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

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 5

Introduction The Problem 1 5 servers, 2 busy 2 new job 3 V 4 5

Introduction The Problem 1 5 servers, 2 busy 2 new job price p 2 3 V 4 5

Introduction The Problem 1 5 servers, 2 busy 2 new job price p 2 3 V < p 2 V 4 5

Introduction The Problem 1 5 servers, 3 busy 2 new job price p 2 3 V ≥ p 2 V 4 5

Introduction The Problem 1 5 servers, 3 busy 2 3 4 5

Introduction The Problem 1 5 servers, 3 busy 2 new job price p 3 3 V 4 5

Introduction The Problem 1 5 servers, 3 busy 2 new job price p 3 3 V 4 p 0 , p 1 , .., p 4 ? 5

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

System Model System Model exp( µ ) 1 2 Poisson λ 3 i.i.d. V ∼ G 4 5 price vector ( p 0 , p 1 , p 2 , p 3 , p 4 )

System Model State Evolution 1 2 3 0 4 State 0 5

System Model State Evolution 1 2 3 1 4 State 1 5

System Model State Evolution 1 1 2 2 if V ≥ p 1 arrival 3 3 value V 4 4 5 5

System Model State Evolution 1 1 2 2 service completion 3 3 4 4 5 5

System Model State Evolution 1 1 2 2 λ P ( V ≥ p 1 ) 3 3 1 2 2 µ 4 4 5 5

System Model State evolution λ 0 λ 1 λ 2 λ 3 λ 4 0 1 2 3 4 5 µ 2 µ 3 µ 4 µ 5 µ λ i = λ P ( V ≥ p i ) = λ (1 − G ( p i )) = λ G ( p i ) Gives stationary distribution π

System Model Revenue i busy servers λ price p i joins w . p . G ( p i ) w . p . π i Revenue = λ � K − 1 i =0 π i G ( p i ) p i

System Model The Infinite Server Case 1 Infinite Servers Revenue = λ � ∞ 2 i =0 π i G ( p i ) p i 3 ≤ λ G ( p ∗ ) p ∗ � i π i p ∗ = arg max pG ( p ) 4 5

System Model A Sub-Optimal Scheme Uniform Pricing 1 All states have same price 2 3 5 = arg max pG ( p )(1 − π 5 ( p )) p ∗ 4 5

Optimal Solution and its properties Optimal Price 2 1 . 5 Price 1 p ∗ 5 0 . 5 0 1 2 3 4 number of busy servers

Optimal Solution and its properties Properties of the Optimal Solution 0 . 4 8 Revenue/arrival rate 0 . 35 6 Revenue 0 . 3 4 0 . 25 2 0 0 . 2 10 15 20 25 30 10 15 20 25 30 arrival rate arrival rate

Optimal Solution and its properties Properties of the Optimal Solution 9 6 8 5 Revenue/service rate Revenue 7 4 3 6 2 5 1 2 3 4 5 1 2 3 4 5 service rate service rate

Optimal Solution and its properties Properties of the Optimal Solution 9 2 8 1 . 8 Revenue/server Revenue 1 . 6 7 1 . 4 6 3 4 5 6 7 1 . 2 3 4 5 6 7 number of servers number of servers

Optimal Solution and its properties Properties of the Optimal Solution p 0 2 p 1 p 2 1 . 8 1 . 6 Price 1 . 4 1 . 2 p ∗ 1 3 4 5 6 7 number of servers

Optimal Solution and its properties Revenue Gain P p ∗ p ∗ 5 Revenue Rate 6 4 2 0 0.5 1 5 10 load Figure: Revenue rate as a function of load

Summary Summary Analysis of system with heterogeneous customers Solution to the server pricing problem for revenue maximization Uniform pricing is optimal for infinite server system Analytical MDP solution to obtain the optimal pricing for a finite server system Two simple heuristic algorithms for pricing a finite server system Properties of optimal pricing for finite server systems Performance comparison between the optimal and heuristic algorithms for finite server systems