Variance of Departure Process in Two-Node Tandem Queue with
Unreliable Servers and Blocking
Yang Woo Shin
1
and Dug Hee Moon
2
1
Department of Statistics, Changwon National University, Changwon, Gyeongnam 51140, Korea
2
School of Industrial Engineering and Naval Architecture, Changwon National University,
Changwon, Gyeongnam 51140, Korea
Keywords:
Variance, Departure Process, Tandem Queue, Finite Buffers, Blocking, Markovian Arrival Process.
Abstract:
This paper provides an effective method for evaluating the second moments such as variance and covariance
for the number of departures in two-node tandem queue with unreliable servers. The behavior of the system
is described by a level dependent quasi-birth-and-death process and the departure process is modeled by a
Markovian arrival process. Algorithms for the transient behavior, the variance and covariance structure for the
output process and the time to the nth departure are developed. We show that the results can be applied to
derive approximate formulae for the due-date performance and the distribution of the number of outputs in a
time interval.
1 INTRODUCTION
There is an extensive literature for the analysis of
manufacturing systems with finite buffers and unre-
liable servers. Most of the works related to the per-
formance evaluation of manufacturing systems have
been focused on analyzing the first order measures
such as average production rates and average buffer
levels in steady-state e.g. see the monographs (Buza-
cott and Shanthikumar, 1993; Gershwin, 1994), the
survey papers (Dallery and Gershwin, 1992; Pa-
padopoulos and Heavey, 1996; Li et al., 2009) and
the references therein. The first order measures can be
used to get information about the capabilities of a pro-
duction system in the long run. However, there may
be tremendous variability from a time period to pe-
riod (Gershwin, 1994, Section 3.2; Tan, 1999a). Thus
the second order measures such as the variance of the
number of parts produced in a given time period and
the inter-departure times and covariancebetween con-
secutive inter-departure times are also very useful to
design and control production systems in a more ef-
fective way. The information about the time depen-
dent second order measures can especially be useful
to respond short-term and long-term requirements in
an effective and timely way.
Studies on variance of the output process in a se-
rial production line have been presented during the
last decades, for a review of recent studies on the
variance of the output for production systems, one
can refer to the papers (Tan,2000; Tan, 2013; Lager-
shausena and Tan, 2015). For discrete material flow
production systems with finite buffers, Tan (1999b,
2000) use a Markov reward model to calculate the
variance of the number of parts produced in a given
time period in a two-station production line with finite
buffer capacities and deterministic processing times
and geometrically distributed failure and repair times.
Our approach to be developedin this paper is to model
the output process by a Markovian arrival process
(MAP) and to use the closed formulae for the tran-
sient behavior and the variance and covariance struc-
ture for the number of outputs during a period (0, t]
and the nth departure time in the literature.
This paper is aimed on providing an effective
method for evaluating the second moments of the
number of outputs and inter-departure times and in-
vestigating the effects of the system parameters to the
second moments. The results can be applied to the
practical problem such as due-time performance in
manufacturing system and are basis on analyzing the
long line. This paper concerns to the two-station sys-
tem with finite buffer capacities. A model of a two-
node system is simple, but it helps us to understand
the behavior of the system and gives some insights
of the more complicated system. The approach can
also be used as building block for analyzing the more
complex system with multiple nodes.
258
Shin Y. and Moon D.
Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking.
DOI: 10.5220/0006119302580264
In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 258-264
ISBN: 978-989-758-218-9
Copyright
c
2017 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
This paper is organized as follows. In Section 2,
the model is described in detail. The moment formu-
lae for MAP are reviewed and algorithms for the per-
formance measures are presented in Sections 3 and 4,
respectively. In section 5, numerical results are pre-
sented. Concluding remarks are given in Section 6.
2 MODEL
We consider a tandem queueing network that consists
of two service stations S
1
and S
2
and one buffer of
finite size b between them. Each station S
i
has an
unreliable server M
i
, i = 1,2. Assume the following
system characteristics.
BAS blocking mechanism : Blocking after service
(BAS) rule is adopted, that is, if the buffer is full upon
a completion of service at the first station, the server
M
1
is blocked and the customer is held at the station
where it just completed its service until the station S
2
can accommodate it.
Open and saturated system: In many manufac-
turing system, it has been assumed that the first sta-
tion the first station is never starved and the last sta-
tion is never blocked. For potential applications of
the method and results to developing approximation
method of more complicated system, we assume that
the server M
1
in the first station is never starved and it
starts newservice immediately after a service comple-
tion unless the server is blocked and the server M
2
in
the second station is never blocked and the customer
at M
2
leaves the system immediately after completing
its service.
