SRPT Applied to Bandwidth Sharing Networks Samuli Aalto TKK Helsinki University of Technology, Finland Urtzi Ayesta LAAS-CNRS, France

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Bandwidth sharing network BS network = Flow-level model of a data network Resources: Links l Î L with bandwidth = capacity cl (bps) Traffic: Routes r Î R (set of links) used by elastic (TCP) flows i with arrival time Ai size Si (bits to be transferred) route ri Control: Link bandwidth allocation to flows inter-route bw allocation intra-route bw allocation

Example: Linear network with L = 4 route 1 route 2 route 3 route 4 link 1 link 2 link 3 link 4 route 0

Example: Symmetric linear network with L = 2 route 1 route 2 link 1 link 2 route 0 Symmetric with unit capacities:

Bandwidth allocation Inter-route bandwidth allocation fr = total bandwidth allocated to the flows with route r feasible allocation satisfies the capacity constraints: Intra-route bandwidth allocation tells how the total bandwidth fr is shared among the flows with route r for example: PS = Processor-Sharing = bandwidth is shared evenly among the flows with the same route

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Static setting Fixed number of flows, n = (nr; r Î R) saturated flows of infinite size Problem: Fair bandwidth sharing Solutions (with PS intra-route discipline): MMF: max-min fairness [a ® ¥]; trad., Jaffe (1981) PF: proportional fairness [a = 1]; Kelly (1997) PDM: potential delay minimization [a = 2]; Massoulié & Roberts (1999) TM: throughput maximization [”a ® 0”]; Massoulié & Roberts (1999) alpha-fairness [a Î (0, ¥)]; Mo & Walrand (1998,2000) U-utility maximization; Ye et al. (2003,2005) BF: balanced fairness; Bonald & Proutière (2003)

Fairness in symmetric linear network with L = 2 TM [a ® 0] PF [a = 1] = BF (in this case, not generally) alpha-fairness [a Î (0, ¥)] MMF [a ® ¥]

Fairness in symmetric linear network with L = 2 Note: Throughput maximization does not specify a unique bandwidth allocation when n1 = 0 or n2 = 0 TM as a limit a ® 0 TM* with preemptive priority to routes 1 and 2

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Dynamic setting Randomly varying number of flows Poisson arrivals, generally distributed flow sizes Necessary stability conditions: Definition: Bandwidth allocation policy is stable if the necessary stability conditions are also sufficient Primary concern: stable bandwidth sharing Secondary concern: (mean) delay optimization among stable bandwidth allocations

Single (bottleneck) link M/G/1 queue Fair bandwidth sharing = PS (Processor-Sharing) Stability = WC (Work-Conserving disciplines) Anticipating mean delay optimization = SRPT (Shortest-Remaining-Processing-Time); Schrage (1966) Non-anticipating mean delay optimization for DHR service times = FB (Foregroud-Background) = LAS (Least-Attained-Service); Yashkov (1987)

Stable bw allocations for multiple link networks Massoulié & Roberts (1998,2000): PF stable in linear networks De Veciana et al. (1999,2000): MMF stable in linear networks Bonald and Massoulié (2001): alpha-fair bw allocations stable for any topology (a > 0) Ye et al. (2003,2005): U-utility maximization bw allocations stable for any topology Bonald and Proutière (2003): BF stable for any topology Note: Above, intra-route discipline always PS

Stable bw allocations for multiple link networks Verloop et al. (2006): PR0 and PR12 stable in symm. linear network PR0 gives preemptive priority to class 0 whenever nonempty PR12 gives preemptive priority to classes 1 and 2 whenever both of them are nonempty; otherwise preemptive priority is given to class 0 Intuitive argument: Both policies ensure that the whole capacity of each link used whenever there are flows loading the link Note: PR12 ¹ TM* which gives preemptive priority to classes 1 and 2 whenever at least one of them is nonempty

Unstable bw allocations for multiple link networks Bonald and Massoulié (2001): TM* (with preemptive priority to routes 1 and 2) unstable in linear network TM* stable in symmetric linear network only if Verloop et al. (2005): global SRPT unstable in linear network global LAS unstable in linear network Note: In all these cases, the whole capacity of a link is not necessarily used while there are flows loading the link

Delay optimization among stable bw allocations Yang & de Veciana (2002,2004): optimal allocation: fr(n) = 0 or 1 (depending on state n) in symmetric linear network Verloop et al. (2006): determined optimal non-anticipating allocation in symmetric linear network with exponential flow sizes if m1 + m2 £ m0, then PR0 optimal if m1, m2 £ m0 and m1 + m2 ³ m0, then PR12 optimal Bonald and Proutière (2003): BF insensitive for any topology

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Theoretical setup Consider a BS network with a general topology, Poisson arrivals, and generally distributed flow sizes P = family of stable bw allocation policies p

Delay reduction by local SRPT’ P° = family of stable bw allocation policies p for which where Zr(t) = total bw allocated to class r at time t Nr(t) = number of flows on route r at time t N(t) = (Nr(t); r Î R) Note: All fair bw allocation policies mentioned above Î P°

Delay reduction by local SRPT’ Let p Î P° be fixed. p = a modified policy with the same inter-route bw allocation process, but the intra-route disciplines may be different from the original ones p’ = the modified policy that applies SRPT as the intra-route discipline p* = the policy for which the inter-route bw allocation process is and that applies SRPT as the intra-route discipline ~

Delay reduction by local SRPT’ Note the difference between p’ and p* Theorem 1: Let p Î P°, r Î R and t ³ 0. For any modification p (including p ), Corollary 1: Let p Î P°. For any modification p (including p ), Here T refers to delay (= total transfer time) ~ ~

Delay reduction by local SRPT* P b = family of stable bw allocation policies p for which where Zr(t) = total bw allocated to class r at time t Br(t) = 1(Nr(t) > 0) = busy period indicator B(t) = (Br(t); r Î R) Note: Policies PR0 and PR12 Î P b

Delay reduction by local SRPT* Proposition: Let p Î P b, r Î R and t ³ 0. Then Theorem 2: Let p Î P b, r Î R and t ³ 0. For any modification p (including p ), Corollary 2: Let p Î P b. For any modification p (including p ), ~ ~

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Simulation setup Symmetric linear network with L = 2 and unit capacities Poissson arrivals with constant total rate l = 1 Flow size distribution with mean b = 0.8 deterministic exponential: m = 1/b hyperexponential: p1 = 0.9, m1 = 9/b; p2 = 0.1, m2 = 1/9b Comparison between p, p’ and p* using basic policies BF PR0 PR12

Deterministic flow sizes p p’ p* BF PR12 PR12

Exponential flow sizes p p’ p* BF PR12 PR12

Hyperexponential flow sizes p p’ p* BF PR12 BF

Outline Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results Observations

Observations Mean delay improved by p’ for each class as Thm 1 predicts As Thm 2 tells, for basic policies PR0 and PR12 In all numerical cases, for basic policy BF In all numerical cases, for all basic policies (BF, PR0, PR12) Basic policy PR12 is better than BF for deterministic and exponential flow sizes but worse for hyperexpontial Delay reduction of BF very similar for exponential and hyperexponential flow sizes

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