Super-Fast Delay Tradeoffs for Utility Optimal Scheduling in
Wireless Networks Michael J. Neely University of Southern
California http://www-rcf.usc.edu/~mjneely/ *Sponsored by NSF OCE
Grant 0520324 Slide 2 A multi-node network with N nodes and L
links: t 0 1 2 3 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 Slide 3 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 ] Slide 4 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. Slide 5 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 Slide 6 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 = ) Slide 7 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 = ) Slide 8 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)
Slide 9 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) Slide 10 = 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) Slide 11 Network
Utility Optimization: Static Optimization: (Lagrange Multipliers
and convex duality) Kelly, Maulloo, Tan [J. Op. Res. 1998] Xiao,
Johansson, Boyd [Allerton 2001] Julian, Chiang, ONeill, 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) Slide 12 Network Utility
Optimization: Static Optimization: (Lagrange Multipliers and convex
duality) Kelly, Maulloo, Tan [J. Op. Res. 1998] Xiao, Johansson,
Boyd [Allerton 2001] Julian, Chiang, ONeill, 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) Slide 13 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! Slide 14 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! Slide 15 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!
Slide 16 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! Slide 17 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! Slide 18 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! Slide 19 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!
Slide 20 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! Slide 21 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] Slide 22 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)) Slide 23 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)) Slide 24
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))
Slide 25 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)) Slide 26 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)) Slide 27 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)) Slide 28 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) - Slide 29
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). Slide 30 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
Slide 31 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 Slide 32 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)
Slide 33 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)
Slide 34 Achieving Optimal Logarithmic Utility-Delay Tradeoffs:
Slide 35 Automatically satisfied if we stabilize the network. Slide
36 Achieving Optimal Logarithmic Utility-Delay Tradeoffs: Difficult
to achieve super-fast logarithmic delay tradeoffs working Directly
with this constraint. Slide 37 Achieving Optimal Logarithmic
Utility-Delay Tradeoffs: However: For any queueing system (stable
or not): U n (c) (t) (actual bits transmitted) Slide 38 Achieving
Optimal Logarithmic Utility-Delay Tradeoffs: However: For any
queueing system (stable or not): U n (c) (t) (actual bits
transmitted) Slide 39 Achieving Optimal Logarithmic Utility-Delay
Tradeoffs: U n (c) (t) Want to Solve: We Know: Also: IF EDGE
EFFECTS SMALL: Slide 40 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) Slide
41 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] Slide 42 (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) Slide 43 (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). Slide 44 (2)
Routing: Observe neighbors 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)). Slide 45 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 Slide 46 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) Slide 47 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 2 1 2 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)). Slide 48 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.
