Super-Fast Delay Tradeoffs for Utility Optimal Scheduling in Wireless Networks Michael J. Neely University of Southern California *Sponsored by NSF OCE Grant    

A multi-node network with N nodes and L links: t … Slotted time t = 0, 1, 2, … Traffic (A n (c) (t)) and channel states S(t) i.i.d. over timeslots.     Control for Optimal Utility-Delay Tradeoffs…

1) Flow Control:      A i (c) A n (c) (t) = New Commodity c data during slot t (i.i.d)  R i (c) (t)  A n (c) (t)] =  n (c) , ( n (c) ) = Arrival Rate Matrix R n (c) (t) = Flow Control Decision at (i,c): R n (c) (t) < min[L n (c) (t) + A n (c) (t), R max ]

2) Resource Allocation: Channel State Matrix: S(t) = (S ab (t)) Transmission Rate Matrix:  (t) = (  ab (t)) Resource allocation: choose  (t)  S(t)      S (t) = Set of Feasible Rate Matrices for Channel State S.

3) Routing:  ab (c) (t) = Amount of commodity c data transmitted over link (a,b)  ab (c) (t) <  ab (t) c  ab (c) (t) = 0 if (a,b) L c L c = Set of all links acceptable for commodity c traffic to traverse Examples…

3) Routing:  ab (c) (t) = Amount of commodity c data transmitted over link (a,b)  ab (c) (t) <  ab (t) c  ab (c) (t) = 0 if (a,b) L c L c = All network links Example 1: (commodity c = )

3) Routing:  ab (c) (t) = Amount of commodity c data transmitted over link (a,b)  ab (c) (t) <  ab (t) c  ab (c) (t) = 0 if (a,b) L c L c = a directed subset Example 2: (commodity c = )

3) Routing:  ab (c) (t) = Amount of commodity c data transmitted over link (a,b)  ab (c) (t) <  ab (t) c  ab (c) (t) = 0 if (a,b) L c L c = Specifies a one-hop network Example 3: downlink uplink (no routing decisions)

3) Routing:  ab (c) (t) = Amount of commodity c data transmitted over link (a,b)  ab (c) (t) <  ab (t) c  ab (c) (t) = 0 if (a,b) L c L c = Specifies a one-hop network Example 4: one-hop ad-hoc network (no routing decisions)

 = Capacity region (considering all control algs.)     r g n (c) (r) Utility functions r n (c) = Time average of R n (c) (t) admission decisions. GOAL: (Joint flow control, resource allocation, and routing)

Network Utility Optimization: Static Optimization: (Lagrange Multipliers and convex duality) Kelly, Maulloo, Tan [J. Op. Res. 1998] Xiao, Johansson, Boyd [Allerton 2001] Julian, Chiang, O’Neill, Boyd [Infocom 2002] P. Marbach [Infocom 2002] Steven Low [TON 2003] B. Krishnamachari, Ordonez [VTC 2003] M. Chiang [Infocom 2004] Stochastic Optimization: Lee, Mazumdar, Shroff [2005] (stochastic gradient) Eryilmaz, Srikant [Infocom 2005] (fluid transformations) Stolyar [Queueing Systems 2005] (fluid limits) Neely, Modiano [2003, 2005] (Lyapunov optimization)

Network Utility Optimization: Static Optimization: (Lagrange Multipliers and convex duality) Kelly, Maulloo, Tan [J. Op. Res. 1998] Xiao, Johansson, Boyd [Allerton 2001] Julian, Chiang, O’Neill, Boyd [Infocom 2002] P. Marbach [Infocom 2002] Steven Low [TON 2003] B. Krishnamachari, Ordonez [VTC 2003] M. Chiang [Infocom 2004] Stochastic Optimization: Lee, Mazumdar, Shroff [2005] (stochastic gradient) Eryilmaz, Srikant [Infocom 2005] (fluid transformations) Stolyar [Queueing Systems 2005] (fluid limits) Neely, Modiano [2003, 2005] (Lyapunov optimization)

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff! any rate vector!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Cross-Layer Control Algorithm (with control parameter V>0): Achieves: [O(1/V), O(V)] utility-delay tradeoff! any rate vector!

Our Previous Work (Neely, Modiano, Li Infocom 2005): r g n (c) (r) Utility functions Achieves: [O(1/V), O(V)] utility-delay tradeoff! any rate vector! Uses theory of Lyapunov Optimization [Neely, Modiano 2003, 2005] Generalizes classical Lyapunov Stability results of: -Tassiulas, Ephremides [Trans. Aut. Control 1992] -Kumar, Meyn [Trans. Aut. Control 1995] -McKeown, Anantharam, Walrand [Infocom 1996] -Leonardi et. al., [Infocom 2001]

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Question: Is [O(1/V), O(V)] the optimal utility-delay tradeoff? Results: For a large class of overloaded networks, we can do much better by achieving O(log(V)) average delay.   V Avg. Delay O(log(V))

Overloaded and Fully Active Assumptions:     Assumption 1 (Overloaded): Optimal operating point r* has all positive entries, and the input rate matrix is outside of the capacity region and strictly dominates r*. That is, there exists an  >0 such that:  < r n (c) < n (c) - 

Overloaded and Fully Active Assumptions:     *Assumption 2 (Fully Active): All queues U n (c) (t) that can be positive are also active sources of commodity c data. *Used implicitly in proofs of conference version (Infocom 2006) but not stated explicitly. Described in more detial in JSAC 2006 (on web).

