Universal Scheduling for Networks with
Arbitrary Traffic, Channels, and Mobility
B
Primary Path
Alternate Paths
Michael J. Neely, University of Southern California
Proc. 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
Want to optimally react to unexpected events.
Example 1: Failure at Node B
A
B
C
A
B
C
D
Primary Path
D
Alternate Paths
Example 2: Opportunity via Mobility
mobile node
Primary Path
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B
C
D
Example 2: Opportunity via Mobility
Primary Path
mobile node
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Example 2: Opportunity via Mobility
Primary Path
mobile node
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Example 2: Opportunity via Mobility
Primary Path
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mobile node
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C
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Example 2: Opportunity via Mobility
Primary Path
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mobile node
B
C
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Example 2: Opportunity via Mobility
Primary Path
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B
C
D
mobile node
Example 2: Opportunity via Mobility
Primary Path
A
B
C
D
mobile node
Example 2: Opportunity via Mobility
Primary Path
A
B
C
D
mobile node
Example 2: Opportunity via Mobility
Primary Path
A
B
C
D
mobile node
Example 2: Opportunity via Mobility
Primary Path
A
B
C
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mobile node
Example 2: Opportunity via Mobility
Primary Path
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B
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mobile node
Example 2: Opportunity via Mobility
Primary Path
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mobile node
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 Lookahead
algorithm, 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).
4
Nodes: N = 8
5
1
8
2
7
6
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).
4
Nodes: N = 8
Flows: M = 3
5
1
8
2
7
6
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).
4
1
Nodes: N = 8
Flows: M = 3
• Flow 1: 13
5
1
8
2
7
6
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).
4
1
1
8
2
2
Nodes: N = 8
Flows: M = 3
• Flow 1: 13
• Flow 2: 73
5
7
6
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).
4
1
5
1
7
Nodes: N = 8
Flows: M = 3
• Flow 1: 13
• Flow 2: 73
• Flow 3: 56
8
2
2
3
6
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).
4
1
5
1
7
Nodes: N = 8
Flows: M = 3
• Flow 1: 13
• Flow 2: 73
• Flow 3: 56
8
2
2
3
6
3
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
State Information and Control Decisions:
•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))
Am(t)
Node i
State Information and Control Decisions:
•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))
Am(t)
Xm(t)
Node i
Dropm(t)
State Information and Control Decisions:
•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))
Am(t)
Xm(t)
Sij(t)
Node i
Dropm(t)
Node j
Sik(t)
Node k
State Information and Control Decisions:
•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))
Am(t)
Xm(t)
Dropm(t)
Sij(t)
Sik(t)
Utility Maximization with T-Slot Lookahead:
•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!
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:
Achieved Utility over RT slots
R-1 opt
≥ (1/R)∑r=0
φ [r] – “Fudge Factor”
BT + CV
(i) “Fudge Factor” =
V
RT
(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)
O(1/V), O(V) utility-backlog tradeoff when time horizon R infinity
Example Mobile Network:
D1
S1
Five Mobility Groups:
S2
•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
Example Mobile Network:
D1
S1
Five Mobility Groups:
S2
•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
Example Mobile Network:
D1
S1
Five Mobility Groups:
S2
•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
Example Mobile Network:
D1
S1
Five Mobility Groups:
S2
•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
Example Mobile Network:
D1
S1
Five Mobility Groups:
S2
•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
Example Mobile Network: Sim. 1– Change Social Contacts
D1
S1
Backlog Bound for D1 in a sample RED node
15
10
5
Series1
0
0
50
100
150
200
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-5
Backlog Bound for S1 in a sample RED node
15
10
S2
5
Series1
0
0
50
100
150
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250
-5
Social Contacts:
•Source 1: S1D1 (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 stability
Use V=10, so guarantee no more that 11 source 2 packets
in any queue!
Example Mobile Network: Sim. 1– Change Social Contacts
D1
S1
Moving Average thruput:S2D1
0.5
0.4
0.3
0.2
Series1
0.1
0
-0.1 0
0.5
50
100
150
200
250
Moving Average thruput:S2S1
0.4
0.3
S2
0.2
Series1
0.1
0
-0.1 0
20000
40000
60000
80000
100000 120000
Social Contacts:
•Source 1: S1D1 (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 stability
Use V=10, so guarantee no more that 11 source 2 packets
in any queue!
Example Mobile Network: Sim. 1– Change Social Contacts
D1
S1
Moving Average thruput:S2D1
0.5
0.4
0.3
0.2
Series1
0.1
0
-0.1 0
0.5
50
100
150
200
250
Moving Average thruput:S2S1
0.4
0.3
S2
0.2
Series1
0.1
0
-0.1 0
20000
40000
60000
80000
100000 120000
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!
Example Mobile Network: Sim. 2– Intermittent Jamming
D1
S1
JAM!
JAM!
Time
S2
Social Contacts:
•Source 1: S1D1 (constant rate = 0.07 packets/slot)
•Source 2: S2  S1 (Goal to maximize its throughput)
Intermittent Interference during 2 intervals of the simulation
That 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.
Example Mobile Network: Sim. 2– Intermittent Jamming
D1
S1
JAM!
JAM!
Time
S2
Social Contacts:
•Source 1: S1D1 (constant rate = 0.07 packets/slot)
•Source 2: S2  S1 (Goal to maximize its throughput)
Intermittent Interference during 2 intervals of the simulation
That 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.
Conclusion Slide:
D1
S1
20
Backlog Bound for D1 in a sample RED node
10
Series1
0
-10 0
20
50
100
150
200
250
Backlog Bound for S1 in a sample RED node
10
Series1
0
-10
S2
0.6
0
50
100
150
200
250
Moving Average Thruput of Source 2
0.4
0.2
Series1
0
-0.2
0
50
100
150
200
250
•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)