Vertical and horizontal fluid queues in heavy and low traffic
The paper considers vertical and horizontal fluid queueing systems with consecutive
 service. The workload processes in these systems satisfy the Langevin equations with
 Poisson input. The objective is to investigate the main stationary characteristics in heavy and low traffic.
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| Cite this: | Vertical and horizontal fluid queues in heavy and low traffic / O.K. Zakusilo, N.P. Lysak // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 162–170. — Бібліогр.: 6 назв.— англ. |
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| author | Zakusilo, O.K. Lysak, N.P. |
| author_facet | Zakusilo, O.K. Lysak, N.P. |
| citation_txt | Vertical and horizontal fluid queues in heavy and low traffic / O.K. Zakusilo, N.P. Lysak // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 162–170. — Бібліогр.: 6 назв.— англ. |
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| description | The paper considers vertical and horizontal fluid queueing systems with consecutive
service. The workload processes in these systems satisfy the Langevin equations with
Poisson input. The objective is to investigate the main stationary characteristics in heavy and low traffic.
|
| first_indexed | 2025-12-07T17:25:26Z |
| format | Article |
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Theory of Stochastic Processes
Vol. 12 (28), no. 1–2, 2006, pp. 162–170
UDC 519.21
O. K. ZAKUSILO AND N. P. LYSAK
VERTICAL AND HORIZONTAL FLUID
QUEUES IN HEAVY AND LOW TRAFFIC
The paper considers vertical and horizontal fluid queueing systems with consecutive
service. The workload processes in these systems satisfy the Langevin equations with
Poisson input. The objective is to investigate the main stationary characteristics in
heavy and low traffic.
Introduction
Investigating the fluid queues became popular during the last decade. Contrary to
ordinary queueing system, the input flow to a fluid queue has continuous structure. A
fluid queue is the input-output system, whose input flow consists of substance flowing into
a reservoir and flowing out from it with random speed. Fluid models play an important
role in analyzing the operating characteristics of high-speed networks and repetition
work systems when a huge amount of small tasks are processed. They are actively
used in the sphere of telecommunications. Studying the fluid queues with priorities is
motivated by their usefulness in analyzing the effectiveness of ATM-commutators and
IP-routers which support classes of traffic with different qualities of service [6]. Involving
the fluid queues with priorities is effective for checking the overload in modern high-speed
integrated networks, such as Internet. Fluid queues are also used in the dam theory and
in transport systems for simulating the flow of transport facilities on crossroads.
1. Statement of the problem
The matters of investigation are two fluid systems which will be referred to as vertical
and horizontal. The choice of these names is explained by the peculiarities of their
functioning.
Vertical system is a fluid system with consecutive service consisting of n servers. The
input flow to the first server is given by a generalized Poisson process z1 (t) with parameter
λ and jumps η1
1 = η1, η2
1 , . . . , η
i
1, . . .. The output from any server constitutes the input
flow to the next server. The service speed on the k-th server is proportional to the value
of incomplete work on this server. Any demand entering the system has to pass through
all servers. To be served, the i-th demand needs ηi
1 units of work. If xk (t), k = 1, n,
denotes the value of incomplete work on the k-th server at the moment t, then it serves
with the speed μkxk (t), k = 1, n, and the vector of incomplete work
x (t) = (x1 (t) , x2 (t) , . . . , xn (t))T ∈ RRn
in this system satisfies the Langevin equation
dx (t) = Ax (t) dt + dz (t), (1)
2000 AMS Mathematics Subject Classification. Primary 60K25.
Key words and phrases. Fluid queueing system, heavy and low traffic.
162
VERTICAL AND HORIZONTAL FLUID QUEUES IN HEAVY AND LOW TRAFFIC 163
where z (t) = (z1 (t) , 0, . . . , 0)T ∈ R
n,
A =
∥∥∥∥∥∥∥∥∥
−μ1 0 0 . . . 0 0
μ1 −μ2 0 . . . 0 0
0 μ2 −μ3 . . . 0 0
. . . . . . . . . . . . . . . . . .
