Resource Allocation in Multi

Transcript Resource Allocation in Multi

Resource Allocation in Wireless Networks: Dynamics and Complexity

R. Srikant Department of ECE and CSL

University of Illinois at Urbana-Champaign

Outline

• A simple three-node example: Internet versus wireless networks • Joint scheduling, routing and congestion control for general static multihop wireless networks (Eryilmaz, S.) • Connection-level models and stability (Lin, Shroff, S.) • Complexity of the MAC algorithm: simple distributed algorithms (Wu, S., Perkins)

Three-Node Internet (Kelly)

1 (

1 ) User 1 c a =1 c b =1 User 0

0 (

0 ) User 2

2 (

2 ) subject to

Solution

Functional Decomposition

Queue Lengths (Prices): Congestion Control:

1 (

1 ) User 1

Wireless Network

c A =1 c B =1 User 0

0 (

0 ) User 2

2 (

2 ) subject to

Lagrange Multipliers

Decomposition

Congestion control: MAC or Scheduling (MaxWeight):

Alternative Formulation

1 (

1 ) User 1 c A =1 c B =1 User 0

0 (

0 ) User 2

2 (

2 ) subject to

Decomposition

Congestion control (per-flow queues): MAC or Scheduling (Backpressure):

Differences in the Two Formulations

• Arrivals instantaneously arrive at all nodes in the route versus node-by-node queueing behavior • Sources react to sum of queue lengths queue length?

versus Sources react to entry queue length • Why is it sufficient to react to only the entry – Back-pressure algorithm

Outline

• A simple three-node example: Internet versus wireless networks • Joint scheduling, routing and congestion control for general static multihop wireless networks • Connection-level models and stability • Complexity of the MAC algorithm: simple distributed algorithms

I. Wireless Network Model

• The network is represented by a graph: i 

(i,n)

n 

(n,v)



(n,m)



(v,n)

v  m 

(m,j)

j 

(m,w)



(w,m) (m,v)

w •  = set of link rates that are allowable in a time slot, i.e., we have: 

[t] 2

,

8 t.

Slot 1 Slot 2 time

•

Traffic Model

: The set of flows that share the network.

• Each flow is described by a source-destination pair: No predefined routes.

i b(f)=i flow f e(f)=j j • Let

f x f

denote the rate of flow flow g n v m flow h w • Let  denote the set of flow rates for which the corresponding link rates lie in .

•

U f ( x f )

is a strictly concave function that measures the utility of flow

as a function of

x f

Problem Statement

• Design a mechanism that • guarantees stability of the queues, • allocates flow rates,

{ x f },

that satisfy: •

* denotes the optimizer of the above problem, call it the

fair allocation .

Related Work

• Joint Congestion Control, Routing and MAC • Lin and Shroff (’04, ’05) • Neely, Modiano and Li (’05) • Stolyar (’05) • Eryilmaz and S. (’05, ’06) • Scheduling and Routing • Tassiulas (backpressure policy)

Node Model

• Each node maintains a queue for each destination node.

(j) s (i,n) q n,j (j) s (n,m)

s (k) (i,n) q n,k (k) s (n,v)

•

Node n

In general, the evolution of a queue length is described by v

Primal-Dual Congestion Controller

• At the beginning of each time slot t, each flow, say f, has access to the queue length of its first node , denoted by

q b(f) [t].

• Congestion Control: {y} m M projects the value of y into [m,M] Increase rate when queue length is small Decrease rate when queue length is large

Back-pressure Scheduler [Tassiulas]

• Assign a weight to each edge; find a feasible set of edges with the maximum sum weight •The differential backlog of link is given by

(n,m)

for destination d •Differential backlog of the link is

W (n,m)

max

[t]

: the maximum value among all destinations •Then, choose the rate vector  [t] 2  that satisfies:

Node n

An example:

5 7 2

Node m

1 2 5

W (n,m) max d (n,m) = 2 = (max{5-1,7-2,2-5}) + =5 W (n,k) max d (n,k) =  = (max{5-6,7-8,2-4}) + =0

6 8 4

Node k

Queue Stability

• Define the Lyapunov function where q * 2 K  * .

Drift analysis results in

Theorem 1:

For some finite constant c, we have

Fair Allocation Theorem 2:

There exists a finite B, such that for all

• For large K , the average rate allocation is fair •Tradeoff between delays and fairness • Perhaps use virtual queues to control delays?

Stochastic Models

• The set of allowable rates at each time instant can be time-varying • Don’t need to know the statistics of the channel • Can model randomness in the arrival processes • The proof involves showing that the conditional mean drift of the Lyapunov function has the form shown in the previous page

Stochastic model



Fluid model

• Intuition: M/M/1 queue where the arrival rate decreases with the queue length.

K K K/2 K/(q-1) K/q 0 1 2 . . .

q . . .