ODF rule : Each server is either up (operational)
or under repair (broken-down) at any time. Operation
dependent failure (ODF) is assumed. That is, a server
can fail only while the server is working and a server
never fails while the server is blocked or starved.
Exponential distributions of service time, failure
time and repair time: We define the failure time by
the operation time in units between two successive
failures (from a repair to a failure). The failure time
does not contain the time period while the server is
being blocked, starved or repaired. Service time, fail-
ure time and repair time of M
i
are assumed to be of
exponential with rates µ
i
, ν
i
and η
i
, respectively.
Let X(t) be the number of customers in the buffer
and at station S
2
and the customer blocked at station
S
1
. The state space of X(t) is {0, 1,···,K}, where
K = b + 2. Let J
i
(t) be service phase of the server M
i
at time t denote the states of J(t) by
J
i
(t) =
w, M
i
is working
s, M
i
is starved
b, M
i
is blocked
f, M
i
is failed.
The state space of the stochastic process Z
Z
Z =
{Z(t), t 0} with Z(t) = (X(t),J
1
(t), J
2
(t)) is
S =
K
n=0
S
n
,
where
S
0
= {(0, w, s),(0, f,s)},
S
n
= {(n, j
1
, j
2
) : j
1
, j
2
{w, f}}, 1 n K 1,
S
K
= {(K, b,w),(K,b, f)}.
The stochastic process Z
Z
Z = {Z(t), t 0} forms a
Markov chain with generator of the form
Q =
B
0
A
0
C
1
B
1
A
1
.
.
.
.
.
.
.
.
.
C
K1
B
K1
A
K1
C
K
B
K
.
The matrices B
n
, A
n
, C
n
are as follows:
B
n
=
ν
2
ν
1
0
η
2
0 ν
1
η
1
0 ν
2
0 η
1
η
2
, 1 n K 1,
B
0
=
ν
1
η
1
, B
K
=
ν
2
η
2
,
A
n
=
µ
1
0 0 0
0 µ
1
0 0
0 0 0 0
0 0 0 0
, 1 n K 2,
A
0
=
µ
1
0 0 0
0 0 0 0
,
A
K1
=
µ
1
0
0 µ
1
0 0
0 0
,
C
n
=
µ
2
0 0 0
0 0 0 0
0 0 µ
2
0
0 0 0 0
, 2 n K 1,
C
1
=
µ
2
0
0 0
0 µ
2
0 0
, C
K
=
µ
2
0 0 0
0 0 0 0
,
where the diagonal entries of B
n
are determined by
Qe = 0 and e is a column vector of appropriate size
whose elements are all 1.
Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking
259
3 DEPARTURE PROCESS
Let T be the first time until a customer leaves the sys-
tem and
F
zz
(t) = P(Z(T) = z
,T t |Z(0) = z), z,z
S .
Then T is the same as the absorbing time of a Markov
chain with rate matrix of the form
Q
T
=
D
0
D
1
0 0
,
where
D
0
=
B
0
A
0
B
1
A
1
.
.
.
.
.
.
B
K1
A
K1
B
K
,
D
1
=
O
0
C
1
O
1
C
2
O
2
.
.
.
.
.
.
C
K
O
K
.
The matrix F(t) = (F
zz
(t)) is given by
F(t) =
Z
t
0
exp(D
0
u)duD
1
, t 0
which is the inter arrival time of a Markovian arrival
process (MAP) with representation MAP(D
0
,D
0
), see
Lucantoni et al. (1990).
Let N(t) be the number of customers that leave
the system during an interval (0, t] and P(n,t) =
(P
zz
(n,t)) be the square matrix of size |S | whose
(z,z
)-component is
P
zz
(n,t) = P(N(t) = n, Z(t) = z
|Z(0) = z).
It follows from the Kolomogorov equations that
d
dt
P(n,t) = P(n,t)D
0
+ P(n1,t)D
1
, n 1, t 0
(1)
and P(0,0) = I the identity matrix. The matrix gener-
ating function P
(w,t) =
n=0
w
n
P(n,t) is given by
P
(w,t) = exp[(D
0
+ wD
1
)t] , |w| 1, t 0.
