![](bg2.png)
Figure 1: Tandem queue with parts assembly.
tion method is described briefly. Numerical examples
are given in Section 4. Finally, concluding remarks
are given in Section 5.
2 THE MODEL
We consider a transfer line L in which
¯
N = N + 1
workstations W
i
, i = 0, 1, ··· , N are linked in series
and there is a buffer B
i
for customers (products) of
capacity 0 ≤ c
i
< ∞ between W
i−1
and W
i
as de-
picted in Figure 1. The station W
i
, 1 ≤ i ≤ N − 1
has a server (machine) M
i
and a buffer D
i
of capacity
0 ≤ d
i
− 1 < ∞ for parts to be assembled and denote
the station i by the pair W
i
= (M
i
, D
i
).
The customers are processed along the line and
leave the system immediately after service comple-
tion at W
N
. When a server completes its service at a
stage, if the buffer for the customers of next station is
full at that time, then the server is forced to stop its
service and the customer is held at the station where
it just completed its service until the destination can
accommodate it. A server M
i
is said to be starved if
there are no customers to be served on the server M
i
and the station is said to be lacked if there are no parts
in W
i
. The station W
i
, 1 ≤ i ≤ N − 1 does not work if
it is in starved, lacked and blocked.
The initial station W
0
and the last station W
N
are
for preparation and investigation of final product, re-
spectively, and assume that they consist of a server M
0
and M
N
without buffers for parts, respectively, and
they process their work without parts. We assume
that the initial server M
0
is never starved and never
lacked and it starts new service immediately after a
service completion unless the server is blocked. The
last server M
N
is never blocked and never lacked, and
the customer at M
N
leaves the system immediately af-
ter completing its service.
Parts are supplied to each station W
i
according to
independent Poisson processes with rate λ
i
and arriv-
ing parts enter the server M
i
if there is an available
space for the part in M
i
. Note that the maximum num-
ber of parts that can be stored in the station W
i
is d
i
,
1 ≤ i ≤ N − 1. If the buffer D
i
is full, that is, if there
are d
i
parts in the station W
i
, the arrivals of the parts
to D
i
is forced to stop and begin again at the epoch
when there is an available space in W
i
.
Service time distribution of M
i
is of phase
type with representation PH(α
α
α
i
, T
i
), where α
α
α
i
=
(α
i
(1), · ·· , α
i
(h
i
)) is a probability distribution and T
T
T
i
is a nonsingular matrix of size h
i
with negative diago-
nal elements and nonnegative off-diagonal elements.
Let T
T
T
0
i
= −T
T
T
i
e = (t
0
i
(1), · ·· ,t
0
i
(h
i
))
t
, where e is the
column vector of appropriate size whose components
are all 1. See Neuts (1981) for phase type distri-
bution. Transportation times of customers through
buffers and servers are assumed to be negligible com-
paring to service time.
Stochastic processes. Let D
i
(t) be the number
of parts in W
i
at time t. The state space of D
i
(t) is
{0, 1, ··· , d
i
}. Define the state M
i
(t) of M
i
at time t
by
M
i
(t) =
s, M
i
is starved
j, M
i
is working with service phase j
b, M
i
is blocked.
The state space of W
i
(t) = (M
i
(t), D
i
(t)) of the
station W
i
= (M
i
, D
i
), 1 ≤ i ≤ N − 1 is W
i
=
{(0, 0),s
s
s,w
w
w,b
b
b}, where (0, 0) is the state that X
i
(t) ≥
1 and D
i
(t) = 0, s
s
s = {(s, k), k = 0, 1, · ·· , d
i
},
w
w
w = {( j, k), k = 1, 2, ··· , d
i
, j = 1, 2, · ·· , h
i
}, b
b
b =
{(b, k), k = 1, 2, ··· , d
i
} and let w
w
w
∗
= {(0, 0),w
w
w}.
Since W
0
and W
N
consist of only one server M
0
and
M
1
and have no buffers for part, the state space of
W
0
(t) = M
0
(t) and W
N
(t) = M
N
(t) are given by W
0
=
{b} ∪ { j, j = 1, 2, ·· · , h
0
} and W
N
= {s} ∪ { j, j =
1, 2, ··· , h
N
}.
Let X
i
(t) be the total number of customers wait-
ing in the buffer B
i
, the customers that are being
served, lacked or blocked at M
i
, and the customers
blocked at M
i−1
at time t. Then X
i
(t) takes values on
{0, 1, ··· , K
i
}, where K
i
= c
i
+ 2. Note that X
i
(t) = K
i
is equivalent to M
i−1
(t) = b.
Approximation method. Approximation is based
on decomposition approach. The first step is to de-
compose the N + 1 station system into a set of sub-
systems L
i
, i = 1, 2, ·· · , N. Each subsystem L
i
is con-
sists of upstream station W
i−1
, downstream station W
i
and a buffer B
i
between them. Model the subsys-
tem L
i
with a Markov chain with generator, say Q
i
and the unknowns in Q
i
are calculated by an itera-
tion method. Finally, performance measure such as
throughput is calculated with the stationary distribu-
tion of L
N
. Since W
i
is a downstream station in L
i
and upstream server in L
i+1
, denote the downstream
server in L
i
by W
d
i
and the upstream server in L
i+1
by
W
u
i
, if necessary to distinguish them.
Approximate Analysis of Transfer Line with PH-service Time and Parts Assemble
213