Recent Progress on a Statistical Network Calculus

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1 Recent Progress on a Statistical Network Calculus Jorg Liebeherr Deartment of Comuter Science University of Virginia Collaborators Almut Burchard Robert Boorstyn Chaiwat Oottamakorn Stehen Patek Chengzhi Li

2 Contents R. Boorstyn, A. Burchard, J. Liebeherr, C. Oottamakorn. Statistical Service Assurances for Packet Scheduling Algorithms, IEEE Journal on Selected Areas in Communications. Secial Issue on Internet QoS, Vol. 18, No. 12, , December A. Burchard, J. Liebeherr, and S. D. Patek. A Calculus for End to end Statistical Service Guarantees. (2nd revised version), Technical Reort, May J. Liebeherr, A. Burchard, and S. D. Patek, Statistical Per-Flow Service Bounds in a Network with Aggregate Provisioning, Infocom C. Li, A. Burchard, J. Liebeherr, Calculus with Effective Bandwidth, July Service Guarantees Sender Switch Receiver A deterministic service gives worst-case guarantees Delay d A statistical service rovides robabilistic guarantees Pr[ Delay d ] ε or Pr[ Loss l ] ε

3 Multilexing Gain Resource needed to suort QoS for N flows << Resource needed N to suort QoS for 1 flow Sources of multilexing gain: Traffic Conditioning (Policing, Shaing) Scheduling Statistical Multilexing of Traffic Scheduling Scheduling algorithm determines the order in which traffic is transmitted

4 Exected case Probable worstcase Deterministic worst-case Designing Networks for Multilexing Gain Scheduling Earliest-Deadline-First GPS Static Priority FCFS Average Rate Deterministic service Statistical service Peak Rate Allocation Service / Admission Control Peak Rate Token Bucket Multile Buckets Traffic Characterization/ Conditioning

5 Designing Networks for Multilexing Gain Scheduling Earliest-Deadline-First GPS Static Priority FCFS Deterministic service Statistical service Average Rate Peak Rate Allocation Service / Admission Control Peak Rate Token Bucket Multile Buckets Traffic Characterization/ Conditioning Designing Networks for Multilexing Gain Earliest-Deadline-First GPS Static Priority FCFS Scheduling By now: The design sace for deterministic guarantees is well understood. Peak Rate Allocation Deterministic service Statistical service Average Rate Service / Admission Control Peak Rate Token Bucket Multile Buckets Traffic Characterization/ Conditioning

6 Designing Networks for Multilexing Gain Earliest-Deadline-First GPS Static Priority FCFS Scheduling Still oen: Is there an elegant framework to reason about statistical guarantees? Statistical Network Calculus Deterministic service Statistical service Average Rate Peak Rate Allocation Service / Admission Control Peak Rate Token Bucket Multile Buckets Traffic Characterization/ Conditioning Related Work (small subset) Effective Bandwidth: J. Hui 88 Guerin et.al. 91 Kelly `91 Gibbens, Hunt `91 Deterministic network calculus Cruz, 1991 Effective bandwidth in network calculus Chang 94 Service Curves Cruz 95 (min,+) algebra for det. networks: Agrawal et.al. 99 Chang 98 LeBoudec Motivation for our work on statistical network calculus: (1) Maintain elegance of deterministic calculus (2) Exloit know-how of statistical multilexing

7 Source Assumtions Arrivals Aj(t,t+τ) are random rocesses Deterministic Calculus: (A1) Additivity: For any t 1 < t 2 < t 3, we have: (A2) Subadditive Bounds: Traffic Aj is constrained by a subadditive deterministic enveloe A * j as follows with (P,σ,ρ) A*=min (Pt,σ+ρt) τ Source Assumtions Statistical Calculus: (A1) +(A2) (A3) Stationarity: The A j are stationary random variables (A4) Indeendence: The A i and A j (i j) are stochastically indeendent (No assumtions on arrival distribution!)

