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A Statistical Network Calculus for Computer Networks Jorg Liebeherr Department of Computer Science University of Virginia

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Collaborators Almut Burchard Robert Boorstyn Chaiwat Oottamakorn Stephen Patek Chengzhi Li Florin Ciucu

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R. Boorstyn, A. Burchard, J. Liebeherr, C. Oottamakorn. “Statistical Service Assurances for Packet Scheduling Algorithms”, IEEE Journal on Selected Areas in Communications. Special Issue on Internet QoS, Vol. 18, No. 12, pp. 2651-2664, December 2000. A. Burchard, J. Liebeherr, and S. D. Patek. “A Calculus for End–to–end Statistical Service Guarantees.” (2nd revised version), Technical Report CS-2001-19, May 2002. J. Liebeherr, A. Burchard, and S. D. Patek, “Statistical Per-Flow Service Bounds in a Network with Aggregate Provisioning”, Infocom 2003. C. Li, A. Burchard, J. Liebeherr, “Calculus with Effective Bandwidth”, Technical Report CS-2003-20, November 2003. F. Ciucu, A. Burchard, J. Liebeherr, ",A Network Service Curve Approach for the Stochastic Analysis of Networks”, ACM Sigmetrics 2005, to appear. Papers

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“Toy Models” in Computer Networking Learn from Physics: Wide use of toy models … that capture key characteristics of studied system … that permit back-of-the-envelope calculations … that are usable by non-theorists Simple models have played a major role in the evolution and development of data networks Queueing Networks Effective Bandwidth (Deterministic) Network Calculus

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(Product Form) Queueing Networks Jackson (50’s), Kelly, BCMP (70’s) Flow of “jobs” in system of queues and servers Applications: Provided motivation for packet-switching (Kleinrock’s PhD thesis) Main result: Steady state probability of queue occupancey n = (n 1, n 2, …, n k ) : P(n ) = P(n 1 ) P(n 2 ) … P(n k ) Limitations: Limited to Poisson traffic Limited scheduling algorithms

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Effective Bandwidth Hui, Mitra, Kelly (90s) Describes bandwidth needs of complex traffic by a number Application: admission control in ATM networks Can consider: service guarantees wide variety of traffic (incl. LRD) statistical multiplexing Limitations: not well suited for scheduling Peak rate Mean rate effective bandwidth

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Network Calculus Cruz, Chang, LeBoudec (90’s) Worst case delay and backlog bounds for fluid flow traffic Application: design of new schedulers (WFQ) new services (IntServ). Sender Receiver S3S3 S1S1 S2S2 S net Limitations: No random losses No statistical multiplexing, therefore pessimistic Main result: If S 1, S 2 and S 3 describes the service at each node, then S net = S 1 * S 2 * S 3 describes the service given by the network as a whole.

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State-of-the-art No analysis methodology is widely used today. Today, a lot of networking research relies on simulation and measurements to validate new designs Simulation and measurement are generally not suitable for evaluation of radically new designs RequirementsQueueing networks Effective bandwidth Network calculus Traffic classes (incl. self-similar, heavy-tailed) LimitedBroad Broad (but loose) Scheduling LimitedNoYes QoS (bounds on loss, throughput, delay) Very limited Loss, throughput Deterministic Statistical Multiplexing SomeYesNo

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Motivation: Develop network calculus into new “Toy Model” Today, fundamental progress in networking is hampered by the lack of methods to evaluate how radically new designs will perform. Opportunity: Simple (`toy') models that permit fast (`back-of-the-envelope') evaluations can become an enabling factor for breakthrough changes in networking research Approach: Probabilistic version of network calculus (stochastic network calculus) is a candidate for a new class of toy models for networking

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Rate Variance Envelope Knightly `97 Effective Bandwidth: J. Hui ’88 Guerin et.al. ’91 Kelly `91 Gibbens, Hunt `91 Deterministic network calculus Cruz `91 Effective bandwidth in network calculus Chang `94 (min,+) algebra for det. networks: Agrawal et.al. `99 Chang `98 LeBoudec `98 Service Curves Cruz `95 Cruz calculus with probabilistic traffic Kurose `92 Exponentially/stochasti- cally. bounded burstiness Yaron/Sidi `93 Starobinski/Sidi `99 Stochastically bounded service curve Qiu et.al.`99 1985199019952000 Our goals: (1)Maintain elegance of deterministic calculus (2)Exploit statistical multiplexing (3)Try to express other models 2005 Related Work (small subset)

