Analysis of a Yield Management Model Analysis of a Yield Management Model for for On DemandOn Demand IT Services IT Services

Parijat DubeParijat DubeIBM Watson Research CenterIBM Watson Research Center

with Laura Wynter andwith Laura Wynter and Yezekael HayelYezekael Hayel

On Demand computing servicesOn 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 modelsfor IT hardware.

It is also an alternative to purchasing software licensesfor use on proprietary hardware.

It means paying for use only, of IT hardware, software and networking resources.

On Demand computing servicesOn Demand computing services

On Demand takes advantage of network speed and sophisticated “middleware”, which allows seamlessoperation of IT resources, remotely.

On Demand is a win-win proposition, for the providerof 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 savingsof the scale economies passed on by the provider.

Features of On DemandFeatures of On Demand

Temporary (very short term) increases and Temporary (very short term) increases and decreases in resource needs can be satisfied decreases in resource needs can be satisfied instantaneously,instantaneously,

Neither space nor human resources need be Neither space nor human resources need be consumed, or reassigned when no longer needed,consumed, or reassigned when no longer needed,

There is opportunity to pool resources.There is opportunity to pool resources.

Why Yield Mgmt. for On DemandWhy Yield Mgmt. for On Demand

Marginal cost of providing On Demand services is Marginal cost of providing On Demand services is very low,very low,

Market for On Demand services is segmentable, Market for On Demand services is segmentable, with different job requirements and urgencies,with different job requirements and urgencies,

While mainly large players (IBM, HP,Sun) are While mainly large players (IBM, HP,Sun) are touting On Demand now, field will grow to a large touting On Demand now, field will grow to a large number of mid-size providers -> synchronization of number of mid-size providers -> synchronization of pricing is inevitable.pricing is inevitable.

Yield management vs. no Y.M.Yield management vs. no Y.M.

Consider charging a single price. Consider charging a single price. Given a demand curve, one can find the profit-Given a demand curve, one can find the profit-

maximizing single price. maximizing single price. Revenue = 25; Market share = 50%Revenue = 25; Market share = 50%

Yield management vs. no Y.M.Yield management vs. no Y.M.

Now consider charging a different price for each Now consider charging a different price for each segment.segment.

Based on the same demand curve,Based on the same demand curve,

Yield management vs. no Y.M.Yield management vs. no Y.M.

Determine the optimal quantities to offer at each Determine the optimal quantities to offer at each price segment. price segment.

Revenue = 40; Market share = 80%.Revenue = 40; Market share = 80%.

Yield management vs. no Y.M.Yield management vs. no Y.M.

Revenue increases with the number ofRevenue increases with the number of

segments used, under some conditions.segments used, under some conditions.

Yield management: PropertiesYield management: Properties

We have the following properties of the Yield We have the following properties of the Yield Management segmented prices:Management segmented prices:

Theorem 1Theorem 1: Let demand be any monotone, decreasing, and nonnegative : Let demand be any monotone, decreasing, and nonnegative function, d(p) of price p. Suppose that as the number of price 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 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 levels are maintained, and a new price level is added as the i+1 st segment. segment. Then, the revenue increases as the number of price Then, the revenue increases as the number of price segments increases.segments increases.

Sketch of proofSketch of proof::

Show that as N increases, R increases.Show that as N increases, R increases.

R(N) = R(N) = i=1..Ni=1..Nppii ( d(p ( d(pii) – d(p) – d(pi+1i+1) ) = ) ) = i=1..Ni=1..N(p(pii – p – pi-1i-1 ) d(p ) d(pii) . ) .

R(N+1) = R(N+1) = i=1..N+1i=1..N+1ppii (d(p (d(pii) – d(p) – d(pi+1i+1)) = R(N) + (p)) = R(N) + (pN+!N+! – p – pNN ) d(p ) d(pN+!N+!) ,) ,

price difference is positive by assumption as is d(.).price difference is positive by assumption as is d(.).

Yield management: PropertiesYield management: Properties

Theorem 2Theorem 2: Let demand be any monotone, : Let demand be any monotone, decreasing, and nonnegative function, d(p) of decreasing, and nonnegative function, d(p) of price p. Then if the price levels are set price p. Then if the price levels are set optimally, the optimally, the revenue increases as the number revenue increases as the number of price segments increasesof price segments increases, irrespective of , irrespective of whether price levels are maintained or not.whether price levels are maintained or not.

Sketch of proofSketch of proof::Let R*(N) be the revenue with N optimally set Let R*(N) be the revenue with N optimally set prices, and R(N) the revenue with N, possibly prices, and R(N) the revenue with N, possibly suboptimal, prices. Then we have thatsuboptimal, prices. Then we have that

R*(N+1) R(N+1) R*(N),R*(N+1) R(N+1) R*(N),where the first inequality follows from the where the first inequality follows from the optimality of the prices, and the second from optimality of the prices, and the second from Thrm.1.Thrm.1.

