TWO PROBLEMS IN NETWORK PROBING DARRYL VEITCH The University of Melbourne 1 Temporal Loss and Delay Tomography 2 Optimal Probing in Convex Networks Paris Networking 27 Juin 2007
TEMPORAL LOSS AND DELAY TOMOGRAPHY VIJAY ARYA, NICK DUFFIELD, DARRYL VEITCH To appear, Performance 2007
NETWORK TOMOGRAPHY! Several interesting inverse problems " Link level inferences from path measurements Multicast inference of network characteristics (MINC) " Path level inferences from link measurements Origin-destination traffic matrix estimation Loss rate IP Network Probe Packets Tomos End-to-end measurements Graphia Link level characteristics 1
NETWORK TOMOGRAPHY! Previous models for estimation " Assume temporal (and spatial) independence Estimated link parameters: loss rates, delay distribution! Limitation " Packet traffic is bursty and exhibits temporal dependence " Previous models support estimation of only average loss rates and delay distributions of links " Performance of applications also depends on temporal parameters of links, such as durations of loss bursts and bursts of large delays! This work " Relaxes temporal independence assumption " Estimation of temporal characteristics of links from end-to-end measurements 2
TEMPORAL LOSS CHARACTERISTICS! Loss-run distribution Probability [ Loss-run length! k ] Mean loss-run length Pr 1 2 3 Loss-run length Loss/pass bursts! Importance " Impacts delay sensitive applications like VoIP (FEC tuning) " helps characterizing bottleneck links better 3
TEMPORAL DELAY CHARACTERISTICS Link delay states 2 1 0 Packets! Run distributions of delays Probability [ Duration of a state! k ] Mean duration of high delay states 4
OUTLINE! MINC! Loss model! Estimation! Experiments! Conclusions
OUTLINE! MINC! Loss model! Estimation! Experiments! Conclusions
MINC! Multicast-based Inference of Network Characteristics... multicast probes! 1 Loss rates in logical Tree! 7 1 0 1 1 1 1 1 0... 1 0 0 1 MINC Loss Estimator 5
MINC! Multicast-based Inference of Network Characteristics... multicast probes! 1 Loss rates in logical Tree! 7 1 0 1 1 1 1 1 0... 1 0 0 1 MINC Loss Estimator Temporal Loss Estimator? 5
LOSS MODEL! Probes losses on links are modeled using two discrete-time binary processes! Bookkeeping process : records which probes reach node! Link loss process : determines which probes successfully cross the link 1 1 0 0 1 1 0 0 -&'#).",,)%*"+$,, 1 0 0 0!""##$$%&'()%*"+$,,! Loss dependence structure! Spatial: Loss processes of links are mutually independent (as before)! Temporal: Loss processes are stationary and ergodic (previously Bernoulli) 6
ACCESSING TEMPORAL PARAMETERS! Sufficiency of joint link passage probabilities e.g. " Mean Loss-run length " Loss-run distribution 7
OUTLINE! MINC! Loss model! Estimation! Experiments! Conclusions
JOINT PASSAGE PROBABILITIES " Sample: (set of receiver nodes) To estimate joint passage probabilities for all links " Probe index set " Joint link passage probability ( vector notation) by stationarity e.g. " Joint path passage probability Sufficient to estimate these 8
ESTIMATION: JOINT PATH PASSAGE PROBABILITY No of equations equal to degree of node k computed recursively over index sets = 0 for 9
ESTIMATION: JOINT PATH PASSAGE PROBABILITY! Estimation of in general trees " Requires solving polynomials with degree equal to the degree of node k Numerical computations for trees with large degree " Recursion over smaller index sets! Simpler temporal loss estimators " Subtree-partition Requires solutions to only linear or quadratic equations No loss of samples Also simplifies existing MINC estimators...... " Avoid recursion over index sets by considering only subsets of receiver events which imply 10
ESTIMATOR PROPERTIES! Consistent s rational functions of s! Variance Experiments No explicit expressions 11
OUTLINE! MINC! Loss model! Estimation! Experiments! Conclusions
EXPERIMENTS! Setup " Model-based simulations " Link losses simulated using two families of loss processes Discrete-time Markov chains On-off processes: pass-runs geometric, loss-runs Zipf (finite support) " Estimation Passage probability of a probe, Joint passage probability for a pair of consecutive probes, Mean loss-run length, " Relative error = 12
EXPERIMENTS! Estimation for shared path in case of two-receiver binary trees Markov chain On-off process 13
EXPERIMENTS! Estimation for shared path in case of two-receiver binary trees Markov chain On-off process 14
EXPERIMENTS! Estimation of for larger trees Link 1 Trees taken from router-level map of AT&T network produced by Rocketfuel (2253 links, 731 nodes) Random shortest path multicast trees with 32 receivers. Degree of internal nodes from 2 to 6, maximum height 6 15
