Time Varying and Multi-Phase Throughput
Diagram (RBD) models are generally used to describe the reliability-wise
configuration of a system but they can also be used to represent the
system's production/processing sequence. In such cases, we can take system
analysis through RBDs a step further and utilize them to study the
productivity or yield of the system via throughput analysis. This is
applicable when the system can process (or make) something, such as
manufacturing systems, chemical processes and oil refineries. Throughput
analysis has been discussed before in this e-magazine; in this article we
discuss the modeling and analysis of phased and time varying throughput with
the help of
This article combines concepts of
throughput analysis and Reliability Phase Diagrams (RPDs). For more
information about throughput analysis, click
To find out more about RPDs, click
here. You can also read an
about RPD analysis in the latest issue of the Reliability Edge.
Applications for This Type of Analysis
Time varying and multi-phase throughput can be found in many industry
applications. For example, consider a textile factory that receives
different quantities of raw materials during different seasons. These
seasons could be treated as different phases.
Another example is the start up period in a
processing plant. When the plant stops operating, due to planned or
unplanned maintenance, the equipment requires a warm up period before
reaching its maximum production capacity. The methodology used for variable
throughput is discussed next.
Variable (Time Varying) Throughput
In BlockSim 7, time varying throughput can be specified at the phase
level through the Variable Throughput
property of an operational phase. Variable throughput permits modeling of
scenarios where the throughput changes over time. Three general models for
variable throughput are available in BlockSim. Each of these models
has two parameters, which are specified by the user. These models are
1) Linear model:
y = ax
This model describes the change in
throughput y as a linear function of time x. The total
throughput processed between any two points of time
is obtained by integration of the linear function as:
2) Exponential model: y = beax
This model describes the change in throughput
y as an exponential function of time x. The total throughput
processed in a period of time between any two points
is obtained as:
3) Power model: y = bxa
This model describes the change in throughput
y as a power function of time x. The total throughput processed
between two points of time
is obtained as:
All of the above models also
have a user-defined maximum throughput capacity value. If this maximum
throughput capacity value is reached, the throughput per unit time becomes
constant at that value. In this situation, the variable throughput model
then acts as a constant throughput model. The above models may at first
glance seem limited, but they do in fact provide ample modeling flexibility.
This flexibility is achieved by using these functions as building blocks for
more complex functions. For example, a step model can be easily created by
using multiple phases, each with a constant throughput. A ramp model would
use phases with linearly increasing functions in conjunction with constant
phases, and so forth. In addition, decreasing throughput can be modeled by
manipulation of the parameters.
We will use a simple example to illustrate the use of BlockSim 7 to
model phase and time varying throughput.
A production machine follows a Weibull distribution with β = 1.55 and
η = 250.4 days. When an outage happens, it takes about 12 days to
restore the machine.
The above figure represents the Reliability
Block Diagram (RBD) of the machine. The failure properties are shown next.
Every time the machine fails, the machine is
repaired then the throughput of the machine goes through a warm up period of
5 days until it reaches normal production capacity again. The throughput in
units per day can be described as follows:
y(x) =20x (within warm
t <= 5)
y(x) = 100 (during normal operation, t > 5)
The normal operation throughput is specified in the machine's properties
window, as shown next (no backlog is specified for this example).
To model the two-phase operation (warm up and
normal operation) of this machine, a Reliability Phase Diagram (RPD) is
used, as shown next.
Both the Warm Up Period and Normal Operation
phases in the above RPD are linked to the same Diagram Sheet, which contains
the machine's RBD. The following figures show the phase properties of these
first two phases. In addition to the linked diagram, the phase properties
also specify the duration of the phase and what action is to be taken if a
system failure occurs. In this case, the action upon system failure is to go
to a maintenance phase, which is specified in the next field. Once the
maintenance phase has been executed, the system will then go through warm up
The warm up phase has a variable throughput,
specified as follows.
The normal operation phase has a constant
throughput, as shown next.
The maintenance phase is defined in
BlockSim 7 as a template that contains the blocks to be fixed in the RBD
and their associated repair properties (such as repair times, resources,
etc.) Note that while repairs may be performed during an operational phase,
a maintenance phase is exclusively dedicated to the execution of maintenance
The repair properties for the block in the maintenance template are shown
The system is simulated for 1092 days (3 years)
with 156 increments for weekly results. On the Throughput Settings
tab of the Maintainability/Availability Simulation window, Report
Throughput Point Results
is selected, as shown next.
A System Overview summary of the simulation is
shown next (note that your results may vary slightly since they were
obtained via simulation).
The mean availability of the system for the
next three years is estimated to be 95.41% and the expected throughput of
the plant at the end of three years is expected to be 102,719.31 units.
For an overview of the plants availability as
a function of time, you can generate a point availability plot like the one
The point results of the throughput as a
function of time can also be obtained from the System Point Results
report. The following is a plot of the cumulative throughput over time.
The following is a plot of the weekly