0 0 0 . . . μn−1 −μn
∥∥∥∥∥∥∥∥∥
,
μk > 0, k = 1, n.
Horizontal systems only differ from vertical in the matrices A. For horizontal systems,
processes xi (t) satisfy the system of differential equations
dx1 (t) = −μ1 (x1 (t) − x2 (t)) dt + dz1 (t),
dxk (t) = −μk (xk (t) − xk+1 (t)) dt + μk−1 (xk−1 (t) − xk (t)) dt,
k = 2, . . . , n − 1,
dxn (t) = −μnxn (t) dt + μn−1 (xn−1 (t) − xn (t)) dt,
where μk > 0, k = 1, 2, . . . , n.
Hence, the vector x (t) of incomplete work in this system satisfies the Langevin equa-
tion (1) with the same process z(t), but with the matrix
A =
∥∥∥∥∥∥∥∥∥∥∥
−μ1 μ1 . . . 0 0 0
μ1 − (μ1 + μ2) . . . 0 0 0
0 μ2 . . . 0 0 0
. . . . . . . . . . . . . . . . . .
0 0 . . . μn−2 − (μn−2 + μn−1) μn−1
0 0 . . . 0 μn−1 − (μn−1 + μn)
∥∥∥∥∥∥∥∥∥∥∥
,
where μk > 0, k = 1, n.
We establish conditions for the stationary regimes of these systems to exist and inves-
tigate the behavior of their stationary characteristics in heavy (λ → ∞) and low (λ → 0)
traffic.
2. Conditions of existence of stationary regimes
It follows from [1] that the process x(t) satisfying Eq. (1) possesses a limit distribution
as t → ∞ which does not depend on the initial value x0 = x(0) if and only if
a) all eigenvalues of A have negative real parts,
b) E(ln η1; η1 > 1) < ∞.
If these conditions are satisfied, then the limit distribution is the unique stationary
distribution of x(t) with the characteristic function
Ξξ (s) = exp{−λ
∫ ∞
0
(1 − ϕ(exp{uAT }s))du}, (3)
where ϕ(s) = E{exp i(s, η)}, s = (s1, s2, . . . , sn), η = (η1, 0, . . . , 0).
Note that, for a vertical system, condition a) is equivalent to
μ1 > 0, μ2 > 0, . . . , μn > 0.
164 O. K. ZAKUSILO AND N. P. LYSAK
3. Behavior of the stationary characteristics
of a vertical system in heavy traffic
Let the stationary distribution of the process
x (t) = (x1 (t) , x2 (t) , . . . , xn (t))T
coincide with the distribution of the vector ξ = (ξ1, ξ2, . . . , ξn)T . Let also
F (y1, y2, . . . , yn)
be the distribution function of η = (η1, 0, . . . , 0)T , Eη = (Eη1, 0, . . . , 0)T , Eη1 = m1.
Lemma 1. If Eη1 = m1 < ∞, then λ−1 (ξ1, ξ2, . . . , ξn)T converges in probability to
m1
(
μ−1
1 , μ−1
2 , . . . , μ−1
n
)T
as λ → ∞.
D. ue to condition a), ∫ ∞
0
exp
{
uAT
}
du = − (AT
)−1
,
and (3) implies
lim
λ→∞
Ξλ−1ξ (s) = exp
{
i
(
s1m1μ
−1
1 + s2m1μ
−1
2 + · · · + snm1μ
−1
n
)}
.
The following lemma is obvious.
Lemma 2. If Eη1 = m1 < ∞, then λ−1 (z1 (t) , 0, . . . , 0)T converges in probability to
(m1t, 0, . . . , 0)T as λ → ∞.
Assume that a demand entering the system at the moment t0 needs y units of work
to be served and that
x (t0) = (x1 (t0) , x2 (t0) , . . . , xn (t0)) = (x∗
1 + y, x∗
2, . . . , x
∗
n).
