Throughput Analysis (Part
Throughput analysis allows you to analyze a reliability block diagram (RBD) such that each component within the system is designed to process (make) something while it is in operation. Part I of this article will introduce some throughput metrics and terminology. In addition, some basic analysis using throughput and how this can be accomplished in BlockSim 6 will also be discussed. Part II will be presented in next month's issue of the HotWire and will explore more complex scenarios regarding throughput analysis.
As an example, consider the case shown next with two components in series and each processing/producing 10 and 20 items per unit time (ipu), respectively.
In this case, the system configuration is not only the system's reliability-wise configuration but also its production/processing sequence. In other words, the first component processes/produces 10 ipu and the second component can process/produce up to 20 ipu. However, the second component can only process/produce items it receives from the block before it and it is only receiving 10 items in this case. If we assume that neither component can fail, then the maximum items processed from this configuration would be 10 ipu. If the system operates for 100 time units, then the throughput of this system will be 1000 items (10*100).
Terminology for Throughput Analysis
To examine throughput, consider the following two scenarios.
Consider the system shown in Figure 1.
Figure 1: Throughput RBD of system for Scenario 1
Blocks A through I produce a number of items per unit time, as identified next to each letter (e.g. A: 100 implies 100 units per time unit for A). The connections shown in the RBD display the physical path of the items through the process (or production line). For the sake of simplicity, also assume that the blocks can never fail and that items are routed equally to each path.
Therefore, this implies that over a single time unit, the following occurs:
Thus, Table 1 would represent the throughput and excess capacity of each block after 1 time unit.
Table 1: Block throughput summary for Scenario 1
Now consider the same system presented in Scenario 1, but both E and H have failed, as shown in Figure 2.
Figure 2: Throughput RBD of system for Scenario 2
Table 2 presents the block throughput summary for Scenario 2.
Table 2: Block throughput summary for Scenario 2
Table 3 presents the utilization summary for Scenario 2.
Table 3: Utilization summary for Scenario 2
It can be seen from these results that the "bottlenecks" in the system are blocks F and G.
Figure 3: Block throughput settings
Table 4: Block throughput summary of Scenario 2 with F and G with limited backlog (1 time unit)
After two time units, the results are presented in Table 5.
Table 5: Block throughput summary of Scenario 2 with F and G with limited backlog (2 time units)
Note that in this example, the units will never be able to process the backlog. However, if we were to observe the system for a longer time of operation and through failures and repairs of the other blocks, there would be opportunities for the blocks to process their backlogs and catch up. It is very important to note that when simulating a system with failures and repairs in BlockSim, you must define the block as one that operates through system failure if you wish for backlog to be processed. If this is not defined, then the block will not operate through system failure and, thus, will not be able to process any backlog when components that cause system failure (from an RBD perspective) fail.
Before running a throughput simulation, you can also specify the allocation scheme across multiple paths (i.e. equal or weighted) and decide if the items should be sent to failed parts. These options are shown in Figure 4.
Figure 4: BlockSim's throughput settings for simulation
To explain these settings, consider the following system.
If the Weighted allocation across paths option is selected, then the 60 items made by A will be allocated to B, C and D based on their throughput capabilities. Specifically, the portion that each block will receive, Pi, is:
The actual amount is then the portion times available units. In this case, B's portion is , C's portion is and D's portion is . When a total of 60 units are processed through A, B will get 10 units, C will get 20 units and D will get 30 units. The results would then be as shown in Table 6.
Table 6: Weighted allocation results
If the Allocate equal share to all paths option is selected, then 20 units will be sent to B, C and D, regardless of their processing capacity, yielding the results shown in Table 7.
Table 7: Equal allocation results
If the Send units to failed blocks option is not selected, the throughput units are allocated only to operational blocks. Otherwise, if the option is selected, units are allocated to failed blocks and they become part of the failed block's backlog.
In the special case that a block (or blocks) fail causing a disruption in the path and the Send Units to Failed Blocks option not selected, then the blocks immediately before the failed block(s) will not be able to process any items since the items cannot be sent to the forward block. In this case, these blocks will keep all items received in their backlog. As an example, consider the system in Figure 5. If E has failed (and E cannot accept any items in its backlog while failed) then B, C and D cannot forward any items to it. Thus, they will not process any items sent to them from A. Items sent from A will be placed in the backlog of items B, C and D.
Figure 5: Illustrating a special backlog case
In next month's issue of the HotWire, we will continue the discussion of throughput analysis and how the reliability and maintenance properties of the blocks and the system are taken into account.
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