A Non-Monetary Mechanism for Optimal Rate Control Through Effjcient - - PowerPoint PPT Presentation

A Non-Monetary Mechanism for Optimal Rate Control Through Effjcient Delay Allocation

Tao Zhao1 Korok Ray2 I-Hong Hou1

1Department of ECE

Texas A&M University

2Mays School of Business

Texas A&M University

WiOpt’17

1

Motivation

Optimal Rate Control

A wireless network with multiple clients Individual utility: function of request arrival rate Problem: Find optimal rates that maximize total utility

Game theory is needed.

Clients: selfjsh and strategic Individual utility: private

• 2

Motivation

Existing Work

Auction: e.g. VCG auction Direct payment between client and server

Issues of Monetary Mechanisms

Monetary exchange requires addtional infrastructure. Pricing every packet? Impractical. Non-monetary mechanism!

3

Motivation

Existing Work

Auction: e.g. VCG auction Direct payment between client and server

Issues of Monetary Mechanisms

Monetary exchange requires addtional infrastructure. Pricing every packet? Impractical. Non-monetary mechanism!

3

How Non-Monetary?

Observation

Each client sufgers disutility based on experienced delay. Server can control delay by scheduling.

Our Approach

Use delay as the currency!

Main Contribution

A non-monetary mechanism by effjcient delay allocation

4

How Non-Monetary?

Observation

Each client sufgers disutility based on experienced delay. Server can control delay by scheduling.

Our Approach

Use delay as the currency!

Main Contribution

A non-monetary mechanism by effjcient delay allocation

4

System Model

One server: Average request service rate µ Client i = 1, 2, . . . , N:

Average request arrival rate λi: adjustable Utility Ui(λi): increasing, twice difgerentiable, concave Average request delay Di(λi, λ−i)

µ Server: max

∑

iUi(λi) − λiDi

Client 1 λ1 Client i: max Ui(λi) − λiDi(λi, λ−i) λi Client N λN . . .

5

System Model

Total average delay

Function of total average request arrival rate, Λ := ∑

i λi

Increasing and convex Fitted by a (N − 2)-order polynomial C(Λ)

Assume feasible λ := [λi] satisfjes Λ < (1 − ϵ)µ, λi > λδ > 0

µ Server: max

∑

i Ui(λi) − ΛC(Λ)

where Λ =

∑

i λi

Client 1 λ1 Client i: max Ui(λi) − λiDi(λi, λ−i) λi Client N λN . . .

5

Game Between Clients and Server

Client chooses its arrival rate λi Server allocates delays [Di(λi, λ−i)] Server enforces delays by scheduling Client observes its own delay Di

6

Nash Equilibrium and Effjciency

Defjnition

A vector ˜ λ := [˜ λi] is said to be a Nash Equilibrium if ˜ λi = argmaxλi Ui(λi) − λiDi(λi, ˜ λ−i), ∀i.

Defjnition

A rule of allocating delays, [Di(·)], is said to be effjcient if the vector that maximizes the total net utility, λ∗ := [λ∗

i ], is the only

Nash Equilibrium.

Remark

Server’s problem is to fjnd and enforce the rule that allocates delays, [Di(·)], to induce optimal choices of [λi].

7

Non-Monetary Mechanism for Optimal Rate Control

1

Effjcient Delay Allocation Rule

2

Scheduling Policy to Enforce Allocated Delays

3

Distributed Rate Control Protocol

8

Non-Monetary Mechanism for Optimal Rate Control

1

Effjcient Delay Allocation Rule

2

Scheduling Policy to Enforce Allocated Delays

3

Distributed Rate Control Protocol

9

Property of Effjcient Delay Allocation Rule

Server

λ∗ is the solution to max ∑

i

Ui(λi) − ΛC(Λ). Hence, U′

i(λ∗ i ) = ∂

∂λi Λ∗C(Λ∗)

Client

λ∗ is the solution to max Ui(λi) − λiDi(λi, λ∗

−i).

Hence, U′

i(λ∗ i ) = ∂

∂λi λ∗

i Di(λ∗ i , λ∗ −i)

Observation

Want ΛC(Λ) − λiDi(λi, λ−i) =: Ri(λ−i), the external disutility, independent of λi

10

Delay Allocation Rule

Delay Allocation Rule

λiDi(λi, λ−i) = ΛC(Λ) − Ri(λ−i) Ri(λ−i) = ∑N−1

j=1 βj i

βj

i = cj

∑

p∈Pj

i

N−1 N−G(p) j! p1!···p

N!λp1

1 · · · λp

N

N

cj: j-th order coeffjcient of polynomial ΛC(Λ) Pj

i := {p = [pn] | pn ∈ Z∗, ∑N i=1 pn = j, pi = 0}

G(p) be the number of nonzero coordinates of p

Theorem

Our rule of delay allocation [Di(·)] is effjcient.

11

An Example of Delay Allocation Rule

Example (N = 3)

βj

i

j = 1 j = 2 i = 1 c1(λ2 + λ3) c2(λ2

3 + 4λ2λ3 + λ2 2)

i = 2 c1(λ1 + λ3) c2(λ2

3 + 4λ1λ3 + λ2 1)

i = 3 c1(λ2 + λ1) c2(λ2

1 + 4λ2λ1 + λ2 2)

External disutility Ri (row sum) is independent of λi Allocated disutility λiDi = ΛC(Λ) − Ri Total disutility ∑

i λiDi = 3ΛC(Λ) − ∑ i Ri = ΛC(Λ)

12

Non-Monetary Mechanism for Optimal Rate Control

1

Effjcient Delay Allocation Rule

2

Scheduling Policy to Enforce Allocated Delays

3

Distributed Rate Control Protocol

13

Scheduling Policy

Problem

How to enforce target delay Di(λi, λ−i) for client i ?

