Fix survival chance per timestep calculation (#287)

TF consistently interprets the 'momentary failure rate' curve as
'instantaneous hazard rate' - the proportional failure rate of systems
still working at the current time - for display purposes (e.g. the
'Reliability' and 'MTBF' displays are calculated as if it is the hazard
rate). However, the conversion to a failure chance per update cycle
wasn't done correctly, and the actual failure rate over time didn't
match the instantaneous MTBF when the failure rate is not constant.

Now, the cycle reliability curve directly specifies the hazard rate h(t)
(which is the same as the 'failure rate' λ(t)), and the chance to fail
over a timestep Δt is h(t)*Δt.

Author: Nazfib

TestFlightAPI/TestFlightAPI/TestFlightReliability.cs

@@ -18,8 +18,6 @@ public class TestFlightReliabilityBase : PartModule, ITestFlightReliability
         public FloatCurve reliabilityCurve;
         [KSPField(isPersistant=true)]
         public float lastCheck = 0;
-        [KSPField(isPersistant=true)]
-        public float lastReliability = 1.0f;
 
 
         public List<ConfigNode> configs = new List<ConfigNode>();
@@ -173,6 +171,7 @@ public override void OnUpdate()
             if (operatingTime < lastCheck + 1f)
                 return;
 
+            double timestep = operatingTime - lastCheck;
             lastCheck = operatingTime;
             double baseFailureRate = core.GetBaseFailureRate();
             MomentaryFailureRate momentaryFailureRate = core.GetWorstMomentaryFailureRate();
@@ -183,19 +182,11 @@ public override void OnUpdate()
             else
                 currentFailureRate = baseFailureRate;
 
-            // Given we have been operating for a given number of seconds, calculate our chance of survival to that time based on currentFailureRate
-            // This is *not* an exact science, as the real calculations are much more complex than this, plus technically the momentary rate for
-            // *each* second should be accounted for, but this is a simplification of the system.  It provides decent enough numbers for fun gameplay
-            // with chance of failure increasing exponentially over time as it approaches the *current* MTBF
-            // S() is survival chance, f is currentFailureRate
-            // S(t) = e^(-f*t)
-
-            float reliability = Mathf.Exp((float)-currentFailureRate * (float)operatingTime);
-//            double survivalChance = Mathf.Pow(Mathf.Exp(1), (float)currentFailureRate * (float)operatingTime * -0.693f);
-            double survivalChance = reliability / lastReliability;
-            lastReliability = reliability;
-//            float failureRoll = Mathf.Min(UnityEngine.Random.Range(0f, 1f), UnityEngine.Random.Range(0f, 1f));
-//            float failureRoll = UnityEngine.Random.Range(0f, 1f);
+            // The momentary failure rate, as implemented in TF, is more accurately called the 'hazard rate', h(t). This is defined as the
+            // proportional failure rate of systems still functioning at time t.
+            // For a working system at time t and a small interval dt, the chance to fail in the next dt seconds is approximately h(t)*dt; this
+            // approximation is exact in the limit where dt goes to 0.
+            double survivalChance = 1 - currentFailureRate * timestep;
             double failureRoll = core.RandomGenerator.NextDouble();
             if (verboseDebugging)
             {
@@ -206,7 +197,7 @@ public override void OnUpdate()
 //                Debug.Log(String.Format("TestFlightReliability: Survival Chance at Time {0:F2} is {1:f4} -- {2:f4}^({3:f4}*{0:f2}*-1.0)", (float)operatingTime, survivalChance, Mathf.Exp(1), (float)currentFailureRate));
                 if (verboseDebugging)
                 {
-                    Log($"Part has failed after {operatingTime:F1} secodns of operation at MET T+{vessel.missionTime:F2} seconds with roll of {failureRoll:F4}");
+                    Log($"Part has failed after {operatingTime:F1} seconds of operation at MET T+{vessel.missionTime:F2} seconds with roll of {failureRoll:F4}");
                 }
                 core.TriggerFailure();
             }