconcave, nonlinear nonconcave, nonlinear Degree of nonconcavity related primarily to the choice of sojourn time function for each job, Degree of nonconcavity related primarily to the choice of sojourn time function for each job, Some linear cases do exist, Some linear cases do exist, When sojourn times are exogenous, nonconvexity of model is minimal, can in practice be solved to global optimization by NLP code When sojourn times are exogenous, nonconvexity of model is minimal, can in practice be solved to global optimization by NLP code

A Simplified Model: an example Logit Probability function: Logit Probability function: A simple two class, single period model A simple two class, single period model Utility of each class Utility of each class Optimization Problem Optimization Problem

Model Analysis Assume total workload is same for different customer classes Assume total workload is same for different customer classes Need to solve in one variable: Need to solve in one variable: Solution of fixed point equation: Solution of fixed point equation: Observe that and Observe that and Further is decreasing and is increasing in Further is decreasing and is increasing in

Analytical Solution Assume Assume We can approximate We can approximate by its Taylor’s expansion by its Taylor’s expansion Need to find the solution of Need to find the solution of quadratic in

Solution Efficiency An Example: An Example: We get: revenue We get: revenue Approximation: revenue Approximation: revenue Error: Error: Revnue error is practically zero. Revnue error is practically zero.

Yield management: Model Properties Model objective with exogenous sojourn times, multinomial logit choice function.

Induced Demand Curve The expected quantity that would subscribe to the IT service based on multi-variate logit model at a given price and quality, all other data being fixed. The expected quantity that would subscribe to the IT service based on multi-variate logit model at a given price and quality, all other data being fixed.

Optimal Yield Management Solution Increase in revenue as the number of price segments increases Increase in revenue as the number of price segments increases Tradeoff in increasing complexity due to a high number of price segments is balanced by a little increase in revenue. Tradeoff in increasing complexity due to a high number of price segments is balanced by a little increase in revenue.

Yield Management for Transactions at a Service Center Total demand over time; Revenue with a single (high, med, low) price vs. 5 price segments

Optimal Number of Price Segments Vs. Demand

Optimal Number of Price Segments Vs. Demand (contd.) Optimal number of price segments is not monotone in demand Optimal number of price segments is not monotone in demand Yield management system should be re-run as new and better demand data become available Yield management system should be re-run as new and better demand data become available

Summary and conclusions Revenue theoretically increases in this type of market with an increasing number of price segments. Revenue theoretically increases in this type of market with an increasing number of price segments. In the optimization model, with discrete choice preference functions (instead of a single demand curve, d(p), behavior is more complex: In the optimization model, with discrete choice preference functions (instead of a single demand curve, d(p), behavior is more complex: Ideal number of segments varies with demand; Ideal number of segments varies with demand; Program must be rerun periodically to optimize revenue. Program must be rerun periodically to optimize revenue. Additional work needed to smooth end-user price over usage horizon; various financial instruments (options, futures) may be of value. Additional work needed to smooth end-user price over usage horizon; various financial instruments (options, futures) may be of value.