Formations

[Zook (OP)]

Do you use the Escort orders to create formations? If so, how do they look? What distances and offsets?

If you have any good advice, I might put it in the wiki.

[Erik L]

I have.

I usually put 2-3 destroyer escort squads are -45, 0, +45 degrees offset from the main body at 100-120k.

[Zook (OP)]

I still don't have much experience with ship combat, but it seems to me that most players don't bother with formations and instead use "Empire Formations", having everything and the kitchen sink in the same spot. I'm still not giving up on the subject, but the advantages of wider formations (ships at an angle to the main body) seems questionable to me now. Instead I'm toying with a column formation:

Code: [Select]

(TF heading this way)
    ^
    |
    |
Small anti-missile corvettes  (2,000 tons and harder to spot than the rest)
    |
    |
Anti-missile frigates
    |
    |
Beam area defense (by being ahead of the main body, they can fire their lasers at missiles flying past them, thereby doubling their firing opportunities)
    |
    |
Main Body (missile destroyers and colliers)
    |
    |
Sensor ship(s) - easy to spot, should be protected by every defense in the TF
    |
    |
Support ships (tankers, jump ships etc.) - waaay behind the main body

Does that make sense?

[TheDeadlyShoe]

It's mostly down to the mechanics of anti-missile beam defence.

beam area defence is mostly ineffectual against missiles of the same tech level. at best, you get about the same results as if you had built for final fire instead.

And final fire works best in Empire State formation.

Seperating out anti missile ships can work in order to give them more time to fire anti-missiles, but there are still three problems with this:

*your AMMs need to be significantly faster than enemy ASMs (not usually true on similar tech levels)
*you lose final fire defence on your AMM ships
*it only actually makes a difference if you are limited by your # of launchers rather than your stowage.

The last point is the most important. Usually, you are more limited by your AMM capacity than your fire rate.  There are a few scenarios where this is not the case,  however.

[Steve Walmsley]

It can be useful to switch into a formation from Empire State when you come under missile attack. The enemy missiles are already en route so their targeting won't be affected by the formation. If you turn away the main body and detach AMM ships to take up positions to the rear of the formation and either side of the approaching missiles you can have a much longer engagement window. Placing them to the side is useful because you tell if missiles are heading for the main body or the escorts and react accordingly.

Formations can also be useful to protect certain units by placing them 180 degrees from the threat.

Steve

[Theokrat]

Just to expand a bit on TheDeadlyShoe’s point analytically: There are two problems with formations.

The first is that you need to make sure that the advance picket does not become a target itself. Being the first in line of the enemy means the rest of the fleet cant properly support its defence, so you need to make sure it can not be targeted. As you say, one way to achieve this is to make it small. You can include some ECM or cloaking to make that more viable.

However, the larger problem is that area-defence for beam-weapons is quite costly and thereby inefficient. For a weapon that fires every 5s you can use the following formula to determine how many missiles an advance picket is expected to intercept:

P = r/(5*v) (2 – r/R) * A

Where r is the range of the beam-weapon, v is the relative speed of the missile, R is the maximum range of the firecontroll, and A is a constant factor that is comes from non-range considerations (tracking speed, crew grade, tracking bonus, missile ECM etc.). The formula assumes that r < R (i.e. the Fc has a longer range than the beam), and that the picket is exactly on the attack line and at least at a distance of r in front of the missiles intended target. Also A<1 is assumed (and usually the case).

Whereas for final fire you can use:

Q = (1-10/L)*A

Where L is the Range of the FC in thousand km (analogous to R above, just to avoid confusion).

Let’s plug in the numbers from the Wiki for an early beam ship: r=90k km, R = 192k km (i.e. we will want a very long range and pick the longest-reaching FC for the area defence version), L=48k km. v=24k km/s (pretty conservative). Let us say we have 1,400t worth of payload that we could use as beam defence. I think that is quite high, because with engines, fuel etc that would make a ship of about 2,500t and that is probably on the relatively large side for an advance picket.

