Regenerative estimator of the overflow probability in a tandem network Irina Dyudenko, Evsey Morozov, Michele Pagano Institute of Applied Mathematical Research, Karelian Research Center and Petrozavodsk University Dept. of Information Engineering, University of Pisa, Italy Rennes – RESIM 2008 September 2008
Outlines 2 ☞ Network scenario ➳ Quality of Service (QoS) ➳ Bandwidth allocation techniques ☞ Effective Bandwidth (EB) ➳ Definition and physical meaning ➳ Scaled Cumulant Generating Function (SCGF) ☞ Effective Bandwidth estimators ➳ Standard estimator: the batch-mean approach ➳ Refined estimator for input traffic with a regenerative structure ☞ Performance Comparison ➳ Tandem network with 2 queues and deterministic service rate ➳ Variance of the estimator ☞ Conclusions Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Network scenario 3 ☞ Evolution of network services and architectures ➳ Quality of Service (QoS) requirements ➳ Packet loss (usually approximated through the overflow probability) as a typical QoS guarantee ➳ Traffic Engineering ☞ Necessity of dynamic control mechanisms ➳ Bandwidth allocation techniques ➳ Admission Control ☞ The notion of effective bandwidth has emerged as a powerful metric to quantify the amount of resources needed by connections in order to guarantee a required QoS level ☞ Theorethical background: Large Deviation Theory ➳ For a wide class of buffered systems the overflow probability decreases exponentially fast as buffer size increases ➳ The rate-function allows to calculate the required exponent ☞ Open issue: estimation of the effective bandwidth from traffic measurements Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Workload process in a single server queue 4 X n C W n ☞ Stationarity condition: E X n < C ☞ Workload process: � u � � W n = sup X i − C · u u ≤ n i =1 ☞ Under mild assumptions, stationary workload W n ⇒ W exists and satisfies a Large Deviation Principle (LDP) P ( W > b ) ≈ e − δ b Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Large Deviation Principle for the workload process 5 1 lim x log P ( W > x ) = − δ x →∞ where ☞ the exponent is defined as δ = δ ( C ) = sup { θ > 0 : Λ( θ ) < Cθ } ☞ the quantity 1 n log E e θ P n ∆ i =1 X i Λ( θ ) = lim = n →∞ Λ n ( θ ) lim n →∞ is known as Limiting Scaled Cumulant Generating Function (LSCGF) In case of an i.i.d. sequence X n , the previous limit leads to Λ( θ ) = log E e θX Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Effective Bandwidth (EB) 6 ☞ In real networks, it is reasonable to assume that the buffer size b is fixed, while the capacity C can be varied ☞ Goal: determine the minimum service capacity C Γ which guarantees an overflow probability ≤ Γ C Γ = min( C : e − δ ( C ) b ≤ Γ) ☞ Hence, we obtain δ ( C Γ ) = θ ∗ = − log(Γ) b and, since δ = δ ( C ) = sup { θ > 0 : Λ( θ ) < Cθ } we can express C as a function of θ ∗ (provided that Λ( θ ) is known): C Γ = C (Γ , b ) = Λ( θ ∗ ) θ ∗ ☞ Given the QoS parameter Γ and the buffer size b , the value C Γ = C (Γ , b ) is called the effective bandwidth of the incoming traffic Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
EB Estimation: batch-mean approach 7 ☞ If the samples X i were i.i.d., then we could estimate the LSCGF for θ = θ ∗ by means of the sample mean estimator n Λ n ( θ ∗ ) = log 1 � e θ ∗ X � e θ ∗ X i → log E ˆ � n i =1 ☞ Aggregation into blocks of size B X 1 X 2 X 3 X B X B +1 X B +2 X 2 B X ( k − 1) B +1 X kB X 1 X 2 X 3 X B X B +1 X B +2 X 2 B X ( k − 1) B +1 X kB B 2 B kB ˆ � ˆ � ˆ � X 1 = X i X 2 = X i X k = X i i =1 i = B +1 i =( k − 1) B +1 ☞ If B is large enough, the dependence between blocks should be negligible , and ˆ X i constitute (approximately) an i.i.d. sequence Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
