Storage processes in Poisson approximation scheme
Discrete storage processes, given by a sum of random variables on Markov and semi-Markov processes, are approximated by the Poisson compound processes on increasing time intervals.
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irk-123456789-45692009-12-08T12:00:34Z Storage processes in Poisson approximation scheme Koroliuk, V.S. Discrete storage processes, given by a sum of random variables on Markov and semi-Markov processes, are approximated by the Poisson compound processes on increasing time intervals. 2008 Article Storage processes in Poisson approximation scheme / V.S. Koroliuk // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 67-76. — Бібліогр.: 7 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4569 en Інститут математики НАН України |
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Discrete storage processes, given by a sum of random variables on Markov and semi-Markov processes, are approximated by the Poisson compound processes on increasing time intervals. |
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Storage processes in Poisson approximation scheme / V.S. Koroliuk // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 3-4. — С. 67-76. — Бібліогр.: 7 назв.— англ. |
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Theory of Stochastic Processes
Vol.14 (30), no.3-4, 2008, pp.67-76
VOLODYMYR S. KOROLIUK
STORAGE PROCESSES IN POISSON
APPROXIMATION SCHEME
Discrete storage processes, given by a sum of random variables on
Markov and semi-Markov processes, are approximated by the Poisson
compound processes on increasing time intervals.
Introduction
Renewal storage process (RSP) defined by a sum of independent identi-
cally distributed random variables αn, n ≥ 1 taking values in Euclidean
space Rd
ρ(t) = u+
ν(t)∑
n=1
αn, t ≥ 0,
where the counting renewal process ν(t) = max
{
n : τn ≤ t
}
, t ≥ 0,
is defined by renewal moments τn n ≥ 0, (τ0 = 0) on real line R+ =[
0, +∞ )
.
RSP has various interpretations in applications [1-3]. The main prob-
lem is to investigate the behavior of the RSP on increasing time inter-
vals as t → ∞ . An effective method is to introduce the parameter series
ε→ 0 (ε > 0) in such a way that the limit theorems for stochastic pro-
cesses may be used [1-5].
Asymptotic analysis of random evolution process is the most effective
approach to get limit result for RSP in the series scheme. The theorem
of Poisson approximation for RSP is realized under different assumptions
for the renewal process ν(t), t ≥ 0, driven by Markov or semi-Markov
processes.
2000 Mathematics Subject Classifications. 60G50.
Key words and phrases. Discrete storage processes, sum of random variables, Markov
and semi-Markov processes, approximation, Poisson compound process.
67
68 VOLODYMYR S. KOROLIUK
1.1. Renewal processes with Poisson jumps
Storage processes (SP) in the series scheme with small parameter series
ε → 0 (ε > 0) are given by relation
ρε(t) = u+
ν(t/ε)∑
n=1
αεn , t > 0 (1.1)
where the counting process
ν(t) = max {n : τn ≤ t} , τn =
∑n
k=1 θk, n ≥ 0, τ0 = 0,
defined by i.i.d. random variables θk, k ≥ 0 with the distribution func-
tion G(t) = P (θk ≤ t) , G(0) = 0. The random variables αεn, n ≥ 1,
take values at the real line (or in Rd, d > 1). The Poisson approximation
conditions (PAC) (see [1, Ch. 7]) are given for the distribution functions
Φε(u) = P {αεn < u} , u ∈ R .
PAC 1: Approximation of distribution functions:∫
R
g(u)Φε(du) = ε
[
Φg + θεg
]
, g(u) ∈ C3 (R)
C3 (R) is the measure determining class:
Φg =
∫
R
g(u)Φ(du).
PAC 2: Approximation of mean values:∫
R
uΦε(du) = ε [a + θεa] ,
∫
R
u2Φε(du) = ε [c+ θεc ] .
The negligible terms |θε•| → 0 when ε→ 0.
