Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 890243, 16 pages
doi:10.1155/2012/890243
Research Article
Further Research on the M/G/1 Retrial Queueing
Model with Server Breakdowns
Ehmet Kasim and Geni Gupur
College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China
Correspondence should be addressed to Ehmet Kasim, ehmetkasim@163.com
Received 13 March 2012; Accepted 28 June 2012
Academic Editor: Song Cen
Copyright q 2012 E. Kasim and G. Gupur. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We study spectral properties of the operator which corresponds to the M/G/1 retrial queueing
model with server breakdowns and obtain that all points on the imaginary axis except zero belong
to the resolvent set of the operator and 0 is not an eigenvalue of the operator. Our results show that
the time-dependent solution of the model is probably strongly asymptotically stable.
1. Introduction
There has been considerable interest in retrial queueing systems, see Atencia et al. 1, 2,
Choi et al. 3, Djellab 4, Gupur 5, 6, Kasim and Gupur 7, Li et al. 8, Li and Wang 9,
Wang et al. 10, 11, and Yang and Templeton 12. Many researchers studied the M/G/1
retrial queueing systems with server breakdowns in the steady-state case, see Atencia et al.
1, 2, Choi et al. 3, Djellab 4, Li and Wang 9, and Wang et al. 10. Only few researchers
studied transient solutions of M/G/1 retrial queueing systems, see Wang et al. 11, Gupur
5, 6, Kasim and Gupur 7. And Wang et al. 11 studied the transient solution of the M/G/1
retrial queueing system with server failure by using Laplace transform and obtained the
expression of the probability-generating function. In other words, they studied the existence
of the time-dependent solution of the model. Gupur 5, 6, 13, 14, Gupur et al. 15, and Kasim
and Gupur 7 did dynamic analysis for several queueing models including retrial queueing
models by using functional analysis and obtained the existence and uniqueness of the timedependent solution of several queueing models and asymptotic behavior of their timedependent solutions. In this paper, by using the idea in Gupur 6 and Gupur et al. 15,
we study the asymptotic behavior of the time-dependent solution of the M/G/1 retrial
queueing system with server breakdowns in which the failure states of the server are
absorbing states. This queueing system was studied by Wang et al. 10 in 2001. By using
2
Journal of Applied Mathematics
the supplementary variable technique they established the corresponding queueing model
and obtained explicit expressions of some reliability indices such as the availability, failure
frequency for steady-state cases under the following hypothesis: “the time-dependent solution of the model converges to a steady-state solution.” By reading their paper we find that
the above hypothesis, in fact, implies the following two hypotheses.
Hypothesis 1. The model has a nonnegative time-dependent solution.
Hypothesis 2. The time-dependent solution of the model converges to a steady-state solution.
In 2010, Gupur 6 studied the above two hypotheses. Firstly, he converted the model
into an abstract Cauchy problem by selecting a suitable Banach space and defining the underlying operator which corresponds to the model and its domain. Next, by using the HilleYosida theorem and the Phillips theorem he proved that the model has a unique non-negative
time-dependent solution and therefore obtained that Hypothesis 1 holds. Then, when the
service completion rate is a constant, he studied the asymptotic behavior of its timedependent solution. By studying resolvent set of of the adjoint operator of the operator which
corresponds to the model in this case, the M/G/1 retrial queueing model with server breakdowns is called the M/M/1 retrial queueing model with server breakdowns he obtained
the resolvent set of the operator on the imaginary axis: all points on the imaginary axis
except zero belong to its resolvent set. And he proved that 0 is not an eigenvalue of the
operator. Thus, he suggested that the time-dependent solution of the model probably strongly
converges to zero. Until now, any other results about this model have not been found in
the literature. In this paper, we try to study the asymptotic behavior of the time-dependent
solution of the above model when the service completion rate is a function. Firstly, we convert
the model into an abstract Cauchy problem by Gupur 6 and we determine the resolvent set
of the adjoint operator of the operator corresponding to the model when the service
completion rate satisfies a certain condition. Also, we prove that 0 is not an eigenvalue of the
operator. If we can verify 0 is not in the residue spectrum of the operator, then from the above
results we conclude that the time-dependent solution of the model strongly converges to zero.
Naturally, the results obtained in 6 are now special cases of our results.
