The 1996 Mathematics Awareness Week poster shows a mathematical
model of a high speed production floor with an automated material
handling system which utilizes an overhead conveyor system to move
raw material and finished goods. The model, provided by F&H
Simulations, determines for the user where bottlenecks will occur,
and what will happen to the throughput rate as system variables
such as conveyor speeds, AGV (automated guidance vehicle) speeds
and sorting rules are changed. The model takes into account the
stochastic nature of shop floor production and stock replenishment.
All discrete event simulation models work on the same set of
principles. Mathematical variables can be used to quantify the
characteristics of the system. For example, the cycle time for a
part on a machine might be represented by the variable CTIME. If
the variable PRODUCED is used to represent the total number of
parts produced in a 1000 hour period, then PRODUCED=1000/CTIME so
long as the CTIME is a constant value for every part.
Computer simulation models handle the dynamic and random nature of
a real life system by breaking the process down into discrete
occurring events. The computer evaluates one event at a time,
updating the system at each event. The computer model then carries
out the process according to user-defined rules with the updated
variables until the next event in time when the process is repeated
all over again.
Deterministic mathematical formulas have been derived for simple
queuing systems. The Pollaczek-Kyntchin equation determines the
average waiting time for any element within this type of system
(more specifically, an M/G/1 queuing system - one which has an
exponentially distributed arrival rate, at a normally distributed
service time, and only one server):
AverageWaitingTime = (1/2)*(1+CV^2)*(U/(1-U))*Pavg
- where:
- CV = standard deviation / mean
U = server utilization (busy time / total time)
Pavg = average processing time
Note that with this equation, the waiting time increases
exponentially as the utilization approaches 100 percent.
