Universal Scheduling for Networks with Arbitrary Traffic, Channels, and Mobility

Michael J. Neely, University of Southern CaliforniaProc. IEEE Conf. on Decision and Control (CDC), Atlanta, GA, Dec. 2010

PDF of paper at: http://www-bcf.usc.edu/~mjneely/Sponsored in part by the NSF Career CCF-0747525, ARL Network Science Collaborative Tech. Alliance

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Want to optimally react to unexpected events.Example 1: Failure at Node B

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Example 2: Opportunity via Mobility

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Assumptions and Main Questions:

Assumptions:•Arbitrary mobility, traffic, channels.•Little or no probability models known in advance.•Any sample path is possible (non-ergodic).•Future is unknown.

Questions:•Can we design “universal” scheduling algorithms that work on general time-varying networks? •Can we optimize without knowing the future?

Main Results:

•We use a backpressure/max-weight algorithm that does not know future.

•Define a “T-Slot Lookahead” Utility as that obtained by an “ideal” algorithm that has perfect knowledge of the future up to T slots.

•For any T, our algorithm can achieve utility that is arbitrarily close to the utility of the ideal T-slot Lookaheadalgorithm, with tradeoff in convergence time and queue backlog.

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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Nodes: N = 8

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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Nodes: N = 8Flows: M = 3

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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1 Nodes: N = 8Flows: M = 3• Flow 1: 13

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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Nodes: N = 8Flows: M = 3• Flow 1: 13• Flow 2: 73

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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3 Nodes: N = 8Flows: M = 3• Flow 1: 13• Flow 2: 73• Flow 3: 56

Problem Formulation:•Timeslotted system, slots t = {0, 1, 2, …}.•N network nodes (possibly mobile).•M Data Flows (each with source-destination).•No pre-specified routes (we learn them).

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4 5

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3 Nodes: N = 8Flows: M = 3• Flow 1: 13• Flow 2: 73• Flow 3: 56

Network Queueing:

a b a

•Each node keeps queues for each separate commodity (“commodity” = “destination”).•For commodity c (say, green commodity):

Qa(c)(t+1) = Qa

(c)(t) – Transmit out + Endogenous Arrivals + Exogenous Arrivals

•A(t) = (A1(t), …, AM(t)) = New Arrivals. •X(t) = (X1(t), …, XM(t)) = Flow Control Decisions.•S(t) = “Topology State” observed on slot t.•(μij

(c)(t)) = Transmission Decisions (in set Γ(S(t))

State Information and Control Decisions:

Node iAm(t)

•A(t) = (A1(t), …, AM(t)) = New Arrivals. •X(t) = (X1(t), …, XM(t)) = Flow Control Decisions.•S(t) = “Topology State” observed on slot t.•(μij

(c)(t)) = Transmission Decisions (in set Γ(S(t))

State Information and Control Decisions:

Node iAm(t)Xm(t)

Dropm(t)

•A(t) = (A1(t), …, AM(t)) = New Arrivals. •X(t) = (X1(t), …, XM(t)) = Flow Control Decisions.•S(t) = “Topology State” observed on slot t.•(μij

(c)(t)) = Transmission Decisions (in set Γ(S(t))

State Information and Control Decisions:

Node iAm(t)Xm(t)

Dropm(t)

Node j

Node k

Sij(t)

Sik(t)

•A(t) = (A1(t), …, AM(t)) = New Arrivals. •X(t) = (X1(t), …, XM(t)) = Flow Control Decisions.•S(t) = “Topology State” observed on slot t.•(μij

(c)(t)) = Transmission Decisions (in set Γ(S(t))

State Information and Control Decisions:

Am(t)Xm(t)

Dropm(t)

Sij(t)

Sik(t)

•φm(x) = concave utility function for flow m•Segment timeline into T-slot frames.•φopt[r] = optimal sum utility over frame r, assuming future is known in frame!

Utility Maximization with T-Slot Lookahead:

Frame 0 Frame 1 Frame 2

•Value of φopt[r] can be written as a non-linear program (assuming future A(t), S(t) known)…

Utility Maximization with T-Slot Lookahead:

Frame r•Value of φopt[r] can be written as a non-linear program (assuming future A(t), S(t) known):

Ω(t) = set of rates possible under S(t)

Analytical Approach:•Lyapunov Function for queues: L(Q) = ∑ [Qi

(c)]2

•New sample path “T-slot” Lyapunov Drift:

ΔT(t) = L(Q(t+T)) – L(Q(t))

•Every slot “greedily” minimize drift-plus-penalty:

Δ1(t) + V x Penalty(t) , Penalty(t) = -φ(γ(t)) •Results in a joint backpressure and flow control alg similar to those defined for ergodic systems in: [Neely, Modiano, Li -- INFOCOM 2005] [Georgiadis, Neely, Tassiulas -- F&T 2006]

Performance Result:Theorem: For any R>0, T>0:

(ii) Worst Case Queue Backlog = O(V).

