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Super-Fast Delay Tradeoffs for Utility Optimal Scheduling in Wireless Networks. l. e. e. e. e. Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely/. *Sponsored by NSF OCE Grant 0520324. A multi-node network with N nodes and L links:. l. e. e. e. - PowerPoint PPT Presentation
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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
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) = 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 = 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) Lc
Lc = 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) Lc
Lc = 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) Lc
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)
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!
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!
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!
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!
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!
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!
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!
any rate vector!
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!
any rate vector!
Our Previous Work (Neely, Modiano, Li Infocom 2005):
r
gn(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
. D
elay 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
. D
elay 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
. D
elay 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
. D
elay 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
. D
elay 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
. 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) -
Overloaded and Fully Active Assumptions:
*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).
Overloaded and Fully Active Assumptions:
*Assumption 2 (Fully Active): All queues Un(c)(t) that
can be positive are also active sources of commodity cdata.
*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 Un(c)(t) that
can be positive are also active sources of commodity cdata.
*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 Un(c)(t) that
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)
Overloaded and Fully Active Assumptions:
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:
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 workingDirectly with this constraint.
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
However: For any queueing system (stable or not):
Un(c)(t) (actual bits
transmitted)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
However: For any queueing system (stable or not):
Un(c)(t) (actual bits
transmitted)
Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
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.