Theorem 1. Under the conditions PAC 1-2 the weak convergence
ρε(t) ⇒ ρ0(t), ε→ 0
takes place. The limit compound Poisson process
ρ0(t) = u+ bt+
ν0(t)∑
n=1
α0
n , t ≥ 0. (1.2)
The distribution functions Φ0(u) = P (α0
n < u) of i.i.d. random variables
α0
n, n ≥ 1, defined by the relation
Eg
(
α0
k
)
=
∫
R
g(u)Φ0(du) = Φg/Φ (R) . (1.3)
STORAGE PROCESSES 69
The counting Poisson process ν0(t), t ≥ 0, is given by the intensity
Eν0(t) = q0t, q0 = qΦ (R) = qΛ, q = 1/Eθ, Λ := Φ (R) (1.4)
The parameter of counting drift
b = q
(
a− Λa0
)
, a0 = Eα0
k. (1.5)
Remark 1. The limit compound Poisson process (1.2) can be represented
as follows
ρ0(t) = u+ qat+
ν0(t)∑
n=1
α̃n, α̃n = α0
n − a0. (1.6)
Example 1. Φε(au) =
{
εΛ, du = a0,
1 − εΛ du = εa0,
Eαεn = ε (Λa0 + a0) + εθεa,
a = a0 + Λa0, Φg = Λg (a0) , Φ0(g) = Φ(g)/Λ, α0
n = a0, b = q(a−
−ΛEα0
n) = q (a− Λa0) = qa0.
Remark 2. The intensity q0 = qΛ is proportional to average intensity of
the renewal moments and the intensity Λ of big jumps of the sum (1.1).
Remark 3. Under the conditions PAC 1-2 the small jumps (εa0) are trans-
formed into continuous drift, and the big jumps (a0) are get as jumps of the
limit compound Poisson process.
1.2. Predictable characteristics of storage process
It is easy to calculate the predictable characteristics of the storage pro-
cess (1.1) [4]: Bε(t) = εν (t/ε) [a + θεb ], C
ε(t) = εν (t/ε) [c+ θεc ], Φε
g(t) =
εν (t/ε)
[
Φg + θεg
]
. According to renewal theorem [7, Ch. 9]: εν (t/ε) ⇒ qt,
ε→ 0, q = 1/Eθ .
Under the conditions of Theorem 1 we have the following limit re-
sults for ε → 0 : Bε(t) ⇒ qat, Cε(t) ⇒ qct, Φε
g(t) ⇒ qΦgt = qΛΦ0
gt.
Φε
g(t) ⇒ qΦgt = q
∫
R
g(u)Φ (du) t = qΛ
∫
R
g(u)Φ0(du)t. Here Φ0(du) =
Φ (du) /Φ(R); Λ := Φ(R).
Now the predictable characteristics B0(t) = qat, C0(t) = qct, Φ0
g(t) =
qΛΦ0
gt define the limit compound Poisson process with the drift:
ρ0(t) = u+ aqt+
ν0(t)∑
n=1
(
α0
n − a0
)
, t ≥ 0
70 VOLODYMYR S. KOROLIUK
or, another form, is (1.2) with
b = q (a− Λa0) , a0 = Eα0
n =
∫
R
uΦ0(du)
2.1. Storage process at Markov process
Markov storage process (MSP) in a series scheme is defined as follows
ρε(t) = u+
ν(t/ε)∑
n=1
αεn (κn) , t ≥ 0 (2.1)
where Markov process κ (t) , t ≥ 0 , at a standard phase space (E, ε) is
given by a generator [1, Ch. 1]
Qϕ(x) = q(x)
∫
E
P (x, dy) [ϕ(y) − ϕ(x)] . (2.2)
Counting process
ν(t) := max {n : τn ≤ t} , τn+1 = τn + θn+1, n ≥ 0 (2.3)
and renewal moments θn are defined by conditional distribution functions
Gx(t) := P (θx ≤ t) = P {θn+1 ≤ t |κn = x} =
= 1 − e−q(x)t, t ≥ 0, x ∈ E. (2.4)
The embedded Markov chain (EMC) κn, n ≥ 0 is defined by a stochastic
kernel
P (x,B) = P (κn+1 ∈ B |κn = x) , x ∈ E, B ∈ ε. (2.5)
We suppose that the EMC is uniformly ergodic with the stationary distri-
bution ρ (B) , B ∈ ε. The family of random variables αεn(x), x ∈ E, n ≥ 1
is defined by a family of distribution functions
Φε
x(u) = P (αεn(x) < u) , u ∈ R, x ∈ E. (2.6)
The conditions of Poisson approximation are also supposed [1, Ch. 7]:
PAC1:
∫
R
g(u)Φε
x(du) = ε
[
Φg(x) + θεg(x)
]
, g(u) ∈ C3 (R) ,
PAC2:
∫
R
uΦε
x(du) = ε [a(x) + θεa(x)],
∫
R
u2Φε
x(du) = ε [c(x) + θεc(x)]
with the negligible terms sup
x∈E
|θε•(x)| → 0 , ε→ 0.
