Utility Optimization for Dynamic Peer-to-Peer Networks with Tit-for-Tat Constraints Michael J. Neely, Leana Golubchik University of Southern California Proc. IEEE INFOCOM, Shanghai, China, April 2011 PDF of paper at: Sponsored in part by the NSF Career CCF , ARL Network Science Collaborative Tech. Alliance Network Cloud

N nodes. Each node n has download social group G n. G n is a subset of {1, …, N}. Each file f is in some subset of nodes N f. Each node n can request download of a file f from any node in G n N f

Slotted time t in {0, 1, 2, …}. S(t) = “topology state” on slot t. µ ab (t) = transmission rate from a to b on slot t.  S(t) = set of matrices (µ ab (t)) allowed under S(t). Transmissions are supported by the network cloud. Transmission Decision: Every slot t, observe S(t). Then choose (µ ab (t)) in  S(t). “One-Hop” Network Transmission Model µ 12 (t) µ 34 (t)

“Internet Cloud” Example 1: Network Cloud Uplink capacity C 1 uplink S(t) = Constant (no variation). ∑ b µ nb (t) ≤ C n uplink for all nodes n. This example assumes uplink capacity is the bottleneck.

“Internet Cloud” Example 2: Network Cloud S(t) specifies a single supportable (µ ab (t)). No “transmission rate decisions.” The allowable rates (µ ab (t)) are given to the peer-to-peer system from some underlying transport and routing protocol.

“Wireless Basestation” Example 3: = base station = wireless device Wireless device-to-device transmission increases capacity. (µ ab (t)) chosen in  S(t). Transmissions coordinated by base station.

Network File Request Model Each node desires at most 1 new file per slot. Assume G n N n (t) is non-empty. Assume 0 ≤ A n (t) ≤ A max. Files larger than A max packets can be treated as separate files that come in successive slots. n (A n (t), N n (t)) A n (t) = size of desired file on slot t. N n (t) = subset of other nodes that have it. Get help from nodes in: G n N n (t)

“Commodities” for Request Allocation Each file corresponds to a subset of nodes. Queueing files according to subsets would result in O(2 N ) queues. (complexity explosion!). Instead of that, without loss of optimality, we use the following alternative commodity structure…

“Commodities” for Request Allocation Use subset info to determine the decision set. n (A n (t), N n (t)) j k m G n N n (t)

“Commodities” for Request Allocation Use subset info to determine the decision set. Choose which node will help download. n (A n (t), N n (t)) j k m G n N n (t)

“Commodities” for Request Allocation Use subset info to determine the decision set. Choose which node will help download. That node queues the request: Q mn (t+1) = max[Q mn (t) + R mn (t) - µ mn (t), 0] Subset info can now be thrown away. n (A n (t), N n (t)) j k m Q mn (t)

Stochastic Network Optimization Problem: Maximize: ∑ n g n ( ∑ a r an ) Subject to: (1)Q mn < infinity (Queue Stability Constraint) (2)α ∑ a r an ≤ β + ∑ b r nb for all n (Tit-for-Tat Constraint)

Maximize: ∑ n g n ( ∑ a r an ) Subject to: (1)Q mn < infinity (Queue Stability Constraint) (2)α ∑ a r an ≤ β + ∑ b r nb for all n (Tit-for-Tat Constraint) concave utility function Stochastic Network Optimization Problem:

Maximize: ∑ n g n ( ∑ a r an ) Subject to: (1)Q mn < infinity (Queue Stability Constraint) (2)α ∑ a r an ≤ β + ∑ b r nb for all n (Tit-for-Tat Constraint) concave utility function time average request rate Stochastic Network Optimization Problem:

Maximize: ∑ n g n ( ∑ a r an ) Subject to: (1)Q mn < infinity (Queue Stability Constraint) (2)α ∑ a r an ≤ β + ∑ b r nb for all n (Tit-for-Tat Constraint) concave utility function time average request rate α x Download rate Stochastic Network Optimization Problem:

Maximize: ∑ n g n ( ∑ a r an ) Subject to: (1)Q mn < infinity (Queue Stability Constraint) (2)α ∑ a r an ≤ β + ∑ b r nb for all n (Tit-for-Tat Constraint) concave utility function time average request rate α x Download rate β + Upload rate Stochastic Network Optimization Problem:

Solution Technique Use “Drift-Plus-Penalty” Framework for Stochastic Network Optimization [Georgiadis, Neely, Tassiulas, F&T 2006] [Neely, Morgan & Claypool 2010] No Statistical Assumptions on [S(t); (A n (t), N n (t))] Quick Advertisement: New Book: M. J. Neely, Stochastic Network Optimization with Application to Communication and Queueing Systems. Morgan & Claypool, T007 PDF also available from “Synthesis Lecture Series” (on digital library) Lyapunov Optimization theory (including universal scheduling, renewals) Detailed Examples and Problem Set Questions.

Use “Drift-Plus-Penalty” Framework: Virtual queue for each TFT constraint: α ∑ a r an ≤ β + ∑ b r nb Virtual queue H n (t) for each concave utility function. L(t) = ∑ Q mn (t) 2 + ∑F n (t) 2 + ∑H n (t) 2. Δ(t) = L(t+1) – L(t). Drift-Plus-Penalty Algorithm: Every slot t, choose action to greedily minimize: F n (t) α ∑ a R an (t)β + ∑ b R nb (t) Δ(t) – V x Utility(t)

Resulting Algorithm: (Auxiliary Variables) For each n, choose an aux. variable γ n (t) in interval [0, A max ] to maximize: Vg n (γ n (t)) – H n (t)g n (t) (Request Allocation) For each n, observe the following value for all m in { G n N n (t)}: -Q mn (t) + H n (t) + (F m (t) – αF n (t)) Give A n (t) to queue m with largest non-neg value, Drop A n (t) if all above values are negative. (Scheduling) Choose (µ ab (t)) in  S(t) to maximize: ∑ nb µ nb (t)Q nb (t)

How the Incentives Work for node n: F n (t) α x Receive Help(t)β + Help Others(t) -Q mn (t) + H n (t) + (F m (t) – αF n (t)) Node n can only request downloads from others if it finds a node m with a non-negative value of: F n (t) = “Node n Reputation” (Good reputation = Low value)

How the Incentives Work for node n: F n (t) α x Receive Help(t)β + Help Others(t) -Q mn (t) + H n (t) + (F m (t) – αF n (t)) Node n can only request downloads from others if it finds a node m with a non-negative value of: F n (t) = “Node n Reputation” (Good reputation = Low value) BoundedCompare Reputations!

How the Incentives Work for node n: F n (t) α x Receive Help(t)β + Help Others(t) -Q mn (t) + H n (t) + (F m (t) – αF n (t)) Node n can only request downloads from others if it finds a node m with a non-negative value of: F n (t) = “Node n Reputation” (Good reputation = Low value) BoundedCompare Reputations!

Concluding Theorem: For any arbitrary [S(t); (A n (t), N n (t))] sample path, we guarantee: a)Q mn (t) ≤ Q max = O(V) for all t, all (m,n). b)All Tit-for-Tat constraints are satisfied. c)For any T>0: liminf K  inf [Achieved Utility(KT)] ≥ liminf K  inf (1/K)∑ i=1 [“T-Slot-Lookahead-Utility[i]”]- BT/V Frame 1 Frame 2Frame 3 0T2T3T K