Asynchronous flow line: two stations with a
finite buffer
An asynchronous flow line comprising two processing stations
and a buffer. Processing times follow an exponential distribution with means
1/my1 and 1/my2. Interarrival times at station 1 are exponential distributed
with mean 1/lam. Alternatively, it may be assumed, that station 1 is never starved
(infinite arrival rate). The buffer size is limited (sizes between 0 and 2 are
possible). Station 2 is never blocked. Production blocking (blocking-after-service)
is assumed. Workpieces finding station 1 busy (or blocked) are rejected.
The flow line is modelled as a continuous time markov chain.
First the steady-state balance equations are constructed.
The state of the system is described with the tupel (x,y), where
x=state of station 1 and y=state of station 2.
Possible
states of station 1:
Possible
states of station 2:
| 0 |
idle |
|
1 |
busy |
|
2 |
busy and one workpiece in buffer |
|
3 |
busy
and twoworkpieces in buffer |
The column RS is the right-hand side of the system of balance
equations.
After the balance equations have been built up, the system is
solved. The probabilities of the states are analysed to find the throughput
of the system and the mean work-in-process.
Symbols:
| lam |
arrival rate
at station 1 |
|
my1 |
processing rate (1/mean processing time) at station 1 |
|
my2 |
processing rate (1/mean processing time) at station 2 |
|
L |
work-in-process at a station |
|
u(i,j) |
transition rate between states i and j |
| X |
throughput
of the system |
| RS
|
right-hand
side of the system of balance equations |
Literature:
- Papadopoulos/Heavey/Browne
(1993), p. 55-56
- Viswanadham/Narahari (1992), p. 423-425
|