According to Wang et al. 10, the M/G/1 retrial queueing system with server breakdowns can be described by the following system of partial differential equations with integral
boundary conditions:
∞
dpI,0,0 t
−λpI,0,0 t μxpw,0,1 x, tdx,
dt
0
∞
dpI,i,0 t
−λ iθpI,i,0 t μxpw,i,1 x, tdx, i ≥ 1,
dt
0
∂pw,0,1 x, t ∂pw,0,1 x, t
− λ α μx pw,0,1 x, t,
∂t
∂x
∂pw,i,1 x, t ∂pw,i,1 x, t
− λ α μx pw,i,1 x, t
∂t
∂x
λpw,i−1,1 x, t, i ≥ 1,
pw,i,1 0, t λpI,i,0 t i 1θpI,i1,0 t,
pI,0,0 0 1,
pI,i,0 0 0,
i ≥ 1;
i ≥ 0,
pw,j,1 x, 0 0,
j ≥ 0.
1.1
Journal of Applied Mathematics
3
Here x, t ∈ 0, ∞×0, ∞; pI,i,0 t i ≥ 0 represents the probability that the server is idle and
there are i customers in the retrial group at time t; pw,i,1 x, t represents the joint probability
that at time t there are i customers in the retrial group and the server is up and customer is
being served with elapsed service time x; λ is the arrival rate of customers; α is the server
failing rate; θ is the successive interretrial times of customers; μx is the service completion
rate at time x satisfying
∞
μx ≥ 0,
μxdx ∞.
1.2
0
In this paper, we use the notations in Gupur 6:
⎛
λ
⎜0
⎜
⎜
0
Γ⎜
⎜
⎜0
⎝
..
.
θ
λ
0
0
..
.
0
2θ
λ
0
..
.
0
0
3θ
λ
..
.
0
0
0
4θ
..
.
⎞
···
· · ·⎟
⎟
⎟
· · ·⎟.
⎟
· · ·⎟
⎠
..
.
1.3
Take a state space as follows:
X
1
p
p
,
p
,
p
,
.
.
.
∈l
I
I,0,0
I,1,0
I,2,0
pw pw,0,1 , pw,1,1 , pw,2,1 , . . .
pI , pw ∈ L1 0, ∞ × L1 0, ∞ × L1 0, ∞ × · · ·
∞
∞
pI , pw pI,i,0 pw,i,1 1
<∞
L 0,∞
⎧
⎪
⎪
⎪
⎪
⎨
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎭
i0
.
1.4
i0
It is obvious that X is a Banach space and also a Banach lattice. In the following we define
operators and their domains:
⎛
⎜⎛
⎜ −λ
0
0
0
⎜
⎜⎜ 0 −λ θ
0
0
⎜⎜
⎜⎜ 0
0
−λ
2θ
0
⎜
⎜
A pI , pw ⎜⎜
⎜⎜ 0
0
0
−λ 3θ
⎜⎝
..
..
..
..
⎜
⎜
.
.
.
.
⎝
⎛
⎞⎛
⎞
pI,0,0
···
⎜
⎟
· · ·⎟
⎟⎜pI,1,0 ⎟
⎟⎜
⎟
· · ·⎟⎜pI,2,0 ⎟,
⎟⎜
⎟
· · ·⎟⎜pI,3,0 ⎟
⎠⎝
⎠
..
..
.
.
⎞
⎞
d
0
0
0 · · ·⎟⎛
⎜− dx
⎞⎟
⎜
⎟ pw,0,1 x ⎟
d
⎜
⎟
⎟
⎜ 0 −
0
0 · · ·⎟⎜pw,1,1 x⎟⎟
⎜
⎜
⎟
⎟
⎟
dx
⎜
⎟⎜
⎟⎟
d
pw,2,1 x⎟⎟,
⎜
⎜ 0
⎟
0 · · ·⎟⎜
0 −
⎜
⎟⎟
⎜
⎟⎜pw,3,1 x⎟⎟
dx
⎜
⎟
⎝
⎠⎟
d
..
⎜ 0
⎟
· · ·⎟
0
0 −
⎜
⎟
⎟
.
dx
⎝
⎠
⎠
..
..
..
..
..
.
.
.
.
.
4
Journal of Applied Mathematics
⎧
⎪
⎪
⎨
pw,i,1 xi ≥ 0 are absolutely continuous
functions and satisfy
pw 0 ΓpI ,
DA pI , pw ∈ X ∞ dpw,i,1 ⎪
⎪
⎩
<∞
dx 1
n0
L 0,∞
⎛⎛
0
⎜
⎜
⎜⎜0
U pI , pw ⎜⎜0
⎝⎝
..
.
⎛
u
⎜λ
⎜
⎜0
⎝
..
.
0
0
0
..
.
0
0
0
..
.
⎞
⎞⎛
···
pI,0,0
⎟
⎜
· · ·⎟
⎟⎜pI,1,0 ⎟
,
⎟
⎜
· · ·⎠⎝pI,2,0 ⎟
⎠
.
..
..