For later use, define the following notation. Let π
π
π =
(π(x),x S ) be the stationary distribution of Q and
Π
Π
Π = eπ
π
π, Ψ = (eπ
π
πQ)
1
, λ = π
π
πD
1
e
c = π
π
πD
1
Ψ, d = ΨD
1
e.
It can be easily seen that π
π
πΨ = π
π
π, Ψe = e and ce =
λ = π
π
πd.
The following theorem can be found in (Neuts,
1989, Theorems 5.4.1 and 5.4.2; Artalejo et al, 2010).
Theorem 3.1. In stationary state, that is, π(x) =
P(Z(0) = x), mean µ(t) = E[N(t)], variance
σ
2
(t) = Var[N(t)] and the covariance Cov(t,u,v) =
Cov[N(t),N(v) N(u)] (0 < t u < v) are given as
follows:
µ(t) = λt,
σ
2
(t) =
˜
σ
2
(t) + 2c[exp(Qt) Π
Π
Π]d,
Cov(t,u,v) = π
π
πD
1
[I exp(Qt)] exp[Q(ut)]
×[I exp(Q(vu))]Ψd.
where
˜
σ
2
(t) = 2(λ
2
cd) + (λ2λ
2
+ 2cD
1
e)t. (2)
Remark 1. It is well known that as t
exp(Qt) = Π
Π
Π+ O(t
r1
e
ηt
), (3)
where η is the real part of η
, the non-zero eigen
value of Q with maximum real part, and r is the mul-
tiplicity of η
, see e.g. (Narayana and Neuts, 1992).
It can be easily seen from Theorem 3.1 and (3) that
Cov(t,u,v) 0 as ut .
Remark 2. It can be seen from Theorem 3.1 that
the variance rate is given by the closed formula
V = lim
t
Var[N(t)]
t
= (λ 2λ
2
+ 2cD
1
e).
Tan (1999b) use numerical result of the asymptotic
variance rate V to determine the variance σ
2
(t) Vt
for large t. We can see that
˜
σ
2
(t) provides more ac-
curate approximation of σ
2
(t) than that of Vt and it is
easy to compute
˜
σ
2
(t).
Let ξ
n
, n = 0,1,2, ··· be the nth transition time
of N
N
N with ξ
0
= 0 and set τ
n
= ξ
n
ξ
n1
and Z
n
=
Z(ξ
n
+ 0), n = 1,2,··· with Z
0
= Z(0). The tran-
sition probability matrix of {Z
n
,n = 0, 1,2, ···} is
P = (D
0
)
1
D
1
and the stationary distribution p
p
p of
P is given by
p
p
p =
1
λ
π
π
πD
1
.
The following theorem can be found in Artalejo et al.
(2010).
Theorem 3.2. Assume that Z(0) has a distribution
a = (a(x),x S ) with a(x) = P(Z(0) = x). The mean,
variance and covariance of τ
n
are given as follows:
E[τ
n
] = aP
n1
d
0
,
Var[τ
n
] = 2aP
n1
(D
0
)
2
e
aP
n1
d
0
2
,
Cov(τ
k
,τ
n
) = aX
k,n
e(E[τ
k
])(E[τ
n
]), 1 k < n,
where
d
0
= (D
0
)
1
e,
X
k,n
= P
k1
(D
0
)
1
P
nk
(D
0
)
1
P, 1 k < n.
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
260
Remark 3. Since lim
n
P
n
= ep
p
p, it can be easily
seen that for each k = 1, 2,···,
lim
n
Cov(τ
k
,τ
n
) = 0.
Remark 4. The mean and variance of ξ
n
are as
follows
E[ξ
n
] =
n
i=1
E[τ
i
],
Var[ξ
n
] =
n
i=1
Var[τ
i
] + 2
n1
i=1
n
j=i+1
Cov(τ
i
,τ
j
).
Remark 5. Assuming a = p
p
p, the mean, variance
and covariance of τ
n
are as follows:
E[τ
n
] =
1
λ
,
Var[τ
n
] =
2
λ
π
π
πd
0
1
λ
2
,
Cov(τ
k
,τ
n
) =
1
λ
π
π
πP
n
d
0
1
λ
2
,n = 1,2,···.
4 ALGORITHMS
It is necessary to π
π
π, Ψ = (eπ
π
πQ)
1
and exp(Qt) for
the variance, covariance of N(t) and τ
n
. In this sec-
tion, some algorithms for computing π
π
π, Ψ = (eπ
π
π
Q)
1
and exp(Qt) are presented.
1. Algorithm for stationary distribution π
π
π of Q.