•B, C are known constants. •V = “knob” to turn to affect the tradeoff•R = Running Time (number of T-slot frames)

V RT

(i) “Fudge Factor” = BT + CV

O(1/V), O(V) utility-backlog tradeoff when time horizon R infinity

Achieved Utility over RT slots ≥ (1/R)∑r=0 φopt[r] – “Fudge Factor”R-1

Example Mobile Network:

Five Mobility Groups:

•10 nodes Group 1 (upper left) •10 nodes Group 2 (upper right)•10 nodes Group 3 (lower right)•10 nodes Group 4 (lower left)•1 node Group 5

Group 1 nodes: Random Walk on Upper Left Region

S1

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Example Mobile Network:

Five Mobility Groups:

•10 nodes Group 1 (upper left) •10 nodes Group 2 (upper right)•10 nodes Group 3 (lower right)•10 nodes Group 4 (lower left)•1 node Group 5

Group 2 nodes: Random Walk on Upper Right Region

S1

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Example Mobile Network:

Five Mobility Groups:

•10 nodes Group 1 (upper left) •10 nodes Group 2 (upper right)•10 nodes Group 3 (lower right)•10 nodes Group 4 (lower left)•1 node Group 5

Group 3 nodes: Random Walk on Lower Right Region

S1

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Example Mobile Network:

Five Mobility Groups:

•10 nodes Group 1 (upper left) •10 nodes Group 2 (upper right)•10 nodes Group 3 (lower right)•10 nodes Group 4 (lower left)•1 node Group 5

Group 4 nodes: Random Walk on Lower Left Region

S1

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Example Mobile Network:

S1

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Five Mobility Groups:

•10 nodes Group 1 (upper left) •10 nodes Group 2 (upper right)•10 nodes Group 3 (lower right)•10 nodes Group 4 (lower left)•1 node Group 5

Group 5 node: Periodically cycles about the clockwise orbit

Social Contacts:•Source 1: S1D1 (constant rate = 0.07 packets/slot) •Source 2: S2 S1 (for first half of simulation) S2 D1 (for second half of simulation)Goal: Maximize Throughput of Source 2 subject to stabilityUse V=10, so guarantee no more that 11 source 2 packetsin any queue!

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Backlog Bound for D1 in a sample RED node

Backlog Bound for S1 in a sample RED node

Example Mobile Network: Sim. 1– Change Social Contacts

Social Contacts:•Source 1: S1D1 (constant rate = 0.07 packets/slot) •Source 2: S2 S1 (for first half of simulation) S2 D1 (for second half of simulation)Goal: Maximize Throughput of Source 2 subject to stabilityUse V=10, so guarantee no more that 11 source 2 packetsin any queue!

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Example Mobile Network: Sim. 1– Change Social Contacts

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Moving Average thruput:S2D1

Moving Average thruput:S2S1

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Example Mobile Network: Sim. 1– Change Social Contacts

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Moving Average thruput:S2D1

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Overall Performance is Seamless: •Backlog no more than 11 packets in any queue for Source 1 data•Backlog no more than 15 packets in any queue for Source 2 data•Overall Thruput of Source 2 is maintained at near-optimal over the change, even though the routes must fundamentally change!

S1

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Example Mobile Network: Sim. 2– Intermittent Jamming

Social Contacts:•Source 1: S1D1 (constant rate = 0.07 packets/slot) •Source 2: S2 S1 (Goal to maximize its throughput)Intermittent Interference during 2 intervals of the simulationThat completely cut interaction between the groups 1-4.Can only use the cyclic mobile node at these times!Max Thruput of Source 2 during interference ~= 0.03.

Time

JAM! JAM!

S1

S2

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Example Mobile Network: Sim. 2– Intermittent Jamming

Social Contacts:•Source 1: S1D1 (constant rate = 0.07 packets/slot) •Source 2: S2 S1 (Goal to maximize its throughput)Intermittent Interference during 2 intervals of the simulationThat completely cut interaction between the groups 1-4.Can only use the cyclic mobile node at these times!Max Thruput of Source 2 during interference ~= 0.03.

Time

JAM! JAM!

S1

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Conclusion Slide:

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Backlog Bound for D1 in a sample RED node

Backlog Bound for S1 in a sample RED node

Moving Average Thruput of Source 2

•Overall Seamless Operation•Throughput During Jamming goes down, but is close to optimal value of 0.03. •Fudge Factor = BT/V + CV/RT•Worst Case Queue Backlog = O(V)•Framework useful for stock market trading! (Thursday @ 10:20am)