MRQ Scheduling Policy

Let Qi(t) be the queue length of client i at time t, and gi := λiDi. At time t, the MRQ policy schedules the client with the maximum relative queue length, defjned as Qi(t)/gi.

Intuition

Eventually all relative queue lengths are equal on average in steady state, or equivalently, average queue length (delay) = target queue length (delay).

14

State Space Collapse

Theorem (State Space Collapse)

The effjcient delay allocation rule is enforced by the MRQ scheduling policy in the heavy traffjc regime.

Remark

Heavy traffjc: Λ → µ Show the deviation of the limiting queue length vector from the target queue length vector approaches 0 Lyapunov drift based technique

15

Non-Monetary Mechanism for Optimal Rate Control

1

Effjcient Delay Allocation Rule

2

Scheduling Policy to Enforce Allocated Delays

3

Distributed Rate Control Protocol

16

How Distributed?

We already know

Our delay allocation rule is effjcient. Our MRQ scheduling policy enforces the delay allocation rule.

Problem

How are the clients supposed to update their request rates distributedly to converge to the Nash Equilibrium?

Idea

Projected gradient method: Centralized How to make it distributed?

17

Centralized → Distributed

Centralized update: ˆ λ(k + 1) = λ(k) + κ(k) η(k)∇ [∑ Ui(λi) − ΛC(Λ) ] , λ(k + 1) = P(ˆ λ(k + 1))

κ(k): step size at the k-th iteration η(k): Euclidean norm of the gradient P: projection to the feasible region s.t. λi > λδ and Λ < (1 − ϵ)µ k k k and d

C d

are the same for all clients: Broadcast!

λ(k) ˆ λ(k + 1) λ(k + 1)

18

Centralized → Distributed

Distributed update: ˆ λi(k + 1) = λi(k) + κ(k) η(k) [ U′

i(λi(k)) − d[ΛC(Λ)]

dΛ ] , λ(k + 1) = P(ˆ λ(k + 1))

κ(k): step size at the k-th iteration η(k): Euclidean norm of the gradient P: projection to the feasible region s.t. λi > λδ and Λ < (1 − ϵ)µ k k k and d

C d

are the same for all clients: Broadcast!

λ(k) ˆ λ(k + 1) λ(k + 1)

18

Centralized → Distributed

Distributed update: ˆ λi(k + 1) = λi(k) + κ(k) η(k) [ U′

i(λi(k)) − d[ΛC(Λ)]

dΛ ] , λi(k + 1) = min{max{ˆ λi(k + 1), λδ}, λi(k)(1 − ϵ)µ Λ(k) }

κ(k): step size at the k-th iteration η(k): Euclidean norm of the gradient P: projection to the feasible region s.t.

i

and k k k and d

C d

are the same for all clients: Broadcast!

λ(k) ˆ λ(k + 1) λ(k + 1)

18

Centralized → Distributed

Distributed update: ˆ λi(k + 1) = λi(k) + κ(k) η(k) [ U′

i(λi(k)) − d[ΛC(Λ)]

dΛ ] , λi(k + 1) = min{max{ˆ λi(k + 1), λδ}, λi(k)(1 − ϵ)µ Λ(k) }

κ(k): step size at the k-th iteration η(k): Euclidean norm of the gradient P: projection to the feasible region s.t.

i

and Λ(k), κ(k), η(k), and d[ΛC(Λ)]

dΛ

are the same for all clients: Broadcast!

λ(k) ˆ λ(k + 1) λ(k + 1)

18

Simulations

Validate our non-monetary mechanism

Polynomial approximation assumption State space collapse in scheduling Optimality of distributed rate control protocol

Baseline mechanism

FIFO (fjrst-in-fjrst-out) scheduling policy Centralized projected gradient method for rate control

Two systems: M/M/1 v.s. M/D/1

N = 10 clients Poisson arrivals: Λ = 0.99 × 103 s−1 Exponential/Deterministic service time: µ = 1 × 103 s−1

19

Polynomial Approximation

0.95 0.96 0.97 0.98 0.99 1 50 100 150 200

M/M/1 Theory M/M/1 Approx M/D/1 Theory M/D/1 Approx

Total disutility ΛC(Λ) v.s. Normalized total request rate Λ/µ

20

State Space Collapse

0.2 0.4 0.6 0.8 Time (s) 0.05 0.1 0.15 0.2 0.25 0.3 Relative difference of queue lengths

Same rate, M/M/1 Same rate, M/D/1 Diff rates, M/M/1 Diff rates, M/D/1

Normalized difgerence of relative queue lengths v.s. Time

21

Nash Equilibrium

20 40 60 80 100 120 Index of iteration 4.58 4.581 4.582 4.583 4.584 4.585 4.586 Total net utility 10 4 MRQ, Dist MRQ, Cent FIFO

M/M/1 system

20 40 60 80 100 120 Index of iteration 4.59 4.5902 4.5904 4.5906 4.5908 Total net utility 10 4 MRQ, Dist MRQ, Cent FIFO

M/D/1 system

22

Summary

Non-Monetary Mechanism for Optimal Rate Control

Effjcient delay allocation rule MRQ scheduling policy Distributed rate control protocol

23

Summary

Non-Monetary Mechanism for Optimal Rate Control

Delay = Currency Time = Money

23

Thank you!

Tao Zhao alick@tamu.edu