Theorem 2. Under the conditions PAC 1-2 the storage process (2.1) con-
verges weakly to a compound Poisson process
ρ0(t) = u+ bt+
ν0(t)∑
n=1
α0
n, t ≥ 0. (2.7)
STORAGE PROCESSES 71
Distribution function Φ0(u) = Φ(u)/Φ(R) = P (α0
n < u) of i.i.d. random
variables α0
n, n ≥ 1, is defined as
Φ(u) = q
∫
E
ρ(dx)Φx(u), Φg(x) =
∫
R
g(u)Φx(du), g ∈ C3 (R) .
The compound Poisson process ν0(t), t ≥ 0 is given by the intensity
Eν0(t) = q0t, q0 = qΛ, Λ := Φ(R).
The velocity of continuous drift b = q (a− Λa0) , a0 = Eα0
n . The average
intensity of Markov process
q =
∫
E
π(dx)q(x), π (dx) q(x) = qρ(dx),
where π(B), B ∈ ε is the stationary distribution of Markov process κ(t),
t ≥ 0.
2.2. Predictable characteristics of Markov storage process
(MSP)
According to the theorem about the representation of semimartingale
(see [4, Ch. 2]), predictable characteristics of MSP are given as:
Bε(t) =
ν[t/ε]∑
n=1
E [αn (κn) |Fn−1 ] ,
Cε(t) =
ν[t/ε]∑
n=1
E
[
α2
n (κn) |Fn−1.
]
, (2.8)
Φε
g(t) =
ν[t/ε]∑
n=1
E [g (αεn (κn)) |Fn−1 ] ,
where Fn−1 := σ
{
κr, r ≤ n− 1
}
, n ≥ 1 is a family of σ-algebras.
According to the main assumptions PAC 1-2, the predictable character-
istics of MSP have the following form
Bε(t) = Bε
0(t) + θεb(t), C
ε(t) = Cε
0(t) + θε0(t), Φε
g(t) = Φε
g,0(t) + θεg(t), (2.9)
where the main parts are normalized increment processes
Bε
0(t) = ε
ν(t/ε)∑
n=1
a (κn) , C
ε
0 = ε
ν(t/ε)∑
n=1
c (κn) , Φε
g,0(t) = ε
ν(t/ε)∑
n=1
Φε
g(κn).
72 VOLODYMYR S. KOROLIUK
Now the weak convergence of predictable characteristics (2.8) is equiva-
lent to the weak convergence of normalized processes with increments (2.10)
that follows from the Theorem 3.2 [1] . Limit predictable characteristics are
the following:
B0(t) = a0t, C0(t) = c0t, Φ0
g(t) = Φ0
gt, (2.11)
where
a0 = qa, c0 = qc, Φ0
g = qΦg, Φg = Φ0
gΛ, (2.12)
a =
∫
E
ρ(dx)a(x), c =
∫
E
ρ(dx)c(x), Φg =
∫
E
ρ(dx)Φg(x). (2.13)
Predictable characteristics (2.11)-(2.13) define the limit compound Pois-
son process (2.7).
3.1. Semi-Markov storage process (SMSP)
SMSP in a series scheme is defined by a correlation (as in (2.1))
ρε(t) = u+
ν(t/ε)∑
n=1
αεn (κn) , t ≥ 0 (3.1)
with semi-Markov switching process κ(t), t ≥ 0, that is defined by a semi-
Markov kernel [1, Ch. 1]
Q (x,B, t) = P (x,B)Fx(t) , x ∈ E, B ∈ ε, t ≥ 0 (3.2)
Stochastic kernel P (x,B) , x ∈ E, B ∈ ε defines the transition proba-
bilities of embedded Markov chain κn, n ≥ 0.
Counting process
ν(t) = max
{
n : τn ≤ t
}
, n ≥ 0 (3.3)
is defined by renewal moments
τn+1 = τn + θn+1, n ≥ 0
where the times between renewing θn+1, n ≥ 0 are defined by conditional
distribution functions
Fx(t) = P (θn+1 ≤ t |κn = x) =: P (θx ≤ t.) (3.4)
The main assumption is that SMP κ(t), t ≥ 0 is uniformly ergodic
with stationary distribution π (B) , B ∈ ε, that satisfies the correlation
π(dx)q(x) = qρ(dx), q =
∫
E
π(dx)q(x), (3.5)
STORAGE PROCESSES 73
where the averaged intensity
q(x) = 1/m(x), m(x) =
∫∞
0
F̄x(t)dt, F̄x(t) := 1 − Fx(t). (3.6)
Stationary distribution ρ (dx) of EMC κn, n ≥ 0 satisfies the corre-
lation
ρ(B) =
∫
E
ρ(dx)P (x,B) , B ∈ ε, ρ (E) = 1.
