Short-Term Fairness and Long- Term QoS Lei Ying ECE dept, Iowa State University, Joint work with Bo Tan, UIUC and R. Srikant, UIUC

Resource allocation for the Internet Resource allocation algorithm for the Internet are designed to ensure fairness among users present in the network Assume the number of users is fixed (static model) In reality, the users arrive, bringing in a certain amount of work in the form of a file to be transferred, and depart when the work is completed (connection-level model)

Resource allocation for the Internet The stability of the network when there are file arrivals and departures has been studied in a number of papers (Robert&Massoulie’98, Veciana et al’01, Bonald&Massoulie’01, Lin et al’07) The network is stochastically stable under the proportional-fairness if Connection-level performance beyond stability?

Network and flow model Consider a network with L links and R routes File arrivals of each type: Poisson, rate r File size of each type: Exponential, parameter  r Capacity of each link = c l The capacity of each link is divided among the files using the link A file departs after it has transferred its data

Resource allocation and backlog n r (t): number of files of type r x r (t): rate allocated to flows of type r at time t Backlog is affected by the rate allocation  Backlog:

Resource allocation and backlog Proportionally-fair resource allocation on the backlog  Proportionally-fairness can be implemented in a distributed fashion  Support the maximum connection-level stability  Doesn’t maximize the departure rate at each time slot

Line network example r =  r = , c l =1 n 1 [t]=n 2 [t]=n 3 [t] ) x 1 [t]=x 2 [t]=x 3 [t]=0.5 ) overall departure rate is 1.5  x 2 [t]=x 3 [t]=1 ) overall departure rate is 2 

Long-term QoS Goal: Study the impact of proportionally-fair resource allocation on the backlog Obtain an upper-bound on the backlog under proportional fairness Find the optimal resource allocation strategy to minimize the backlog Obtain a lower bound on the backlog under the optimal strategy Compare the upper and lower bound in the heavy-traffic regime:  r  r ! 1

Long-term QoS: Line network Optimal policies for a line network with two links were proposed by Verloop et al’ 06. The delay-performance of the optimal policies and the proportionally-fair policy were compared using simulations, and it was shown that the gap is less than 20%.

Optimal resource allocation: Star network If all the 3 file types are non- empty  Serve each of them at rate 0.5 If only 2 file types are non- empty  Serve the file type with more files at rate 1 If only 1 file type is non-empty  Serve it at rate 1 Recall each link has capacity 1

Intuition behind optimality x=(0.5,0.5,0.5) maximizes total service rate,  Feasible only when all file types are non-empty. If only 2 file types are non- empty, serve the one with the larger number of files  This would increase the likelihood that all file types are non-empty in the future Motivated by Verloop et al (2005) for 2-link, 3-flow network

Proof of optimality Use uniformization to convert to discrete-time problem Consider the objective Prove the optimality of the scheme for all N  Use induction and dynamic programming

Performance of the optimal scheme Largest 2 file types behave like a single queue: total service rate for them = 1 Suggests the Lyapunov function: m 1 (t) m 2 (t) 2 1 m 3 (t)

Optimal scheme vs proportional fairness Lower bound for optimal scheme: Heavy-traffic limit

Performance of proportional fairness Lyapunov function E[W[t+1] – W[t] ] = 0 in steady-state Upper bound on steady-state backlog Compare with upper bound upper bound / lower bound = 1.5

Simulation results

Upper bound for general networks Lyapunov function

Upper bound for general networks

Upper bound for general networks Upper bound This result complements the work of Kang, Kelly, Lee, Williams (2007)  Their model assumes each link has a dedicated flow;  Letting the load due to local flows go to zero leads to a heuristic upper bound

Line network Our upper bound Upper bound by Kang, Kelly, Lee, Williams (2007)

Star network Our upper bound Upper bound by Kang, Kelly, Lee, Williams (2007)

Summary Derived an upper bound for general networks, which linearly increases with the number of routes in the network. Tighter lower bound?