Denote, by T , the time to complete servicing this demand by all servers and, by Wi, the
time that passes till the moment when this demand enters the i-th server (i = 1, n). In-
troduce the process βi (t) which is equal to the amount of work executed by the i-th server
at the moment t ≥ t0 (without loss of generality, put t0 = 0). Then dβi (t) = μixi (t) dt,
βi (0) = 0. It follows from the definition of βi (t), that
{T > t} = {x∗
1 + x∗
2 + · · · + x∗
n + y > βn (t)},
{Wi > t} = {x∗
1 + x∗
2 + · · · + x∗
i > βi (t)}.
Rewriting system (1) in the form⎧⎪⎨
⎪⎩
dx1 (t) = dz1 (t) − μ1x1 (t) dt,
dx2 (t) = μ1x1 (t) dt − μ2x2 (t) dt,
. . .
dxn (t) = μn−1xn−1 (t) dt − μnxn (t) dt
and adding all equations of this system, we obtain
μnxn (t) dt = dz1 (t) − dx1 (t) − dx2 (t) − · · · − dxn (t),
dβn (t) = dz1 (t) − dx1 (t) − dx2 (t) − · · · − dxn (t),∫ t
0
dβn (u) =
∫ t
0
dz1 (u) −
∫ t
0
dx1 (u) −
∫ t
0
dx2 (u) − · · · −
∫ t
0
dxn (u),
βn (t) = z1 (t) − x1 (t) + x1 (0) − x2 (t) + x2 (0) − · · · − xn (t) + xn (0)
= z1 (t) − x1 (t) − x2 (t) − · · · − xn (t) + x∗
1 + x∗
2 + · · · + x∗
n + y.
VERTICAL AND HORIZONTAL FLUID QUEUES IN HEAVY AND LOW TRAFFIC 165
Hence,
P {T > t} = P {x1 (t) + x2 (t) + · · · + xn (t) > z1 (t)}. (4)
In a similar manner, we get
P {Wi > t} = P {x1 (t) + x2 (t) + · · · + xi (t) > z1 (t) + y} . (5)
If the system operates in a stationary regime, we use notations T s and W s
i for the analogs
of the above-mentioned characteristics. Their distributions are given by formulas (4) and
(5), respectively, provided that the distribution of x (t) = (x1 (t) , x2 (t) , . . . , xn (t))T is
stationary and coincides with that of ξ = (ξ1, ξ2, . . . , ξn)T [2].
Theorem 1. If Eη1 = m1 < ∞, then(
μ−1
1 + μ−1
2 + · · · + μ−1
n
)−1
T s w.⇒ 1
as λ → ∞.
E. quality (4) implies
P {T s > t} = P {ξ1 + ξ2 + · · · + ξn > z1 (t)} = P
{
λ−1 (ξ1 + ξ2 + · · · + ξn) > λ−1z1 (t)
}
.
Lemmas 1 and 2 give
P {T s > t} w.⇒P
{
m1μ
−1
1 + m1μ
−1
2 + · · · + m1μ
−1
n > m1t
}
= P
{
μ−1
1 + μ−1
2 + · · · + μ−1
n > t
}
=
{
1, t < t1
0, t ≥ t1
as λ → ∞, where t1 = μ−1
1 + μ−1
2 + · · · + μ−1
n . This suffices to prove the theorem.
Theorem 2. If Eη1 = m1 < ∞, then(
μ−1
1 + μ−1
2 + · · · + μ−1
i
)−1
W s
i
w.⇒ 1
as λ → ∞.
Theorems 1 and 1 allow us to investigate the following characteristics:
time Ti spent by a demand in the system till completing the servicing at the i-th server,
time T ′
i spent by a demand in the system since it enters the i-th server till the moment
of its output from the system,
time Vi spent by a demand at the i-th server,
time Ki spent by a demand in the queue to the i-th server,
time Si of servicing a demand by the i-th server provided that the system operates in
the stationary regime.
The following results are given in [3].
