# Development Cycle Modeling: Resource Estimation
## 4 Results
### SAbMDE Resource Utilization  
Begin an estimate of the resource utilization associated with a DPath by assigning a price to each vocabulary item and relation. Then, with an appropriate function, assign a price to the act of their composition. Similarly, assign a price to their decomposition. The definitions and equations below describe a simple pricing system. \[P_{v} =\text{vocabulary item price, actual or normalized}\] \[P_{r} =\text{relation price, actual or normalized}\] \[g_{cp}(P_{v},P_{r}) =\text{composition pricing function}\] \[g_{dp}(P_{v},P_{r}) =\text{decomposition pricing function}\] \[f_{b} =\text{backtrack factor}\] \[l_{b} =\text{backtrack length}\] \[n =\text{number of decision re-tries}\]  
\[g_{cp}(P_{v},P_{r}) =(P_{v}+P_{r}) \tag{10}\] \[g_{dp}(P_{v},P_{r}) =f_{b}g_{cp}(P_{v},P_{r})\] (11) \[P =g_{cp}+(n-1)l_{b}g_{dp} \tag{12}\]  
Finally, calculate a DPath's resource utilization by summing the composition prices for each product in the DPath. Such a summation for as-yet untraversed portions of the DPath is an estimate of the resources to be utilized. As noted previously, only a perfect agent can make the perfect decision set to reach a DEP. All other agents will make incorrect decisions. Of course, agents must attempt to complete a project; so when an agent discovers an incorrect decision, they will likely undo that decision and try again. The number of retries is a function of the agent's skill index and the number of alternative choices as defined by the DSpace. An agent retries by selecting one of the alternatives, evaluating the  
\[P_{r} =\text{relation price, actual or normalized}\]0 Probability of sequential decision success. selection, and again discarding incorrect selections. The hypergeometric distribution [40] in (13) can describe this process. \[p_{c}(k_{m})=\frac{\left( \begin{array}{c}K_{M}\\ k_{m}\end{array} \right)\left( \begin{array}{c}M-K_{M}\\ m-k_{m}\end{array} \right)}{\left( \begin{array}{c}M\\ m\end{array} \right)} \tag{13}\]  
where  
\[M=|V||R|,\ K_{M}=M_{p_{skill}},\ k_{m}=0\]  
The (13) variables are interpreted as follows. For each skill index, there is a probability, \(p_{skill}\), of making a correct decision. For a decision population size, \(M\), \(p_{skill}\) is equivalent to having a correct decision population size, \(K_{M}\). Given these values, the hypergeometric distribution can calculate the probability that a sample of size m will have \(k_{m}\) correct decisions. Setting \(k_{m}=0\) and choosing a small probability criterion, \(p_{c}(k_{m})\), corresponds to calculating an upper limit on the number of retries, \(n=k_{m}\), that will guarantee a correct decision. Table 1 is an excerpted example of this calculation for a \(p_{skill}\) corresponding to skill index=0, \(M=100\), \(K_{M}=1\), and a 0.1 probability criterion, \(p_{c}(k_{m})\). The table shows that as many as 90 retries are required to ensure a 90% chance of decision success. When an agent makes an incorrect decision, the agent may continue on the incorrect DPath for some number of compositions. Once the error is discovered, the agent will likely return to the last correct decision and retry; and the no longer needed compositions will be decomposed during the return trip. Equations (11) and (12) handle the added resource utilization with the backtrack length and backtrack factor parameters. The latter assumes that the price of composition and decomposition differ. Figure 4 shows resource utilization estimates calculated as described above. These estimates were calculated using skill index 7, with vocabulary item and relation prices of 1 and 10, respectively, and with backtrack length and backtrack factor values of 1 and 1.5, respectively. In keeping with the last Markov Chain implication noted above, estimates are repeated at each composition index of the 20-composition project. For each graph trace, the resource utilization value on the far right (\(l=19\)) is the estimate for the remainder of the project as viewed from the composition index of the origin of the trace.