.
0
u
λ
..
.
0
0
u
..
.
⎞⎞
⎞⎛
···
pw,0,1 x
⎟⎟
⎜
· · ·⎟
⎟⎜pw,1,1 x⎟⎟
⎜pw,2,1 x⎟⎟,
· · ·⎟
⎠⎠
⎠⎝
..
..
.
.
⎫
⎪
⎪
⎬
,
⎪
⎪
⎭
DU X,
1.5
where u −λ α μx.
⎛⎛ ∞
E p I , pw 0
⎜⎜ ∞
⎜⎜
⎜⎜ 0
⎜⎜ ∞
⎜⎜
⎝⎝ 0
⎞
⎛ ⎞
0
⎟
⎟
⎜ ⎟⎟
μxpw,1,1 xdx⎟
⎟ ⎜0⎟⎟
⎟, ⎜0⎟⎟,
⎝ ⎠⎟
μxpw,2,1 xdx⎟
⎠
.. ⎠
..
.
.
μxpw,0,1 xdx
⎞
DE X.
1.6
Then the above system of 1.1 can be expressed as an abstract Cauchy problem:
d pI , pw t
A U E pI , pw t, ∀t ∈ 0, ∞,
dt
⎛⎛ ⎞ ⎛ ⎞⎞
1
0
⎜⎜0⎟ ⎜0⎟⎟
pI , pw 0 ⎝⎝ ⎠, ⎝ ⎠⎠.
..
..
.
.
1.7
Gupur 6 have obtained the following results.
Theorem 1.1. If μ supx∈0,∞ μx < ∞, then A U E generates a positive contraction C0 -semi
group T t. And the system 1.7 has a unique nonnegative time-dependent solution pI , pw x, t T tpI , pw 0 satisfying pI , pw ·, t ≤ 1, ∀t ∈ 0, ∞.
2. Asymptotic Behavior of the Time-Dependent Solution of the System
1.7
In this section, firstly we determine the expression of A U E∗ , the adjoint operator of
A U E, next we study the resolvent set of A U E∗ , through which we deduce the
resolvent set of A U E on the imaginary axis. Thirdly, we prove that 0 is not an eigenvalue
of A U E. Thus, we state our main results in this paper.
Journal of Applied Mathematics
5
It is easy to see that X ∗ , the dual space of X, is as follows see Gupur 6:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
∗
∗
X ,
qI∗ , qw
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∗
∗
∗
qI∗ qI,0,0
, qI,1,0
, qI,2,0
, . . . ∈ l∞ ,
∗
∗
∗
∗
qw
qw,0,1
, qw,1,1
, qw,2,1
,...
∈ L∞ 0, ∞× L∞ 0, ∞ × L∞ 0, ∞ × · · · ∗ ∗ ∗ ∗ <∞
qI , Qw sup supqI,i,0
, supqw,i,1
∞
L 0,∞
i≥0
i≥0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
2.1
It is not difficult to verify that X ∗ is a Banach space.
Lemma 2.1. A U E∗ , the adjoint operator of A U E, is as follows:
∗
∗
L G H qI∗ , qw
,
A U E∗ qI∗ , qw
∗
∀ qI∗ , qw
∈ DL,
where
⎛⎛
−λ
0
0
⎜⎜ 0 −λ θ
0
⎜⎜
∗
L qI∗ , qw
⎜
⎜
0
−λ 2θ
⎜⎝ 0
⎝
..
..
..
.
.
.
⎞⎛q∗ ⎞
···
I,0,0
⎜q∗ ⎟
· · ·⎟
⎟
⎟⎜
⎜ I,1,0
⎟,
∗
· · ·⎟
⎟
qI,2,0
⎠⎜
⎝
⎠
..
..
.
.
⎞
⎛ d ⎞
⎛ ∗
⎞
− λ α μx
0
···
⎟ qw,0,1 x ⎟
⎜ dx
⎟⎜ ∗
⎜
⎟⎟
d x⎟⎟
⎜q
⎜
− λ α μx · · ·⎟
0
⎟⎝ w,1,1 ⎠⎟,
⎜
..
dx
⎠
⎝
⎠
..
..
.
..
.
.
.
⎛⎛ ⎞ ⎛
⎞⎛q∗ ⎞⎞
0
μx 0
0 ···
I,0,0
⎜ 0⎟ ⎜ 0 μx 0 · · ·⎟⎜q∗ ⎟⎟
⎟
⎜
∗ ∗ ⎜⎜
⎜ ⎟ ⎜
⎟⎜ I,1,0 ⎟
⎟,
G qI , qw ⎜
⎟, ⎜ 0
⎟⎜q∗ ⎟
0
0
μx
·
·
·
⎟
⎜⎜
⎠⎝ I,2,0 ⎠⎟
⎠
⎝⎝ ⎠ ⎝
..