Theorem 3. If Eη1 = m1 < ∞, then(
μ−1
i + · · · + μ−1
n
)−1
T ′
i
w.⇒ 1
as λ → ∞.
Theorem 4. If Eη1 = m1 < ∞, then
μiKi
w.⇒ 1
as λ → ∞.
166 O. K. ZAKUSILO AND N. P. LYSAK
Theorem 5. If Eη1 = m1 < ∞, then
μiVi
w.⇒ 1
as λ → ∞.
Denote exp {tA} = ‖aij(t)‖n
i,j=1. The following results indicate the speed of conver-
gence of T s and W s
n to μ−1
1 + μ−1
2 + · · · + μ−1
n .
Depending on the values of μk, k = 1, n, consider the following cases:
1. μ1 = μ2 = · · · = μn = μ. In this case ani (t) = 1
(n−i)!μ
n−itn−i exp {−μt}, i = 1, . . . , n,
ai1 (t) = 1
(i−1)!μ
i−1ti−1 exp {−μt}, i = 1, . . . , n.
2. All μk, k = 1, n, are different. In this case ai j+1 (t) = 0, i = 1, . . . , n−1, j = i, . . . n−1,
ai i (t) = exp {−μit} , i = 1, . . . , n, ai 1 (t) =
∑i
k=1 ci
k1 exp {−μkt}, i = 2, . . . , n,
anj (t) =
n∑
k=j
cn
kj exp {−μkt}, j = 1, . . . , n − 1
(here, ci
kj is the coefficient of exp {−μkt} in the i-th row and the j-th column of the
matrix exp {tA}).
3. The characteristic polynomial of A has the form
χ (γ) = (γ + μ1)
r1 (γ + μ2)
r2 . . . (γ + μs)
rs ,
μi �= μj , i �= j, r1 + r2 + · · · + rs = n (we do not indicate the elements of exp {tA}, since
we do not consider this case for brevity).
In case 1, we have the following results.
Theorem 6. Let F (x) belong to the domain of attraction of a stable law with exponent α,
1 < α ≤ 2. Then there exists a function f (λ) > 0 regularly varying with exponent 1
α − 1
such that the distributions of the variables
(
T s − n
μ
)
1
f(λ) and
(
W s
n − n
μ
)
1
f(λ) weakly
converge as λ → ∞ to the distribution of the sum of independent random variables w1
and w2 with the characteristic functions
χw1 = exp
⎧⎨
⎩
∫ ∞
0
ln χ
⎛
⎝s1
n∑
k=1
n−k+1∑
j=1
(
nμ−1
)n−k+1−j
μn−j
(n − k + 1 − j)! (k − 1)!
uk−1 exp {− (n + μu)}
⎞
⎠du
⎫⎬
⎭
and
χw2 = exp
{∫ n
μ
0
ln χ
(
s1
n∑
k=1
μn−kun−k
(n − k)!
exp {−μu} − 1
)
du
}
,
where
χ (s1) = exp
{
− |s1|α
(
1 − i
s1
|s1| tg
πα
2
)}
.
This distribution is stable with exponent α.
The following theorem considers the case m1 = ∞.
Theorem 7. Let F (x) belong to the domain of attraction of a stable law with exponent
α, 0 < α < 1. Then
lim
λ→∞
P {T s < t} = lim
λ→∞
P {W s
n < t} = P {v1 + v2 < 0},
VERTICAL AND HORIZONTAL FLUID QUEUES IN HEAVY AND LOW TRAFFIC 167
where the random variables v1 and v2 are independent and have Laplace transforms
exp
{−sα
1
∫∞
0
qα
1 (u)du
}
and exp
{−sα
1
∫∞
0
qα
2 (u) du
}
, respectively,
q1 (u) =
n∑
k=1
n−k+1∑
j=1
tn−k+1−jμn−j
(n − k + 1 − j)! (k − 1)!
uk−1 exp {−μ (t − u)},
q2 (u) =
n∑
k=1
μn−kun−k
(n − k)!
exp {−μu} − 1.
Further, we denote 1
μ1
+ · · · + 1
μn
= c.
