SRPT Applied to Bandwidth Sharing Networks to appear

SRPT Applied to Bandwidth Sharing Networks (to appear in Annals of Operations Research) Samuli Aalto TKK Helsinki University of Technology, Finland Urtzi Ayesta LAAS-CNRS, France BSnetworks. ppt TKK/Com. Net Research Seminar, 7. 3. 2008 1

Outline • • • Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations 2

Resource sharing • Single resource – loss system (e. g. Erlang model M/G/n/n) • one customer per server, no buffer • model for telephone traffic at the connection level – queueing system (e. g. M/G/1 -FCFS) • customers served one-by-one, infinite buffer • model for data traffic at the packet level – sharing system (e. g. M/G/1 -PS) • customers share the server, infinite buffer • model for (elastic) data traffic at the flow level • Multiple resources – loss networks (e. g. Ross) – queueing networks (e. g. Jackson, BCMP, Kelly, . . . ) – bandwidth sharing networks (Massoulié & Roberts) 3

Bandwidth sharing network • Characterization: – Flow-level model of a data network loaded with elastic flows (such as file transfers using TCP) • Resources: – Links l with bandwidth cl (bps) – Routes r (route = collection of consecutive links) • Traffic: – Elastic flows i with arrival time (possibly random), size (bits to be transferred), and route (fixed) • Control: – Bandwidth allocation to flows • inter-route bw allocation • intra-route bw allocation 4

Example: Symmetric linear network with L = 2 route 1 route 2 link 1 link 2 route 0 Symmetric with unit capacities: 5

Bandwidth allocation • Inter-route bandwidth allocation – fr = total bandwidth allocated to the flows on route r – feasible allocation satisfies the capacity constraints: • Intra-route bandwidth allocation – tells how the total bandwidth fr is shared among the flows using route r – an example: • PS = Processor-Sharing = bandwidth is shared evenly among the flows with the same route 6

Outline • • • Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations 7

Static setting • Characterization: – Fixed number of saturated flows, n = (nr; r Î R) • 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) – TM: throughput maximization [a ® 0]; Massoulié & Roberts (1999) – PDM: potential delay minimization [a = 2]; 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) 8

Fairness in symmetric linear network with L = 2 • alpha-fairness [a Î (0, ¥)] • TM [a ® 0] • PF [a = 1] = BF (in this case, not generally) • MMF [a ® ¥] 9

Fairness in symmetric linear network with L = 2 • Note: Throughput maximization does not specify a unique bandwidth allocation when n 1 = 0 or n 2 = 0 • TM as a limit a ® 0 • TM* with preemptive priority to routes 1 and 2 10

Outline • • • Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations 11

Dynamic setting • Randomly varying number of flows – Poisson arrivals, generally distributed flow sizes • Necessary stability conditions: • Definition: – Bandwidth allocation policy is (maximally) stable if the necessary stability conditions are also sufficient • Primary concern: – stable bandwidth sharing • Secondary concern: – (mean) delay optimization among stable bandwidth allocations 12

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 (1968) • Non-anticipating mean delay optimization for DHR (Decreasing Hazard Rate) service times: FB (Foregroud. Background) = LAS (Least-Attained-Service); Yashkov (1987) 13

Stable bw allocations for multilink networks • Exponential flow sizes – PF stable in linear networks; Massoulié & Roberts (1998) – MMF stable in linear networks; De Veciana et al. (1999) – alpha-fair bw allocations stable for any topology; Bonald and Massoulié (2001) – U-utility maximization bw allocations stable for any topology; Ye et al. (2003, 2005): • General flow sizes – BF stable for any topology; Bonald and Proutière (2003) – MMF stable for any topology; Bramson (2005) – PF stable for any topology; Massoulié (2005) – alpha-fair bw allocation stable for tree topologies; Gromoll and Williams (2006) • Note: Above, intra-route discipline always PS 14

Stable bw allocations for multilink networks • Verloop et al. (2006): – PR 0 and PR 12 stable in symm. linear network • PR 0 gives preemptive priority to class 0 whenever nonempty • PR 12 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 is used whenever there are flows loading the link • Note: PR 12 ¹ TM* which gives preemptive priority to classes 1 and 2 whenever at least one of them is nonempty 15

Unstable bw allocations for multiple link networks • Bonald and Massoulié (2001): – TM* not maximally stable in linear network – TM* stable in symmetric linear network only if • Verloop et al. (2005): – global SRPT not maximally stable in linear network – global LAS not maximally stable 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 16

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 m 1 + m 2 £ m 0, then PR 0 optimal – if m 1, m 2 £ m 0 and m 1 + m 2 ³ m 0, then PR 12 optimal • Bonald and Proutière (2003): – BF insensitive for any topology 17

Outline • • • Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations 18

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 19

Delay reduction by local SRPT’ • P n = 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 n 20

Delay reduction by local SRPT’ • Let p Î P n 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’ • p* = the modified policy – that applies SRPT as the intra-route discipline = the policy – for which the inter-route bw allocation process is – and that applies SRPT as the intra-route discipline 21

Delay reduction by local SRPT’ • Note the difference between p’ and p* • Proposition 1: – Let p Î P n, r Î R and t ³ 0. ~ For any modification p (including p ), • Theorem 1: – Let p Î P n. ~ For any modification p (including p ), • Here T refers to delay (= total transfer time) 22

Delay reduction by local SRPT* • P w = family of stable bw allocation policies p for which where – Zr(t) = total bw allocated to class r at time t – Wr(t) = total workload on route r at time t – W(t) = (Wr(t); r Î R) • Note: Policies PR 0 and PR 12 Î P w 23

Delay reduction by local SRPT* • Now there is no difference between p’ and p* • Proposition 2: – Let p Î P w, r Î R and t ³ 0. ~ For any modification p (including p ), • Theorem 2: – Let p Î P w. ~ For any modification p (including p ), 24

Outline • • • Introduction: Bandwidth sharing networks Background: Static setting Background: Dynamic setting Theoretical results: Delay reduction by applying SRPT Numerical results and observations 25

Simulation setup • Symmetric linear network with L = 2 and unit capacities – Poisson arrivals with constant total rate l = 1 – Flow size distribution with mean b = 0. 8 • deterministic • exponential: m = 1/b • hyperexponential: p 1 = 0. 9, m 1 = 9/b; p 2 = 0. 1, m 2 = 1/9 b • Now, PR 12 is the optimal non-anticipating allocation policy • Mean delay comparison between basic policy p and its SRPTmodifications p’ and p* using basic policies – BF = PF, PR 0, PR 12 26

Deterministic flow sizes p p’ p* BF PR 12 27

Exponential flow sizes p p’ p* BF PR 12 28

Hyperexponential flow sizes p p’ p* BF PR 12 BF 29

Increasing variability BF PR 0 PR 12 30

Observations • Mean delay improved by p’ for each class as Prop 1 predicts • As Prop 2 implies, for the basic policies PR 0 and PR 12 • In all numerical cases, for the basic policy BF • In all numerical cases, for all basic policies (BF, PR 0, PR 12) • Basic policy PR 12 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 31

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