Overview and notation
The shared notation table used by every formula.
Every number the tool produces comes out of the seven formulas in this section. They are written in standard mathematical notation so you can cross-check them against the academic literature, and each formula has a hand-calculated example so you can verify it yourself.
Notation table
| Symbol | Meaning | Unit |
|---|---|---|
| V | The set of all steps in the process | — |
| tv | The duration of one run of step v | minutes |
| cv | The cost of one run of step v | money |
| nv | The expected number of runs of step v in one pass of the process | runs |
| pi | The probability of entering branch i of a gateway | 0…1 |
| P | The transition probability matrix between steps | — |
| e0 | The indicator vector of the start point (1 at the first step, 0 elsewhere) | — |
| k | The number of branches of a parallel gateway | — |
Seven formulas, one thread
| # | Formula | Used for |
|---|---|---|
| 1 | Tseq = ∑ ti | Sequential chain |
| 2 | TAND = max Tj | Parallel block |
| 3 | CAND = ∑ Cj | Parallel block |
| 4 | E[TXOR] = ∑ pi Ti | Exclusive gateway |
| 5 | nT(I − P) = e0T | Loops |
| 6 | E[T] = ∑ nv tv | Putting it together |
| 7 | P90, P95 | Simulation |
Two calculation routes, to check each other
The tool computes every process by two independent methods:
| Analytical | Monte Carlo simulation | |
|---|---|---|
| How it works | Solving a system of linear equations | Running thousands of random trial passes |
| Result | One exact expected value | A whole distribution: median, P90, P95 |
| Loops | Counts the full infinity of turns | Samples according to the real probabilities |
| Parallel blocks | Uses the max of the per-branch expected values (an approximation) | Takes the max of the simulated values (exact) |
| Repeatability | Always gives the same result | Fixed by the seed |
Use the gap between them as a test
These two numbers should be close. A significant gap (over 5–10%) is usually a sign that the process has parallel branches where each branch itself contains randomness — in that case max E[·] < E[max ·] and the simulated number is the trustworthy one. See Monte Carlo simulation.
These two numbers should be close. A significant gap (over 5–10%) is usually a sign that the process has parallel branches where each branch itself contains randomness — in that case max E[·] < E[max ·] and the simulated number is the trustworthy one. See Monte Carlo simulation.
Anything unclear or missing?