In case 2, we have the following results.
Theorem 8. Let F (x) belong to the domain of attraction of a stable law with exponent
α, 1 < α ≤ 2. Then there exists a function f (λ) > 0 regularly varying with exponent
1
α − 1 such that the distributions of the variables (T s − c) 1
f(λ) and (W s
n − c) 1
f(λ) weakly
converge as λ → ∞ to the distribution of the sum
n∑
k=1
cn
k1
μk
exp {−μkc} v1 +
n∑
k=2
cn
k2
μk
exp {−μkc}
2∑
j=1
c2
j1vj + . . .
+
n∑
k=n−1
cn
kn−1
μk
exp {−μkc}
n−1∑
j=1
cn−1
j1 vj +
1
μn
exp {−μkc}
n∑
j=1
cn
j1vj +
n∑
k=1
cn
k1
μk
wk.
This distribution is stable with exponent α. The sum consists of summands involving the
independent random variables vj, wj, j = 1, . . . , n with the characteristic functions
χ
1
αμj (s1), exp
{
|s1|α
(
1 + i
s1
|s1| tg
πα
2
)∫ c
0
(exp {−μju} − 1)α
du
}
, j = 1, . . . , n,
respectively.
Theorem 9. Let F (x) belong to the domain of attraction of a stable law with exponent
α, 0 < α < 1, then limλ→∞ P {T s < t} = limλ→∞ P {W s
n < t} =
P
⎧⎨
⎩
n∑
k=1
cn
k1
μk
exp {−μkt}w∗
1 +
n∑
k=2
cn
k2
μk
exp {−μkt}
2∑
j=1
c2
j1w
∗
j + · · ·+
n∑
k=n−1
cn
kn−1
μk
exp {−μkt}
n−1∑
j=1
cn−1
j1 w∗
j +
1
μn
exp {−μkt}
n∑
j=1
cn
j1w
∗
j +
n∑
k=1
cn
k1
μk
v∗k < 0
⎫⎬
⎭,
where random variables w∗
j , v∗j , j = 1, . . . , n, are independent and have the Laplace trans-
forms exp
{
− sα
1
μjα
}
, exp
{
−sα
1
∫ t
0 (1 − exp {−μju})α
du
}
, j = 1, . . . , n, respectively.
4. Behavior of the stationary characteristics
of a horizontal system in heavy traffic
Assume that a demand entering a horizontal system at the moment t0 needs y units
of work to be served. Let Ss denote the time to complete servicing this demand by all
servers and let V s
n be the time that passes till the moment when this demand enters the
n-th server provided that the system operates in the stationary regime.
As earlier, F (y1, y2, . . . , yn) is the distribution function of η = (η1, 0, . . . , 0)T ,
Eη = (Eη1, 0, . . . , 0)T ,
168 O. K. ZAKUSILO AND N. P. LYSAK
Eη1 = m1, ξ = (ξ1, ξ2, . . . , ξn)T is a vector, whose distribution is stationary for x(t).
The main results can be formulated as the following statements [5].
Lemma 3. If Eη1 = m1 < ∞, then λ−1 (ξ1, ξ2, . . . , ξn)T converges in probability to
m1
(∑n
i=1 μ−1
i ,
∑n
i=2 μ−1
i , . . . , μ−1
n
)T
as λ → ∞.
Theorem 10. If Eη1 = m1 < ∞, then(
μ−1
1 + 2μ−1
2 + · · · + nμ−1
n
)−1
Ss w.⇒ 1
as λ → ∞.
Theorem 11. If Eη1 = m1 < ∞, then(
μ−1
1 + 2μ−1
2 + · · · + nμ−1
n
)−1
V s
n
w.⇒ 1
as λ → ∞.
Investigating the speed of convergence of Ss and V s
n to μ−1
1 + 2μ−1
2 + · · · + nμ−1
n
turned out to be a troublesome problem. The proofs of limiting theorems contain cum-
bersome expressions even in the case of two servers with μ1 = μ2 = μ, and the theorems
themselves have unattractive forms. At the same time, changing the processes x (t)
and z (t) by some linear transformations x̃ (t) and z̃ (t) allows us to establish the limit
theorems in the n-dimensional case.