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Multiplexing gain is the raison d’être for packet networks. Sources of multiplexing gain: Traffic characterization and conditioning Scheduling Statistical Multiplexing Multiplexing Gain

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Traffic Conditioning Traffic conditioning is typically done at the network edge Reshaping traffic increases delays and/or losses Traffic Conditioning

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Scheduling algorithm determines the order in which traffic is transmitted Examples: Different loss priorities priority scheduling Traffic with rate guarantees rate-based scheduling (WFQ, WRR) Delay constraints deadline-based scheduling (EDF) Scheduling

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Multiplexing Gain Flow 1 Worst case arrivals Flow 2 Flow 3 Time Without statistical multiplexing Backlog Worst-case backlog Flow 1 Flow 2 Flow 3 Time Backlog Arrivals With statistical multiplexing Backlog

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Deterministic worst-case Expected case Probable worst- case

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Example of Statistical Multiplexing: Retirement Savings Life expectancy: Normal( =75, =10) years Retiring Age:65 years Interest: 0% Withdrawal: $50,000 per year How much money does a person need to save (with confidence of 95% or 99%)? Life expectancy in a group of N people is Normal( , N). N=1 person (Individual Savings): 95% confidence: 10 + 2 = 30 years $1.5 Mio. 99% confidence: 10 + 2 = 40 years $2 Mio. N=100 people (Pooled Savings): 95% confidence: 10 + 2 = 12 years $600,000 99% confidence: 10 + 2 = 13 years $650,000

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The importance of Statistical Multiplexing At high data rates, statistical multiplexing gain dominates the effects of scheduling and traffic characterization

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Arrivals from a flow j are a random process Stationarity: The are stationary random processes Independence: The and are stochastically independent Traffic Characterization

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Leaky Buckets: Regulated arrivals Each flow is regulated Buffer with Scheduler Flow 1 Flow N C...... Traffic is constrained by a subadditive deterministic envelope such that Regulated Arrivals

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Definition: Definition: Effective envelope for is a function such that Note: Effective envelope is not a sample path bound. Often, we need a stronger version of the effective envelope! Effective envelope Define a function that bounds traffic with high probability “Effective Envelope”

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Sample Paths and Envelopes Sample paths Note: All envelopes are non-random functions Effective envelope At any time, at most one sample path is violated Stronger effective envelope At most one sample path is violated Deterministic envelope Never violated

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Probabilistic Sample Path Bound A strong effective envelope for an interval of length is a function which satisfies Relationship between the envelopes is established as follows: with

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Arrivals from multiple flows: Deterministic Network Calculus: Deterministic Network Calculus: Worst-case of multiple flows is sum of the worst-case of each flow Regulated arrivals Traffic Conditioning Buffer with Scheduler Flow 1 Flow N C...... Aggregating Arrivals

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Effective Envelopes for aggregated flows Stochastic Calculus: Stochastic Calculus: Exploit independence and extract statistical multiplexing gain by calculating For example, using the Chernoff Bound, we can obtain

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Type 1 flows: P =1.5 Mbps =.15 Mbps =95400 bits Type 2 flows: P = 6 Mbps =.15 Mbps = 10345 bits Type 1 flows strong effective envelopes Effective vs. Deterministic Envelope Envelopes

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Traffic rate at t = 50 ms Type 1 flows Effective vs. Deterministic Envelope Envelopes

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Scheduling Algorithms Work-conserving scheduler with unit rate that serves Q classes Class-q traffic has delay bound d q Scheduling algorithm: Scheduler...... Static Priority (SP) : Earliest Deadline First (EDF) : Deterministic Service Never a delay bound violation if: Statistical Service Delay bound violation with if:

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Statistical Multiplexing vs. Scheduling Statistical multiplexing makes a big difference Scheduling has small impact Example: MPEG videos with delay constraints at C= 622 Mbps Deterministic service vs. statistical service ( = 10 -6) Thick lines: EDF Scheduling Dashed lines: SP scheduling d terminator =100 ms d lamb =10 ms