8 Aggregating Arrivals Flow 1 Flow N A * ( τ 1 ) Regulator A1 ( t, t + τ ).. C. A * N ( τ A ( t, t + τ ) ) N Regulated arrivals Buffer with Scheduler Arrivals from multile flows: Deterministic Calculus: Worst-case of multile flows is sum of the worst-case of each flow Aggregating Arrivals 2000 Statistical Calculus: To bound aggregate arrivals we define a function that is a bound on the sum of multile flows with high robability Effective Enveloe Effective enveloes are non-random functions effective enveloe : strong effective enveloe :

9 Obtaining Effective Enveloes with with Effective vs. Deterministic Enveloe Enveloes A*=min (Pt, σ+ρt) Tye 1 flows: P =1.5 Mbs ρ =.15 Mbs σ =95400 bits Tye 2 flows: P = 6 Mbs ρ =.15 Mbs σ = bits Tye 1 flows

10 Effective vs. Deterministic Enveloe Enveloes Traffic rate at t = 50 ms Tye 1 flows Scheduling Algorithms Consider a work-conserving scheduler with rate R Consider class-q arrival at t with t+d q : Arrivals from class Class- arrivals from class which Are transmitted before tagged arrival. d q t τˆ t t τ Tagged Limit arrival (Scheduler Deendent) Tagged arrival has no delay bound violation if su ˆ τ AC ( t ˆ, τ t + τ ) R( ˆ τ + d t +d + q Deadline of Tagged arrival q ) 0

11 Scheduling Algorithms Arrivals from class d q with t τˆ t t + τ t +dq su A ( ˆ, ) ( ˆ C t τ t + τ τ + dq ) 0 ˆ τ FIFO: SP: EDF: τ τ = 0. ˆ τ ( > q), 0 ( = q), d ( < q). = q τ = max{ ˆ, τ d d }. q Admission Control for Scheduling Algorithms 2000 with Deterministic Enveloes: su ˆ τ with Effective Enveloes: ε / Q su GC ( τ + ˆ) τ ˆ τ d ˆ τ with Strong Effective Enveloes: l, ε / Q su HC ( τ + ˆ) τ ˆ τ d ˆ τ A * j ( τ + ˆ) τ ˆ τ d q q q

12 Effective vs. Deterministic Enveloe Enveloe C= 45 Mbs, ε = 10-6 Delay bounds: Tye 1: d 1 =100 ms, Tye 2: d 2 =10 ms, Statistical multilexing makes a big difference Scheduling has small imact Effective Enveloes and Effective Bandwidth 2002 Effective Bandwidth (Kelly, Chang) Given α(s,τ), an effective enveloe is given by

13 Effective Enveloes and Effective Bandwidth Now, we can calculate statistical service guarantees for schedulers and traffic tyes C= 100 Mbs, ε = 10-6 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 Statistical Network Calculus with Min-Plus Algebra A(t) delay=w(s) backlog=b(s) D(t) s A(t) D(t) S(t)

14 Convolution and Deconvolution oerators Convolution oeration: f g (t) = inf τ [0,t] f ( t τ ) + g( τ ) g(t) Deconvolution oeration f(t) f g (t) = su f ( t + τ ) g( τ ) τ [0,t] Imulse function: τ f*g(t) t Service Curves (Cruz 1995) A (minimum) service curve for a flow is a function S such that: D( t) A S ( t), t 0 Examles: Constant rate service curve: Service curve with delay guarantees: S ( t) = c t S ( t) = δ d (t)

15 Network Calculus Main Results (Cruz, Chang, (Cruz, Chang, LeBoudec) 1. Outut Enveloe: A * S is an enveloe for the deartures: A * S ( t) D ( t +τ ) - D ( τ ) 2. Backlog bound: A * S (0) is an uer bound for the backlog B 3. Delay bound: An uer bound for the delay is d max inf τ [0,t] * { d 0 t 0 : A ( t d) S( t) } Network Service Curve (Cruz, Chang, Traffic Conditioning (Cruz, Chang, LeBoudec) Sender S 1 S 2 S net S 3 Receiver Network Service Curve: If S 1, S 2 and S 3 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.