Consider the case where the matrix A is similar to a diagonal matrix D, i.e.
A = TDT−1,
where D = ‖δijλi‖n
i,j=1, λi, i = 1, . . . , n are the eigenvalues of A (real and negative) and
T is a non-singular matrix, T = ‖ tij‖n
i,j=1, T−1 =
∥∥ t′ij
∥∥n
i,j=1
.
If x̃ (t) = (x̃1 (t) , x̃2 (t) , . . . , x̃n (t))T =T−1x (t), then
dx̃ (t) = Dx̃ (t) dt + dz̃ (t),
where
z̃ (t) = (z̃1 (t) , z̃2 (t) , . . . , z̃n (t))T = T−1z (t) = (t′11z1 (t) , t′21z1 (t) , . . . , t′n1z1 (t))T
.
Lemma 4. If Eη1 = m1 < ∞, then λ−1ξ̃ = λ−1T−1ξ = λ−1
(
ξ̃1, ξ̃2, . . . , ξ̃n
)T
converges
in probability to m1
(∑n
j=1 t′1j
∑n
i=j μ−1
i ,
∑n
j=1 t′2j
∑n
i=j μ−1
i , . . . ,
∑n
j=1 t′nj
∑n
i=j μ−1
i
)T
as λ → ∞.
Denote c =
∑n
j=1 t′j1λ
−1
j
∑n
k=1 tkj
(
μn
∑n
j=1 tnjt
′
j1λ
−1
j
)−1
.
Theorem 12. Let F (x) belong to the domain of attraction of a stable law with exponent
α, 1 < α ≤ 2. Then there exists a function f (λ) > 0 regularly varying with exponent
1
α − 1 such that the distributions of the variables (Ss − c) 1
f(λ) and (V s
n − c) 1
f(λ) weakly
converge as λ → ∞ to the distribution of the sum
n∑
j=1
t′j1
(
n∑
k=1
tkj − μntnjλ
−1
j (exp {λjt} − 1)
)
vj + μn
n∑
j=1
tnjt
′
j1λ
−1
j wj ,
This distribution is stable with exponent α. The sum consists of summands involving the
independent random variables vj, wj, j = 1, . . . , n with characteristic functions
χ
− 1
αλj (s1), exp
{
|s1|α
(
1 + i
s1
|s1| tg
πα
2
)∫ c
0
(1 − exp {λju})α
du
}
, j = 1, . . . , n,
VERTICAL AND HORIZONTAL FLUID QUEUES IN HEAVY AND LOW TRAFFIC 169
respectively, where χ (s1) = exp
{
− |s1|α
(
1 − i s1
|s1| tg
πα
2
)}
.
Theorem 13. If F (x) belongs to the domain of attraction of a stable law with exponent
α, 0 < α < 1, then
lim
λ→∞
P {Ss < t} = lim
λ→∞
P {V s
n < t}
= P
⎧⎨
⎩
n∑
j=1
t′j1
(
n∑
k=1
tkj − μntnjλ
−1
j (exp {λjt} − 1)
)
v∗j + μn
n∑
j=1
tnjt
′
j1λ
−1
j w∗
j < 0
⎫⎬
⎭ ,
where the random variables v∗j , w∗
j , j = 1, . . . , n, are independent and have the Laplace
transforms exp
{
sα
1
λjα
}
, exp
{
−sα
1
∫ t
0 (1 − exp {λju})α
du
}
, j = 1, . . . , n, respectively.