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Scheduling has small impact C= 45 Mbps, = 10 -6 Delay bounds: Type 1: d 1 =100 ms, Type 2: d 2 =10 ms, Thick lines: EDF Scheduling Thin lines: SP scheduling Scheduling vs. Statistical Multiplexing Statistical multiplexing makes a big difference

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More interesting traffic types So far: Traffic of each flow was regulated Next: Consider different traffic types: On-Off traffic Fraction Brownian Motion (FBM) traffic Approach: Exploit literature on Effective Bandwidth Describes traffic in terms of a function Expressions have been derived for many traffic types

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Effective Envelopes and Effective Bandwidth Effective Bandwidth (Kelly 1996) Given, an effective envelope is given by

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Comparisons of statistical service guarantees for different schedulers and traffic types Schedulers: SP- Static Priority EDF – Earliest Deadline First GPS – Generalized Processor Sharing Traffic: Regulated – leaky bucket On-Off – On-off source FBM – Fractional Brownian Motion C= 100 Mbps, = 10 -6 Effective Envelopes and Effective Bandwidth

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S(t) A(t) D(t) Statistical Network Calculus with Min-Plus Algebra

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Convolution operation: Deconvolution operation Convolution and Deconvolution operators

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1.Output Envelope: is an envelope for the departures 2.Backlog bound: is an upper bound for the backlog 3.Delay bound: An upper bound for the delay is Cruz `95: A service curve for a flow is a function S such that: (min,+) results (Cruz, Chang, LeBoudec ) Deterministic (min,+) Network Calculus

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1.Output Envelope: is an envelope for the departures with probability 2.Backlog bound: is an upper bound for the backlog with probability 3.Delay bound: An upper bound for the delay with probability is An effective service curve for a flow is a function such that: (min,+) results Stochast Network Calculus

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Given: Service guarantee to aggregate ( C ) is known Total Traffic is known What is a lower bound on the service seen by a single flow? Allocated capacity C Sender Receiver Statistical Per-Flow Service Bounds

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Can show: is an effective service curve for a flow where is a strong effective envelope and is a probabilistic bound on the busy period Allocated capacity C Sender Receiver Statistical Per-Flow Service Bounds

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Type 1 flows: Goal: probabilistic delay bound d=10ms Number of flows that can be admitted

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Sender Receiver S3S3 S1S1 S2S2 (Cruz, Chang, LeBoudec) Deterministic Network Service Curve (Cruz, Chang, LeBoudec) : If are service curves for a flow at nodes, then S net = S 1 * S 2 * S 3 is a service curve for the entire network. S net Network Service Curves

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Network Service Curve: If S 1, , S 2, … S H, are effective service curves for a flow, then for all. Unfortunately, this network service is not very useful! Finding a suitable network service curve has been a longstanding open problem. A solution is presented in an upcoming ACM Sigmetrics 05 paper. Network Service Curve in a Stochastic Calculus

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Effective Network Service Curve Revise the definition of the effective service curve to Define Theorem: A network service curve is given by with where are free parameters

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Application of Network Service Curve Analyze end-to-end delay of through flows for Markov Modulated On-Off Traffic Compare delay with network service curve to a summation of per-node bounds

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Example Peak rate: P = 1.5 Mbps Average rate: = 0.15 Mbps T= 1/ + 1/ = 10 msec C = 100 Mbos Cross traffic = through traffic = 10 -9 Addition of per- node bounds grows O(H 3 ) Network service curve bounds grow O(H log H)

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Conclusions Presented aspects of stochastic network calculus Preserves much (but not all) of the deterministic calculus Can express many existing results on: Deterministic calculus Effective bandwidth Other models (EBB, not shown) Many open issues

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Conclusions Requirements Queueing networks Effective bandwidth Network calculus Traffic classes (incl. self-similar, heavy-tailed) LimitedBroad Broad (but loose) Scheduling LimitedNoYes QoS (bounds on loss, throughput delay) Very limited Loss, throughput Deterministic Statistical Multiplexing SomeYesNo Stochastic network calculus Broad Yes

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