admission controls as pricing schemes for shared computer services the token bucket mechanisms...

Admission Controls Admission Controls as Pricing Schemes for as Pricing Schemes for

Shared Computer Services Shared Computer Services

The Token Bucket Mechanisms ExamplesThe Token Bucket Mechanisms Examples

Opher Baron Dirk Beyer Gabriel BitranThe Rotman School of Management HP Labs The Sloan School of ManagementUniversity of Toronto Palo-Alto MIT

4th Annual INFORMS Revenue Management and Pricing Section

ConferenceJune 10-11, 2004

Sloan School of Management, MIT

04/19/2304/19/23 Admission Controls as Pricing Schemes for Shared Services, the Token Bucket ExamplesAdmission Controls as Pricing Schemes for Shared Services, the Token Bucket Examples

ThanksThanks

• J. Altman, S. Jean, S. Singhal, A. Zhang –from HP

• S. Graves, Y. Wang –from MIT


Admission Controls as Pricing Schemes Admission Controls as Pricing Schemes for Shared Computer Servicesfor Shared Computer Services

Using admission controls as Using admission controls as pricing schemespricing schemes could help in could help in

the provisioning of shared servicesthe provisioning of shared services


AgendaAgenda• Motivation

• Shared computer services are here !

• Some challenges• Pricing• Service level• Admission control

• A Solution• Coordination using admission controls as pricing schemes• Buyers and seller need to overcome coordination problems

• Token bucket admission controls as pricing schemes


Motivation: Shared Computer Motivation: Shared Computer Services Are Here!Services Are Here!

• HP (http://www.hp.com/large/infrastructure/utilitydata/overview/)“… Utility Data Center … creates and runs virtual IT environments as a highly automated service. Simplified delivery of that service optimizes asset utilization and reduces staffing loads”

• IBM (http://www-1.ibm.com/grid/)“Grid Computing powers e-business on demand … integrated as a powerful single system. Grid Computing is a business reality”

• Sun Microsystems (http://wwws.sun.com/software/grid/)“Companies of all sizes and across many industries have employed Grid Computing solutions …”


Challenges in the Provisioning of Shared Challenges in the Provisioning of Shared Computer ServicesComputer Services

• Security and privacy• Definition of

accountable resources• Resources planning

• Reliability• High set-up cost• Fear of change• Ties buyer to seller

• Pricing

• Service level

• Admission control


Pricing Schemes in the Pricing Schemes in the Literature and in PracticeLiterature and in Practice

• Literature is mainly on Internet pricing:• Congestion pricing as the smart market

(MacKie-Mason and Varian 1994)• Other pricing as flexible service plan

(Altman 2001)• Pricing and capacity decisions

(Maglaras and Zeevi 2003)

• In practice, fixed plus variablefixed plus variable costcost per:• Average demand• 95/5

• Literature is complicated practice is simple


Motivation and ChallengesMotivation and Challenges

• Shared services are here

• Pricing and admission control are important challenges

• Combination of admission control and pricing might be beneficial


AgendaAgenda• MotivationMotivation• Some challengesSome challenges• A Solution

• Coordination using admission controls as pricing schemes• Buyers and seller need to overcome coordination problems

• Token bucket admission controls as pricing schemes• Coordination using token bucket admission controls as

pricing schemes• Provide solutions/approximations for coordination problems


Coordinating Between Buyers Coordinating Between Buyers and a Sellerand a Seller

DemandResources

Seller

Service provider (IBM, HP, …)

Buyer

IT department (MIT, University of Toronto,

…)

Coordination

Admission control and pricing

Seller: 4. Parameters’

prices

Given resources and performance

Buyer: 2. Parameters’

values

Given demand, prices, and performance

requirement

Performance: 1. Service level3. Effective demand

Given parameters and demand


AgendaAgenda• MotivationMotivation• Some challengesSome challenges• A SolutionA Solution