..
..
..
..
..
.
.
.
.
.
.
⎛⎛
λ
⎜⎜θ
⎜⎜
⎜⎜ 0
∗
⎜
H qI∗ , qw
⎜
⎜⎜
⎜⎜ 0
⎝⎝
..
.
⎛
0
⎜0
⎜
⎜
⎜0
⎜
⎜0
⎝
..
.
0
λ
2θ
0
..
.
λ
0
0
0
..
.
0
0
λ
3θ
..
.
0
λ
0
0
..
.
0
0
λ
0
..
.
0
0
0
λ
..
.
0
0
0
λ
..
.
⎞⎛ ∗
⎞
qw,0,1 0
···
⎜ ∗
⎟
· · ·⎟
⎟⎜qw,1,1 0⎟
⎟⎜ ∗
⎟
· · ·⎟⎜qw,2,1 0⎟,
⎟⎜ ∗
⎟
· · ·⎟⎜qw,3,1 0⎟
⎠⎝
⎠
..
..
.
.
⎞⎛ ∗
⎞⎞
qw,0,1 x
···
⎜ ∗
⎟⎟
· · ·⎟
⎟⎜qw,1,1 x⎟⎟
⎟⎜ ∗
⎟⎟
· · ·⎟⎜qw,2,1 x⎟⎟,
⎟⎜ ∗
⎟⎟
· · ·⎟⎜qw,3,1 x⎟⎟
⎠⎝
⎠⎠
..
..
.
.
2.2
6
Journal of Applied Mathematics
∗
∗ ∗
∗ qw,i,1 x are absolutely continuous
,
DL qI , qw ∈ X ∗
and satisfy qw,i,1
∞ η, i ≥ 0
DG DH X ∗ .
2.3
Here, η in DL is a nonzero constant which is irrelevant to i.
Proof. By using integration by parts and the boundary conditions on pI , pw ∈ DA, we
∗
∈ DL,
have, for any qI∗ , qw
∞
∞ ∗ ∗ ∗
μxpw,i,1 xdx qI,i,0
−λ iθpI,i,0 A U E pI , pw , qI , qw 0
i0
∞
dpw,0,1 x ∗
− λ α μx pw,0,1 x qw,0,1
−
xdx
dx
0
∞ ∞
dpw,i,1 x − λ α μx pw,i,1 x λpw,i−1,1 x
−
dx
i1 0
∗
× qw,i,1
xdx
∞
∗
− λ iθpI,i,0 qI,i,0
i0
∞ ∞
−
∗
μxpw,i,1 xqI,i,0
dx
dpw,i,1 x ∗
qw,i,1 xdx
dx
∞
∗
λ α μx pw,i,1 xqw,i,1
xdx
0
∞ ∞
0
i1
−
0
∞ ∞
i0
i0
0
i0
∞ ∞
∗
λpw,i−1,1 xqw,i,1
xdx
− λ i0
∗
iθpI,i,0 qI,i,0
∞ ∞
i0
0
∗
pw,i,1 xμxqI,i,0
dx
∞
∗
dqw,i,1
x
x ∞
∗
−pw,i,1 xqw,i,1
dx
pw,i,1 x
x
x0
dx
0
i0
∞ ∞
! "
∗
pw,i,1 x − λ α μx qw,i,1
x dx
∞
i0
0
i0
0
∞ ∞
∞
∗
pw,i,1 xλqw,i1,1
xdx
− λ ∗
iθpI,i,0 qI,i,0
i0
∞ ∞
i0
∞
∗
pw,i,1 0qw,i,1
0
i0
0
∗
pw,i,1 xμxqI,i,0
dx
Journal of Applied Mathematics
7
∞ ∞
i0
0
i0
0
∞ ∞
∞
pw,i,1 x
∗
dqw,i,1
x
dx
∗
− λ α μx qw,i,1
x dx
∗
pw,i,1 xλqw,i1,1
xdx
∗
− λ iθpI,i,0 qI,i,0
∞ ∞
i0
i0
0
∗
pw,i,1 xμxqI,i,0
dx
∞
#
$ ∗
λpI,i,0 i 1θpI,i1,0 qw,i,1
0
i0
∞ ∞
i0
i0
pw,i,1 x
dx
0
∗
− λ α μx qw,i,1
x dx
∗
pw,i,1 xλqw,i1,1
xdx
∗
− λ iθpI,i,0 qI,i,0
i0
∗
dqw,i,1
x
0
∞ ∞
∞
∞ ∞
i0
0
pw,i,1 x
∗
dqw,i,1
x
dx
∗
− λ α μx qw,i,1
x
∗
∗
dx
λqw,i1,1
x μxqI,i,0
∞
∞
∗
∗
pI,i,0 λqw,i,1
0 pI,i1,0 i 1θqw,i,1
0
i0
i0
∗
pI , pw , L G H qI∗ , qw
.