16 Statistical Network Calculus 2001 A (minimum) service curve for a flow is a function S such that: D( t) A S ( t), t 0 A (minimum) effective service curve for a flow is a function S ε such that: Pr ε [ D( t) A S ( t) ] 1-ε, t 0 Statistical Network Calculus Theorems Outut Enveloe: * ε A S is an enveloe for the deartures: Pr * ε [ A S ( t) D( t + τ) -D( τ) ] 1-ε, t, τ 0 2. Backlog bound: A * S ε (0) is an uer bound for the backlog Pr * ε [ B( t) A S (0)] 1-ε, t 0 3. Delay bound: A robabilistic uer bound for the delay, i.e., d max inf τ [0,t] * ε { d 0 t 0 : A ( t d) S ( t) } [ W ( t) d ] 1-ε, t 0 Pr max

17 Effective Network Service Curve 2002 Network Service Curve: If S 1,ε, S 2,ε S H,ε are effective service curves for a flow at nodes, then. Pr [ D( t) A ( S 1, ε S 2, ε... S H, ε δ H a )( t)] 1- Htε / a Unfortunately, this network service is not very useful! A good network service curve can be obtained by working with a modified service curve definition What is the cause of the roblem with the network effective service curve? A 1 D 1= A 2 D 2 Sender S 1, ε S 2, ε Receiver In the convolution D 2 ( t) A 2 2, ε 2 S ( t) = A ( t τ ) + S inf τ [0,t] 2, ε ( τ ) the range [0,t] where the infimum is taken is a random variable that does not have an a riori bound.

18 Statistical Per-Flow Service Bounds 2002 Sender Service available to aggregate S C Receiver Given: Service guarantee to aggregate (S C ) is known Total Traffic is known What is a lower bound on the service seen by a single flow? Statistical Per-Flow Service Bounds 2002 Sender Service available to aggregate S C Receiver Can show: is an effective service curve for a flow where is a strong effective enveloe and is a robabilistic bound on the busy eriod

19 Effective service curve of a single flow Tye 1 flows: Bandwidth needed by a erflow allocation to meet a delay bound of d=10ms Number of flows that can be admitted 2002 Tye 1 flows: Goal: robabilistic delay bound d=10ms

20 Conclusions Convergence of deterministic and statistical analysis with new constructs: Effective enveloes Effective service curves Preserves much (but not all) of the deterministic calculus Oen issues: So far: Often need bound on busy eriod or other bound on relevant time scale. Many roblems still oen for multi-node calculus Adative service curves Modified convolution oeration A t o g (t) = min g(t - t 0), B(t 0) +, inf A( t0, t τ ) + g( τ ) τ [0,t-t0 ] traffic arrivals B(t 0 ) deartures 0 t 0 t time

21 Adative service curves D( t 0, t) A S( t) t o Many service curves are adative ( Cruz/Okino, LeBoudec) Obtain service curve with t 0 =0 adative service curve: t, t 0 0 l l-adative service curve: D( t 0, t) A S ( t) t o t, t 0, t t l-adative effective l, ε service curve: [ D( t, t) A t S ( t) ] o strong (l-adative) effective Pr 0 1-ε t, t0 0, t -t0 service curve: [ D(, ) ( ), [, ] ] l, ε t t A T t t t I l l Pr 0 t 0 l 1-ε o 0-0 Effective Network Service Curve Traffic Conditioning A* Sender T 1,l,ε T 2,l,ε T net,l,3 ε T 3,l,ε Receiver Network Service Curve: If T 1,l,ε, T 2,l,ε, and T 3,l,ε, are strong effective service curves for a flow at nodes, then T net,l, 3ε = T 1,l,ε * T 2,l,ε * T 3,l,ε is a service curve for the entire network.

22 Recover original effective setwork curve Given a strong effective service curve T l,ε. If the backlog clears in any time interval of length l with robability ε 1, i.e, [ t [ t l, t]:b( t ) = 0] Pr ε Then S ε+ε 1 is an effective service curve