     The steady-state mean and the variance of the above chain are both Θ(K).

Outline

Connection-Level Model

• Assume a fixed route for each source (can be generalized) • Files arrive according to a for route r Poisson process of rate  r • Each connection for route r is a file whose size is drawn independently from an exponential distribution with mean 1/  r • Large variance can be modelled by allowing file sizes to be mixtures of exponentials

Necessary Condition for Stability

• For each link l, the total load on the link should be less than its capacity:  r: l 2 r  r < c l where  r =  r /  r • c 1 , c 2 ,... should lie in the capacity region  • Is this also sufficient? Yes, for a large class of utility functions.

Models of Fairness

• Mo-Walrand ‘00:          !

, max-main fairness 2, TCP-Reno 1, proportional fairness 0, maximize total throughput We assume 1

Prior Work: Fair-Sharing Policies

• Assume TCP converges instantaneously, i.e., the optimization is solved instantaneously (RM ’98) • This is a time-scale separation assumption • dLK ’99, BM ’00: The system is stable if load on each link is less than its capacity for fair resource allocation policies

A Fluid Model

• n r : number of files using route r • Does n r go to zero?

• Depending on the decomposition price model different (maxweight or backpressure)

Stability

• Lyapunov function

• (n,p) forms a Markov chain

• A drift argument for the Markov chain implies positive recurrence < 0

Take-Away Message

• The Lyapunov function is a linear combination of the Bonald-Massoulie Lyapunov function and a quadratic Lyapunov function of the queue lengths • The congestion control may not have time to converge (the fixed-user Lyapunov function plays no role) • Lin-Shroff ’05 has an example of a network where the fixed-user congestion controller does not converge but the connection-level system is stable

Outline

Network model

• Network is represented by a graph: • Link if nodes are within communication range of each other •Time is slotted.

• Each time slot is long enough to transmit a single packet.

Spectrum-Sharing Model

• Graph matching

Simple Collision Model

• No collision at receivers

802.11 Model

• Data/ack-based model i j • All nodes in a two-hop have to be silent.

Interference Set

• Associated with each link is an interference set •The • • interference set E Link l belongs to E Symmetry: l l associated with link l satisfies the following properties: If link k belongs to E l , then link l belongs to E k • Symmetry does not imply link k and link l have the same interference sets • If link l is scheduled then no other link in other link in E l is scheduled E l can be scheduled. Link l cannot be scheduled if any

Schedule and capacity region

• Scheduling policy: a rule to determine the set of links which can be ON during a time slot such that the interference constraints is satisfied. • Think of a schedule as a vector of 0’s and 1’s: 0 for a link that is not scheduled, 1 otherwise.

• : set of all feasible schedules.

• Assume that link l has an arrival rate  l • The capacity region: • Achieving 100% throughput requires the use of MaxWeight policy: too complex to implement

Greedy scheduling

• Each node attempts to independently schedule transmission over one of its backlogged links.

A X B C D X E X F • Any backlogged link will be scheduled if no other link in its inhibited edge set is scheduled.

• A maximal number of non-empty links in the network will be scheduled. • Similar to maximal matching in graph theory.

Related work

• MM scheduling can achieve at least half of the capacity region in switches: Weller and Hajek ‘97, Dai and Prabhakar ’00 • The bound of ½ is tight in the sense that for some traffic load pattern, MM scheduling does take twice as long as the optimal scheduling.

• Wireless Networks: Lin and Shroff ’05 for a network where the only constraint is that a node cannot transmit and receive simultaneously • Chaporkar, Kar and Sarkar ’06: Results similar to this talk for the single link case and a different approach for the multi-hop case. Rate stability established previously in ’05.

Stability condition for single-hop routes

• Property of greedy schedule: – if q l ≥ 1, either link l is scheduled and another link in E l is scheduled • Stability condition: Total load on each interference set should be less than 1.

Throughput Loss

• View each interference set as follows Arrival rate Number of links that can be scheduled • Arrival rate cannot be larger than service rate for • Given topology constraint (max number of schedulable links in an interference set), can construct networks and set of arrival sets such that the bound is tight • Example: Loss in throughput=1/2 under maximal

Controlling the arrival rates

• Source algorithm: U’(  s )=p s • Path price: p s =  l q El • Link algorithm: Compute q El • Asynchronous version of this algorithm converges (joint work with Bui, Eryilmaz & Wu, ‘06)

Open Problems

• MAC: Tradeoff between protocol overhead, • Distributed algorithm for more complicated interference models (e.g., the power used at all the nodes determines the probability of success) • Implications of the assumption that arrivals occur instantaneously at all nodes (controlled version of the Kumar-Seidman, Rybko-Stolyar examples) • Connection-level performance without the time-scale separation assumption • Connection-level stability of  -fair policies at the connection level for  <1 • Connection-level stability for primal-dual algorithms