2.4
From the definition of the adjoint operator and 2.4 we know that the assertion of this lemma
is true.
Lemma 2.2. Assume that there exist two positive constant μ, μ such that
0 < μ inf μx ≤ μ sup μx < ∞.
x∈0,∞
x∈0,∞
2.5
If μ < α μ, then
⎧
⎧
⎨ λ
⎪
⎪
λμ
λ
iθ
λ
⎪
⎪
sup , sup ,
,
⎪
⎪
⎩
⎪
Re
γ
λ
αμ
γ
λ
γ
λ
iθ
γ λ Re γ λ α μ
⎪
i≥1
⎪
⎪
⎧
⎫⎫
⎨
⎨
⎬⎬
γ ∈ C
λ iθμ
λ
⎪
sup
< 1,
⎪
⎪
Re γ λ α μ ⎭⎭
⎪
i≥1 ⎩ Re γ λ α μ γ λ iθ ⎪
⎪
⎪
⎪
⎪
⎪
⎩
Re γ λ α μ > 0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
2.6
8
Journal of Applied Mathematics
belongs to the resolvent set of A U E∗ . In particular, all points on the imaginary axis except zero
belong to the resolvent set of A U E∗ , which implies that all points on the imaginary axis except
zero belong to the resolvent set of A U E.
∗
∗
∗
∈ X ∗ , consider the equation γI − L − GqI∗ , qw
HyI∗ , yw
,
Proof. For any given yI∗ , yw
that is,
∗
∗
λyw,0,1
γ λ qI,0,0
0,
2.7
∗
∗
∗
θyw,0,1
γ λ θ qI,1,0
0 λyw,1,1
0,
2.8
∗
∗
∗
iθyw,i−1,1
γ λ iθ qI,i,0
0 λyw,i,1
0, i ≥ 1,
2.9
∗
dqw,i,1
x
dx
∗
∗
∗
γ λ α μx qw,i,1
− λyw,i1,1
x − μxqI,i,0
x,
∗
qw,i,1
∞ η,
2.10
i ≥ 0,
i ≥ 0.
2.11
By solving 2.7–2.10, we have
∗
qI,0,0
∗
qI,i,0
λ
y∗ 0,
γ λ w,0,1
iθ
λ
∗
yw,i−1,1
y∗ 0,
0 γ λ iθ
γ λ iθ w,i,1
2.12
i ≥ 1,
2.13
x
∗
qw,i,1
x ai e 0 γλαμτdτ
x
e 0 γλαμτdτ
x!
" τ
∗
∗
−μτqI,i,0
− λyw,i1,1
τ e− 0 γλαμsds dτ
0
2.14
x
ai e 0 γλαμτdτ
x
− e 0 γλαμτdτ
x!
" τ
∗
∗
μτqI,i,0
λyw,i1,1
τ e− 0 γλαμsds dτ,
i ≥ 0.
0
Through multiplying e−
e
−
x
0
x
γλαμτdτ
0 γλαμτdτ q ∗
w,i,1 x
to two sides of 2.14, we obtain
x!
" τ
∗
∗
μτqI,i,0
λyw,i1,1
ai −
τ e− 0 γλαμsds dτ,
0
i ≥ 0.
2.15
By combining 2.11 with 2.15, we deduce
ai ∞!
" τ
∗
∗
μτqI,i,0
λyw,i1,1
τ e− 0 γλαμsds dτ,
i ≥ 0.
0
By inserting 2.16 into 2.14, we get
∗
qw,i,1
x
e
x
0
γλαμτdτ
∞!
x
" τ
∗
∗
μτqI,i,0
λyw,i1,1
τ e− 0 γλαμsds dτ
2.16
Journal of Applied Mathematics
x
e 0 γλαμτdτ
9
∞
x
x
λe 0 γλαμτdτ
∗
μτqI,i,0
e−
∞
x
τ
0 γλαμsds
∗
yw,i1,1
τe−
dτ
τ
0 γλαμsds
dτ
2.17
⇒
∗ qw,i,1 L∞ 0,∞
∞
≤
0
λ
x
e 0 Re γλαdτ
∞
∞
x
x
e 0 Re γλαdτ
0
Re γ λ α μ
∞
x
μ
≤ ∗ − 0τ Re γλαds − xτ μsds
μτqI,i,0
e
dτ dx
e
τ
τ
∗
μτyw,i1,1
τe− 0 Re γλαds e− x μsds dτ dx
∗ qI,i,0
λ
∗
,
yw,i1,1 ∞
L 0,∞
Re γ λ α μ
i ≥ 0.
