A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
A Queueing Theory Investigation of Phase
Transitions in Complex Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Neal Mc Bride
[email protected]
Supervisor: Prof. John Bulava
Trinity College Dublin
September 13, 2013
Table of Contents
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
1 Introduction
Queuing Network
Complex Network
Phase Transitions
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
2 Algorithm Design
3 Scaling
Network Generation
Traffic Simulation
4 Results
Network of Queues
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Number of queues connected by customer routing.
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Customers are serviced at one node.
Choice: Proceed to another node or leave network.
Jackson Network
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Open network of J nodes.
Queue of customers at each node.
Customers may enter system, reroute to another queue
and leave.
Product-Form Equilibrium Distribution
π(x) = ΠJi=1 P(Yi = xi )
where Yi are independent random variables.
Scaling
Network
Generation
Traffic
Simulation
Results
∀ x ∈ ZJ+
Jackson Network: Limitations
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Poisson process arrival/departure from network.
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Single class of customer.
Infinite queue capacity.
Complex Network: Visual Aid
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Complex Network Connectivity Graph (10 Nodes)
Complex Network: Description
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Graph Structure: i nodes with ki edges
Power Law Degree Distribution
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
P(k) = k −γ
γ=3
Barabàsi-Albert Model: Growth and Preferential
Attachment
ki
Π(ki ) = P
j kj
Complex Network: Degree Distribution
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Power Law Degree Distribution
Complex Network: Graph Scaling
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Connectivity Graph (40 Nodes)
Phase Transitions: Definition
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Change in state of system resulting in rapid shift of order
parameter from around zero to some non-zero value.
Order Parameter: Job Density
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
ρ = lim
t→inf
N(t + τ ) − N(t)
τ Np
N = number of jobs, t = simulation time, τ = observation
time, N = number of nodes, p = job creation rate
First Order: Discontinuity in first derivative.
Second Order: Continuous in first derivative.
Algorithm Design: Queue Network
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Queue Array
Algorithm Design: Growth and Preferential
Attachment
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Figure: Roulette Wheel Neighbour Selection
Scaling
Network
Generation
Traffic
Simulation
Results
ki
Π(ki ) = P
j
kj
Algorithm Design: Remembering Neighbours
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Linked List
Algorithm Design: Routing Mechanism
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Leaky Bucket
Algorithm Design: Data Fidelity
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Buffer Transfer
Algorithm Design: Job Routing
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Serial Communication
Algorithm Design: Function Profiling
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Function Call Graph
Algorithm Design: Parallel Job Routing
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Parallel Communication
Scaling: Network Generation
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Weak Scaling Behaviour
Scaling: Traffic Simulation
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Weak Scaling Behaviour
Scaling: Traffic Simulation
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Strong Scaling Behaviour
Results: Equilibrium Distribution
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Stationary to Non-Stationary Eq. Dist.
Results: Density Plot
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Global Measure of Rate of Job Number Increase in System
Results: Phase Transition
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Free to Congested Phase Transition
Results: Low Traffic Density
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Global Measure of Rate of Job Number Increase for Low
Traffic Conditions
Results: Complex Network Core Visual Aid
A Queueing
Theory
Investigation
of Phase
Transitions in
Complex
Networks
Outline
Introduction
Queuing
Network
Complex
Network
Phase
Transitions
Algorithm
Design
Scaling
Network
Generation
Traffic
Simulation
Results
Figure: Complex Network: Core Connected Nodes