So for the area-defence version we would have to spend 800t on the firecontroll (size 16, 4x range, 4x tracking). That would leave space for four 10cm-Lasers. Therefore we would expect about 4*P = 4* 90/120* (2-90/192) * A = 4.6 * A missiles being shot down.

Conversely, what happens for the final-fire version? The firecontroll is only 200t (size 4, 1x range, 4x tracking), which leaves 1200t for eight 10cm-Lasers. Therefore we would expect to shoot down 8*Q= 8*(1-10/48)*A= 6.3 * A missiles.

In other words: Spending the same tonnage on the final-fire version means we shoot down ~40% more missiles! And that solution means you don’t have to muck about problem 1 either (avoiding that the advance picket becomes a target itself). Additonally, while both setups use the same tonnage, the final-fire version will likely be much cheaper because it has the smaller tonnage devoted to the firecontroll (which are disproportionately expensive). Moreover: There are altogether better alternatives to lasers when it comes for anti-missile defence like railguns and gaussguns. These alternatives share a rather reduced range, which means they can hardly be employed in area defence either.

So bottom line: Final fire is by far the most efficient way to deal with incoming missiles through beam weapons. The prime reason for this is that you get a guaranteed shot at a short distance which conveys the smallest penalty for the distance-modifier. “Don't fire until you see the whites of their eyes”

[Charlie Beeler]

Theokrat there are a couple of very important things you do not appear to addressing correctly. 

• beam fire control hit probability at intercept range
• tracking bonus is not a constant, it's cumulative

[Theokrat]

Theokrat there are a couple of very important things you do not appear to addressing correctly. 

• beam fire control hit probability at intercept range

I am not sure what you mean. I do explicitly cover the effect of range on the hit probabilities. That is after all the entire point that makes a difference between final-fire and area-defence.  The whole derivation of the hitchances quoted above shoves everything else into the “A” factor; the rest of the formulas is only concerned with the hit probability due to range.

Take final fire for example: Intercept occurs at 10,000 km, so the hit probability due to the range of the firecontroll is P =  1 – 10,000 km / R, where R is the range of the firecontroll. So for the example above its P = 1 – 10,000km / 48,000km = 79%. Of course this is not the final hitchance, which would include other effects (most importantly tracking speed), but it’s the separable factor due to the range of the BFC.

For area defence the derivation is a bit more complex, as essentially you have to take an integral over the hitchance at range x (P = 1 – x/R) times the probability density function ( uniform distribution, i.e. rho(x) = 1/(5v)). With a bit of algebra you get to the formula above.

• tracking bonus is not a constant, it's cumulative

Could you explain that in more detail? I was under the impression that the hit chance is computed as follows P = p_range * p_trackingspeed * p_crewgrade * p_trackingbonus * p_missileECM. So I basically wrote that as  P = p_range * EverythingElse in order to compare final fire versus area defence. If the tracking bonus is an additive bonus rather than a multiplicative one, then that would change the calculations (in favour of area defence). Is that what you are saying?

I understand that the tracking bonus must accumulate over time until it reaches the maximum possible amount at the current tech level. So I implicitly assumed that this highest possible level had been reached at the maximum range of the area-defence variant (and that thereafter the tracking bonus was constant). That is prudent I would say, as its quite easy to have missile search sensors with a range of millions of km, while the maximum range of the 10cm laser from above was a mere 90k km. Moreover it is a conservative assumption in the sense that the assumption favours the area-defence variant (which still loses out).

[Zook (OP)]

Sweet Jesus. The man has a degree in auroralogy. Anyway, that's great stuff. I'll post it to the wiki when I understand it.

But although I know that area defense is less effective, there are two reasons I still build laser turrets:
1) for the unlikely event that I came across an unarmed target (civilian, crippled ship, etc.) and
2) for the time when my missile magazines have run dry, or exploded.