Yield management: PropertiesYield management: Properties

Corollary 1Corollary 1: For the case where d(p) = ap+b, with : For the case where d(p) = ap+b, with a < 0 and b > 0, i.e., the demand is affine and a < 0 and b > 0, i.e., the demand is affine and decreasing in price, the optimal price levels for i decreasing in price, the optimal price levels for i price segments is:price segments is:

Theorem 3Theorem 3: Let demand be any monotone, : Let demand be any monotone, decreasing, and nonnegative function, d(p) of decreasing, and nonnegative function, d(p) of price p. Then if the price levels are set price p. Then if the price levels are set optimally, the optimally, the maximum revenuemaximum revenue is obtained is obtained when the number of segments goes to infinity, when the number of segments goes to infinity, and and is given by the integral under the demand is given by the integral under the demand curvecurve over the region in which d(p)>0. over the region in which d(p)>0.

.,2,1,1

* ijai

jbP ij

Yield management: PropertiesYield management: Properties

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

segments and a market size of segments and a market size of and similarly and similarly for and for and

21 dd

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1*2 NR

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Yield management: Opt. ModelYield management: Opt. Model

The model to determine optimal yield mgmt. quantities The model to determine optimal yield mgmt. quantities on the IT utility takes as input:on the IT utility takes as input:

User (random) discrete choice preference function User (random) discrete choice preference function describing the probability of a user with workload describing the probability of a user with workload type accepting a YM offeringtype accepting a YM offering

Probability that an arriving job is of that typeProbability that an arriving job is of that type Random workload, storage req. of jobsRandom workload, storage req. of jobs Parallelizability of jobsParallelizability of jobs Characteristics of the resources (node speeds, Characteristics of the resources (node speeds,

storage available, memory and CPU available)storage available, memory and CPU available)

Yield management: Yield management: Optimization Model Optimization Model

Cc ti Nk Qq

ciikikikqikqikqi snWPspnrcnWT..1 ..1 ..1 ..1

),,())(,,(max

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 cc=customer type, i=time, k=fee, q=machine type

c

Yield management: Model FeaturesYield management: Model Features

nonconcave, nonlinearnonconcave, nonlinear

Degree of nonconcavity related primarily to the Degree of nonconcavity related primarily to the choice of sojourn time function for each job,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 When sojourn times are exogenous, nonconvexity of model is minimal, can in practice be solved to global model is minimal, can in practice be solved to global optimization by NLP codeoptimization by NLP code

A Simplified Model: an exampleA Simplified Model: an example Logit Probability function:Logit Probability function:

A simple two class, single period modelA simple two class, single period model Utility of each classUtility of each class

Optimization ProblemOptimization Problem

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Model AnalysisModel Analysis

Assume total workload is same for different customer Assume total workload is same for different customer classesclasses

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

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Analytical SolutionAnalytical Solution

Assume Assume

We can approximate We can approximate by its Taylor’s expansion by its Taylor’s expansion

Need to find the solution ofNeed to find the solution of quadratic in quadratic in

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Solution EfficiencySolution Efficiency

An Example: An Example: We get: revenue 2.6007We get: revenue 2.6007 Approximation: revenue 2.6004Approximation: revenue 2.6004

Error: Error: Revnue error is practically zero. Revnue error is practically zero.

2,1,3,2,2,1,05.0 2121 TNrr

1973.0*1 n

2131.0*1 n

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Yield management: Model PropertiesYield management: Model Properties

Model objective with exogenous sojourn times,Model objective with exogenous sojourn times,

multinomial logit choice function.multinomial logit choice function.

Induced Demand CurveInduced Demand Curve

The expected quantity that would subscribe to the The expected quantity that would subscribe to the IT service based on multi-variate logit model at a IT service based on multi-variate logit model at a

given price and quality, all other data being fixed.given price and quality, all other data being fixed.

Optimal Yield Management SolutionOptimal Yield Management Solution Increase in revenue as the number of price Increase in revenue as the number of price

segments increasessegments increases Tradeoff in increasing complexity due to a high Tradeoff in increasing complexity due to a high

number of price segments is balanced by a little number of price segments is balanced by a little increase in revenue. increase in revenue.

Yield Management for Transactions Yield Management for Transactions at a Service Centerat a Service Center

Total demand over time; Revenue with a single (high, Total demand over time; Revenue with a single (high,

med, low) price vs. 5 price segmentsmed, low) price vs. 5 price segments

Optimal Number of Price Segments Optimal Number of Price Segments Vs. DemandVs. Demand

Optimal Number of Price Segments Optimal Number of Price Segments Vs. Demand (contd.)Vs. Demand (contd.)

Optimal number of price segments is not monotone in Optimal number of price segments is not monotone in demanddemand

Yield management system should be re-run as new and Yield management system should be re-run as new and better demand data become availablebetter demand data become available

Summary and conclusionsSummary and conclusions

Revenue theoretically increases in this type of market Revenue theoretically increases in this type of market with an increasing number of price segments.with an increasing number of price segments.

In the optimization model, with discrete choice In the optimization model, with discrete choice preference functions (instead of a single demand preference functions (instead of a single demand curve, d(p), behavior is more complex: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 Program must be rerun periodically to optimize

revenue.revenue. Additional work needed to smooth end-user price Additional work needed to smooth end-user price

over usage horizon; various financial instruments over usage horizon; various financial instruments (options, futures) may be of value.(options, futures) may be of value.