The family of random variables αεn(x), x ∈ E, n ≥ 1 that are in-
dependent in general, is defined by the distribution function Φε
x(du) =
P (αεn(x) ∈ du).
Theorem 3. The conditions of Poisson approximation are the following:
PAC 1:
∫
R
uΦε
x(du) = [a(x) + θεa(x)] ,
∫
R
u2Φε
x(du) = ε [c(x) + θεc(x)] ,
PAC 2:
∫
R
g(u)Φε
x(du) = ε
[
Φg(x) + θεg(x)
]
, g(u) ∈ C3 (R) ,
Φg(x) =
∫
R
g(u)Φx(du).
Under the conditions PAC 1-2 the following weak convergence
ρε(t) ⇒ ρ0(t), ε→ 0
takes place.
The limit compound Poisson process ρ0(t) is defined by its predictable
characteristics
B0(t) = b0t, C0(t) = c0t, Φ0
g(t) = qΦgt (3.7)
where
Φg =
∫
E
ρ(dx)Φg(x) = ΛΦ0
g , b0 = qb, c0 = qc
b =
∫
E
ρ(dx)a(x)c =
∫
E
ρ(dx)c(x),Λ = Φg(R), Φg =
∫
R
g(u)Φg(du). (3.8)
3.2. Predictable characteristics of SMSP
Predictable characteristics of SMSP (3.1) have the following form:
Bε(t) =
ν(t/ε)∑
n=1
E
[
αεn (κn)
∣∣Fn−1
]
74 VOLODYMYR S. KOROLIUK
Cε(t) =
ν(t/ε)∑
n=1
E
[
(αεn (κn))
2
∣∣Fn−1
]
(3.9)
Φε
g(t) =
ν(t/ε)∑
n=1
E
[
g (αεn (κn))
∣∣Fn−1
]
.
According to the assumptions PAC 1-2 predictable characteristics (3.9)
are the following
Bε(t) = Bε
0(t) + θεb(t),C
ε(t) = Cε
0(t) + θεc(t),Φ
ε
g(t) = Φε
g,0(t) + θεg(t), (3.10)
where
Bε
0(t) = ε
ν(t/ε)∑
n=1
a (κn), C
ε
0(t) = ε
ν(t/ε)∑
n=1
c (κn)Φ
ε
g,0(t) = ε
ν(t/ε)∑
n=1
Φg (κn) (3.11)
and negligible terms |θε•(t)| → 0 when ε → 0 .
Now the process of increments (3.11) at Markov chain κn, n ≥ 0 con-
verges weakly at ε → 0 according to Theorem 3.2. [1, Ch. 1]
Bε
0(t) ⇒ ât, Cε
0(t) ⇒ ĉt, Φε
0(t) ⇒ Φgt (3.12)
.
Under the conditions PAC 1-2 and main assumptions the following weak
convergence of predictable characteristics takes place:
Bε(t) ⇒ b0(t), Cε(t) ⇒ c0t, Φε
g(t) ⇒ Φgt
where b0, c0 and Φg are defined in (3.7)-(3.8).
The limit predictable characteristics define the limit compound Poisson
process ρ0(t) in Theorem 3 with predictable characteristics (3.7).
4. Storage processes superposition of two renewal processes.
4.1. The superposition of two renewal processes is given by two sequences
of sums (see [2, Ch. 1])
τ
(i)
n =
∑n
k=1 θ
(i)
k , n ≥ 1, τ
(i)
0 = 0, i = 1, 2 (4.1)
of i.i.d. positive random variables θ
(i)
k , k ≥ 1, i = 1, 2 , defined by dis-
tribution functions Pi(t) = P
{
θ
(i)
k ≤ t
}
, Pi(0) = 0, i = 1, 2 .
The superposition of two renewal processes is defined by a sum
ν(t) = ν1(t) + ν2(t) (4.2)
STORAGE PROCESSES 75
where νi(t) = max
{
n : τ
(i)
n ≤ t
}
, i = 1, 2 .