I'm fighting only my second battle as I type this, but #2 has happened to me the first time, and my trusty triple lasers still shot down a few missiles, if only with 19% hit chances. Better tech should raise this to about 40% with the next generation ships. I'm also adding a line of 3,000-ton gauss-turret corvettes, as the last line of defense.

Speaking of the wiki, what's the exact rule for final fire? And what's that missile tracking bonus that you can research?

[Redshirt]

Reason 3 for laser turrets- nebulae 
They're also good system defense ships that never have to worry about ordinance. Sure, they might be severely outclassed by an invasion force, but they should probably be able to deal with scouts and enemy survey ships quite handily.

[Zook (OP)]

Right! Three cheers for beam defenders! OK, I haven't seen nebulae yet, but they sound nasty.

One more question: if I remember correctly, the last time I fought I could reload missiles from a collier that wasn't even in the same location as my destroyers. The distance wasn't great, but instant reloads came in very handy. Is there a maximum distance for that?

And while we're at it, the percentage displayed next to an enemy missile's speed, heading etc. - is that the tracking bonus? Is it applied to beam or missile defenses, or both?

[Charlie Beeler]

I am not sure what you mean. I do explicitly cover the effect of range on the hit probabilities. That is after all the entire point that makes a difference between final-fire and area-defence.  The whole derivation of the hitchances quoted above shoves everything else into the “A” factor; the rest of the formulas is only concerned with the hit probability due to range.

Take final fire for example: Intercept occurs at 10,000 km, so the hit probability due to the range of the firecontroll is P =  1 – 10,000 km / R, where R is the range of the firecontroll. So for the example above its P = 1 – 10,000km / 48,000km = 79%. Of course this is not the final hitchance, which would include other effects (most importantly tracking speed), but it’s the separable factor due to the range of the BFC.

For area defence the derivation is a bit more complex, as essentially you have to take an integral over the hitchance at range x (P = 1 – x/R) times the probability density function ( uniform distribution, i.e. rho(x) = 1/(5v)). With a bit of algebra you get to the formula above.

Could you explain that in more detail? I was under the impression that the hit chance is computed as follows P = p_range * p_trackingspeed * p_crewgrade * p_trackingbonus * p_missileECM. So I basically wrote that as  P = p_range * EverythingElse in order to compare final fire versus area defence. If the tracking bonus is an additive bonus rather than a multiplicative one, then that would change the calculations (in favour of area defence). Is that what you are saying?

I understand that the tracking bonus must accumulate over time until it reaches the maximum possible amount at the current tech level. So I implicitly assumed that this highest possible level had been reached at the maximum range of the area-defence variant (and that thereafter the tracking bonus was constant). That is prudent I would say, as its quite easy to have missile search sensors with a range of millions of km, while the maximum range of the 10cm laser from above was a mere 90k km. Moreover it is a conservative assumption in the sense that the assumption favours the area-defence variant (which still loses out).

Whether beam point defense is set to area or final the formula for hit probability is the same.  The only things that are different are intercept range and tracking time.

Your baseline hit probability of target range/max fire control range is correct.  

your failing to account for is the modifier for fire control and/or turret tracking speed being below the target speed.  Against missiles, unless you have a significant tech advantage, there will always be a negative modifier.  That is what the tracking bonus, if the tech has been developed, offsets.  The modifier is tracking speed/target speed.  

Tracking bonus accumulates per cycle up to the bonus.  This starts from when the missile salvo is first detected and increments 2% each  5 second cycle until the bonus is reached, the speed penalty is zeroed, the missile intercepts it's target, or the missile is intercepted whichever comes first.  

When building a detailed explanation of game mechanics never assume.

Area defense mode is the least likely to be effective.  But it is a function of beam range vs the distance a missile travels in a single game cycle.  It's leftover from when missile speeds were segnificantly slower and you could reasonably expect to have 2 or more chances to engage missiles with lasers prior to intercept.

