Optimal Pricing in Finite Server Systems Ashok Krishnan K.S. a , - - PowerPoint PPT Presentation

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Jul 24, 2023 457 likes •732 views

Optimal Pricing in Finite Server Systems Ashok Krishnan K.S. a , Chandramani Singh a , Siva Theja Maguluri b , Parimal Parag a a Indian Institute of Science b Georgia Institute of Technology Introduction The Problem 1 5 servers, 2 busy 2 3 4

SLIDE 1

Optimal Pricing in Finite Server Systems

Ashok Krishnan K.S.a, Chandramani Singha, Siva Theja Magulurib, Parimal Paraga

aIndian Institute of Science bGeorgia Institute of Technology

SLIDE 2

Introduction

The Problem

1 2 3 4 5 5 servers, 2 busy

SLIDE 3

Introduction

The Problem

1 2 3 4 5 5 servers, 2 busy new job V

SLIDE 4

Introduction

The Problem

1 2 3 4 5 5 servers, 2 busy new job V price p2

SLIDE 5

Introduction

The Problem

1 2 3 4 5 5 servers, 2 busy new job V price p2 V < p2

SLIDE 6

Introduction

The Problem

1 2 3 4 5 5 servers, 3 busy new job V price p2 V ≥ p2

SLIDE 7

Introduction

The Problem

1 2 3 4 5 5 servers, 3 busy

SLIDE 8

Introduction

The Problem

1 2 3 4 5 5 servers, 3 busy new job V price p3

SLIDE 9

Introduction

The Problem

1 2 3 4 5 5 servers, 3 busy new job V price p3 p0, p1, .., p4?

SLIDE 10

Introduction

Previous Work and Our Setting

Social optimum vs Revenue Maximization

Naor ’69, Chen ’01, Borgs ’14

Homogeneous vs heterogeneous customers

Whang ’90, Shimkin ’00, Mandelbaum ’02

Single vs Multi Servers

Haviv ’94, Bradford ’96, Dumas ’11

Our setting:

Revenue maximization Heterogeneous customers Multi server system

SLIDE 11

System Model

1 2 3 4 5 Poisson λ i.i.d. V ∼ G exp(µ) price vector (p0, p1, p2, p3, p4)

SLIDE 12

System Model

State Evolution

1 2 3 4 5 State 0

SLIDE 13

System Model

State Evolution

1 2 3 4 5 1 State 1

SLIDE 14

System Model

State Evolution

1 2 3 4 5 arrival value V 1 2 3 4 5 if V ≥ p1

SLIDE 15

System Model

State Evolution

1 2 3 4 5 1 2 3 4 5 service completion

SLIDE 16

System Model

State Evolution

1 2 3 4 5 1 2 3 4 5 1 2 λ P(V ≥ p1) 2µ

SLIDE 17

System Model

State evolution

1 2 3 4 5 λ0 λ1 µ λ2 2µ λ3 3µ λ4 4µ 5µ λi = λP(V ≥ pi) = λ(1 − G(pi)) = λG(pi) Gives stationary distribution π

SLIDE 18

System Model

Revenue

λ i busy servers price pi w.p. πi joins w.p. G(pi) Revenue =λ K−1

i=0 πiG(pi)pi

SLIDE 19

System Model

The Infinite Server Case

1 2 3 4 5 Infinite Servers Revenue =λ ∞

i=0 πiG(pi)pi

≤ λG(p∗)p∗

i πi

p∗ = arg max pG(p)

SLIDE 20

System Model