super-fast delay tradeoffs for utility optimal scheduling in wireless networks

Super-Fast Delay Tradeoffs for Utility Optimal Scheduling in Wireless Networks

Michael J. NeelyUniversity of Southern California

http://www-rcf.usc.edu/~mjneely/*Sponsored by NSF OCE Grant 0520324

http://www-rcf.usc.edu/~mjneely/

A multi-node network with N nodes and L links:

t0 1 2 3 …

Slotted time t = 0, 1, 2, …

Traffic (An(c)(t)) and channel states S(t) i.i.d. over timeslots.

Control for Optimal Utility-Delay Tradeoffs…

1) Flow Control:

Ai(c)

An(c)(t) = New Commodity c data during slot t (i.i.d)

Ri(c)(t)

An(c)(t)] = n

(c) , (n(c)

) = Arrival Rate Matrix

Rn(c)(t) = Flow Control Decision at (i,c):

Rn(c)(t) < min[Ln

(c)(t) + An(c)(t) , Rmax]

2) Resource Allocation:

Channel State Matrix: S(t) = (Sab(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) Lc

Lc = Set of all linksacceptable forcommodity c trafficto traverse

Examples…

3) Routing:



ab(c)(t) < ab(t) c


Lc = All network links

Example 1:

(commodity c = )

3) Routing:



ab(c)(t) < ab(t) c


Lc = a directed subset

Example 2:

(commodity c = )

3) Routing:



ab(c)(t) < ab(t) c


Lc = Specifies a one-hop network

Example 3: downlink uplink

(no routing decisions)

3) Routing:



ab(c)(t) < ab(t) c


Lc = Specifies a one-hop network

Example 4:one-hop ad-hoc network

(no routing decisions)

= Capacity region (considering all control algs.)

r

gn(c)(r)

Utility functions

rn(c) = Time average of Rn

(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)

Our Previous Work (Neely, Modiano, Li Infocom 2005):

r

gn(c)(r)

Utility functions

Cross-Layer Control Algorithm (with control parameter V>0):

Achieves: [O(1/V), O(V)] utility-delay tradeoff!


r

gn(c)(r)

Utility functions

Cross-Layer Control Algorithm (with control parameter V>0):


any rate vector!


r

gn(c)(r)

Utility functions


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

. D

elay O(log(V))

Overloaded and Fully Active Assumptions:

Assumption 1 (Overloaded): Optimal operating point r* hasall positive entries, and the input rate matrix is outside ofthe capacity region and strictly dominates r*. That is, thereexists an >0 such that:

< rn(c) < n

(c) -


*Assumption 2 (Fully Active): All queues Un(c)(t) that

can be positive are also active sources of commodity cdata.

*Used implicitly in proofs of conference version (Infocom 2006) but not stated explicitly. Described in more detial in JSAC 2006 (on web).




*Natural assumption for overloaded one-hop networks.(Network is defined by all active links)

downlink uplink




*Natural assumption for overloaded one-hop networks.(Network is defined by all active links)

one-hop ad-hoc network



can be positive are also active sources of commodity cdata. Holds for a large class of multi-hop networks.

one-hop ad-hoc network

Example: 1 or more commodities, all nodes are independentsources of each of these commodities (as in “all-to-all” traffic)


1 2

Fully Active assumption can be restrictive in general multi-hop networks with stochastic channels:Logarithmic Utility-Delay Tradeoffs Unknown:

1 2

Logarithmic Utility-Delay Tradeoffs Achievable:

(t)(t)

(t) (t)

Achieving Optimal Logarithmic Utility-Delay Tradeoffs:


Automatically satisfied if we stabilize the network.


Difficult to achieve “super-fast”logarithmic delay tradeoffs workingDirectly with this constraint.


However: For any queueing system (stable or not):

Un(c)(t) (actual bits

transmitted)


Un(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:

Zn(c)(t)

Define the aggregate “bi-modal” Lyapunov Function:

Un(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 Un(c)(t).

Rest of Network

Un(c)(t)

Rn(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 Un(c)(t) > Q then Rn

(c)(t) = 0 (reject all new data)If Un

(c)(t) < Q then Rn(c)(t) = An

(c)(t) (admit all new data)

(1) Flow Control (b): At node n, observe virtual queue Zn(c)(t).

Rest of Network

Un(c)(t)

Rn(c)(t)n

(c)

Utility-Delay Optimal Algorithm (UDOA):(stated here in special case of zero transport layer storage)

Then Update the Virtual Queues Zn(c)(t).

(2) Routing: Observe neighbor’s queue length Un(c)(t), compute:

link (n,b) cnb*(t) = Node n

Define Wnb*(t) = maxmizing weight over all c (where (n,c) Lc) Define cnb*(t) as the arg maximizer.

(This is the best commodity to send over link (n,b)if Wnb*(t) >0. Else send nothing over link (n,b)).

Note: Routing Algorithm is related to the Tassiulas-EphremidesDifferential 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 overloadedAnd 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-hopnetworks 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)D

elay

(slo

ts)

Utility Optimal Throughput point V parameter

V parameter

Thruput 1

Thru

put 2 Bound

Simulationinput rate

Pr[ON] = p1

Pr[ON] = p2

1

2

Two Queue Downlink Simulation:

Observation: The coefficient Q can be reduced by a factor of 30without 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)

Del

ay (s

lots

)

Utility Optimal Throughput point V parameter

V parameter

Thruput 1

Bound

Simulationinput 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.

super-fast delay tradeoffs for utility optimal scheduling in wireless networks

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