5. Limit theorems for the solution to
the Langevin equation in low traffic
Consider Eq. (1) in a wider case where a matrix A has the form A = UJU−1, J is a
Jordan matrix, U = ‖uij‖n
i, j=1 is a non-singular matrix, and
z(t) = (z1(t), z2(t), . . . , zn(t))T ∈ R
n
is a generalized Poisson process with parameter λ and jumps η1, η2, . . . , ηj , . . . . In this
section, we investigate the limit behavior, as λ → 0, of x̃ = (x̃1, . . . , x̃n)T = U−1x(·, λ)
provided that x(·, λ) is in a stationary regime.
Denote η̃j = (η̃j
1, . . . , η̃
j
n)T = U−1ηj , j = 1, 2, . . . , pr = P
{
η̃j
r = 0
}
, p+
r =
P{η̃j
r > 0}, sgn z = (sgn z1, . . . , sgn zn)T , if z = (z1, . . . , zn)T ∈ R
n, J = {J1, . . . , Jm},
where Ji is a Jordan cell of dimension ki related to the eigenvalue λi, i = 1, m, of A
(some λi can coincide),
∑m
i=1 ki = lm, m = 1, n, ln = n.
Since the components of x̃ = (x̃1, . . . , x̃n)T are determined by Jordan cells, we restrict
ourselves by the description of the part of x̃ which corresponds to Ji. Denote it by
(x̃li−1+1, x̃li−1+2, . . . , x̃li)T . Let also (η̃j
li−1+1, η̃
j
li−1+2, . . . , η̃
j
li
)T , j = 1, 2, . . . , be the
part of η̃j which corresponds to Ji, A1 = {η̃1
li
�= 0}, B1 = {η̃1
li−1 �= 0}, P{A1} =
p, P{B1} = q. (All processes and random variables are supposed to be determined on
the same probability space.)
Consider the following cases:
I. λi < 0. If this is the case, then x̃li−1+1, . . . , x̃li are real.
II. λi = ai + ibi, (ai < 0, bi �= 0). If this is the case, then x̃li−1+1, . . . , x̃li are
complex. So, x̃li−1+1 =
∣∣x̃li−1+1
∣∣ exp
{
iϕli−1+1
}
, . . . , x̃li = |x̃li | exp {iϕli}, where
ϕli−1+1 = arg x̃li−1+1, . . . , ϕli = arg x̃li , ϕli−1+1, . . . , ϕli ∈ (0 , 2π).
Let us introduce the notation νi = λλ−1
i , if λi is real, and κi = λa−1
i , if λi = ai + ibi.
In the rest of the paper, α stays for a random variable which does not depend on other
variables and is uniformly distributed on (0, 1).
In case I, we have the following theorems [4].
Theorem 14. If pli = 0, then the distribution of
( ∣∣x̃li−1+1
∣∣−νi
, . . . , |x̃li |−νi , sgn (x̃li−1+1, . . . , x̃li)
)
weakly converges as λ → 0 to the distribution of (α, . . . , α, sgn (η̃1
li
, . . . , η̃1
li
)).
170 O. K. ZAKUSILO AND N. P. LYSAK
Theorem 15. If 0 < pli < 1, then the distribution of( ∣∣x̃li−1+1
∣∣−νi
, . . . , |x̃li |−νi , sgn x̃li−1+1, . . . , sgnx̃li
)
weakly converges as λ → 0 to the distribution of
(
α
1
p , . . . , α
1
p , γ, . . . , γ
)
, where γ takes
values 1 and −1 with probabilities p+
li
and 1 − p+
li
, respectively.
Theorem 16. If pli = 1, pli−1 = 0, then the distribution of(∣∣x̃li−1+1
∣∣−νi
, . . . , |x̃li−1|−νi , |x̃li | , sgnx̃li−1+1, . . . , sgnx̃li−1, sgnx̃li
)
weakly converges as λ → 0 to the distribution of
(
α, . . . , α, 0, sgnη̃1
li−1, . . . , sgnη̃1
li−1, 0
)
.