[Theokrat]

Sweet Jesus. The man has a degree in auroralogy. Anyway, that's great stuff. I'll post it to the wiki when I understand it.

Hehe, I am merely an analyst, so breaking things down to numbers is kind of what I do. Although it greatly helps that in Aurora systems come with a number and formula tag attached. In real life the painful thing is trying to get people to give you a reasonable answer how much an increase of this or that variable by x% would help them. Here is a rather remarkable paper on system analysis. Ok I am getting quite off-topic.

But although I know that area defense is less effective, there are two reasons I still build laser turrets:
1) for the unlikely event that I came across an unarmed target (civilian, crippled ship, etc.) and
2) for the time when my missile magazines have run dry, or exploded.

Both of these would also apply to other final fire weapons, like railguns and gaussguns though.

Whether beam point defense is set to area or final the formula for hit probability is the same.  The only things that are different are intercept range and tracking time.

Yes, the formula for hit probability of a single shot is the same. But your list of “the only things that are different” between final fire and area defence is incomplete: Area defence has the potential to engage the same salvo more than once, while final fire has one shot only. When you compare the effectiveness of both methods you have to take that into account.

We should not be interested in the hit probability of a single shot, we should be interested in the total missiles that we can expect to shot down. For final fire there is no difference, because you only get one shot (at a fixed predetermined distance). So for final fire you can just use the hit-probability to determine the expected shot down missiles.

The same is not true for area defence. Area defence can sometimes shot more than once, so you cannot simply take the hit chance of a single shot in order to determine the expected shot down missiles. The formula of total expected shot down missiles for area defence is different from the hit probability for a single shot, because area defence is not about single shots but about a probability-weighted aggregate of engaging missiles at different distances, potentially multiple times. If you consider all this correctly you get the formula shown above.

I am not sure what you mean. I do explicitly cover the effect of range on the hit probabilities. That is after all the entire point that makes a difference between final-fire and area-defence.  The whole derivation of the hitchances quoted above shoves everything else into the “A” factor ; the rest of the formulas is only concerned with the hit probability due to range.

Take final fire for example: Intercept occurs at 10,000 km, so the hit probability due to the range of the firecontroll is P =  1 – 10,000 km / R, where R is the range of the firecontroll. So for the example above its P = 1 – 10,000km / 48,000km = 79%. Of course this is not the final hitchance, which would include other effects (most importantly tracking speed), but it’s the separable factor due to the range of the BFC.

For area defence the derivation is a bit more complex, as essentially you have to take an integral over the hitchance at range x (P = 1 – x/R) times the probability density function ( uniform distribution, i.e. rho(x) = 1/(5v)). With a bit of algebra you get to the formula above.

Could you explain that in more detail? I was under the impression that the hit chance is computed as follows P = p_range * p_trackingspeed  * p_crewgrade *  p_trackingbonus  * p_missileECM. So I basically wrote that as  P = p_range * EverythingElse  in order to compare final fire versus area defence. If the tracking bonus is an additive bonus rather than a multiplicative one, then that would change the calculations (in favour of area defence). Is that what you are saying?

I understand that the tracking bonus must accumulate over time until it reaches the maximum possible amount at the current tech level. So I implicitly assumed that this highest possible level had been reached at the maximum range of the area-defence variant (and that thereafter the tracking bonus was constant). That is prudent I would say, as its quite easy to have missile search sensors with a range of millions of km, while the maximum range of the 10cm laser from above was a mere 90k km. Moreover it is a conservative assumption in the sense that the assumption favours the area-defence variant (which still loses out).

your failing to account for is the modifier for fire control and/or turret tracking speed being below the target speed.  Against missiles, unless you have a significant tech advantage, there will always be a negative modifier.  That is what the tracking bonus, if the tech has been developed, offsets.  The modifier is tracking speed/target speed.

