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) User 0 User 1 User 2 c a =1 c b =1 subject to

Solution

Functional Decomposition Queue Lengths (Prices): Congestion Control:

Wireless Network User 0 User 1 User 2 c A =1 c B =1 subject to

Lagrange Multipliers

Decomposition Congestion control: MAC or Scheduling (MaxWeight):

Alternative Formulation User 0 User 1 User 2 c A =1 c B =1 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 versus Sources react to entry queue length Why is it sufficient to react to only the entry queue length? –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 nm v w j  (i,n)  (n,m)  (n,v)  (v,n)  (m,v)  (m,j)  (m,w)  (w,m) time Slot 1Slot 2  = set of link rates that are allowable in a time slot, i.e., we have:  [t] 2  8 t.

Traffic Model : The set of flows that share the network. Each flow is described by a source-destination pair: No predefined routes. i nm v w j flow g flow h flow f Let x f denote the rate of flow f 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 f as a function of x f. b(f)=i e(f)=j

Problem Statement Design a mechanism that guarantees stability of the queues, allocates flow rates, { x f }, that satisfy: x * 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 i m v s (i,n) q n,j q n,k s (n,m) s (n,v) Each node maintains a queue for each destination node. In general, the evolution of a queue length is described by (j) (k) Node n

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 (n,m) for destination d is given by 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:

An example: Node n Node m Node k W (n,m) max = (max{5-1,7-2,2-5}) + =5 d (n,m) = 2 W (n,k) max = (max{5-6,7-8,2-4}) + =0 d (n,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 f 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 q  KKK/2K/q The steady-state mean and the variance of the above chain are both Θ(K).  K/(q-1)

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

Connection-Level Model Assume a fixed route for each source (can be generalized) Files arrive according to a Poisson process of rate r for route 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:   !1, max-main fairness   ! 2, TCP-Reno   ! 1, proportional fairness   ! 0, maximize total throughput  We assume 

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 will be different, scheduling algorithm will be different (maxweight or backpressure)

Stability Lyapunov function For appropriately chosen constants, < 0 (n,p) forms a Markov chain A drift argument for the Markov chain implies positive recurrence

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 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

Network model Link if nodes are within communication range of each other Time is slotted. Each time slot is long enough to transmit a single packet. Network is represented by a graph:

Spectrum-Sharing Model Graph matching

Simple Collision Model No collision at receivers

Model Data/ack-based model ij All nodes in a two-hop have to be silent.

Interference Set Associated with each link is an interference set The interference set E l associated with link l satisfies the following properties: Link l belongs to E l Symmetry: 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 E l can be scheduled. Link l cannot be scheduled if any other link in E l is scheduled

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 Schedule and capacity region

Greedy scheduling 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. Each node attempts to independently schedule transmission over one of its backlogged links. A B CD E F X X X

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.

View each interference set as follows Arrival rate cannot be larger than service rate for stability 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 matching Throughput Loss Arrival rate Number of links that can be scheduled

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

Open Problems MAC: Tradeoff between protocol overhead, complexity, throughput 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