2.18
Equation 2.12 gives
λ ∗ ∗ λ ∗
≤
.
0
y
y
qI,0,0 ≤ γ λ w,0,1 L∞ 0,∞
γ λ w,0,1
2.19
From 2.13 we estimate
λ
iθ
∗ ∗
∗
yw,i−1,1
yw,i,1
0 0
qI,i,0 ≤ γ λ iθ
γ λ iθ
iθ
λ
∗ ∗
≤ y
y
∞
w,i−1,1
w,i,1
γ λ iθ
γ λ iθ
L∞ 0,∞
L 0,∞
λ iθ
∗
sup
≤ y
w,i,1 L∞ 0,∞ ,
γ λ iθ i≥0
i ≥ 1.
By inserting 2.19 and 2.20 into 2.18, we have
∗ qw,0,1 L∞ 0,∞
λ ∗
y
∞
w,0,1
γ λ
L 0,∞
Re γ λ α μ
≤ ⎧
⎨
μ
λ
∗ yw,1,1 ∞
L 0,∞
Re γ λ α μ
⎫
⎬
λ
≤ ⎩ γ λ Re γ λ α μ
Re γ λ α μ ⎭
λμ
2.20
10
Journal of Applied Mathematics
∗ × supyw,i,1
L∞ 0,∞
i≥0
∗ qw,i,1 L∞ 0,∞
,
λ iθ
∗
sup
y
w,i,1 L∞ 0,∞
γ λ iθ i≥0
Re γ λ α μ
μ
≤ λ
∗ supyw,i,1
∞
L 0,∞
Re γ λ α μ i≥0
⎧
⎨
⎫
⎬
λ iθμ
λ
⎩ Re γ λ α μ γ λ iθ Re γ λ α μ ⎭
∗ × supyw,i,1
i≥0
L∞ 0,∞
,
i ≥ 0.
2.21
By combining 2.19 and 2.20 with 2.21, we derive
⎧
⎨ λ
∗ ∗ λ
iθ
q , qw ≤ sup , sup ,
I
⎩ γ λ i≥1 γ λ iθ
λμ
λ
,
Re γ λ α μ
γ λ Re γ λ α μ
2.22
⎫⎫
⎬⎬
λ iθμ
λ
.
sup Re γ λ α μ ⎭⎭
i≥1 ⎩ Re γ λ α μ γ λ iθ ⎧
⎨
This shows that I − γI − L − G−1 H−1 exists and is bounded when γ belongs to the set
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
γ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎧
⎨ λ
λμ
λ iθ
λ
sup , sup ,
,
γ λ i≥1 γ λ iθ ⎩
Re γ λ α μ
γ λ Re γ λ α μ
⎧
⎫⎫
⎨
⎬⎬
λ iθμ
λ
∈ C
sup < 1,
Re γ λ α μ ⎭⎭
i≥1 ⎩ Re γ λ α μ γ λ iθ Re γ λ α μ > 0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
2.23
∗
∗
yI∗ , yw
for any given
Through discussing the solution of the equation γI − L − GqI∗ , qw
−1
∗
∗
∗
yI , yw ∈ X , it is not difficult to verify γI − L − G exists and is bounded when γ satisfies
Journal of Applied Mathematics
11
2.23. Therefore, by the resolvent equation
#
$−1 %
−1 "&−1
!
γI − L G H
γI − L − G I − γI − L − G H
!
−1 "−1 −1
I − γI − L − G H
γI − L − G ,
2.24
we know that γI − L − G − H−1 exists and is bounded when γ belongs to the set 2.23, which
means that 2.23 belongs to ρL G H.
In particular, if γ ia, a ∈ R \ {0}, i2 −1, then all γ’s belong to 2.23. In fact, by using
the condition on this lemma, we have
√
λ
a2 λ2
< 1,
λ kθ
sup '
< 1,
k≥1
a2 λ kθ2
(
μ < α μ ⇒ λμ < α μ
a2 λ2
(
(
(
⇒ λμ λ a2 λ2 < α μ
a2 λ2 λ a2 λ2
(
(
⇒ λμ λ a2 λ2 < λ α μ
a2 λ2
⇒ √
λμ
λ
< 1,
λαμ
a2 λ2 λ α μ
'
μ < α μ ⇒ λ kθμ < α μ
a2 λ kθ2
'
'
'
⇒ λ kθμ λ a2 λ kθ2 < α μ
a2 λ kθ2 λ a2 λ kθ2
'
'
⇒ λ kθμ λ a2 λ kθ2 < λ α μ
a2 λ kθ2 ,
⎧
⎪
⎨
⎫
⎪
⎬
λ kθμ
λ
< 1.