I do not “fail to account for this modifier”. I have marked the points in blue where I explicitly talk about this factor in my post that you quoted.

The point is that this factor is the same for area defence and final fire – the target speed is certainly the same, and the tracking speed is the same. So if we want to compare the relative effectiveness of final fire vs. area defence we do not need to consider the factors that both components share (like crew grade, tracking speed), but only need to consider the factors that set them apart (engagement range, ability to engage multiple times).

That is why we can aggregate the things that we are not interested in (for this comparison) into the “A” factor, while continuing the discussion on the relevant factors.

So what about the “tracking bonus” that is due to the timing of tracking. Yes I ignored that in my example in the sense that I assumed this bonus would be equal for final fire and area defence engagements. Is that a fair assumption? I would say yes, and here is why:

The example is based on relatively early tech (the beam-warship tutorial from the wiki). So realistically one might assume that the max_tracking_bonus is 20%, i.e. the first technology stage. This bonus is achieved after tracking the missile for 50 seconds (2%/5s). The missile in the example had a relative speed of 24,000 km/s, so in 50 seconds it could fly 1.2 million km. It is very plausible to build a missile search sensor with a range of 1.3 million km. So effectively the tracking bonus will be maxed out to 20% even before the missiles reached the engagement range (0.1 million km) of the area-defence variant. Consequently it will also be maxed out at the very same 20% when it reaches the final fire variant’s engagement range (0.01 million km) – Presto the same number for both variants, so in order to compare their relative effectiveness we can ignore this effect.

But even if you would contest my assumptions about the tech level or the employed search sensor, things would not be radically different. Just how much of an effect would be there if the bonus was not maxed out already? In the example the area defence variant used a laser with a range of 96,000 km versus a missile going 24,000km/s. So in a 5s interval the missile will go 120,000 km. So effectively the “extra time” that we are talking about during which more of a tracking  bonus could accumulate is –maybe- 5 seconds. Worth –maybe- 2% tracking bonus.

This really does not change the overall finding that final fire is ~40% more effective, does it? Moreover it is slightly in favour of the final fire variant. If the first stage of the calculations showed that final fire is way superior and area defence is really not worth bothering, why would we want to go through great length to make a minor second-order adjustment that will only increase the difference? Especially, since I made more significant assumptions in favour of area defence, like the relatively slow missile. Anyway, glad we discussed.

Area defense mode is the least likely to be effective.  But it is a function of beam range vs the distance a missile travels in a single game cycle.

Yeah. And if you have a close look at the formula above you will see that these factors are in there: Beam range (r), distance a missile travels in a single game cycle (5*v), and even range of the beam fire-controll (R).

[Charlie Beeler]

You’re right, I am only addressing a single intercept and when your formula is broken down it is as well.  Don’t believe me?  It’s simple, by using only a single fixed intercept range value not a variable modifier to account for changes as the missile crosses the intercept envelope.

Here is the correct formula for calculating beam hit probability:  hC = (1-r/fR)(sV/tV+tB)(1-(tE-sE))(cG)

• hC –   hit chance
• bR –   maximum beam range
• mR –   Minimum beam range (10,000km)
• r  –   range to target
• fR –   maximum fire control range
• sV –   ship tracking speed
• tV –   target speed
• tB –   current tracking bonus vs missiles (if applicable)
• tE –   target ECM%
• sE –   ship/fire control ECCM%
• cG –   crew grade modifier

There are limiters to this formula:

• range to target must be <= beam maximum range
• tracking speed modifier is never a true bonus to hit probability (ie if ship tracking is greater than target speed there is no modifier)
• tracking bonus vs missiles can only neutralize the tracking speed modifier.  It is never a true bonus to hit probability
• ship ECCM can only neutralize target ECM.  It is a never true bonus to hit probability

(missile armor can also be a factor, but is a secondary calculation for missiles that have been intercepted and not a modifier to the hit probability.  It’s not included here after reviewing the distributed database and of the 12 NPR’s none are using missile armor.)

