Oblivious AQM and Nash Equilibria Dutta, Goal and Heidmann In Proceedings of the IEEE Infocom, pages 106-113, San Francisco, California, USA, March 2003. IEEE. Presented by ZHOU Zhen cszz COMP 670O 1-1
Oblivious AQM and Nash Equilibria Today’s Internet • There are indications that the amount of non-congestion-reactive traffic is on the rise. Most of this misbehaving traffic does not use TCP. e.g. Real-time multi-media, netork games. • The unresponsive behavior can result in both unfairness and con- gestion collapse for the Internet. • The network itself must now participate in controlling its own resource utilization March 2006 2-1
Oblivious AQM and Nash Equilibria Active Queue Management A congestion control protocol (e.g. TCP) operates at the end-points and uses the drops or marks received from the Active Queue Manage- ment policies (e.g. Drop-tail, RED) at routers as feedback signals to adaptively modify the sending rate in order to maximize its own good- put. • Oblivious (stateless) AQM: a router strategy that does not differ- entiate between packets belonging to different flows. Easier to implement • Stateful schemes: e.g. Fair Queuing Gateways maintain separate queues for packets from each indi- vidual source. The queues are serviced in a round-robin manner. March 2006 3-1
Oblivious AQM and Nash Equilibria Oblivious AQM Scheme – Drop Tail Buffers as many packets as it can and drops the ones it can’t buffer • Distributes buffer space unfairly among traffic flows. • Can lead to global synchronization as all TCP connections ”hold back” simultaneously, hence networks become under-utilized. March 2006 4-1
Oblivious AQM and Nash Equilibria Oblivious AQM Scheme – Random Early Detection Monitors the average queue size and drops packets based on statistical probabilities • If the buffer is almost empty, all incoming packets are accepted; As the queue grows, the probability for dropping an incoming packet grows; When the buffer is full, the probability has reached 1 and all incoming packets are dropped. • Considered more fair than tail drop – The more a host transmits, the more likely it is that its packets are dropped. • Prevents global synchronization and achieves lower average buffer occupancies. March 2006 5-1
Oblivious AQM and Nash Equilibria Oblivious AQM and Nash Equilibria The paper studies the existence and quality of Nash equilibria imposed by oblivious AQM schemes on selfish agents: • Motivation • Markovian Internet Game Model • Existence • Efficiency • Achievability • Summary March 2006 6-1
Oblivious AQM and Nash Equilibria Game Setting • Players: n selfish end-point traffic agents. Model player i ’s traffic arrival by Poison process ( λ i ). • Strategy: increase or decrease the average sending rate λ i . • Utility: U i = goodput µ i = successful rate . total rate • Rules: oblivious AQM policy with dropping probability p . Model the system as M/M/1/K queue. Poisson arrivals/Exponentially distributed service/one server/finite capacity buffer March 2006 7-1
Oblivious AQM and Nash Equilibria Symmetric Nash Equilibrium Condition • No selfish agent has any incentive to unilaterally deviate from its current state. ∂U i ∀ i, ∂λ i = 0 March 2006 8-1
Oblivious AQM and Nash Equilibria Symmetric Nash Equilibrium Condition • No selfish agent has any incentive to unilaterally deviate from its current state. ∂U i ∀ i, ∂λ i = 0 • Every agent has the same goodput at equilibrium. µ i = µ j and λ i = λ j = λ ∀ i, j n March 2006 8-2
Oblivious AQM and Nash Equilibria Symmetric Nash Equilibrium Condition • No selfish agent has any incentive to unilaterally deviate from its current state. ∂U i ∀ i, ∂λ i = 0 • Every agent has the same goodput at equilibrium. µ i = µ j and λ i = λ j = λ ∀ i, j n • Hence functions of router state (drop probability, queue length) are independent in i . ∂ d ∀ i, ∂λ i = dλ March 2006 8-3
Oblivious AQM and Nash Equilibria Symmetric Nash Equilibrium Condition • No selfish agent has any incentive to unilaterally deviate from its current state. ∂U i ∀ i, ∂λ i = 0 • Every agent has the same goodput at equilibrium. µ i = µ j and λ i = λ j = λ ∀ i, j n • Hence functions of router state (drop probability, queue length) are independent in i . ∂ d ∀ i, ∂λ i = dλ • Utility fucntion for each player at N.E. U i = µ i = λ i (1 − p ) . March 2006 8-4
