Markovian Arrival Processes

68 Pages·2011·1.03 MB·English

by  Giuliano Casale[-10pt]

Building Accurate Workload Models Using
Markovian Arrival Processes
Giuliano Casale
DepartmentofComputing
ImperialCollegeLondon
Email:[email protected]
ACM SIGMETRICS Tutorial – June 7th, 2011
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Outline
Main topics covered in this tutorial:
• Phase-type (PH) distributions
• Moment matching
• Markovian arrival processes (MAP)
• Inter-arrival process fitting
Important topics not covered in this tutorial:
• Queueing applications, matrix-geometric method, ...
• Non-Markovian workload models (e.g., Pareto, matrix
exponential process, ARMA processes, fBm, wavelets, ...)
• Maximum-Likelihood (ML) methods, EM algorithm, ...
• ...
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1. PH DISTRIBUTIONS
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Continuous-Time Markov Chain (CTMC) Notation
• m states
• λ ≥ 0: (exponential) transition rate from state i to j
i,j
• λ = (cid:80)m λ : total outgoing rate from state i
i j=1 i,j
• Infinitesimal generator matrix:
   
−λ λ ... λ 1
1 1,2 1,m
λ2,1 −λ2 ... λ2,m 1
Q =  ... ... ... ... , Q1 = 0, 1 = ...
λ λ ... −λ 1
n,1 n,2 m
• π(t) = π(0)eQt: state probability vector at time t
• π (t): probability of the CTMC being in state i at time t
i
• π(0): initial state probability vector
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Example 1.1: CTMC Transient Analysis
Initial state:
π(0) = [0.9,0.0,0.1,0.0]
 
−4 4 0 0
 4 −7 2 1  Transient analysis:
Q =  
 2 3 −5 0  (cid:88)∞ (Qt)k
2 0 0 −2 π(t) = π(0)eQt = π(0)
k!
k=0
A Sample Path Transient Probabilities
1
4 0.8
current state23 state probabilities00..46
0.2
1
0 5 10 15 20 tim2e5 30 35 40 45 00 0.5 1 t1im.5e 2 2.5 3
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Phase-Type Distribution (PH)
• Q: CTMC with m = n+1 states, last is absorbing (λ = 0)
n+1
 
−λ λ ... λ λ
1 1,2 1,n 1,n+1
λ −λ ... λ λ 
(cid:20) (cid:21)  2,1 2 2,n 2,n+1
T t  
Q = 0 0 =  ... ... ... ... ... 
 
λ λ ... −λ λ 
n,1 n,2 n n,n+1
0 0 ... 0 0
• T: PH subgenerator matrix, T1 < 0
• t = −T1: exit vector
• “No mass at zero” assumption: π(0) = [α,0], α1 = 1
References:[Neu89]
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Example 1.2: Absorbing State Probability
Initial state: π(0) = [α,0.0] = [0.9,0.0,0.1,0.0]
 
−4 4 0 0    
−4 4 0 0
4 −7 2 1
Q =  ,T =  4 −7 2 ,t = −T1 = 1
 2 3 −5 0
2 3 −5 0
0 0 0 0
1 1
state probabilities000...468 orbing state probability000...468
0.2 bs0.2
a
0 0
0 5 10 15 0 5 10 15
time time
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PH: Fundamental Concepts
Basic idea: model F(t) = Pr(event occurs in X ≤ t time units) as
the probability mass absorbed in t time units in Q.
• Semantics: entering the absorbing state models the occurrence
of an event, e.g., the arrival of a TCP packet, the completion
of a job, a non-exponential state transition in a complex system
• Understanding the absorption dynamics:
 
∞ ∞
(cid:80) (Tt)k (cid:80) Tk−1tkt
π(t) = π(0)eQt = π(0) k! k! 
k=0 k=1 
0 1
• Using the definition t = −T1, we get
π(t) = [αeTt,π (t)], π (t) = 1−αeTt1
n+1 n+1
where π (t) is the probability mass absorbed in t time units.
n+1
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PH: Fundamental Formulas
PH distribution: F(t) = Pr(event occurs in X ≤ t) =def 1−αeTt1
• PH representation: (α,T)
• Probability Density function: f(t) = αeTt(−T)1
• (Power) Moments: E[Xk] = (cid:82)+∞tkf(t)dt = k!α(−T)−k1
0
• (−T)−1 = [τ ] ≥ 0
i,j
• τ : mean time spent in state j if the PH starts in state i
i,j
• Median/Percentiles: no simple form, determined numerically.
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Example 1.3: a 3-state PH distribution
   
−(λ+µ) λ µ 0
α = [1,0,0], T =  0 −λ λ ,t = 0, λ ≥ 0.
0 0 −λ λ
• Case µ → +∞: E[Xk] = k!λ−k (Exponential, c2 = 1).
• Case µ = λ: E[Xk] = (k +1)!λ−k (Erlang-2, c2 = 1/2)
• Case µ = 0: E[Xk] = (k+2)!λ−k (Erlang-3, c2 = 1/3).
2
• No choice of µ delivers c2 > 1
Remarks: Squared coefficient of variation: c2 =def Var[X]/E[X]2
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