Markov

Structure and Drive
Paul A. Jensen Copyright July 20, 2003 A system is made up of several operations with flow passing between them. The structure of the system describes the flow paths from inputs to outputs. In this section we recognize three structure alternatives, line, tree and network. The drive option specifies the cause of flow through the system. Here we have two options, pull and push.For the pull option, flow is pulled from the outputs of operations. For the push option, flow is pushed into the inputs of the operations. For each of the six combinations of structure and drive, given the flows into or out of the structure, one can compute the flows through the system operations. The remainder of this section describes the analysis. Pull Line The line structure, illustrated inFig. 1, is the simplest because flow enters at the first operation and leaves from the last. Here we show the pull line where the flows are determined by the flow leaving the system at operation 5. When there is no change in flow caused by the operations, the flow that enters operation 1 also leaves operation 5 and all operations have the same flow. We analyze the case where some mechanism allowsthe flow to be increased or decreased as it passes through an operation. We assume the operations are numbered in order from 1 to m, with m the last operation. m is the flow withdrawn from operation m. In general we will also allow flow to be withdrawn from the other operations as well, with i defined as the flow pulled from the output of operation i. We call the flows pulled from or pushed intothe system external flows to distinguish them from the flows within the system. Our goal is to find the flow through each operation as a function of the external flows. Define the following notation. Figure 1. The pull line xi = The flow passing into operation i xi’ = The flow passing out of operation i

The flow ratio for an operation is the ratio between the flow leaving the operation and theflow entering. Then ri = xi ′ . xi

There are many situations where the ratio is other than 1. Perhaps the operation does inspection in a manufacturing system and faulty items are removed from the flow. Here the ratio would be less than 1. In another situation, the operation may divide an item into two parts. Every entering item results in two leaving items, so the ratio is 2. We assume theratios are given. The value of xi’ is entirely dependent on the pull flow withdrawn at operation i and the amount required by the following operation, i + 1. xi ′ = xi =
i

+ x i +1
i

xi ′ = ri

+ x i+1 ri

(1)

Since the flow for an operation depends on the flow of its unique following operation, we can compute flows recursively, starting with the xm and continuing for each operation withsequentially decreasing operation index. An example based on Fig. 1 has operations 2 and 4 each causing a 10% loss in flow. The resultant operation flows are in Table 1. In order to pull 100 units from operation 5, more than 100 units must pass through operation 1 to accommodate for the losses.

Table 1. Pull line parameters and operation flows with Index (i)
1 2 3 4 5

5

=100.

PullOut ( i)
0 0 0 0 100

Ratio (ri)
1 0.9 1 0.9 1

Flow (xi)
123.46 123.46 111.11 111.11 100

Push Line The push line is illustrated in Fig. 2. The only difference between the push and the pull lines is the driver for flows. When all flow ratios are 1, there really is no difference between the two drive mechanisms because all operation flows are the same. Again we assume the operations arenumbered in order from 1 to m, with m the last operation. Flows enter operation 1 in the amount 1. In addition to node 1, products may also be pushed into the network at other operations. The flow entering at operation i is ι. Note that push flow enters the process just before the operation, while pull flow leaves the process at the output of the operation. The notation for flow and flow ratios...