Additionally ship tracking speed determination is a little complex. 
Which is greater ship speed or fire control tracking speed tech?  If target speed is greater than tracking speed, is the ship fire control speed > tech speed?  If the fire control is greater and  the beam mount is turreted then the ship tracking speed is the lesser of fire control speed or the turret tracking speed.

So with the formula above there are 2 subsections that can be excluded for basic missile intercept analysis:

• (1-(tE-sE))   since missiles vary rarely have ECM.
• cG             since crewgrade is ship specific, but is more common than ECM for missiles.

You also argue that since tracking speed and target speed (sV/tV) are fixed data points and not variable they can be functionally ignored.  I stipulate that you are wrong.  If preliminary analysis showed that these values made no change to the baseline probability you would be correct.  But since we’re actually talking about a modifier that can substantially reduce the hit probability it must be included.   Especially with you own examples have fire controls with significantly different tracking speeds against the same target speed(3000 vs 12000).

The last modifier you ignore (missile tracking bonus tB) also has a potential for a substantial change in the results.  In fairness to those reading the analysis examples with and within the bonus should be given to demonstrate the impact on results.
The last point of contention is that your “density function” falls short, it only accounts for a single 5 second impulse not the multiples that you assert. 

Since the area defense analysis needs variable intercept range accounted for substitute ((1-bR/fR)+(1-mR/fR))/2 for (1-r/fR).  This is a crude model, but serves.

To account for possible multiple intercepts add (bR/I(5*tV)) where I – 5 second impulses.  If the modeled beam weapon has a ROF that allows multiple shots while the target crosses engagement envelope adjust the value of I for the needed impulses.

This gives P = (bR/(I(5*tV)))(((1-bR/fR)+(1-mR/fR))/2 )(sV/tV+tB))

If bR >= fR or < (I(5*tV)) then (1-bR/fR) is replaced with .01 to represent worst case intercept chance.

The new formula may be multiplied by the number of weapons linked to the fire control to determine minimum intercepts.

Now we plug-in weapon and fire control values for the area defense example:

4*P=(90/(1(5*24)))(((0.01)+(1-10/192))/2)(12/24+0.2) = 1.01 missiles intercepted not 4.6.

The final defense example becomes: 
8*P=(1-10/48)(12/24+0.2) = 4.43 missiles intercepted not 6.3.

Note: tB = .2 to assume that tracking bonus 20% is in place and that an active sensor has the requisite detection range.  Otherwise the intercepts drop to .72 for the area model and 3.17 for final without it.

Something else to note:  The above figures assume that the lasers are turret mounted with turret tracking at least 12,000kps.  If not the tracking speed drops to 3,000kps, even with the advanced fire control , and the numbers tank (.47 vs 1.01 and 2.06 vs 4.43||.18 vs .72 and .79 vs 3.17).  Nor are the area defense and final defense examples equal in hs/tonnage.  Area defense FC and Turret total 32.32 hull spaces while the final defense FC and Turrets total 36.64 hull spaces (assuming quad turrets).  This does not take into account the required power plants.  If the PP tech is Pebblebed each laser needs a 1hs reactor as well taking the area defense suite to 36.32hs and the final to 42.64hs.

These examples still only demonstrate single intercepts.  The problem lays in what it takes to have weapon with sufficient range to cover multiple missile movement impulses and the ROF to take advantage of it.  Turreted Lasers are the only practical choice and even they do not do the job well. 

To develop the tech for an Ion missile (speed 30,000kps for 50%msp/150,000km per impulse) only costs 30,000rp. 

To develop the tech for a 10cm laser with a max range of 150,000km costs 30,000rp for the Far Ultraviolet range modifier alone.  It still needs a C3 Capacitor (6,000rp), Beam Fire Control 50% 24,000km (6,000rp), Fire control speed 3,000kps costs 6,000rp (4,000kps would be better at another 8,000rp),Max Tracking bonus 20%(4,000rp) and turret tracking of 3,000kps at 3,000rp(4,000kps would be better at another 4,000rp). Granted the turret gears could be dispensed with if you’re willing to have massive turrets.  Total of 55,000rp to 67,000rp.

