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 Texas A&M University Texas A&M University WiOpt’17 1 Tao Zhao 1 Korok Ray 2 I-Hong Hou 1 1 Department of ECE 2 Mays School of Business

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 5 One server: Average request service rate µ Client i = 1 , 2 , . . . , N : Average request arrival rate λ i : adjustable Utility U i ( λ i ) : increasing, twice difgerentiable, concave Average request delay D i ( λ i , λ − i ) Client 1 λ 1 . . . λ i Server: µ max ∑ i U i ( λ i ) − λ i D i Client i : max U i ( λ i ) − λ i D i ( λ i , λ − i ) λ N Client N

System Model Increasing and convex Total average delay 5 Function of total average request arrival rate, Λ := ∑ i λ i Fitted by a ( N − 2) -order polynomial C (Λ) Assume feasible λ := [ λ i ] satisfjes Λ < (1 − ϵ ) µ, λ i > λ δ > 0 Client 1 λ 1 . . . Server: λ i µ max i U i ( λ i ) − Λ C (Λ) ∑ where Λ = ∑ i λ i Client i : max U i ( λ i ) − λ i D i ( λ i , λ − i ) λ N Client N

Game Between Clients and Server 6 Client chooses Server allocates delays [ D i ( λ i , λ − i )] its arrival rate λ i Client observes Server enforces its own delay D i delays by scheduling

Nash Equilibrium and Effjciency Defjnition Server’s problem is to fjnd and enforce the rule that allocates Remark Nash Equilibrium. Defjnition 7 A vector ˜ λ := [˜ λ i ] is said to be a Nash Equilibrium if λ i = argmax λ i U i ( λ i ) − λ i D i ( λ i , ˜ ˜ λ − i ) , ∀ i . A rule of allocating delays, [ D i ( · )] , is said to be effjcient if the vector that maximizes the total net utility, λ ∗ := [ λ ∗ i ] , is the only delays, [ D i ( · )] , to induce optimal choices of [ λ i ] .

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 Hence, Observation Hence, Server Client 10 i max λ ∗ is the solution to λ ∗ is the solution to ∑ max U i ( λ i ) − λ i D i ( λ i , λ ∗ − i ) . U i ( λ i ) − Λ C (Λ) . i ) = ∂ i ) = ∂ U ′ i ( λ ∗ Λ ∗ C (Λ ∗ ) U ′ i ( λ ∗ λ ∗ i D i ( λ ∗ i , λ ∗ − i ) ∂λ i ∂λ i Want Λ C (Λ) − λ i D i ( λ i , λ − i ) =: R i ( λ − i ) , the external disutility, independent of λ i

Delay Allocation Rule i Theorem P j N N Delay Allocation Rule 11 i λ i D i ( λ i , λ − i ) = Λ C (Λ) − R i ( λ − i ) R i ( λ − i ) = ∑ N − 1 j =1 β j j ! N − 1 ∑ β j i = c j N ! λ p 1 1 · · · λ p N − G ( p ) p 1 ! ··· p p ∈ P j c j : j -th order coeffjcient of polynomial Λ C (Λ) i := { p = [ p n ] | p n ∈ Z ∗ , ∑ N i =1 p n = j , p i = 0 } G ( p ) be the number of nonzero coordinates of p Our rule of delay allocation [ D i ( · )] is effjcient.

An Example of Delay Allocation Rule i 12 Example ( N = 3 ) β j j = 1 j = 2 c 2 ( λ 2 3 + 4 λ 2 λ 3 + λ 2 i = 1 c 1 ( λ 2 + λ 3 ) 2 ) c 2 ( λ 2 3 + 4 λ 1 λ 3 + λ 2 i = 2 c 1 ( λ 1 + λ 3 ) 1 ) c 2 ( λ 2 1 + 4 λ 2 λ 1 + λ 2 i = 3 c 1 ( λ 2 + λ 1 ) 2 ) External disutility R i (row sum) is independent of λ i Allocated disutility λ i D i = Λ C (Λ) − R i Total disutility ∑ i λ i D i = 3Λ C (Λ) − ∑ i R i = Λ C (Λ)