The superposition of two renewal processes (4.2) may be described using
a semi-Markov process
κ(t), t ≥ 0
at a phase space
E =
{
ix, i = 1, 2, x > 0
}
, θix = θ(i) ∧ x.
The first integer component i stands for an index of renewal moment,
the second continuous component x > 0 stands for the time left till the
moment of renewing with another index. The embedded Markov process
κn = κ (τn) , n ≥ 0 , is defined by a transition probability matrix (see [2,
Par. 1.2.4])
P =
[
P1 (x− dy) P1 (x+ dy)
P2 (x+ dy) P2 (x− dy)
]
. (4.3)
The distinguishing specialty of embedded Markov chain κn, n ≥ 0 ,
with transition probabilities (4.3) is its ergodicity with the stationary dis-
tribution
ρ1(dx) = ρ1P
∗
2 (x)dx, ρ2(dx) = ρ2P
∗
1 (x)dx (4.4)
where by the definition
P ∗
i (x) := P̄i(x)
/
mi, P̄i(x) := 1 − Pi(x) ,
ρ1 = ρm2, ρ2 = ρ,
here mi = Eθ
(i)
k =
∫∞
0
P̄i(x)dx.
The storage process at superposition of two renewal processes is defined
in an ordinary way
ρε(t) = u+
∑ν(t/ε)
n=1 αεn (κn) , t ≥ 0, (4.5)
i.i.d. random variables αεn(x), x ∈ E are defined by distribution func-
tions
Φε
ix(du) = P {αεn(ix) ∈ du} , i = 1, 2
that satisfy Poisson approximation conditions:
PAC1:
∫
R
uΦε
ix(du) = ε [ai(x) + θεai(x)] , i = 1, 2,∫
R
u2Φε
ix(du) = ε [ci(x) + θεci(x)] , i = 1, 2,
PAC2:
∫
R
g(u)Φε
ix(du) = ε
[
Φg(ix) + θεgi(x)
]
, i = 1, 2,
76 VOLODYMYR S. KOROLIUK
where Φg(ix) =
∫
R
g(u)Φix(du), g(u) ∈ C3 (R) .
Corollary 1. Under the conditions PAC 1-2 the weak convergence ρε(t) ⇒
ρ0(t) , ε → 0 takes place.
The limit compound Poisson process ρ0(t), t ≥ 0, is defined by its
predictable characteristics
B0(t) = qb0t, C0(t) = qc0t, Φ0
g(t) = qΦ0
gt,
b0 = ρ1Ea1 (θ∗2) + ρ2Ea2 (θ∗1) , c0 = ρ1Ec1(θ
∗
2) + ρ2Ec2 (θ∗1),
Φ0
g = ρ1
∫ ∞
0
P ∗
2 (x)Φg(1x)dx+ ρ2
∫ ∞
0
P ∗
1 (x)Φg(2x)dx
Conclusions
1) Asymptotic behavior of stochastic storage processes with critical
jumps in random media, described by Markov or semi-Markov processes,
at increasing time intervals are approximated by compound Poisson process
with continuous drift.
2) Critical stochastic events like catastrophes, large payments, etc. take
place by an exponential distribution of event’s time. Thus, in the models
of stochastic storage processes studied here the forecast of critical events is
impossible. Only statistical estimation of the intensity of critical events is
possible.
References
1. Korolyuk, V.S., Limnios, N., Stochastic Systems in Merging Phase Space.
- World Scientific, 2005.
2. Korolyuk, Vladimir S., Korolyuk, Vladimir V. Stochastic Models of Sys-
tems. - Kluwer Academic Publishers, 1999.
3. Anisimov, V.V. Convergence of storage processes with switching, Proba-
bility Theory and Math. Stat., 63 (2000), pp.3-12.
4. Borovskikh, Yu.V., Korolyuk, V.S., Martingale approximation, VSP, 1997.
5. Jacod, J., Shiryaev, A.N., Limit theorems for random processes, - M. Fiz-
matlit, 1994, v. 1,2.
6. Cinlar, E., Jacod, J., Protter, P., Sharpe, M.J., Semimartingales and
Markov processes, Zeitschift, 1980. v.54, pp. 161-219.
7. Feller, W., Introduction in Probability Theory and its Applications, v.2,
M.:Mir, 1967.
Institute of Mathematics, Kyiv, Ukraine
E-mail: korol@imath.kiev.ua
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