• Coordination using admission controls as pricing schemesCoordination using admission controls as pricing schemes• Buyers and seller need to overcome coordination problemsBuyers and seller need to overcome coordination problems

• Token bucket admission controls as pricing schemes• Coordination using token bucket admission controls as

pricing schemes• Provide solutions/approximations for coordination problems


The Token Bucket Admission Control The Token Bucket Admission Control

Definitions

• Unit of work

• Token (stamp)

• A finite queue of tokens - Bucket


Token Bucket Admission Token Bucket Admission Control MechanismControl Mechanism

•A pricing schemeA pricing scheme

Seller

Buyer

Constant Token rate r

Bucket Depth d


Token Bucket With Rate Control Token Bucket With Rate Control Admission Control MechanismAdmission Control Mechanism

Seller

Buyer

Constant token rate r

Bucket depth d

An infinite capacity jobs queue: Backlog instead of lost-sales


AgendaAgenda• MotivationMotivation• Some challengesSome challenges• A SolutionA Solution

• Coordination using admission controls as pricing schemesCoordination using admission controls as pricing schemes• Buyers and seller need to overcome coordination problemsBuyers and seller need to overcome coordination problems

• Token bucket admission controls as pricing schemesToken bucket admission controls as pricing schemes• Coordination using token bucket admission controls as Coordination using token bucket admission controls as

pricing schemespricing schemes• Provide solutions/approximations for coordination problems


Coordinating Between Buyers Coordinating Between Buyers and a Sellerand a Seller Using Token Bucket Using Token Bucket

DemandResourcesToken bucket

admission control and pricing

Seller: 4. Parameters:

R=? D=?

Buyer: 2. Parameters: :

r=? d=?

Performance:1. Service level=?

3. Effective demand=?

Seller


Buyer


…)

Coordination


1. Service Level1. Service Level

• Definition of service level for admission control: Percentage of periods with losses

• AssumptionDemand in each period is iid with a “nice” MGF

• AnalysisThe bucket levels as regulated random walks

?losses with periodsPr r,d


1. Service Level: The Bucket Levels as 1. Service Level: The Bucket Levels as Regulated Random WalksRegulated Random Walks

• It is a two sided regulated random walk• Losses when no tokens in the bucket• Similar to waiting time of a D/GI/1 queue, the Lindely

recursion

ii

iiii

ii

i

r-uL if dd

durL if r-uL

r-uL if

L~

~0

~0

~0

~1

Process LevelBucket The~ L


1. Service Level: Random Walks1. Service Level: Random Walks

-5

0

5

10

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Time

# of

tok

ens

RW One Regulator Two RegulatorsOne sided regulated Two sided regulatedR. walk

d=4

Time


1. Service Level: Upper Bounds1. Service Level: Upper Bounds

• Known bounds and approximations for one-sided regulated random walk (Ross 1974), (Siegmund 1985):

• where s* is the conjugate point of the demand distribution

• P2≤ P1≤ Exp(-s*·d)

dsbxs

bebxeEL

** 1

0inf 0Pr

dsdsdLs

sCeeeEL

***

* 0Pr

C·A· A·

Theorem 1:)(

)()(112

ij

iju

tE

tErFPP


2. The Buyer’s Problem2. The Buyer’s Problem

0,

10~

Pr

..

min,

dr

L

ts

dDrRdr

• Controls both drift and threshold

• Glasserman, 1997 “Bounds and asymptotics for planning critical safety stocks”

r,dr,d


2. The Buyer’s Problem: 2. The Buyer’s Problem: Constrained VersionConstrained Version

Theorem 2:• The conjugate point s* is strictly increasing with r

• 1/s* is strictly convex in r

• The constrained problem is convex FOC• Closed form solutions for exponential & normal demands

0,

1

..

min

*

,

dr

SLeC

ts

dDrR

ds

dr


2. The Buyer’s Problem: Bounds for 2. The Buyer’s Problem: Bounds for Normal(10,1) Demand, D/R=0.1Normal(10,1) Demand, D/R=0.1

