Renewal Processes
Renewal Processes
Bo Friis Nielsen1
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1 DTU
Informatics
02407 Stochastic Processes 7, October 11 2016
Bo Friis Nielsen
Phase type distribuions
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Phase type distribuions
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Conditional Expectation and Martingales
Renewal Processes
Renewal Processes
Bo Friis Nielsen
Renewal Processes
Age, Residual Life, and Total Life (Spread)
F (x) = P{X ≤ x}
Wn = X1 + · · · + Xn
N(t) = max{n : Wn ≤ t}
E(N(t)) = M(t) Renewal function
Bo Friis Nielsen
Renewal phenomena
Renewal Processes
γt
= WN(t)+1 − t( excess or residual life time)
δt
= t − WN(t) ( current life or age)
βt
= δt + γt ( total life or spread
Bo Friis Nielsen
Renewal Processes
Topics in renewal theory
Discrete Renewal Theory (7.6)
1
Elementary renewal theorem M(t)
t → µ
E WN(t)+1 = µ(1 + M(t))
Rt
P
M(t) = ∞
Fn (t) = 0 Fn−1 (t − x)dF (t)
n=1 Fn (t),
Rt
Renewal equation A(t) = a(t) + 0 A(t − u)dF (u)
P{X = k } = pk
SolutionR to renewal equation R
∞
t
A(t) = 0 a(t − u)dM(u) → µ1 0 a(t)dt
A renewal equation. In general
Limiting distribution of residual
life time
R
1 x
limt→∞ P{γt ≤ x} = µ 0 (1 − F (u))du
vn = bn +
M(n) =
pk [1 + M(n − k )]
k =0
= F (n) +
n
X
pk M(n − k )
k =0
n
X
pk vn−k
k =0
The solution is unique, which we can see by solving recursively
Limiting distribution of joint distribution
R ∞ of age and residual life
time limt→∞ P{γt ≥ x, δt ≥ y } = µ1 x+y (1 − F (z))dz
v0 =
Limitng distribution ofR total life time (spread)
x
t dF (t)
limt→∞ P{βt ≤ x} = 0 µ
Bo Friis Nielsen
n
X
v1 =
Renewal Processes
Bo Friis Nielsen
Discrete Renewal Theory (7.6) (cont.)
b0
1 − p0
b1 + p1 v0
1 − p0
Renewal Processes
The Discrete Renewal Theorem
Let un be the renewal density (p0 = 0) i.e. the probability of
having an event at time n
un = δn +
n
X
Theorem
7.1 Page 383 Suppose that 0 < p1 < 1 and that {un } and {vn }
are the solutions
to the renewal equations
P
P
vn = bn + nk =0 pk vn−k and un = δn + nk =0 pk un−k ,
respectively. Then
pk un−k
k =0
Lemma
P
7.1 Page 381 If {vnP
} satisifies vn = bn + nk =0 pk vn−k and un
satisfies un = δn + nk =0 pk un−k then
vn =
n
X
(a) limn→∞ un = P∞1 kp
k
k =0
P∞
(b) if k =0 |bk | < ∞ then limn→∞ vn =
P∞
k =0 bk
P∞
k =0 kpk
bn−k uk
k =0
Bo Friis Nielsen
Renewal Processes
Bo Friis Nielsen
Renewal Processes
Continuous Renewal Theory (7.6)
Expression for M(t)
P{Wn ≤ x} = Fn (x) with
Z x
Fn (x) =
Fn−1 (x − y )dF (y )
P{N(t) ≥ k } = P{Wk ≤ t} = Fk (t)
P{N(t) = k } = P{Wk ≤ t, Wk +1 > t} = Fk (t) − Fk +1 (t)
∞
X
k P{N(t) = k }
M(t) = E(N(t)) =
0
The expression for Fn (x) is generally quite complicated
Renewal equation
Z x
v (x) = a(x) +
v (x − u)dF (u)
k =1
=
0
Z
=
x
a(x − u)dM(u)
v (x) =
∞
X
k =0
∞
X
P{N(t) > k } =
∞
X
P{N(t) ≥ k }
k =1
Fk (t)
k =1
0
Bo Friis Nielsen
Renewal Processes
Bo Friis Nielsen
Poisson Process as a Renewal Process
E[WN(t)+1 ]


N(t)+1
E[WN(t)+1 ] = E 
= µ+
X
j=1
∞
X


N(t)+1
Xj  = E[X1 ] + E 
X
Xj 
j=1
E Xj 1(X1 + · · · + Xj−1
≤ t)
j=2
=µ+
Fn (x) =
∞
X
(λx)n
i=n
∞
X
E Xj E 1(X1 + · · · + Xj−1 ≤ t) = µ + µ
Fj−1 (t)
j=2
n!
e
−λx
=1−
n
X
(λx)n
i=0
M(t) = λt
Xj and 1(X1 + · · · + Xj−1 ≤ t) are independent
∞
X
Renewal Processes
Excess life, Current life, mean total life
j=2
E[WN(t)+1 ] = exp[X1 ]E[N(t) + 1] = µ(M(t) + 1)
Bo Friis Nielsen
Renewal Processes
Bo Friis Nielsen
Renewal Processes
n!
e−λx
The Elementary Renewal Theorem
E[N(t)]
1
M(t)
= lim
=
t→∞
t→∞
t
t
µ
lim
The constant in the linear asymptote
h
µ i σ 2 − µ2
lim M(t) −
=
t→∞
t
2µ2
Asymptotic Distribution of N(t)
When E[Xk ] = µ and Var[Xk ] = σ 2 both finite
(
)
Z x
N(t) − t/µ
1
2
p
√
lim P
≤x =
e−y /2 dy
2
3
t→∞
2π −∞
tσ /µ
Example with gamma distribution Page 367
Bo Friis Nielsen
Renewal Processes
Limiting Distribution of Age and Excess Life
1
lim P{γt ≤ x} =
t→∞
µ
Z
Renewal Processes
Size biased distributions
x
(1 − F (y ))dy = H(x)
0
1
P{γt > x, δt > y } =
µ
Bo Friis Nielsen
Bo Friis Nielsen
Z
fi (t) =
∞
(1 − F (z))dz
x+y
Renewal Processes
t i f (t)
E Xi
The limiting distribution og β(t)
Bo Friis Nielsen
Renewal Processes
Delayed Renewal Process
Stationary Renewal Process
Delayed renewal distribution
R xwhere
P{X1 ≤ x} = Gs (x) = µ−1 0 (1 − F (y ))dy
Distribution of X1 different
t
µ
Prob{γt ≤ x} = Gs (x)
MS (t) =
Bo Friis Nielsen
Renewal Processes
Additional Reading
S. Karlin, H. M. Taylor: “A First Course in Stochastic
Processes” Chapter 5 pp.167-228
William Feller: “An introduction to probability theory and its
applications. Vol. II.” Chapter XI pp. 346-371
Ronald W. Wolff: “Stochastic Modeling and the Theory of
Queues” Chapter 2 52-130
D. R. Cox: “Renewal Processes”
Søren Asmussen: “Applied Probability and Queues” Chapter V
pp.138-168
Darryl Daley and Vere-Jones: “An Introduction to the Theory of
Point Processes” Chapter 4 pp. 66-110
Bo Friis Nielsen
Renewal Processes
Bo Friis Nielsen
Renewal Processes