A Queueing Model for Yield A Queueing Model for Yield Management of ComputingManagement of Computing CentersCenters

Parijat DubeParijat DubeIBM Research, NY, USAIBM Research, NY, USA

Yezekael Hayel Yezekael Hayel IRISA, Rennes, France IRISA, Rennes, France

INFORMS Annual Meeting, San Francisco, Nov. 13-16, 2005INFORMS Annual Meeting, San Francisco, Nov. 13-16, 2005

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: 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 Characteristics of the resources (node speeds, Characteristics of the resources (node speeds,

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

Optimization Model (Dube et al. 2005)Optimization Model (Dube et al. 2005)

Cc ti Nk Qq

ciikikikqikq snWPspnrT..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: choice probability function; : probability of arrival of a customer of type cc=customer type, i=time, k=fee, q=machine type

nonconcave, nonlinearnonconcave, nonlinearDegree of nonconcavity related primarily to Degree of nonconcavity related primarily to

the choice of sojourn time function for each job the choice of sojourn time function for each job the discrete choice model of customer behaviorthe discrete choice model of customer behavior

c

Customer Choice ModelsCustomer Choice Models

Customer (dis)utility with class i: Customer (dis)utility with class i:

Weighted UtilityWeighted Utility

Logit ProbabilityLogit Probability

)()( iiiiii nTnrnU

k

k

ii U

U

KP 1

1

1

K

j

nU

nU

kjj

ik

e

enP

1

)(

)(

Prior WorksPrior Works

P. Dube, Y. Hayel, L. Wynter (2005)P. Dube, Y. Hayel, L. Wynter (2005)

A model for yield management of computational resources A model for yield management of computational resources

with exogenous sojourn timeswith exogenous sojourn times. .

Objective function with two classes and logit probabilityObjective function with two classes and logit probability

A Reduction to a Single Period ProblemA Reduction to a Single Period Problem

At each decision epochs, the market demand and parameters in At each decision epochs, the market demand and parameters in customer choice functions are updated customer choice functions are updated

An optimization problem is solved with new data and the optimal An optimization problem is solved with new data and the optimal allocation of aggregate CPU to different classes is determinedallocation of aggregate CPU to different classes is determined

We neglect any demand overlap between periods We neglect any demand overlap between periods

kk

K

kkkk

Kin

Nn

Prni 1

1,max

:Problemon Optimizati Period Single

Expression for Sojourn TimesExpression for Sojourn Times

We need a characterization of We need a characterization of The probability depends on which in turn The probability depends on which in turn

depends on depends on Intituitively should depend on Intituitively should depend on

the processing speed of class k, i.e., the processing speed of class k, i.e.,

(larger the smaller is )(larger the smaller is ) the fraction of demand seen by k, i.e., the fraction of demand seen by k, i.e.,

We use queueing theoretic formulations to We use queueing theoretic formulations to express as a function of andexpress as a function of and

FIFO service discipline at each class kFIFO service discipline at each class k

kkk PnT

1

kP

kP

kT

kn

kP

kU

kT

kn kT

kT kn

The Fixed Point ProblemThe Fixed Point Problem

For each feasible allocation, the customer choice For each feasible allocation, the customer choice probability can be characterized as a solution to a probability can be characterized as a solution to a system of fixed point equations: system of fixed point equations:

Existence and Uniqueness of Probability vector is Existence and Uniqueness of Probability vector is establishedestablished For both the weighted utility and logit probability For both the weighted utility and logit probability

Single Period Problem (weighted utility)Single Period Problem (weighted utility)

An example An example

1.0,9.0,100,1,10 21 Nrr

Single Period Problem (weighted utility): choice Single Period Problem (weighted utility): choice probabilityprobability

An example An example 1.0,9.0,100,1,10 21 Nrr

Single Period Problem (weighted utility): Single Period Problem (weighted utility): sojourn times sojourn times

Sojourn TimesSojourn Times

Conclusion and Future WorkConclusion and Future Work

Yield management for IT resources Yield management for IT resources Transaction duration has an implicit Transaction duration has an implicit

dependence with the processing speed of the dependence with the processing speed of the class. class.

A model to express the sojourn time as a A model to express the sojourn time as a function of system resources and the market sizefunction of system resources and the market size

The formulation should be generalized to The formulation should be generalized to account for demand dependency across periodsaccount for demand dependency across periods

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-ser price over Additional work needed to smooth end-ser price over

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