10.0

10.2

10.4

10.6

10.8

11.0

11.2

0% 5% 10% 15% 20% 25%

Targeted percentage lost

Co

st

UBCost ApproxCost OptCost LBCost BAppCost

P2 ≤ A·P1 ≤ A·C·Exp(-s*·d)

Exp(-s*·d)

P2=Optimal

C·Exp(-s*·d)

A· C·Exp(-s*·d)

Lower Bound


2. The Buyer’s Problem: Bounds for 2. The Buyer’s Problem: Bounds for Exponential(1) Demand, D/R=0.2Exponential(1) Demand, D/R=0.2

1. 0

1. 2

1. 4

1. 6

1. 8

2. 0

2. 2

2. 4

2. 6

2. 8

0% 5% 10% 15% 20% 25%

Targeted Percentage Lost

Co

st

BoundCost OptCost ApproxCost1sOptCost 2sOptCost BoundCost


3. Effective Demand3. Effective Demand

• Using a Brownian Motion approximation for the bucket level we can express the effective demand

• However, the Brownian Motion approximation ignores the discrete probabilities to be on 0 and d, thus it tends to underestimate the standard deviation of demand

• We improved the Brownian Motion approximation using approximations for these discrete probabilities


SummarySummary• Motivation

• Shared computer services are here !

• Some challenges• Pricing• Service level• Admission control

• A SolutionA Solution• Coordination using admission controls as pricing schemesCoordination using admission controls as pricing schemes• Buyers and seller need to overcome coordination problemsBuyers and seller need to overcome coordination problems

• Token bucket admission controls as pricing schemesToken bucket admission controls as pricing schemes• Coordination using token bucket admission controls as Coordination using token bucket admission controls as

pricing schemespricing schemes• Provide solutions/approximations for coordination problems


Coordinating Between Buyers Coordinating Between Buyers and a Sellerand a Seller Using Token Bucket Using Token Bucket

DemandResourcesToken bucket

admission control and pricing

Seller: 4. Parameters:

R=? D=?

Buyer: 2. Parameters:

r=? d=?

Performance:1. Service level=?

3. Effective demand=?

Seller


Buyer


…)

Coordination


• Token bucket pricing• Aggregate effective demand from TB admission controls

• The seller’s pricing problem

• Secondary tokens market

• Pricing of multiple resources

• Resource planning• Analyze additional admission controls as pricing

schemes

Future WorkFuture Work


Admission Controls as Pricing Schemes Admission Controls as Pricing Schemes for Shared Computer Servicesfor Shared Computer Services

Using admission controls as Using admission controls as pricing schemespricing schemes could help in could help in

the provisioning of shared servicesthe provisioning of shared services


More information:

1. Baron, O., Beyer, D., and Bitran, G. R., (2004) Pricing of Shared Computer Services. Submitted to the Journal of Revenue and Pricing Management.

2. Baron, O., Beyer, D., and Bitran, G. R., (2004). Analysis of Two-Sided Regulated Random Walks. Submitted to the Journal of Applied Probability.

3. Baron, O. (2003). Pricing and Admission Control for Shared Computer Services Using the Token Bucket Mechanism. Ph.D. Thesis, Sloan School of Management, MIT.


Important Attributes of Important Attributes of Pricing Schemes and Admission ControlsPricing Schemes and Admission Controls

Buyer Seller

Is simple to understand and monitor

Is simple to understand, operate, and monitor

Has known costs/ budget Results in known revenues

Supports demand variability Supports resource planning

Supports service level Guarantees

Gives buyers incentives to truthfully report usage and to smooth demand


Token Bucket as a Pricing SchemeToken Bucket as a Pricing SchemeBest > Medium > Worse* - Not in sellers control.

Fixed Cost

Smart Market

Flexible ser. plan

95/5 Token Bucket

Supports demand var.