Yes you could go with a 15cm laser and only near ultraviolet range modifier for only 12,000rp and have a max range of 180,000km.  The required C6 capacitor for ROF 5 cost 59,000rp to develop. (36,000rp vs 71,000rp)

[Theokrat]

You also argue that since tracking speed and target speed (sV/tV) are fixed data points and not variable they can be functionally ignored.  I stipulate that you are wrong.

Hah! And I stipulate that you are wrong! _{/Tongue-in-cheeck}

Ok, seriously though, which bit do you disagree with?
a) Target speed and tracking speed are the same for the final-fire and area-defence variant. The tracking bonus is likely to be the same, or only very marginally different.
b) When all these three input-variables have the same respective value in both scenarios, then the factor computed from these inputs also takes the same value in both scenarios.
b) When the factor is a constant multiplier it does not bear influence on the comparison between the two systems.

For a) I would think we can readily agree that the target speed will be the same, because we should only really compare situations in which we are attacked by the same weapon. Tracking speed is also the same because that is the way both of our examples are constructed. (In another part of your post you wrongfully assert that I choose different tracking speeds for both examples, but more on that later). So the only source of disagreement I can find is the tracking bonus. Now I think I have demonstrated in my last example that it is very conceivable that the tracking bonus is maxed out (and thus equal for both). Even if its not, it will not take a drastically different value for both scenarios, as there is at most a 10s difference in tracking time leading to a 4 percentage point difference in the tracking bonus.

For b) I would hope that this is relatively obvious. It’s the same formula, it’s the same inputs, so it would be same output. It is fair to request an analysis on the effect of a small change in the tracking bonus on the result of the factor, but again I approximate that is not big.

For c) let me repeat that I stipulate that we can functionally ignore the common factor for a relative comparison between the two systems and only for this purpose. If we want to compare X to Y and X = a * s, while Y = b * s, then X/Y= (a*s) / (b*s)=a/b. I.e. for the relative comparison the common constant factor “s” becomes irrelevant. I fully agree that only focusing on “a” or “b” will not give you an answer to the absolute value of X and Y, but that is what I have repeatedly said all along, even before your criticism. I explicitly wrote “this is not the final hitchance, which would include other effects (most importantly tracking speed), but it’s the separable factor due to the range of the BFC.”

Maybe the last bit is the most controversial here, because you write:

But since we’re actually talking about a modifier that can substantially reduce the hit probability it must be included.

Now I could equally assert that you “forgot” the crew grade factor. That can easily be +30%, so its very relevant right? Yes, of course its relevant, but not for the question posed, as both systems would have the same factor.

Now we plug-in weapon and fire control values for the area defense example:

4*P=(90/(1(5*24)))(((0.01)+(1-10/192))/2)(12/24+0.2) = 1.01 missiles intercepted not 4.6.

The final defense example becomes: 
8*P=(1-10/48)(12/24+0.2) = 4.43 missiles intercepted not 6.3.

Well you went through great length through about the tracking speed and was it really worthwhile? We could have saved ourselves the trouble and just abbreviated the factors as “A” and thereby have arrived at the same relative result. Oh wait ;-)

Also, and I want to be quite clear on this: I have never claimed the final-fire defence variant could intercept 6.3 missiles. I have claimed that it can intercept 6.3 * A (yes I did put that in bold) missiles, where “A” is a factor that encompasses other variables that are not down to range. Like for instance tracking speed, which I mentioned as the most important. And if you evaluate this “A” Factor to (12/24+0.2)=0.7 you get 6.3 * 0.7 = 4.41, i.e. your result (minor difference due to rounding).