VARIANCE! Estimation for shared path in case of tertiary tree Standard errors shown for various temporal estimators 16
CONCLUSIONS! Estimators for temporal loss parameters, in addition to loss rates Estimation of any joint probability possible for a pattern of probes! Class of estimators to reduce computational burden Subtree-partition: simplifies existing MINC estimators! Future work Asymptotic variance MLE for special cases (Markov chains) Hypothesis tests Experiments with real traffic 17
OPTIMAL PROBING IN CONVEX NETWORKS FRANÇOIS BACCELLI, SRIDHAR MACHIRAJU, DARRYL VEITCH, JEAN BOLOT Sprint Technical Report ATL-020124
PREVIOUS WORK The Case Against Poisson Probing Zero-sampling bias not unique to Poisson (in non-intrusive case) PASTA talks about bias, is silent on variance PASTA is not optimal for variance or MSE PASTA is about sampling only, is silent on inversion Probe Pattern Separation Rule: select inter-probe (or probe pattern) separations as i.i.d. positive random variables, bounded above zero. Example: Uniform (i.i.d.) separations on [ 0.9!, 1.1! ] Aims for variance reduction Avoids phase locking (leads to sample path `bias") Allows freedom of probe stream design 10
LIMITATIONS Results derived in context of delay only No optimality result (expected to be highly system and traffic dependent) 10
NEW WORK Extended all results to loss case (loss and delay in uniform framework) For convex networks, have universal optimum for variance But: bias problem Give family of probing strategies with Zero bias Variance as close to optimal as desired (tunable) Fast simulation Consistent with spirit of Separation Rule But are networks convex? Some systems when answer is known to be Yes virtual work of M/G/1 loss process of M/M/1/K Insight into when No Real data when answer seems to be Yes 10
TWO PROBLEMS: SAMPLING AND INVERSION Sampling For end-to-end delay Only have probe samples of a probe of size x Inversion From the measured delay data of perturbed network, may want: Unperturbed delays Link capacities Available bandwidth Cross traffic parameters at hop 3 TCP fairness metric at hop 5. Here we focus on sampling only, do so using non-intrusive probing 10
THE QUESTION OF GROUND TRUTH Non-intrusive Delay Delay process using zero sized probes still meaningful Each probe carries a delay: samples available Non-intrusive Loss Losses with x=0 are hard to find.. meaningful but useless Lost probes are not available (where? How?) Probes which are not lost tell us? about loss? Cannot base non-intrusive probing on virtual (x=0) probes 10
GENERAL GROUND TRUTH Approach Define end-to-end process directly in continuous time Must be a function of system in equilibrium - no probes, no perturbation! Non-intrusive probing defined directly as a sampling of this process: Interpretation: what probe would have seen if sent in at time. Note Process may be very general, not just isolated probes but probe patterns! Applies equally to loss or delay This is the only way, even virtual probes can be intrusive! 10
LOSS GROUND TRUTH EXAMPLES 1-hop FIFO, buffer size K bytes, droptail If packet based instead, becomes independent of x! 2-hop Depends on x Other examples: Packet pair jitter Indicator of packet loss in a train Largest jitter in a chain, or largest lost 10
NIPSTA: NON-INTRUSIVE PROBING SEES TIME AVERAGES Empirical averages seen by probe samples converge to true expectation of continuous time process, assuming: stationarity and ergodicity of CT stationarity and ergodicity of PT (easy) independence of CT and PT (easy) joint ergodicity between CT and PT Result follows just as in delay work, using marked point processes, Palm Calculus and Ergodic Theory Not Restricted to Means : Temporal quantities also, eg jitter Probe-train sampling strategies 10
OPTIMAL VARIANCE OF SAMPLE MEAN Covariance function Sample Mean Estimator Periodic Probing 10
PERIODIC PROBING IS OPTIMAL! Compare integral terms : If R convex, then can use Jensen s inequality: No amount of probe train design can beat periodic! Problem: Periodic is not mixing, so joint ergodicity not assured (phase lock) If occurs, get `sample path bias (estimator not consistent) 10
GAMMA RENEWAL PROBING Gamma Law Gamma Renewal Family: As increases, variance drops at constant mean Poisson is beaten once Process tends to periodic as Sensitivity to periodicities increases however Proof: Again uses Jensen s inequality Uses a conditioning trick and a technical result on Gamma densities 10
TEST WITH REAL DATA FOR MEAN DELAY 10
EXAMPLE WITH REAL DATA 10
EXAMPLE WITH REAL DATA 10