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 MRQ Scheduling Policy At time t , the MRQ policy schedules the client with the maximum Intuition Eventually all relative queue lengths are equal on average in steady length (delay). 14 How to enforce target delay D i ( λ i , λ − i ) for client i ? Let Q i ( t ) be the queue length of client i at time t , and g i := λ i D i . relative queue length, defjned as Q i ( t )/ g i . state, or equivalently, average queue length (delay) = target queue

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 Show the deviation of the limiting queue length vector from Lyapunov drift based technique 15 Heavy traffjc: Λ → µ the target queue length vector approaches 0

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

and d 18 P : projection to the feasible region same for all clients: Broadcast! are the d C k k k Centralized update: Centralized → Distributed λ ( k + 1) = λ ( k ) + κ ( k ) [∑ ] ˆ η ( k ) ∇ U i ( λ i ) − Λ C (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 ˆ λ ( k + 1) s.t. λ i > λ δ and Λ < (1 − ϵ ) µ λ ( k + 1) λ ( k )

and d 18 Distributed update: same for all clients: Broadcast! are the d C k k k P : projection to the feasible region Centralized → Distributed λ i ( k + 1) = λ i ( k ) + κ ( k ) [ i ( λ i ( k )) − d [Λ C (Λ)] ] ˆ U ′ , η ( k ) d 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 ˆ λ ( k + 1) s.t. λ i > λ δ and Λ < (1 − ϵ ) µ λ ( k + 1) λ ( k )

and d 18 Distributed update: same for all clients: Broadcast! are the d C k k k and i s.t. P : projection to the feasible region Centralized → Distributed λ i ( k + 1) = λ i ( k ) + κ ( k ) [ i ( λ i ( k )) − d [Λ C (Λ)] ] ˆ U ′ , η ( k ) d Λ λ i ( k + 1) , λ δ } , λ i ( k )(1 − ϵ ) µ λ i ( k + 1) = min { max { ˆ } Λ( k ) κ ( k ) : step size at the k -th iteration η ( k ) : Euclidean norm of the gradient λ ( k + 1) ˆ λ ( k + 1) λ ( k )

18 Distributed update: same for all clients: Broadcast! are the and i s.t. P : projection to the feasible region Centralized → Distributed λ i ( k + 1) = λ i ( k ) + κ ( k ) [ i ( λ i ( k )) − d [Λ C (Λ)] ] ˆ U ′ , η ( k ) d Λ λ i ( k + 1) , λ δ } , λ i ( k )(1 − ϵ ) µ λ i ( k + 1) = min { max { ˆ } Λ( k ) κ ( k ) : step size at the k -th iteration η ( k ) : Euclidean norm of the gradient λ ( k + 1) ˆ λ ( k + 1) Λ( k ) , κ ( k ) , η ( k ) , and d [Λ C (Λ)] λ ( k ) d Λ

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 19 N = 10 clients Poisson arrivals: Λ = 0 . 99 × 10 3 s − 1 Exponential/Deterministic service time: µ = 1 × 10 3 s − 1

Polynomial Approximation 20 200 M/M/1 Theory M/M/1 Approx M/D/1 Theory 150 M/D/1 Approx 100 50 0 0.95 0.96 0.97 0.98 0.99 1 Total disutility Λ C (Λ) v.s. Normalized total request rate Λ/ µ

State Space Collapse Normalized difgerence of relative queue lengths v.s. Time 21 0.3 Same rate, M/M/1 Relative difference of queue lengths Same rate, M/D/1 0.25 Diff rates, M/M/1 Diff rates, M/D/1 0.2 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 Time (s)

Nash Equilibrium M/D/1 system M/M/1 system 22 10 4 10 4 4.586 4.5908 4.585 4.5906 4.584 Total net utility Total net utility MRQ, Dist MRQ, Dist 4.583 MRQ, Cent 4.5904 MRQ, Cent FIFO FIFO 4.582 4.5902 4.581 4.58 4.59 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Index of iteration Index of iteration