1 / 23

Delay-Based Network Utility Maximization

Delay-Based Network Utility Maximization. a 1 (t). S 1 (t). S 2 ( t). a 2 (t). Utility. a K ( t ). S K ( t ). Thruput x k. Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely IEEE INFOCOM, San Diego, March 2010.

kele
Download Presentation

Delay-Based Network Utility Maximization

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Delay-Based Network Utility Maximization a1(t) S1(t) S2(t) a2(t) Utility aK(t) SK(t) Thruputxk Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely IEEE INFOCOM, San Diego, March 2010 *Sponsored in part by the DARPA IT-MANET Program, NSF Career CCF-0747525

  2. ak(t) Ψk(x(t), S(t)) • Network Model: • 1-Hop Network with K Queues--- (Q1(t), …, QK(t)) • Slotted time, t in {0, 1, 2, … } • S(t) = (S1(t), …, SK(t)) = “Channel State Vector” (i.i.d. over slots) • x(t) =(x1(t), …,xK(t)) = “Transmission Decision Vector” • a(t) = (a1(t), …, aK(t)) = “Packet Arrival Vector” (i.i.d. over slots) • Fixed Length Packets: a(t), x(t) are 0/1 vectors. • Arrival Rates: E{a(t)} = (λ1, …, λK). • Reliability is a Function of Transmission Decision: • Observe S(t) every slot. Choose 0/1 transmission vector x(t). • Pr[Success on link k | x(t), S(t)] = Ψk(x(t), S(t))

  3. ak(t) Ψk(x(t), S(t)) Delay = 1 Delay = 2 Delay = 4 Queue k: Hk(t) = 4 Dropsk(t) • Packet Dropping: • We can decide to drop packets at any time. • Packets that fail in transmission can either be retransmitted • or dropped. • Thruput on channel k = yk = λk – drop rate on channel k • Delay-Based Control: • Stamp Head-of-Line (HOL) Packets with their Delays Hk(t). • Utility Maximization Objective: • Maximize: g1( y1 ) + g2( y2 ) + … + gK( yK) where gk(y) are concave, non-decreasing utility functions

  4. Prior work on Stochastic Network Optimization: • Stability: (“Max-Weight” = “Min-Lyapunov-Drift”) • Tassiulas, Ephremides[1992, 1993] (Minimize drift Δ(t)) • Kahale, Wright [1997] • Andrews et. al. [2001] • Neely, Modiano, Rohrs [2003, 2005] • Kobayashi, Caire, Gesbert [2005] • Joint Stability and Utility Optimization: • Neely, Modiano [2003, 2005] (Minimize Δ(t) + V*Penalty(t)) • Georgiadis, Neely, Tassiulas [2006] • Stolyar [2005] (Primal-Dual “Fluid Model” analysis) • Special case of “Infinitely Backlogged Sources”: • Agrawal, Subramanian [2002], Kushner, Whiting [2002] • Eryilmaz, Srikant [2005], Lin, Shroff [2004] All of these references use queue backlog as weights!

  5. Alternative “Delay-Based” Rules that use HOL Delays • as weights are known only for Network Stability: • Mekkittikul, McKeown [1996] • Shakkottai, Stolyar [2002] • Andrews, Kumaran, Ramanan,Stolyar, Vijaykumar, Whiting [2004] Our work fills the gap by developing a Delay-Based Rule for Maximizing Network Utility Subject to Stability. Utility Ψk(x(t), S(t)) ak(t) Thruputxk Delay = 1 Delay = 2 Delay = 4 Dropsk(t)

  6. Challenges: • Prior “Drift-Plus-Penalty” Algorithm Admit/Drops Packets • Immediately when New Packets Arrive: Δ(t) + V*Penalty(t). • This does not directly Affect the HOL Values! • Tricky Correlation Issues between HOL sizes and Decisions! • Key Ideas for this paper: • Queue all packets. Admit/Drop only at HOL. • Use “Drift-Plus-Penalty” with a different queue structure • Use a “concavely extended utility function” • Advantages of Delay-Based Approach: • Provides “Delay-Fairness.” Queue-Based Rules can • leave loner packets stranded. • Provides Worst Case Delay Guarantees. • Disadvantage: Must know (λ1, …, λK) to implement.