Overloaded and Fully Active Assumptions:     *Assumption 2 (Fully Active): All queues U n (c) (t) that can be positive are also active sources of commodity c data. *Natural assumption for overloaded one-hop networks. (Network is defined by all active links) downlink uplink

Overloaded and Fully Active Assumptions:     *Assumption 2 (Fully Active): All queues U n (c) (t) that can be positive are also active sources of commodity c data. *Natural assumption for overloaded one-hop networks. (Network is defined by all active links) one-hop ad-hoc network

Overloaded and Fully Active Assumptions:     *Assumption 2 (Fully Active): All queues U n (c) (t) that can be positive are also active sources of commodity c data. Holds for a large class of multi-hop networks. one-hop ad-hoc network Example: 1 or more commodities, all nodes are independent sources of each of these commodities (as in “all-to-all” traffic)

Overloaded and Fully Active Assumptions: 12  Fully Active assumption can be restrictive in general multi-hop networks with stochastic channels: Logarithmic Utility-Delay Tradeoffs Unknown: 12  Logarithmic Utility-Delay Tradeoffs Achievable:   (t)   (t)   (t) 

Achieving Optimal Logarithmic Utility-Delay Tradeoffs:

Automatically satisfied if we stabilize the network.

Achieving Optimal Logarithmic Utility-Delay Tradeoffs: Difficult to achieve “super-fast” logarithmic delay tradeoffs working Directly with this constraint.

Achieving Optimal Logarithmic Utility-Delay Tradeoffs: However: For any queueing system (stable or not): U n (c) (t) (actual bits transmitted)

Achieving Optimal Logarithmic Utility-Delay Tradeoffs: However: For any queueing system (stable or not): U n (c) (t) (actual bits transmitted)

Achieving Optimal Logarithmic Utility-Delay Tradeoffs: U n (c) (t) Want to Solve: We Know: Also: IF EDGE EFFECTS SMALL:

Introduce a virtual queue [Neely Infocom 2005]: Achieving Optimal Logarithmic Utility-Delay Tradeoffs: Want to Solve: We Know: Also: IF EDGE EFFECTS SMALL: Z n (c) (t)

Define the aggregate “bi-modal” Lyapunov Function: U n (c) Q Designing “gravity” into the system: The Tradeoff Optimal Control Algorithm: Minimize: [Buffer partitioning Concept similar to Berry-Gallager 2002]

(1)Flow Control (a): At node n, observe queue backlog U n (c) (t). Rest of Network U n (c) (t) R n (c) (t) n (c) (where V is a parameter that affects network delay) Utility-Delay Optimal Algorithm (UDOA): (stated here in special case of zero transport layer storage) If U n (c) (t) > Q then R n (c) (t) = 0 (reject all new data) If U n (c) (t) < Q then R n (c) (t) = A n (c) (t) (admit all new data)

(1)Flow Control (b): At node n, observe virtual queue Z n (c) (t). Rest of Network U n (c) (t) R n (c) (t) n (c) Utility-Delay Optimal Algorithm (UDOA): (stated here in special case of zero transport layer storage) Then Update the Virtual Queues Z n (c) (t).

(2) Routing: Observe neighbor’s queue length U n (c) (t), compute: link (n,b) c nb *(t) =Node n Define W nb *(t) = maxmizing weight over all c (where (n,c) L c ) Define c nb *(t) as the arg maximizer. (This is the best commodity to send over link (n,b) if W nb *(t) >0. Else send nothing over link (n,b)).

Note: Routing Algorithm is related to the Tassiulas-Ephremides Differential backlog policy [1992], but uses weights that switch Aggressively and discontinuously ON and OFF to yield optimal delay tradeoffs. (3) Resource Allocation: Observe Channel State S(t). Choose  ab (c) (t) such that

Theorem (UDOA Performance): If the overloaded And fully active assumptions are satisfied, then with Suitable choices of parameters Q,  (as functions of V), we have for any V>0: Theorem (Optimality of logarithmic delay): For one-hop networks with zero transport layer storage space (all admission/rejection decisions made upon packet arrival), then any average congestion tradeoff is necessarily logarithmic in V. (details in paper)

“Super-Fast” Flow Control. (Input Traffic exceeds network capacity). V (Log scale x-axis) Delay (slots) Utility Optimal Throughput point V parameter Thruput 1 Thruput 2 Bound Simulation input rate Pr[ON] = p 1 Pr[ON] = p Two Queue Downlink Simulation: Observation: The coefficient Q can be reduced by a factor of 30 without Effecting edge probability, leading to further (constant factor) reductions in average delay with no affect on utility. Shown below is Reduction by 30 (original Q would have delay multiplied by 30)).

“Super-Fast” Flow Control. (Input Traffic exceeds network capacity). V (Log scale x-axis) Delay (slots) Utility Optimal Throughput point V parameter Thruput 1 Bound Simulation input rate Conclusions: 1)“Super-Fast” Logarithmic Delay Tradeoff Achievable via Dynamic Scheduling and Flow Control. 2) Logarithmic Delay is Optimal for one-hop Networks. Fundamental Utility-Delay Tradeoff: [O(1/V), O(log(V))] 3)Novel Lyapunov Optimization Technique for Achieving Optimal Delay Tradeoffs.