Dynamics of Resource Sharing in Networks Frank Kelly www.statslab.cam.ac.uk/~frank MITACS International Focus Period Network and Internet Economics Workshop, Vancouver, 1 June 2011
Outline • Fairness in networks • Rate control in communication networks (relatively well understood) • Ramp metering (early models) • Energy networks (preliminary remarks)
Network structure - set of resources J - set of routes R = - if resource j is on route r 1 A jr = - otherwise A 0 jr route resource
Notation - set of resources J - set of users, or routes R j ∈ r - resource j is on route r x r - flow rate on route r U r ( x r ) - utility to user r - capacity of resource j C j Ax ≤ C - capacity constraints route resource
The system problem ∑ Maximize U ( x ) SYSTEM( U,A,C ): r r ∈ r R ≤ subject to Ax C ≥ over x 0 Maximize aggregate utility, subject to capacity constraints
The user problem ⎛ ⎞ w ⎜ ⎟ − USER r ( U r ; λ r ): r Maximize U w ⎜ ⎟ λ r r ⎝ ⎠ r ≥ over w 0 r User r chooses an amount to pay per unit time, w r , and receives in return a flow x r =w r / λ r
The network problem ∑ Maximize w log x NETWORK( A,C;w ): r r ∈ r R ≤ subject to Ax C ≥ over x 0 As if the network maximizes a logarithmic utility function, but with constants { w r } chosen by the users
Problem decomposition Theorem: the system problem may be solved by solving simultaneously the network problem and the user problems K 1997, Johari, Tsitsiklis 2005, Yang, Hajek 2006
Max-min fairness Rates { x r } are max-min fair if they are feasible: ≥ , ≤ x 0 Ax C and if, for any other feasible rates { y r }, ∃ > ⇒ ∃ < < r : y x s : y x x r r s s r Rawls 1971, Bertsekas, Gallager 1987
Proportional fairness Rates { x r } are proportionally fair if they are feasible: ≥ , ≤ x 0 Ax C and if, for any other feasible rates { y r }, the aggregate of proportional changes is negative: y r − x r ∑ ≤ 0 x r r ∈ R
Weighted proportional fairness A feasible set of rates { x r } are such that are weighted proportionally fair if, for any other feasible rates { y r }, y r − x r ∑ ≤ w r 0 x r r ∈ R
Fairness and the network problem Theorem: a set of rates { x r } solves the network problem, NETWORK( A,C;w ), if and only if the rates are weighted proportionally fair
Bargaining problem (Nash, 1950) Solution to NETWORK( A,C;w ) with w = 1 is unique point satisfying • Pareto efficiency • Symmetry • Independence of Irrelevant Alternatives (General w corresponds to a model with unequal bargaining power)
Market clearing equilibrium (Gale, 1960) Find prices p and an allocation x such that ≥ ≤ p 0 , Ax C (feasibility) (complementary − = T p ( C Ax ) 0 slackness) ∑ = ∈ (endowments spent) w x p , r R r r j ∈ j r Solution solves NETWORK( A,C;w )
Outline • Fairness in networks • Rate control in communication networks (relatively well understood) • Ramp metering (early models) • Energy networks (preliminary remarks)
End-to-end congestion control receivers senders Senders learn (through feedback from receivers) of congestion at queue, and slow down or speed up accordingly. With current TCP, throughput of a flow is proportional to 1 /( T p ) T = round-trip time, p = packet drop probability. (Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd & Fall 1999)
Network structure route resource - set of resources J - set of routes R ∈ - resource j is on route r j r - flow rate on route r at time t x ( t ) r μ - rate of congestion indication, ( t ) j at resource j at time t
A primal algorithm d ( ) ∑ = κ − μ x ( t ) ( x ( t )) w x ( t ) ( t ) r r r r r j dt ∈ j r ( ) ∑ μ = ( t ) p x ( t ) j j s ∈ s : j s x r (t) - rate changes by linear increase, multiplicative decrease p j (.) - proportion of packets marked as a function of flow through resource
Global stability Theorem: the above dynamical system has a stable point to which all trajectories converge. The stable point is proportionally fair with respect to the weights { w r }, and solves the network problem, when = 0 ≤ ( ) p x x C j j = ∞ > x C j K, Maulloo, Tan 1998
General TCP-like algorithm Source maintains window of sent, but not yet acknowledged, packets - size cwnd cwnd ≈ xT On route r , • cwnd incremented by a r cwnd n on positive acknowledgement cwnd decremented by b r cwnd m • for each congestion indication (m>n) • a r = 1 , b r = 1/2 , m=1, n= -1 corresponds to Jacobson’s TCP
Differential equations with delays − ( ) d x t T = r r x ( t ) r dt T r ( ) − λ − λ n m . a ( x ( t ) T ) ( 1 ( t )) b ( x ( t ) T ) ( t ) r r r r r r r r ( ) ∏ λ = − − μ − ( t ) 1 1 ( t T ) r j jr ∈ j r j ⎛ ⎞ ∑ ⎜ ⎟ μ = − ( t ) p x ( t T ) ⎜ ⎟ j j r rj ⎝ ⎠ ∈ r : j r r + = T T T rj jr r
Equilibrium point − 1 / m n ⎛ ⎞ − λ 1 a 1 = ⎜ ⎟ ∈ r r x r R ⎜ ⎟ λ r ⎝ ⎠ T b r r r • a r = 1 , b r = 1/2 , m=1, n= -1 corresponds to Jacobson’s TCP, and recovers square root formula • But what is the impact of delays on stability? Can we choose m, n,… arbitrarily?