Here, we present an algorithm for stationary distribu-
tion π
π
π of Q. Write π
π
π = (π
π
π
0
,π
π
π
1
,··· ,π
π
π
K
), where π
π
π
i
is
the vector of size l
i
, 0 i K. Let R
1
,··· , R
K
be the
matrices satisfies the following matrix equations
A
n1
+ R
n
B
n
+ R
n
R
n+1
C
n+1
= 0, 1 n K 1,
A
K1
+ R
K
B
K
= 0.
The solutions of the equation are given as follows:
R
K
= A
K1
(B
K
)
1
,
R
n
= A
n1
[(B
n
+ R
n+1
C
n+1
)]
1
, n = K 1,···,1.
Then the stationary distributionπ
π
π of Q is given as fol-
lows
π
π
π
n
= π
π
π
0
R
1
···R
n
, n = 1, 2,···,K
with
π
π
π
0
[B
0
+ R
1
C
1
] = 0
and normalizing condition
π
π
π
0
e+
K
n=1
R
1
···R
n
e
!
= 1.
Once π
π
π is obtained, p
p
p = (p
p
p
0
, p
p
p
1
,··· , p
p
p
K
) can be cal-
culated by
p
p
p
k
=
1
λ
π
π
π
k+1
C
k+1
, k = 0, 1,··· , K 1,
0, k = K.
2. Algorithm for (D
0
)
1
. It can be seen from the
structure of D
0
that (D
0
)
1
is of the form
(D
0
)
1
=
X(0,0) X(0, 1) ··· X(0, K)
X(1,1) ··· X(1,K)
O
.
.
.
.
.
.
X(K,K)
.
The block components X(i, j), 0 i j K are cal-
culated following the algorithm in (Shin, 2009) as fol-
lows :
(1) Compute
G
n
= A
n1
(B
n
)
1
, n = K,K 1,N 2, ··· ,1
and G
0
= (B
0
)
1
.
(2) Compute X(n, k), 0 n K, k = n,n + 1, ··· , K
as follows: For n = 0,1,··· ,K, set X(n,n) =
(B
n
)
1
and
X(n,k) = X(n,k1)G
k
, k = n+ 1,n+ 2,···,K.
3. Algorithm for Ψ = (eπ
π
π Q)
1
. Let E
n
=
(1,··· ,1)
T
(0 n K) be the l
n
-dimensional col-
umn vector whose components are all one and E
1
=
(1,··· ,1)
T
be the (
K
i=1
l
i
)-dimensional column vec-
tor, where l
i
is the number of elements of S
i
. Let
π
π
π
1
= (π
π
π
1
,··· ,π
π
π
K
) and Π[i, j] = E
i
π
π
π
j
, 0 i, j K. De-
note the (i, j) block matrix of a matrix A correspond-
ing to (i, j) block of Q by A[i, j], 0 i, j K. Write
the matrix Q in the block form
Q =
B
0
Q
01
Q
10
Q
11
, eπ
π
πQ =
A
00
A
01
A
10
A
11
,
where
A
00
= E
0
π
π
π
0
B
0
, A
01
= E
0
π
π
π
1
Q
01
,
A
10
= E
1
π
π
π
0
Q
10
, A
11
= E
1
π
π
π
1
Q
11
.
Then the block matrix form of Ψ is given by (e.g.
(Horn and Johnson, 1985, page 18))
Ψ =
A
∗−1
00
A
1
00
A
01
A
∗−1
11
A
∗−1
11
A
10
A
1
00
A
∗−1
11
,
where
A
00
= A
00
A
01
A
1
11
A
10
,
A
11
= A
11
A
10
A
1
00
A
01
.
The matrix Ψ is calculated by the following step:
(1) Calculate (B
0
)
1
using the ordinary algorithm.
(2) Since Q
11
is block tridiagonal matrix, one can use
the algorithm in (Shin, 2009) for (Q
11
)
1
.
(3) For A
1
11
and A
1
00
, one can use the following for-
mula (see Horn and Johnson(1985, page 19))
A
1
11
= (Q
11
)
1
1
1+π
π
π
1
q
11
q
11
π
π
π
1
(Q
11
)
1
,
where q
11
= (Q
11
)
1
E
1
. The inverse matrix
A
1
00
is calculated by usual method.
Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking
261
(4) Calculate A
∗−1
11
by the formula,
A
∗−1
11
= (A
11
+ A
10
(A
1
00
)A
01
)
1
= A
1
11
+ A
1
11
A
10
A
∗−1
00
A
01
A
1
11
,
where A
∗−1
00
is calculated by usual method.
(5) The (i, j) block Ψ[i, j] of Ψ is
Ψ[i, j] =
A
∗−1
00
, i = 0, j = 0,
A
1
00
(A
01
A
∗−1
11
)[ j], i = 0, 1 j K,
(A
∗−1
11
A
10
)[i]A
1
00
, 1 i K, j = 0,
A
∗−1
11
[i, j], 1 i, j K.
4. Calculation of exp(Qt) =
n=0
t
n
n!
Q
n
. We use
the uniformization technique. Let
q = max
zS
([Q]
zz
)
and Θ = I +
1
q
Q. Then
exp(Qt) = Q
M
(t) + E
(M)
(t),
where
Q
M
(t) =
M
n=0
e
qt
(qt)
n
n!
Θ
n
,
E
(M)
(t) =
n=M+1
e
qt
(qt)
n
n!
Θ
n
.
For given ε > 0, let M(ε) be the positive integer such
that
1
M(ε)
n=0
e
qt
(qt)
n
n!
< ε.
For large t, the following addition formula is useful.
First, take an integer n
0
such that t
0
=
qt
n
0
is moderate
with E
(M)
(t
0
)e < ε
0
e. Note that
exp(Qt) = [exp(Qt
0
)]
n
0
=
h
Q
M
(t
0
) + E
(M)
(t
0
)
i
n
0
= [Q
M
(t
0
)]
n
0
+ E
(M)
(t
0
,n
0
),
where
E
(M)
(t
0
,n
0
) =
n
0
k=1
n
0
k
h
E
(M)
(t
0
)
i
n
0
k
[Q
M
(t
0
)]
k
.
Since Q
M
(t
0
)e < (1ε
0
)e and E
(M)
(t
0
)e < ε
0
e, it can
be seen that
E
(M)
(t
0
,n
0
)e < (1(1ε
0
)
n
0
)e.
5. Calculation of P(n,t). Let Θ
0
= I +
1
q
D
0
. Ap-
plying the uniformization technique to P(n,t) and us-
ing the the Kolmogorov equation (1), it can be seen
that
P(k,t) =
n=0
e
qt
(qt)
n
n!
K
(n)
k
, k 1,
where {K
(n)
k
}satisfies the followings: for k = 1,2,···,
K
(n+1)
k
=
1
q
K
(n)
k1
D
1
+ K
(n)
k
Θ
0
, n = 0, 1,2, ··· (4)
with K
(0)
k
= 0, k 1 and K
(0)
0
= I and
K
(n+1)
0
= K
(n)
0
Θ
0
, n 0.
The recursive formula (4) is also given in (Lucantoni,
1991). Let
E
(M)
k
= P(k,t)
M
n=0
e
qt
(qt)
n
n!
K
(n)
k
, k 0.
Note that
exp(Qt) =
k=0
P(k,t) =
n=0
e
qt
(qt)
n
n!
k=0
K
(n)
k
.
It can be seen from
Θ
n
=
k=0
K
(n)
k
, n 0
that K
(n)
k
e < e and hence
E
(M(ε))
k
e < εe, k 0.
5 NUMERICAL RESULTS
We apply the algorithms in section 4 to the system
with two-node tandem queue with a finite buffer and
server breakdown. We consider the system with ser-
vice rates µ
1
= µ
2
= 1.0, failure rates ν
1
= 0.1, ν
2
=
0.04 and repair rates η
1
= 0.5, η
2
= 0.2. The isolated
efficiency of each server is the same as
η
i
ν
i
+η
i
= 0.833.
In this section, we assume that the system is in sta-
tionary state.
1. Speed of convergence to stationary state. We
investigate how fast the distribution of the process
Z
Z
Z converges to the stationary distribution π
π
π. It can
be seen from (3) that
Q
(t) = log||exp(Qt) Π||
is almost linear for large t. We also have seen that
the speed of convergence of exp(Qt) decreases as
buffer size increases. For example, the time t
s
(b) :=
min{t > 0 :
Q
(t) < 5} are t
s
(3) = 74, t
s
(5) = 99,
t
s
(7) = 134.
2. Variance of N(t). Figures 1 and 2 show vari-
ance σ
2
(t) and the difference
σ
2
(t) = σ
2
(t)
˜
σ
2
(t).