Medium Best Good Medium Good

Supports service level guarantees

Worse Best Medium Weak Good

Supports resource planning

Weak Good Good Medium Best

Gives incentives Worse Best (*) Medium Weak Good

Easy to understand and operate

Good Worse Good Good Good

Information requirements

Best Worse Medium Good Weak

Known costs / Revenues

Best Worse Medium Weak Good

Cost of demand’s profile changes

Good Worse Good Good Good$

$

$

$

$

$

$$

$$

$$

$$

$$

$$$$$

$$$$$

$$$$$

*$$$$$

$$$$$

$$$$$

$$$$ $$$$

$$$$

$$$$$$$$

$$$$

$$$$$$$$$$$$$$$$

$$$$

$$$$

$$$$$$$$$$$$$$$$

$$$

$$$$$$

$$$

$$$

$$$

$$$


Problems in Coordination Between Problems in Coordination Between Buyers and a SellerBuyers and a Seller

1. Service level• Define the service level• Characterizes the service level provided• Depends on chosen parameters• Depends on characterization of requested demand

2. The buyer’s problem• Wishes to minimize expenditures • Chooses the admission control parameters • Requires some service level • Knows her demand• Knows prices


Problems in Coordination Between Problems in Coordination Between Buyers and a SellerBuyers and a Seller

3. Effective demand• Characterizes the effective demand accepted by the

admission control (and served by the seller)• Depends on chosen parameters• Depends on characterization of requested demand• Aggregates from different buyers

4. The seller’s problem• Wishes to maximize revenues or profits• Chooses prices of parameters • Considers the effective demand• Knows available resources and the seller’s reaction to

price changes


Regulated Random WalksRegulated Random Walks

-2-1012345

1 6 11 16 21 26 31

T i m e

# of

tok

ens

One Regulator Two Regulators

T1 T2

t11=4 t12=1 t13=1

Down when Ui>r Up when

Ui<r


The Semi-Invariant MGFThe Semi-Invariant MGF


2. Error on the Buyer’s Problem for a 2. Error on the Buyer’s Problem for a 99% Service Level with Normal Demand99% Service Level with Normal Demand

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0.0 0.2 0.4 0.6 0.8 1.0

D/R

(ZB

A-Z

*)/Z

*

STD1 STD2 STD3stdev1 stdev2 stdev3

(Bou

nd

Cos

t-O

ptC

ost)

/Op

tCos

t


3. Effective Demand for Token Bucket: 3. Effective Demand for Token Bucket: the Brownian Motion Approximationthe Brownian Motion Approximation• Conditioning on the bucket level the effective demand is:

• Using a Brownian Motion approximation for the bucket level

• The Brownian Motion approximation ignores the discrete probabilities to be on 0 and d, thus it tends to underestimate demand

uie

ui if ui l i r

l i r if ui

l i r

FUeu

0 for u 0

FUu for 0 u r1 eur

1 ed FUu eur ed

1 ed for r u r d

1 for d r u

where 2r Eu/ 2


The Effective DemandThe Effective DemandToken Bucket: the Enhanced ApproximationToken Bucket: the Enhanced Approximation

• Fill rates for the regulated Brownian motion can be computed

FR 1 Ex|x 0 1 Eu

Lemma 3: The fill rate and percentage-of-periods-with-losses are related according to:

Corollary 4:In the exponential demand case the percentage-of-periods-with-losses of work is equal to the work fill-rate.


