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Population Dynamics in Networks: from Queues to Pdes and Control, from P2P to Deferrable Power Loads

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Population dynamics in networks: from queues to PDEs and control, from P2P to deferrable power loads. Fernando Paganini Universidad ORT Uruguay Joint work with Andrés Ferragut Universidad ORT Uruguay IMA, Sept 2015. Outline 1. Networking research: between queues and fluids. 2. M/G- Processor Sharing queues, point process model and its PDE fluid counterpart. 3. Dynamics of peer-to-peer networks. 4. Dispatching deferrable power loads. Mathematics in networking - a historical view • In the beginning we had queueing theory… – Erlang, circa 1910, dimensioning telephone switches. • Statistical multiplexing inspired packet networks… – Kleinrock, 60s – 70s, Internet pioneer and queueing theorist. • But queueing theory reached its limits in networking: – Limited results in network case, dubious assumptions. • Turn to macroscopic fluid flow models: – Bertsekas/Gallager in late ’70s, Kelly et al in ’90s. – Cover elastic traffic, apply to general networks. – Early ’00s, Network Utility Maximization for multiple layers. • Still, Internet jobs remain statistically multiplexed. – Queuing theory has a comeback. – But fluid models appear also at this scale. Classical M/M/1 queues Poisson (λ ) arrivals exp(µ ) service times A( t ) Equivalently: exponential job size, X( t ) constant service rate. t exp(λ ) t λ Markov chain, state x-1 x is queue occupancy µ λ Stable (ergodic) if ρ = < 1. µ Invariant distribution is geometric: π(x )= ρx (1 − ρ ), x ≥ 0. Networks of Queues and Data networks • Queueing theory concepts influential in the early days of data networks: statistical multiplexing motivates “packet switching”. • But networks of M/M/1 queues are not easy to analyze. The steady-state distribution only known under narrow assumptions (Jackson networks). • Besides, packet networks do not match model assumptions: sending times not Poisson, packet length not exponential. Going from “molecular” to fluid models. Arrivals: rate λ pkt s/sec Real valued x( t ), following ODE λt x( t ) dx A( t ) =λ− µ , saturated to x ()0. t ≥ dt Departures: µ pkt s/sec t Relationship with Markov model for M/M/1 queue: Xm ( mt ) scaling limit x( t )= lim , where X m (0) = m . m→∞ m • Beyond the M/M/1 assumptions, the fluid “tank” is a natural macroscopic first-principles model. Extends to varying rates. • Allows for a simpler transition to the network case. • Leads to way to study feedback control of rates. Fluid view of network congestion control. ri ( t ) feedback • Congestion signals from scarce resources drive input rates. Network Utility Maximization model (Kel ly et al). Decentralized control is primal or dual solution of convex program max r ∑Uri( i ), subject to ∑ Rrclli i ≤ l ∀ . i i CONCAVE UTILITY LINK CAPACITY FUNCTION CONSTRAINTS The network at the scale of connections • The previous fluid models treat connections as permanent. • But connections (flows) arrive, are served and leave, there is also statistical multiplexing at the scale of flows. • Back to queueing theory! Distinguishing features: 1. Service discipline : all flows present are served simultaneously, sharing the network capacity. 2. Job sizes : exponential distribution is too limiting. Heavy tails observed for Internet files: many short transfers, far fewer long connections. Outline 1. Networking research: between queues and fluids. 2. M/G- Processor Sharing queues, point process model and its PDE fluid counterpart. 3. Dynamics of peer-to-peer networks. 4. Dispatching deferrable power loads. The M/G/1-Processor Sharing Queue. i Poisson(λ) arrivals of file transfer jobs i Random file size, general distrib ution. c P x G(σ) =(Si ze > σ ). G(σ ) σ i Processor Sharing: service capacity c (bp s) shared between c x jobs present. Rate per job: r = . Special case of NUM . x i Roberts and Massoulié ’98 proposed this model for TCP traffic, studied it in the case of exponential file sizes. λ λ Markov chain, stable if ρ = < c. µ x-1 x µc i Can we handle a more realistic model of file sizes? M/G/1-PS Queue. i Good news (see Kelly '78): PS queue is i nsen sitiv e: in steady state, the population of jobs has the sam e distribution ge om (ρ) as in exponenti al cas e . i But # of jobs is not the entire state: residual work matters. Compact representation of the state (Gromoll et al. '02, P.Robert '03): 0 σ1 σ2 σ3 σ x σ Point measure on R+ , each point is a job, position is residual work. Arrivals: a new point mass appears, following distribution G(σ ). Service= motion to the left, speed c x . Depart when