Theorem 17. If pli = 1, 0 < pli−1 < 1, then the distribution of(∣∣x̃li−1+1
∣∣−νi
, . . . , |x̃li−1|−νi , |x̃li | , sgnx̃li−1+1, . . . , sgnx̃li−1, sgnx̃li
)
weakly converges as λ → 0 to the distribution of
(
α
1
q , . . . , α
1
q , 0, γ, . . . , γ, 0
)
, where γ
takes values 1 and −1 with probabilities p+
li−1 and 1 − p+
li−1, respectively.
In case II, the following theorems are proved [4].
Theorem 18. If pli = 0, then the distribution of(∣∣x̃li−1+1
∣∣−κi
, . . . , |x̃li |−κi , ϕli−1+1, . . . , ϕli
)
weakly converges as λ → 0 to the distribution of (α, . . . , α, β, . . . , β), where β is uniformly
distributed on (0, 2π).
Theorem 19. If 0 < pli < 1, then the distribution of( ∣∣x̃li−1+1
∣∣−κi
, . . . , |x̃li |−κi , ϕli−1+1, . . . , ϕli
)
weakly converges as λ → 0 to the distribution of
(
α
1
p , . . . , α
1
p , β, . . . , β
)
, where β is uni-
formly distributed on (0, 2π).
These results can be applied in a natural way to vertical and horizontal systems. Note
that a vertical system only uses Theorems 14–17, since its matrix A possesses only real
eigenvalues.
Bibliography
1. Zakusilo O.K., Markov Processes with Semideterministic Parts of Sample Paths., K: FADA
LTD, 2002, pp. 165. (Ukrainian)
2. Wolff R.W., Poisson arrivals see time averages, Oper. Res. 30 (1982), 223-231.
3. Zakusilo O.K., Lysak N.P., On a queueing system with consecutive service, Probability theory
and mathematical statistics 72 (2005), 24-29. (Ukrainian)
4. Zakusilo O.K., Lysak N.P., On a multi-dimensional storage process, Probability theory and
mathematical statistics 71 (2004), 72-81. (Ukrainian)
5. Lysak N.P., Limit theorems for a controlled network, Bulletin of Kyiv University, Ser. Phys.-
Math. Sci. 3 (2005), 328-332. (Ukrainian)
6. Elwalid A.I., Mitra D., Analysis, approximations and admission control of a multi-service
multiplexing system with priorities, Proceedings of INFOCOM’95 2 (1995), 463-472.
E-mail : do@unicyb.kiev.ua, lysak@unicyb.kiev.ua
|
| id | nasplib_isofts_kiev_ua-123456789-4451 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0321-3900 |
| language | English |
| last_indexed | 2025-12-07T17:25:26Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zakusilo, O.K. Lysak, N.P. 2009-11-10T14:54:49Z 2009-11-10T14:54:49Z 2006 Vertical and horizontal fluid queues in heavy and low traffic / O.K. Zakusilo, N.P. Lysak // Theory of Stochastic Processes. — 2006. — Т. 12 (28), № 1-2. — С. 162–170. — Бібліогр.: 6 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4451 519.21 The paper considers vertical and horizontal fluid queueing systems with consecutive
 service. The workload processes in these systems satisfy the Langevin equations with
 Poisson input. The objective is to investigate the main stationary characteristics in heavy and low traffic. en Інститут математики НАН України Vertical and horizontal fluid queues in heavy and low traffic Article published earlier |
| spellingShingle | Vertical and horizontal fluid queues in heavy and low traffic Zakusilo, O.K. Lysak, N.P. |
| title | Vertical and horizontal fluid queues in heavy and low traffic |
| title_full | Vertical and horizontal fluid queues in heavy and low traffic |
| title_fullStr | Vertical and horizontal fluid queues in heavy and low traffic |
| title_full_unstemmed | Vertical and horizontal fluid queues in heavy and low traffic |
| title_short | Vertical and horizontal fluid queues in heavy and low traffic |
| title_sort | vertical and horizontal fluid queues in heavy and low traffic |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/4451 |
| work_keys_str_mv | AT zakusilook verticalandhorizontalfluidqueuesinheavyandlowtraffic AT lysaknp verticalandhorizontalfluidqueuesinheavyandlowtraffic |