Johari, Tan 1999, Delay stability Massoulié 2000, Vinnicombe 2000, Paganini, Doyle, Low 2001 Equilibrium is locally stable if there exists a global constant β such that π ′ < < β n x p ( x ) p ( x ), a ( x T ) β j j r r r 2 condition on condition on sensitivity for aggressiveness each resource j for each route r
Outline • Fairness in networks • Rate control in communication networks (relatively well understood) • Ramp metering (early models) • Energy networks (preliminary remarks)
What we've learned about highway congestion P. Varaiya , Access 27, Fall 2005, 2-9.
Data, modelling and inference in road traffic networks R.J. Gibbens and Y. Saatci Phil. Trans. R. Soc. A366 (2008), 1907-1919.
A linear network ∫ t = + − Λ ≥ m ( t ) m ( 0 ) e ( t ) ( m ( s )) d s , t 0 i i i i 0 queue cumulative metering size inflow rate
Metering policy Suppose the metering rates can be chosen to be Λ = Λ any vector satisfying ( m ) ∑ Λ ≤ ∈ , A C j J ji i j i Λ ≥ ∈ 0 , i I i Λ = = 0 , m 0 i i and such that ∫ t = + − Λ ≥ ≥ m ( t ) m ( 0 ) e ( t ) ( m ( s )) d s 0 , t 0 i i i i 0
Optimal policy? For each of i = I, I-1, …… 1 in turn choose ∫ t Λ ≥ ( m ( s )) d s 0 i 0 to be maximal, subject to the constraints. This policy minimizes ∑ m ( t ) i i for all times t
Proportionally fair metering Λ = Λ ∈ ( m ) ( ( m ), i I ) Suppose is chosen to i ∑ Λ maximize log m i i i ∑ Λ ≤ ∈ A C , j J subject to ji i j i Λ ≥ ∈ 0 , i I i Λ = = 0 , m 0 i i
Proportionally fair metering m Λ = ∈ i ( m ) , i I ∑ i p A j ji j Λ ≥ ∈ where 0 , i I i ∑ Λ ≤ ∈ A C , j J ji i j i ≥ ∈ KKT p 0 , j J j conditions ⎛ ⎞ ∑ − Λ ≥ ∈ ⎜ ⎟ p C A 0 , j J j j ji i ⎝ ⎠ i p - shadow price (Lagrange multiplier) for the j resource j capacity constraint
Outline • Fairness in networks • Rate control in communication networks (relatively well understood) • Ramp metering (early models) • Energy networks (preliminary remarks)
Dynamic demand Use system frequency as a signal to control domestic loads, particularly refrigerators and freezers, to provide operating reserve From: Frequency responsive loads, Jeremy Colandairaj, NIE
Distribution of frequency From: www.dynamicDemand.co.uk (Dynamic Demand is a not ‐ for ‐ profit organisation set up by a grant from the Esmée Fairbairn Foundation)
Hybrid reserve service Operating the Electricity Transmission Networks in 2020, Follow Up Report, National Grid, February 2010
Typical wind turbine power curve Operating the Electricity Transmission Networks in 2020 Initial Consultation, National Grid 2009
Recorded wind load factors 2008 Operating the Electricity Transmission Networks in 2020 Initial Consultation, National Grid 2009
Persistence errors in forecasting wind Operating the Electricity Transmission Networks in 2020 Initial Consultation, National Grid 2009
Matching vehicle charging to the current electricity demand profile Operating the Electricity Transmission Networks in 2020 Initial Consultation, National Grid 2009
British Electricity Transmission System The Transmission System broadly comprises all circuits operating at 400kV and 275kV. In Scotland transmission also includes 132kV networks. The Transmission System is connected via interconnectors to transmission systems in France and Northern Ireland.
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