sup '
λ α μ⎪
⎩ a2 λ kθ2 λ α μ
⎭
k≥1 ⎪
2.25
The above inequalities show that all points on the imaginary axis except zero belong to the
resolvent set of A U E∗ . From the relation between the spectrum of A U E and
spectrum of A U E∗ we know that all points on the imaginary axis except zero belong to
the resolvent set of A U E.
12
Journal of Applied Mathematics
Lemma 2.3. If
0 < μ inf μx ≤ μ sup μx < ∞,
x∈0,∞
2.26
x∈0,∞
then 0 is not an eigenvalue of A U E.
Proof. We consider the equation A U EpI , pw 0, which is equivalent to
λpI,0,0 ∞
μxpw,0,1 xdx,
2.27
0
λ iθpI,i,0 ∞
i ≥ 1,
μxpw,i,1 xdx,
2.28
0
dpw,0,1 x
− λ α μx pw,0,1 x,
dx
2.29
dpw,i,1 x
− λ α μx pw,i,1 x λpw,i−1,1 x,
dx
pw,i,1 0 λpI,i,0 i 1θpI,i1,0 ,
i ≥ 1,
i ≥ 0.
2.30
2.31
It is hard to determine the concrete expressions of all pI,i,0 , pw,i,1 x and to prove pI , pw ∈
DA. In the following we use another method. We introduce the probability-generating
∞
i
i
functions QI z ∞
i0 pI,i,0 z , Qw x, z i0 pw,i,1 xz for |z| < 1. Theorem 1.1 ensures
that QI z, Qw x, z are well-defined. By applying the basic knowledge of power series, the
Fubini theorem and 2.27-2.28, we have
λpI,0,0 ∞
∞
∞ ∞
μxpw,0,1 xdx μxpw,i,1 xzi dx
λ iθpI,i,0 zi 0
i1
)
∞
pI,i,0 zi
⇒ λQI z θz
i1
*
∞
μx
0
i0
d
⇒ λQI z θz QI z dz
∞
0
∞
pw,i,1 xzi dx
2.32
i0
μxQw x, zdx.
0
By 2.29 and 2.30, we deduce
∞
dpw,0,1 x dpw,i,1 x i
z − λ α μx pw,0,1 x
dx
dx
i1
∞
∞
− λ α μx
pw,i,1 xzi λpw,i−1,1 xzi
i1
i1
⇒
∂
∞
i0
∞
∞
pw,i,1 xzi
− λ α μx
pw,i,1 xzi λ pw,i−1,1 xzi
∂x
i0
i1
Journal of Applied Mathematics
13
⇒
∞
∂Qw x, z
− λ α μx Qw x, z λz pw,i,1 xzi
∂x
i0
⇒
∂Qw x, z λz − λ − α − μx Qw x, z
∂x
⇒
x
Qw x, z Qw 0, ze 0 λz−λ−α−μτdτ .
2.33
Equation 2.31 gives
Qw 0, z ∞
∞
∞
pw,i,1 0zi λ pI,i,0 zi i 1θpI,i1,0 zi
i0
i0
2.34
i0
d
λQI z θ QI z.
dz
By combining 2.34 with 2.33 and using 2.32, we calculate
d
λQI z θz QI z dz
∞
μxQw x, zdx
0
∞
x
μxQw 0, ze 0 λz−λ−α−μτdτ dx
0
d
λQI z θ QI z
dz
∞
x
μxe 0 λz−λ−α−μτdτ dx
0
⇒
x
∞
λz−λ−α−μτdτ
dx − 1
dQI z λ 0 μxe 0
x
dz
∞
QI z
θz−
μxe 0 λz−λ−α−μτdτ dx
0
2.35
⇒
x
z ∞
μxe 0 λs−λ−α−μτdτ dx − 1
λ
0
x
QI z QI 0 exp
ds
θ 0 s − ∞ μxe 0 λs−λ−α−μτdτ dx
0
x
z ∞
μxe 0 λs−λ−α−μτdτ dx − 1
λ
0
x
pI,0,0 exp
ds .
θ 0 s − ∞ μxe 0 λs−λ−α−μτdτ dx
0
This together with 2.34 gives
Qw 0, z z−
∞
0
λz − 1
x
μxe 0 λz−λ−α−μτdτ dx
QI z.