EB Estimation: batch-mean approach 8 X 1 X 2 X 3 X B X B +1 X B +2 X 2 B X ( k − 1) B +1 X kB B 2 B kB ˆ � ˆ � ˆ � X 1 = X i X 2 = X i X k = X i i =1 i = B +1 i =( k − 1) B +1 ☞ By using n = kB observations, we obtain a modified expression of the LSCGF 1 k X = 1 n log E e θ ∗ ˆ B log E e θ ∗ ˆ i =1 ˆ n log E e θ ∗ P k X i = lim X Λ( θ ∗ , B ) = lim n →∞ n →∞ ☞ Batch-mean estimator for the LSCGF n/B k Λ n ( θ ∗ , B ) = 1 B log 1 X i = 1 B log B e θ ∗ ˆ e θ ∗ ˆ ˆ � � X i k n i =1 i =1 ☞ Batch-mean estimator for the EB ˆ Λ n ( θ ∗ , B ) ˆ C (Γ , b ) = θ ∗ Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
EB Estimation: regenerative approach 9 α 1 = β 1 − β 0 α 2 = β 2 − β 1 α k = β k − β k − 1 β 0 = 0 β 1 − 1 β 1 β 2 − 1 β 2 β k − 1 β k − 1 β k β 1 − 1 β 2 − 1 β k − 1 ˆ � ˆ � ˆ � X 1 = X i X 1 = X i X k = X i i = β 0 i = β 1 i = β k − 1 ☞ Assumption: regenerative structure of the input with regeneration points β n ➳ randomization of the block size with E α < ∞ ➳ RVs ˆ X n are i.i.d. ☞ Alternative definition of the EB estimator (which uses k regenerative blocks, corresponding to β k observations) k Λ k ( θ ∗ ) = k log 1 e θ ∗ ˆ ˆ � X i β k k i =1 Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Heuristic justification 10 ☞ Considering a large number of time slots n including k regeneration cycles, i.e. where n − β k n = β k + ( n − β k ) → 0 n ☞ Roughly speaking, we have: 1 i =1 X i ≈ 1 X i = k log E e θ ∗ ˆ i =1 ˆ n log E e θ ∗ P n log E e θ ∗ P k X β k β k ☞ From renewal theory k 1 → β k E α ☞ By the Strong Law of Large Numbers k 1 e θ ∗ ˆ X i → E e θ ∗ ˆ � X k i =1 ☞ Hence k Λ k ( θ ∗ ) = k log 1 1 e θ ∗ ˆ E α log E e θ ∗ ˆ X i → ˆ � X β k k i =1 Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Lower bound 11 ☞ Notation: ∆ ➳ k ( n ) = max( k : β k ≤ n ) ➳ E e θ ∗ ˆ ∆ X = a < ∞ ☞ By using conditional expectation and taking into account the independence between k ( n ) and the regenerative blocks 1 i =1 X i ≥ 1 = 1 � E e θ ∗ P k ( n ) ˆ � n log E e θ ∗ P n X i | k ( n ) n log E a k ( n ) n log E i =1 ☞ By Jensen’s inequality log E a k ( n ) ≥ E log a k ( n ) = E k ( n ) log a ☞ From the elementary renewal theorem E k ( n ) 1 → n E α ☞ Hence we obtain the desired lower bound: 1 i =1 X i ≥ log a 1 E α log E e θ ∗ ˆ ∆ n log E e θ ∗ P n X lim inf = E α n →∞ Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Simulation Scenario 12 { t k } C 2 C 2 C 2 C 1 τ k = t k − t k − 1 b 1 ν 1 b 2 ν 2 ☞ The first station is fed by a renewal input ☞ Minimal construction of regenerations for the input to the second station β 0 = 0 t k > β n : ν 1 ( t − � � β n +1 = min k k ) = 0 n ≥ 0 ☞ Since the queue-size is bounded, if P ( τ > 1 /C 1 ) > 0 then the renewal process β = ( β n ) is positive recurrent, that is it has a finite mean E α < ∞ ☞ The target parameter to be estimated is the (constant) service rate C 2 for a given QoS level (overflow probability ≤ Γ ) Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Regeneration Cycle 13 Packet arrivals and departure at the first queue ν 1 ( t ) = 0 ν 1 ( t ) > 0 ν 1 ( t ) = 0 ν 1 ( t ) > 0 β k β k +1 α k = β k +1 − β k α k = β k +1 − β k Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Simulation set–up 14 λ C 2 C 2 C 2 C 1 b 1 ν 1 b 2 ν 2 ☞ The input traffic is Poisson( λ ) , i.e. τ ∈ E ( λ ) ☞ We compare two different estimatators ➳ Regenerative approach ( ˆ C REG ) 2 ➳ Batch-mean approach ( ˆ C BM 2 ) ➠ The block size B is chosen according to U [500; 20000] ➠ For a given value of B , 1000 blocks are considered ➠ 200 estimates are generated for different values of B ☞ Then variances of both estimators are calculated based on 200 sample paths Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
Simulation Results - b 2 = 60 15 C 1 = 0 . 35 , λ = 0 . 2 , b 1 = 100 , Γ = 0 . 0001 Simulation Parameters: C 1 = 0 . 35 , λ = 0 . 2 , b 1 = 100 , Γ = 0 . 0001 C 1 = 0 . 35 , λ = 0 . 2 , b 1 = 100 , Γ = 0 . 0001 250 BatchMean Regeneration 200 150 Variance 100 50 0 0 2 4 6 8 10 12 14 16 18 20 Simulation Trial Regenerative estimator of the overflow probability in a tandem network RESIM 2008 – September 2008 Michele Pagano
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