Figure 1 exhibits that σ
2
(t) increases almost linearly
as t increases as expected in the formula in Theorem
3.1. It can be seen from fig. 2 that
˜
σ
2
(t) can be used
instead of σ
2
(t) for t t
s
. In fact, it follows from
Theorem 3.1 and (3) that for t
s
with
Q
(t
s
) < log
10
ε,
|
σ
2
(t)| < 2|cd|ε, t t
s
.
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
262
Figure 1: The variance σ
2
(t).
Figure 2: Differences
σ
2
(t) = σ
2
(t)
˜
σ
2
(t).
Indeed, for b = 5,
σ
2
(64) < 10
3
,
σ
2
(81) < 10
4
and
σ
2
(99) < 10
5
.
3. Covariance of N(t). Figure 3 de-
picts the Cov[t] = Cov[N(t), N(2t) N(t)] and
lim
t
Cov[t] = π
π
πdλ
2
. Figure 3 shows that Cov[t]
is positive for b = 3 and negative for b = 5.
Figure 3: Covariance Cov[N(t),N(2t) N(t)] for b = 3, 5.
5. Distribution of N(t). The distribution of N(t)
is depicted in figure 4 for t = 30, t = 50, t = 70.
The figures show that the distribution of N(t) visu-
ally resembles the normal distribution. The the pair
(skewness, kurtosis) of N(t) are (0.2599,0.5054),
(0.2393,0.1739) and (0.2141,0.1486) for t = 30,
t = 50 andt = 70, respectively. Here, we approximate
the distribution of N(t) in stationary state with the
normal distribution N(µ(t),σ
2
(t)) with mean µ(t) =
λt and variance σ
2
(t) as Tan (1999b), that is,
P(N(t) n) 1Φ
n0.5λt
p
σ
2
(t)
!
, (5)
where Φ(x) =
R
x
1
2π
exp(y
2
/2)dy is the distribu-
tion function of the standard normal distribution and
n 0.5 is used for correction of the approximation
of discrete random variable using continuous distri-
bution and
˜
σ
2
(t) can be used as an approximation of
σ
2
(t) for large t.
Figure 4: Plot of p(n,t) = P(N(t) = n)] for b = 5.
The approximation errors
N
(t) between
P(N(t) n) and normal approximation are depicted
in figure 5 for t = 30, t = 50, t = 70 and b = 5. The
maximal error of approximation occurs at the mean
λt for each case. Figure shows that the accuracy
increases as t increases.
Figure 5: Error of normal approximation for P(N(t) n).
6. Due time performance. The due-time perfor-
mance of a production line can be measured by a
probability
p = P(N(t
) n
)
of meeting a customer’s order n
on time t
. Some
numerical results for t
for given n
and p are listed
in Table 1.
Table 1: Due time t
p
n
0.5 0.6 0.7 0.8 0.9 0.99
50 72 76 79 84 91 109
70 101 105 110 115 123 143
100 145 150 155 161 170 193
Variance of Departure Process in Two-Node Tandem Queue with Unreliable Servers and Blocking
263
6 CONCLUSIONS
We have provided an algorithm for the transient be-
havior, the variance and covariance structure for the
output process and inter-departure time in two-node
tandem queue. Some numerical results are presented.
We also showed that the results can be applied to
derive approximate formulae for the due-date perfor-
mance and the distribution of the number of outputs
in a time interval.
The algorithm is based on the Markovian arrival
process (MAP) which gives closed formula for vari-
ance and asymptotic variance. This is a different point
from the other methods in the literature for variance
of departure process. The algorithm requires only the
inversions of the block matrices of size 4 in the com-
puting process. Thus the computational complexity of
the algorithm dose not severely depend on the buffer
size of the system. The approach using MAP can be
easily applied to the system with more general ser-
vice, failure and repair time than exponential case.
Although the method developed in this paper is
quite efficiently, it will be limited to apply the method
to the system with multiple nodes due to the rapid in-
crease of the number of states when the number of
stations and the buffer capacities increase. There-
fore developing approximation methods to estimate
the second moment measures in multiple node system
are required. There are many approximation methods
for throughput in a complicated system, for exam-
ple, decomposition method and aggregation method
(Dallery and Gershwin, 1992; Li et al., 2009) that use
the the two-node system. The method of analyzing
the two-node system can be used as a building block
of analyzing the more complex system.
ACKNOWLEDGEMENTS
The authors are very thankful to three anonymous re-
viewers for valuable comments and suggestions.
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