reaching σ = 0. The model applies also to the M/ G / ∞ queue: here, each job present has a firm capaci ty for itself. ⇒ Points move at constant spee d . Fluid models for M/G-PS Queues • At the scale of jobs, stochastic models are very natural. • However, again they are hard to solve except for simple cases. • For a more general bandwidth sharing, (e.g. given by NUM), it is not easy to find steady-state distributions. • Turn again to fluid models: • Job quantities become real numbers • Differential equations replace Markov processes. • How to account for residual work? Fluid distribution on R+ Service= "advection", rate r(σ ) f( t ,σ ) Departures σ Arriving "mass", follows file-size distr ibution PDE using cumulative distributions For the point mass process, X( t ) Φ(t ,σ ) the complementary cumulative distribution is a decreasing, piecewise constant stochastic process Fluid version: deterministic σ1 σ 2 σ 3 σ n σ real-valued f unction F( t ,σ ) x( t ) representin g population o f jobs F( t ,σ ) with more residual wor k tha n σ. Dynamics: ARRIVALS PROPAGATION F( t+ dt ,σ ) = λG() σ dt+ F( t , σ + rdt ) rdt σ ∂Ft(,)σ ∂ Ft (,) σ r :rate per job. ⇒ =λG() σ + rFt (,,) σ ∂t ∂ σ [’P-Tang-Ferragut-Andrew , IEEE TAC '12 ]: first use of PDE to prove a stability conjecture for Internet bandwidth s hari ng with general f ile sizes. Outline 1. Networking research: between queues and fluids. 2. M/G- Processor Sharing queues, point process model and its PDE fluid counterpart. 3. Dynamics of peer-to-peer networks. 4. Dispatching deferrable power loads. BitTorrent-like peer-to-peer networks • Content = set of pieces. S S • Peers: – Seeders own entire content, willing to disseminate it. – Leechers downloading pieces, also contribute by uploading piece s they have. µi • Information on who owns L L which pieces, passed around L L to achieve efficient e xchang e. L L L • “Tit-for-tat” rules give incentives do cooperate . p2p capacity scales with demand We a ssume up load ban dwidth µi p er p eer is t he bottl eneck. Population dynamics of a swarm of peers PEERS (~hundreds) leechers seeders Arrive Trans Depart File PIECES sharing (~thousands) i Coarse model [Yang-DeVe ciana '04, Qiu-Srikant '04]: x leech, y seed. i De tailed mod el [Kesidis et al .07, Massou lié-Vojnov ic '08, Zhu-Hajek '12 ]: pop ulation state per sub set of pi eces. Huge dimens ion, limited results. i Our intermediate model: track profile o f population as a function of residual work σ. Leecher dynamics as an M/G-PS queue i Leechers arrive as Poiss on(λ ), seeking S S download of ra ndom size S, me an E[] S = 1. µ G(σ ) =P( S > σ ) µ σ L L Covers deterministic sizes: G (σ) =1[0,1) ( σ ) µ L L L L (peers want all the file, start with no content). L Populations: xt() leechers. Take for simplicity yty () ≡ 0 seeder s. Assumptions (empirically validated for peers with homo geneous µ): i Efficient use of tota l u pload capacity Rup =µ( x + y 0 ). µ(x+ y ) µ y i Processor shari ng : r =0 =µ + 0 , rate per leecher . x x Leecher dynamics is an M/G/" ∞ + 1"− Processor Sharing queue. From stochastic to fluid model i Exploiting insensitivity, we find the s teady-state distribution of peers ρx+ y 0 λ present in the stochastic queu e: π()x= C() ρ ,0 x ≥. ρ = . (x+ y 0 )! µ i Moreover, in steady state, residual jobs are independently drawn ∞ from ccdf G(σ )= G( s ) ds . [Pechinkin'83, Zachary '07]. For instance, ∫σ deterministic demands yield a u nifo rm download profile in e quilibrium. More information from fluid model: x( t ) F( t ,σ ) ∂Ft(,)σx+ y ∂ Ft (,) σ =λG () σ + µ 0 ∂t x ∂ σ r( F ,, t σ ) σ Note: in the case of exponentially distributed jobs, G(σ ) = e −σ the PDE admits a solution Ft(,)σ = xt ()e−σ , where xt () satisfi es the ODE x (t )=λ − µ ( x + y 0 ). This is the model in [Qiu-Sr ikant '04] . Transient analysis with PDE model Consider a transient sc enario ∂Ft(,)σx+ y ∂ Ft (,) σ = µ 0 with no leecher arrivals. ∂t x ∂ σ Fixed seeders y( t ) ≡ y 0. Initial population of lee chers x0, wi th partial c onten t F (0,σ )= Φ ( σ ) . Theorem: transient time to empty Compare with detailed packet 11 Φ (σ ) simulations (in ns2) of BitTorrent system is T= d σ. ∫0 µΦ( σ ) + y0 1x 1 Bound: T ≤0 ≤ . µx0+ y 0 µ Coarser ODE model: x (t )= −µ ( x + y0 ), x (0) = x 0 Predicts completion time 1 x T '= log 1 + 0 . Pessi mistic ! µ y0 Including arrivals: equilibrium and stability. ∂Ft(,)σx+ y ∂ Ft (,) σ xt()= Ft (,0); =λ + µ 0 ; ∂t x ∂ σ 0≡ F ( t ,1). For concreteness, work with d etermi nistic j ob sizes: G()1σ= [0,1) ( σ ) . Assume seeders alone cannot cope with demand: ρ > y0. PDE h as a unique equilibriu m: x* F * ()σ= ( ρ −Y )(1 − σ ) .
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