Congestion In Large Balanced Fair Links

Transcription

1 Congestion In Large Balanced Fair Links Thomas Bonald (Telecom Paris-Tech), Jean-Paul Haddad (Ernst and Young) and Ravi R. Mazumdar (Waterloo) ITC 2011, San Francisco

2 Introduction File transfers compose much of the traffic of the current Internet Main measures of the quality of service (QoS) are the transfer rates and duration of the file transfer Being able to estimate congestion (when rates are below desired rates) is of great importance to dimensioning capacity to achieve QoS requirements Doing so that is both insensitive to traffic characteristics and tractable will lead to robust engineering rules in designing future networks

3 Scope Of Talk The main focus of this talk will be on congestion in single links that operate under a balanced fair allocation scheme for heterogeneous flows with differing maximum or peak bandwidth requirements Using ideas from local limit large deviations of convolution measures associated, formulas for estimating different measures of congestion that are computationally tractable for large parameters will be presented. A presentation of the mathematical background can be found in: R. R. Mazumdar, Performance Modelling, Loss Networks and Statistical Multiplexing, Series on Communication Networks (J. Walrand, ed.), Morgan and Claypool, 2010.

4 Nomenclature The system is a single link with M classes of traffic Link capacity C Rate limits on individual flows r i, i = 1... M Traffic intensity α i = λ i /µ i, i = 1... M β i = α i /r i, i = 1... M Load ρ = j α j/c Allocated bandwidth φ i, i = 1... M

5 Flow Level Model Introduced by Roberts and Massoulié [4] Ignores the packet level dynamics and models the file transfers as fluid flows The bandwidth allocated to flows of the same class are shared equally In this talk, we will assume that all flows are rate limited and go through a single bottleneck link This can be modeled as letting each class of flow go to separate processor sharing queues but with variable capacity depending on number of flows in system

6 Markov Process Let X be the state process, where the state is the numbers of flows of each class X is modeled as a continuous time jump Markov process λ i y = x + e i State transition rates: q( x, y) = µ i φ i ( x) y = x e i 0 Otherwise

7 Bandwidth Allocation Bandwidth allocation is a fundamental, well studied problem Most popular and studied class of allocations are the Utility based allocations Let x be the state vector whose components x i are the number flows of class i max x j U(φ j ( x)/x j ) φ j s.t. j φ j( x) C φ i ( x) x i r i when U i (x) = log x it is termed proportional fairness.

8 Insensitive Allocations Characterized by Balance Function Φ Allocation is defined as φ i ( x) = Φ( x e i ) Φ( x) Insensitive allocations have the advantage that the stationary distribution π( x) depends on the flow size distribution only through its mean π( x) = π( 0)Φ( x) M i=1 α x i i

9 Balanced Fairness Lemma Introduced by Bonald and Proutière [2]. Most efficient insensitive allocation is Balanced Fairness Consider another positive function Φ such that Φ(0) = 1 and the rate and capacity constraints are satisfied. Then Φ( x) Φ( x) x Z M +. (1) The Balance Function for a single link is: ( ) 1 M Φ( x e i ) Φ( x) = max Φ( x e i ), max C i: x i >0 x i r i i=1 The last constraint i.e. φ i ( x) x i r i is a rate constraint on each flow. If r i = it would reduce to processor sharing.

10 Balanced Fairness Properties The balance function can be simplified to: Φ( x) = 1 C M 1 x i!r x i i=1 i M Φ( x e i ) i=1 if x T r C, Otherwise Lemma i = 1... M, φ i (x) = x i r i iff x T r C This property implies that either all classes get their max rate or none do Theorem Stable iff ρ < 1

11 Balanced Fairness and Proportional Fairness Assuming r i = i, Balanced Fairness coincides with proportional fairness on many topologies and has been empirically shown to approximate Proportional Fairness well in many cases Massoulié [3] proved some very useful theoretical connections between Balanced Fairness and Proportional Fairness Theorem If there exists φ s.t. φ BF i (n x) φ i ( x) as n, then φ( x) = φ PF ( x) Theorem lim n 1 n log πbf (n x) max j φ C Where C is the set of feasible allocations. Conjecture φ BF i (n x) φ PF i ( x) as n x j log(φ j /α j ) s.t. Walton [5] has generalized the results of Massoulié to any max stable (ie. stability condition ρ < 1) insensitive allocation

12 Congestion Metrics We will look at three metrics related to the long run congestion of the system: 1 Probability of congestion P - The long run fraction of time that the system spends in a congested state. 2 Probabilities of congestion P i - The long run probability that an arrival of class i will arrive at a congested system or cause the congestion in link. 3 F i - Fraction of the average sojourn time that a customer of class i does not get its maximum rate while in the system.

13 Congestion Metrics From PASTA and the properties of balanced fairness, one can get a simple characterization of the first two congestion metrics: P = P i = x: x T r>c π( x) x: x T r>c r i π( x)

14 Congestion Metrics Formally, we define F i = [ τi ] E i 1 { X (t) T r>c} dt 0 E i [τ i ] Where τ i is the sojourn time of a class i arrival, X the stationary state process and E i indicates the expectation with respect to the Palm probability of arrivals of class i For our purposes, the metric is not useful in this form and we require an alternative characterization

15 Congestion Metrics Theorem (Swiss Army Formula) [1] [ W0 ] [ ] λ A E A Z(s)dB(s) = 1 t E t π 0 X (s )Z(s)dB(s) 0 Where A is a point process, W n a sequence of marks for A, X, Z non-negative processes and B a non-decreasing process Applying the Swiss Army Formula, we now get x i π( x) F i = x: x T r>c x i π( x) x

16 Congestion Metrics The congestion metrics can be written as a function of far fewer states Lemma and P = M i=1 ρ i B i 1 ρ P i = B i + P with B i = π( x) x: C r i < x T r C

17 Congestion Metrics Lemma For all i, j = 1,..., M, let Q ij = x i π( x), x: C r j < x T r C and Then Q i = x: x T r>c x i π( x). Q i = ρ ip M i 1 ρ + ρ j Q ij 1 ρ, j=1 F i = Q i + Q i x: x T r C x i π( x).