  7. Concavely Extending a Utility Function: gk(yk) fk(yk) slope = ηk slope = ηk yk yk 0 0 -1 1 1 Function gk(yk): Defined over 0 ≤ yk ≤ 1 Function fk(yk): Defined over -1 ≤ yk ≤ 1

  8. Auxiliary Variables and Thruput Variables: yk(t) = λk – Dropsk(t) Time Avg: yk= λk – Dk θk(t), choose s.t. -1 ≤ θk(t) ≤ 1 Time Avg: θk Transformed Stochastic Net Optimization Problem: Maximize: f1( θ1) + f2( θ2) + … + fK( θK) Subject to: (1) E{Qk} < infinity , for all k in {1, …, K} (2) yk ≥ θk , for all k in {1, …, K} (3) -1 ≤ θk(t) ≤ 1 , for all t (4) yk, θk are time averages achievable on the network

  9. Auxiliary Variables and Thruput Variables: yk(t) = λk – Dropsk(t) Time Avg: yk= λk – Dk θk(t), choose s.t. -1 ≤ θk(t) ≤ 1 Time Avg: θk Transformed Stochastic Net Optimization Problem: Maximize: f1( θ1) + f2( θ2) + … + fK( θK) Subject to: (1) E{Qk} < infinity , for all k in {1, …, K} (2) yk ≥ θk , for all k in {1, …, K} (3) -1 ≤ θk(t) ≤ 1 , for all t (4) yk, θk are time averages achievable on the network

  10. Virtual Queue Zk(t) for enforcing constraint (2): Zk(t) θk(t) yk(t) = λk – Dropsk(t) Transformed Stochastic Net Optimization Problem: Maximize: f1( θ1) + f2( θ2) + … + fK( θK) Subject to: (1) E{Qk} < infinity , for all k in {1, …, K} (2) yk ≥ θk , for all k in {1, …, K} (3) -1 ≤ θk(t) ≤ 1 , for all t (4) yk, θk are time averages achievable on the network

  11. Head-of-Line Delay Update Equation (Queue k): Hk(t+1) = 1{k full}(t) max[Hk(t) + 1 – (μk(t)+Dk(t))Tk(t), 0] + 1{k empty}(t) Ak(t) Inter-Arrival Time Tk(t) Service Drops Lyapunov Function: L(t) = (1/2)[H1(t)2 + … + HK(t)2] + (1/2)[Z1(t)2 + … + ZK(t)2] • Drift-Plus-Penalty Approach: • Observe {H1(t), …, HK(t)}, {Z1(t),…,ZK(t)}, {S1(t), …, SK(t)} • Take action to minimize: • Δ(t) – V [f1(θ1(t)) + ... + fK(θK(t))]

  12. Δ(t) – V [f1(θ1(t)) + ... + fL(θL(t))] Resulting Algorithm: (Aux Variables) Each k observes Zk(t). Choose θk(t)to: Maximize : V f(θk(t)) – Zk(t)θk(t) Subject to: -1 ≤ θk(t) ≤ 1 2) (Transmission) Observe S(t), H(t), Z(t). Choose x(t) to: Maximize : xk(t) min[Hk(t), Zk(t)] Ψk(x(t),S(t)) Subject to: (x1(t),…, xK(t)) a 0/1 vector 3) (Packet Dropping) For each queue k with HOL packet that was not successfully transmitted, drop iffZk(t)≤Hk(t). 4) (Update Virtual and Actual Queues) Σk

  13. Focus on Auxiliary Variable Update: Choose θk(t)to: Maximize : V f(θk(t)) – Zk(t)θk(t) Subject to: -1 ≤ θk(t) ≤ 1 Zk(t) < Vηk Zk(t) θk(t) yk(t) = λk – Dropsk(t) -1 1 Key Lemma: If Zk(t) > Vηk, then… …the above chooses θk(t) = -1 …hence, Zk(t) cannot increase on that slot: = -1 ≥ -1 [This is why we concavely extended the utility function over -1 ≤ y ≤ 1 ]

  14. Focus on Auxiliary Variable Update: Choose θk(t)to: Maximize : V f(θk(t)) – Zk(t)θk(t) Subject to: -1 ≤ θk(t) ≤ 1 Zk(t) > Vηk Zk(t) θk(t) yk(t) = λk – Dropsk(t) -1 1 Key Lemma: If Zk(t) > Vηk, then… …the above chooses θk(t) = -1 …hence, Zk(t) cannot increase on that slot: = -1 ≥ -1 [This is why we concavely extended the utility function over -1 ≤ y ≤ 1 ]

  15. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  16. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  17. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  18. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  19. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  20. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  21. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  22. Concluding Theorem: Implementing the above algorithm for any parameter V>0, we have… Delay: Worst Case Delay in Queue k ≤ Vηk + 2 slots. Total Utility Satisfies: g1( y1 ) + … + gK( yK ) ≥ g1( y1*) + … + gK( yK*) – B/V Achieved Utility Optimal Utility Utility Delay V V

  23. Example Utility Function gk( y ) = log( 1 + ηky ) This approximates “proportionally fair” log-utility when ηk is large. The log-utility log( y ) has a singularity at y=0, and so is not always a good choice of utility function. Utility Delay V V

More Related