## 4 PIT Protocol  
For PIT [3], the expected depletion rate \(D\) is _not constant_. The rationale behind PIT is to resolve the shortage issue where PITEC might encounter. The core tasks to embed in the protocol design would be (1) the time to replenish and (2) the quantity of transactions to be replenished. Equilibrium finding within a cycle provides the solutions to the two tasks. To use the verifier resources, one must calibrate the optimum to replenish for the shortage scenarios. In PIT protocol, the randomization is invoked with a basic assumption. The basic assumption is that the expected average transaction validation rate is a stable constant, D, even though the underlying micro-level behavior is random as illustrated in Figure 1.  
When a transaction level \(r\) is reached, this triggers the optimum level, \(Q^{*}\), of fresh transaction release. Let lead time, \(L\), be the time it would take from the time level \(r\) is reached to the time fresh transactions are released. For this model, we have couple variable factors to consider. First \(L\) is random and is predicted by how fast controller could provide the supply. Second, the depletion rates at the evaluators are random. To simplify the analysis, PIT protocol assumes that each cycle is of fixed time length \(T\). A typical layout of such a model may be found in [6] for inventory control system. The useful variables are summarized in Table 1.  
With a fixed cycle time \(T\), this immediately resolves the first task about when to replenish into the system. For the second task, what is left is how to predict the optimum pulsed batch size \(Q^{*}\). In this section, a sketch is provided. Interested readers should refer to the original article [3]. Let \(p(y)\) be a probability mass function that describes the random depletion of transactions. With probability \(p(y)\), \(y\) is the number of transactions verified during the lead time \(T\). The time movement profile of transactions and its random depletion rate is shown in Figure 2.  
Figure 1: Q on the Y axis is the number of transactions to be verified while X axis is the time. T represents the time for a cycle. \(D\) is the mean depletion rate (the dashed line) of random rates. When less than \(r\) transactions left unconsumed, the trigger is activated. During the lead time \(L\), let the average number of verified transactions be  
\[M_{L}=\sum_{y=0}^{\infty}p(y)y, \tag{5}\]  
and let the average volume of the unverified transactions be  
\[\frac{1}{2}(r+\sum_{y=0}^{r}(r-y)p(y)). \tag{6}\]  
The expected number of transactions available to the verifiers after a replenishment until the next reorder trigger is  
\[\frac{1}{2}(\underbrace{(r+Q-M_{L})}_{\text{transactions after replenishment}}+ \underbrace{r}_{\text{transactions at reorder}})=\frac{(2r+Q-M_{L})}{2} \tag{7}\]  
Thus, between replenishment and reorder, the number of transactions consumed is  
\[T_{r}\times D=(Q+r-M_{L})-r=Q-M_{L}, \tag{8}\] \begin{table}
\begin{tabular}{l l} \hline \hline  & **Variable Description** \\ \hline \(A\) & Fixed cost per cycle \\ \(C\) & Cost per cycle per unit time \\ \(F\) & Cost per cycle \\ \(L\) & Lead time: time elapsed between reorder and replenishment \\ \(M_{L}\) & Average volume of verified transactions \\  & during L \\ \(Q^{*}\) & Optimum batch size \\ \(Q\) & Optimum batch size level \\ \(S_{L}\) & Expected number of shortage transactions \\ \(T\) & Time length of a cycle \\ \(T_{r}\) & Time elapsed between replenishment \\  & and reorder \\ \(c_{s}\) & Unit shortage cost during lead time \\ \(r\) & Trigger threshold \\ \(v\) & Cost of carrying released but unverified \\  & transactions to next cycle per transaction \\  & per unit time \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables. Figure 2: Q on the Y axis is the number of transactions to be verified while X axis is the time. T represents the time for a cycle. \(T_{r}\) is the trigger time when \(r\) transactions are left unconsumed. The second plot shows a shortage situation.