Oblivious AQM and Nash Equilibria Symmetric Nash Equilibrium Condition • No selfish agent has any incentive to unilaterally deviate from its current state. ∂U i ∀ i, ∂λ i = 0 • Every agent has the same goodput at equilibrium. µ i = µ j and λ i = λ j = λ ∀ i, j n • Hence functions of router state (drop probability, queue length) are independent in i . ∂ d ∀ i, ∂λ i = dλ • Utility fucntion for each player at N.E. U i = µ i = λ i (1 − p ) . dp 1 − p = ndλ Nash condition : λ March 2006 8-5
Oblivious AQM and Nash Equilibria Efficient Nash Equilibrium Condition • Denote the aggregate throughput ˜ λ n , goodput ˜ µ n , and drop prob- ability ˜ p n at N.E.. • Efficient if the goodput of any selfish agent is bounded below when the throughput of the same agent is bounded above . µ n = ˜ 1. ˜ λ n (1 − ˜ p n ) ≥ c 1 2. ˜ λ n ≤ c 2 where c 1 , c 2 are some constants. • Therefore, ˜ p n is also bounded. March 2006 9-1
Oblivious AQM and Nash Equilibria Outline • Motivation • Markovian Internet Game Model • Existence Are there oblivious AQM schemes that impose Nash equilibria on selfish users? • Efficiency • Achievability • Summary March 2006 10-1
Oblivious AQM and Nash Equilibria Drop-Tail Queuing • Drop probability (from queuing theory) p = λ K (1 − λ ) 1 − λ K +1 Theorem 1 : There is NO Nash Equilibrium for selfish agents and routes implementing Drop-Tail queuing. March 2006 11-1
Oblivious AQM and Nash Equilibria Drop-Tail Queuing • Drop probability (from queuing theory) p = λ K (1 − λ ) 1 − λ K +1 Theorem 1 : There is NO Nash Equilibrium for selfish agents and routes implementing Drop-Tail queuing. Proof : µ i = λ i (1 − p ) = ( λ i λ ) λ (1 − p ) = ( λ i λ ) µ ∂µ i λ ) dµ ∂λ i = µ ∂ ∂λ i ( λ i λ ) + ( λ i dλ > 0 March 2006 11-2
Oblivious AQM and Nash Equilibria Drop-Tail Queuing • Drop probability (from queuing theory) p = λ K (1 − λ ) 1 − λ K +1 Theorem 1 : There is NO Nash Equilibrium for selfish agents and routes implementing Drop-Tail queuing. Proof : µ i = λ i (1 − p ) = ( λ i λ ) λ (1 − p ) = ( λ i λ ) µ ∂µ i λ ) dµ ∂λ i = µ ∂ ∂λ i ( λ i λ ) + ( λ i dλ > 0 µ = λ (1 − λ K ) ∂λ i ( λ i ∂ λ ) = λ − λ i 1 1 − λ K +1 = 1 − λ 2 1+ λ + λ 2 + ... + λ K March 2006 11-3
Oblivious AQM and Nash Equilibria RED • Drop probability (approximate steady state model [Dutta et al ]) 0 if l q < min th p max ( l q − min th ) × if min th ≤ l q ≤ max th p = max th − min th 1 otherwise March 2006 12-1
Oblivious AQM and Nash Equilibria RED • Drop probability (approximate steady state model [Dutta et al ]) 0 if l q < min th p max ( l q − min th ) × if min th ≤ l q ≤ max th p = max th − min th 1 otherwise • Queue length at steady state (from queuing theory) λ (1 − p ) l q = 1 − λ (1 − p ) ≤ max th March 2006 12-2
Oblivious AQM and Nash Equilibria RED Theorem 2 : RED Does NOT impose a Nash equilibrium on uncon- trolled selfish agents. Proof : l q 1+ l q )( 1 1 − p = ( λ ) } l q l q ∂µ i λ ) ∂µ ∂λ i ( λ i ∂ λ ) + ( λ i ∂λ i = ∂λ ( 1+ l q ) > 0 1+ l q µ i = λ i (1 − p ) March 2006 13-1
Oblivious AQM and Nash Equilibria RED Theorem 2 : RED Does NOT impose a Nash equilibrium on uncon- trolled selfish agents. Proof : l q 1+ l q )( 1 1 − p = ( λ ) } l q l q ∂µ i λ ) ∂µ ∂λ i ( λ i ∂ λ ) + ( λ i ∂λ i = ∂λ ( 1+ l q ) > 0 1+ l q µ i = λ i (1 − p ) • RED punishes all flows with the same drop probability. • Misbehaving flows can push more traffic and get less hurt (marginally). • There is no incentive for any source to stop pushing packets. March 2006 13-2
Oblivious AQM and Nash Equilibria Virtual Load RED • Drop probability 0 if l vq < min th l vq − min th p = if min th < l vq < max th max th − min th 1 otherwise λ where l vq = 1 − λ is the M/M/1 queue length. March 2006 14-1
Oblivious AQM and Nash Equilibria Virtual Load RED • Drop probability 0 if l vq < min th l vq − min th p = if min th < l vq < max th max th − min th 1 otherwise λ where l vq = 1 − λ is the M/M/1 queue length. Theorem 3 : VLRED imposes a Nash Equilibrium on selfish agents if min th ≤ √ 1 + max th − 1 . Proof : l vq + l 2 λ dp dλ = vq max th − min th By Nash condition , l 2 vq + ( n + 1) l vq − n max th = 0 . √ The positive root is inde- ( n +1) 2 +4 n max th ˜ − n +1 l vq = pendent of min th . 2 2 l vq ≥ min th , we have min th ≤ √ 1 + max th − 1 . Given that ˜ March 2006 14-2
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