18 Erlang Multirate Loss System The states that are used to calculate the congestion measures are the same states that are used to calculate the blocking formula in an Erlang loss system In fact, for any state x : x T r C, the stationary probability is proportional to the stationary of an associated loss system since π( x) = π( 0) (α i /r i ) x i i x i! Like the loss system counterpart, when parameters are large, the computation becomes onerous Using ideas from local limit large deviations of convolution measures one can get an accurate approximation by scaling the traffic intensities and link capacity

19 Notion of a large system The notion of a large system is obtained by scaling both the capacity and arrival rates by a factor N. Define C(N) = NC and λ k (N) = Nλ k. Note this notion extends to networks In other words the large system can be seen as a N fold scaling of a nominal system where connections arrive at rate λ k, allocated φ k ( x) x k units of bandwidth, and the server capacity is C.

20 Main Results Theorem and for all i = 1... M: P(N) M i=1 ρ i P B i (N) 1 ρ P i (N) P B i (N) + P(N)

21 Where: Pi B (N) e NI e τdɛ(n) d 1 e τr i 2πNσ 1 e τd d is the greatest common divisor of r 1,..., r M, ɛ(n) = NC d NC d, M τ is the unique solution to the equation r i β i e τr i = C, I = Cτ σ 2 = M i=1 M β i (e τr i 1), i=1 r 2 i β i e τr i. i=1

22 Main Results Theorem F i (N) r i NC(1 ρ) P i(n) + M j=1 ρ j 1 ρ PB ij (N) Pij B (N) e NI i e τ i dɛ i (N) d 1 e τ i r j 2πNσi 1 e τ i d

23 Where: d is the greatest common divisor of r 1,..., r M, ɛ i (N) = NC r i d NC ri d, M τ is the unique solution to the equation r j β j e τr j = C, σ 2 = M j=1 τ i = τ r 2 j β j e τr j, r i Nσ 2, I i = ( C r i N ) τi σ 2 i = M j=1 r 2 i β j e τ i r j M β j (e τ i r j 1), j=1 j=1

24 Method of Proof Renormalize the congestion formulas so that they are now computed using the stationary distributions of the associated loss system Show that the normalization constants of the loss system and original system coincide in the limit Apply approximation for loss networks to the formulas for the congestion metrics

25 Numerical Example The system has M = 3 classes of traffic Link capacity C = 10 Rate limits r 1 = 1, r 2 = 2, r 3 = 5 Loads ρ 1 /ρ = 0.5, ρ 2 /ρ = 0.3, ρ 3 /ρ = 0.2

26 Numerical Example Congestion Probabilities Medium load, ρ = 0.6 Exact Approximation N F 1 (N) F 2 (N) F 3 (N) F 1 (N) F 2 (N) F 3 (N) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-12

27 Numerical Example Congestion Probabilities Heavy load, ρ = 0.9 Exact Approximation N F 1 (N) F 2 (N) F 3 (N) F 1 (N) F 2 (N) F 3 (N) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02

28 Numerical Example Time-average congestion rates Heavy load, ρ = 0.9 Exact Approximation N F 1 (N) F 2 (N) F 3 (N) F 1 (N) F 2 (N) F 3 (N) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-02

29 Network Case In general, network case is very difficult to analyze For specific topologies, the techniques from the single link analysis can be applied Of practical interest is a structure occurring in access networks referred to as a parking lot network.

30 Figure: Two Link Parking Lot Network The network has 2 links and 2 routes Route R 1 goes through both links and route R 2 goes through the second link only Each of the M classes of traffic follow one of the two routes Only the case that the capacities of the links satisfy C 1 < C 2 is of interest otherwise, the problem reduces to single link case

31 Numerical Example We conclude the presentation with a numerical example for a parking lot example: The system has M = 4 classes of traffic, two on each route Link capacities C 1 = 5 and C 2 = 9 Rate limits on route R 1 are r 1 = 1, r 2 = 2 Rate limits on route R 2 are r 3 = 1, r 4 = 2 Traffic intensities on route R 1 are α 1 = 2, α 2 = 1 Traffic intensities on route R 2 are α 3 = 2, α 4 = 1

32 Numerical Example Congestion Probability P(N) Exact Approximation N e e e e e e e e-10

33 Concluding Remarks Extension to tree networks is possible Balanced fairness is a good model for insensitive bandwidth sharing in cloud computing Close parallels with VCG auctions Large system means we can approximate balanced fairness via proportional fairness for which a mechanism design exists (primal-dual).

34 References I F. Baccelli and P. Brémaud. Elements of Queueing Theory, volume 47 of Stochastic Modelling and Applied Probability. Springer, Berlin, 2nd edition, T. Bonald and A. Proutière. Insensitive bandwidth sharing in data networks. Queueing Syst. Theory Appl., 44(1):69 100, L. Massoulié. Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab., 17(3): , J. W. Roberts and L. Massoulié. Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems, 15: , 2000.

35 References II N. Walton. Insensitive, maximum stable allocations converge to proportional fairness. Preprint, 2010.