The Effective DemandThe Effective DemandToken Bucket: the Enhanced Algorithm IToken Bucket: the Enhanced Algorithm I

1. Input and initializationBuyer's demand CDF, E(u), σ, r, d, α. Let θ=(r-E(u))/σ>0

2. Estimate E(x|x>0), and E(xT|xT>0)Based on mean, age or large deviations results

3. Approximate the fill ratesIf the fill rates are in the range (0,1), go to 5

4. Alternative derivation of work fill rate Let P(L=0)=1-α. Approximate FR (using E(x|x>0))then compute FRT=E(u)*FR/r


The Effective DemandThe Effective DemandToken Bucket: the Enhanced Algorithm IIToken Bucket: the Enhanced Algorithm II

5. Derivation of work loss probabilityTranslate FR to P(L=0) using E(x|x>0). If the approximation leads to P(L=0)>1-α go to 4

6. Derivation of tokens loss probabilityTranslate the FRT to P(L=d) using E(xT|xT>0)

7. Approximate the bucket level PDF and CDFThese are the enhanced ones

8. Approximate the effective demandThese are the enhanced ones


The Effective DemandThe Effective DemandToken Bucket With Rate ControlToken Bucket With Rate Control

• Much more complicated due to the backlogs

FUeue

0 for ue 0

0

ue

0

xfUyfLy xdydx FUue1 FL0 for ue 0,r

FUr1 FL0 FL0 for ue r

FUue1 FLue r FLue r for ue r,r d

1 for ue r d


TBwRC Effective DemandTBwRC Effective Demand

Ranges for Bucket level L r r L 0 L 0 0 L

Ranges for effective demand

0 ue r NA u L ue u ue u ue

r ue w.p 1 u L r u r u r

r ue r d NA NA NA u ue;L ue r

or u ue;L ue r

r d ue NA NA NA u r d;L d


The Effective DemandThe Effective DemandOutput of Token Bucket Admissions ControlsOutput of Token Bucket Admissions Controls

Results• Characterizes the PDF and CDF of effective demands• Approximates the effective demands• Relates both service level measures• Show equivalence of service measures for the

exponential demand case • Provides an enhancement of the approximations• Plots CDFs and computes moments (error<2%)


Token Bucekt Token Bucket with rate Control

SL D/R Measure Simu ApproxFR Approx EnhanceFR Enhance RealMean Simu Approx Enhance

95 0.9 Mean 9.94 9.44 9.95 9.56 9.97 10.00 9.98 10.06 10.01

STDev 1.94 X 1.90 X 1.93 X 1.94 1.88 1.92

ErrorMean 0.00% -5.05% 0.05% -3.87% 0.21% 0.00% -0.24% 0.62% 0.10%

ErrorSTDev 0.00% X -2.04% X -0.60% X 0.00% -3.23% -1.07%

0.5 Mean 9.94 9.72 9.94 9.82 9.97 10.00 9.97 10.02 9.99

STDev 1.94 X 1.90 X 1.94 X 1.93 1.86 1.93

ErrorMean 0.00% -2.20% 0.01% -1.21% 0.26% 0.00% -0.25% 0.20% -0.06%

ErrorSTDev 0.00% X -1.99% X 0.28% X 0.00% -3.72% 0.26%

99 0.2 Mean 10.00 9.97 9.99 9.97 9.99 10.00 10.01 10.00 10.00

STDev 2.01 X 1.98 X 1.99 X 2.01 1.97 1.99

ErrorMean 0.00% -0.35% -0.15% -0.31% -0.11% 0.00% 0.11% 0.00% 0.00%

ErrorSTDev 0.00% X -1.63% X -1.25% X 0.00% -2.06% -1.14%

0.1 Mean 10.01 9.98 9.99 9.99 9.99 10.00 10.02 10.00 10.00

STDev 1.98 X 1.99 X 1.99 X 1.98 1.98 1.99

ErrorMean 0.00% -0.28% -0.17% -0.26% -0.18% 0.00% 0.16% 0.00% 0.00%

ErrorSTDev 0.00% X 0.35% X 0.44% X 0.00% -0.07% 0.46%

Expectation and Stdev Comparison for Normal(10,2) Demand, Expectation and Stdev Comparison for Normal(10,2) Demand, SL =95% with D/R=0.9, 0.5, and SL=99%with D/R=0.2,0.1SL =95% with D/R=0.9, 0.5, and SL=99%with D/R=0.2,0.1

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