Anyway, down into the tedious bit of who claimed what why that is or is not correct:

Especially with you own examples have fire controls with significantly different tracking speeds against the same target speed(3000 vs 12000).

Actually I am a bit surprised by this statement. Not only is it not true, because my example entails the same exact tracking speed in both scenarios, but also because you also then use the same tracking speed of 12,000 for both and you must be aware of the firecontroll’s speed rating because it enters your calculations on the weight as well. I do not understand how you could interpret my post as to say the firecontrolls have different tracking speeds:

• So for the area-defence version we would have to spend 800t on the firecontroll  (size 16, 4x range, 4x tracking ). […]
  
  Conversely, what happens for the final-fire  version? The firecontroll  is only 200t (size 4, 1x range, 4x tracking ),
  

The last point of contention is that your “density function” falls short, it only accounts for a single 5 second impulse not the multiples that you assert. 

That is not quite true. We can actually compute the number of interceptions by integrating over the probability density function alone: N=2*integral_(0)^(r) dx 1 / (5v) =2r/(5v). Plug in the numbers v=24k, r=90k, so N=1.5. So my formula assumed that you get an expected 1.5 shots versus missiles that stream by. Of course that is an average number; in actual combat you will either get 1 or 2 shots versus any particular missile salvo. Because I integrate over the entire range of the area-defence variant, I do account for multiple interceptions.

Since the area defense analysis needs variable intercept range accounted for substitute ((1-bR/fR)+(1-mR/fR))/2 for (1-r/fR).  This is a crude model, but serves.

Well, you use a crude approximation, while I use an exact formula. When these two do not arrive at the same final value I would not take this as a sign that my formula is broken.

Specifically, the way you account for multiple shots is insufficient. You basically treat it as though you would have multiple shots, all with the hit-probability of the middle of the lasers range. While that may serve as a first approximation it does not account properly for the fact that multiple shots will never occur at an average distance. The example missile flew 120kkm/5s, while the laser had a range of 90kkm (=180kkm in both directions). So you can only get a second shot at the missile if it ended up more than 30kkm from the area-defence variant during the first 5s interval- and the further away it was in the first interval, the closed it will be in the second.

Oh, and somewhat more importantly, you seem to be missing a factor of 2. Remember, the assumption was that the area-defence variant serves as advance picket, i.e. it is at a certain distance of the protected task group and can engage missiles that are streaming towards it, as well as those that have passed the ship and are flying onwards to the task group. So you get to use the full range of the ship twice.

Something else to note:  The above figures assume that the lasers are turret mounted with turret tracking at least 12,000kps.  If not the tracking speed drops to 3,000kps, even with the advanced fire control , and the numbers tank (.47 vs 1.01 and 2.06 vs 4.43||.18 vs .72 and .79 vs 3.17).  Nor are the area defense and final defense examples equal in hs/tonnage.  Area defense FC and Turret total 32.32 hull spaces while the final defense FC and Turrets total 36.64 hull spaces (assuming quad turrets).  This does not take into account the required power plants.  If the PP tech is Pebblebed each laser needs a 1hs reactor as well taking the area defense suite to 36.32hs and the final to 42.64hs.

Yeah true, I ignored the gearing and power plant requirements, both of which should have been considered on a closer look. One rather subtle point though: You do not need such a large reactor for the final fire variant. Implicitly the area-defence calculation assumed that enemy salvos were at least 10s apart (so that you could actually shot at one salvo more than once when the opportunity was there). So it would be sufficient if the final fire variant fired every 10s (or even less frequent) to match that. Hence you could get away with half the powerplants/laser for the final fire variant. ( I recognize you have used 6 PP for the 8 Lasers on the final fire variant, but I am not sure whether this is the reason). This does not invalidate your point at all though, even if it decreases the difference in mass slightly. As a funny coincidence when gearing and powerplants are considered both setups have pretty much the same costs though, so in a way it is still a fair comparison (to some extend)