2.36
14
Journal of Applied Mathematics
From 2.35, we derive
∞
pI,i,0 lim QI z
i0
z→1
x
z ∞
μxe 0 λs−λ−α−μτdτ dx − 1
λ
0
x
lim pI,0,0 exp
ds
z→1
θ 0 s − ∞ μxe 0 λs−λ−α−μτdτ dx
0
x
1 ∞
μxe 0 λs−λ−α−μτdτ dx − 1
λ
0
x
pI,0,0 exp
ds .
θ 0 s − ∞ μxe 0 λs−λ−α−μτdτ dx
0
2.37
x
∞
In the following, by the Rouche theorem we know that s − 0 μxe 0 λs−λ−α−μτdτ dx has a
unique zero point inside the unit circle |z| 1. And by the residue theorem we determine the
above integral.
x
∞
Let fz z and gz 0 μxe−λα−λzx− 0 μτdτ dx. It is well known that fz and
gz are differentiable inside and continuous on the contour |z| 1. And |fz| 1 on |z| 1.
Since, for λ α − λ Re z > 0,
x
∞
−λα−λzx− 0 μτdτ
gz μxe
dx
0
≤−
∞
e−λα−λ Re zx de−
x
μτdτ
0
0
− e
−λα−λ Re zx −
e
x
0
λ α − λ Re z
x∞
x0
μτdτ ∞
x
e−λα−λ Re zx− 0 μτdτ dx
2.38
0
1 − λ α − λ Re z
< 1 fz,
∞
x
e−λα−λ Re zx− 0 μτdτ dx
0
we have |fz| < |gz| on |z| 1. Consequently, all conditions of Rouche’s theorem are
satisfied. Obviously fz − gz has only one zero inside the unit circle |z| 1, since fz has
one. If we denote this zero by z0 , then it is a simple pole of
∞
x
μxe−λα−λzx− 0 μτdτ dx − 1
0
x
.
∞
z − 0 μxe−λα−λzx− 0 μτdτ dx
2.39
Journal of Applied Mathematics
15
By the residue theorem, we calculate
1 ∞
x
μxe−λα−λsx− 0 μτdτ dx − 1
x
ds
∞
0 s−
μxe−λα−λsx− 0 μτdτ dx
0
x
∞
μxe−λα−λzx− 0 μτdτ dx − 1
0
x
lim
z → z0 1 − ∞ λxμxe−λα−λzx− 0 μτdτ dx
0
x
∞
μxe−λα−λz0 x− 0 μτdτ dx − 1
0
x
.
∞
1 − 0 λxμxe−λα−λz0 x− 0 μτdτ dx
0
2.40
Thus, we obtain
x
∞
−λα−λz0 x− 0 μτdτ
∞
dx − 1
λ 0 μxe
x
< ∞.
pI,i,0 pI,0,0 exp
θ 1 − ∞ λxμxe−λα−λz0 x− 0 μτdτ dx
i0
2.41
0
From 2.36, we derive
lim Qw 0, z lim
z→1
z→1z
lim
z→1z
1−
−
−
∞
0
∞
0
∞
0
λz − 1
x
μxe 0 λz−λ−α−μτdτ dx
QI z
λz − 1
x
lim QI z
μxe 0 λz−λ−α−μτdτ dx z → 1
λ1 − 1
μxe−
x
0
αμτdτ
dx
2.42
QI 1
0.
By combining 2.33 with 2.42, we estimate
∞
x
pw,i,1 x lim Qw x, z lim Qw 0, ze 0 λz−λ−α−μτdτ
i0
z→1
lim Qw 0, ze−
z→1
x
0
αμτdτ
2.43
z→1
0.
Since Theorem 1.1 shows that each pw,i,1 x is nonnegative, 2.43 implies pw,i,1 x 0 for
all i ≥ 0. This together with 2.27 and 2.28 gives pI,i,0 0 for all i ≥ 0. In other words,
A U EpI , pw 0 has only zero solution; that is, 0 is not an eigenvalue of A U E.
16
Journal of Applied Mathematics
Lemma 2.3 shows that the system 1.7 does not have nonzero steady-state solution.
If we can prove that 0 is not in the residue spectrum of A U E, then by Theorem 1.1,
Lemmas 2.2 and 2.3, and ABLV Theorem see 16 or 17, we deduce that the timedependent solution of system 1.7 is strongly asymptotically stable. This result is quite
different from other queueing models, see Gupur 13–15 and Kasim and Gupur 7.
Acknowledgments
The authors’ research work is supported by Science Foundation of Xinjiang University no:
XY110101 and the Natural Science Foundation of Xinjiang no: 2012211A023.
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