"Standard" speeds for engine tech; an idea that really needs feedback

[Tree]

sure, now include the cost of tankers and fuel infrastructure, and see if that "optimality" doesn't change a wee bit.

It doesn't change anything.
Warships should be about only two things: payload size, and ship speed. Anything that increases one of those, or the two, is good. Tankers and fuel don't factor into this.

[Iceranger]

Usually, a "smaller" engine with a higher boost modifier and extra fuel storage will be more tonnage-efficient than a bigger engine with a lower boost modifier and less fuel storage, because (a) fuel is pretty space-efficient, and (b) bigger engines require more crew spaces. This may not be a hard and fast rule but it seems to work in my experience. Admittedly, this approach is intended to work with a pretty well-developed fleet logistics arm (notably, pre-positioning tankers ahead of a planned offensive) but you can still get warships with 10-15b km ranges which is usually enough to jump into a hostile system, shoot everything with superior speed, and jump back to wherever your tanker is.

It can be mathematically proven that the optimal engine:fuel ratio, in terms of the minimum space taken, is 3:1 in tonnage/HS, given a range and speed goal. Of course in the game, the engine size/boost and fuel size choices are all discrete, so the practical optimal may sway a bit from the theoretical 3:1 ratio, but not by much unless you hit a boundary of the above design parameters.

A corollary to this is that, if your engine:fuel ratio is much smaller than 3:1, it means you can use larger but less boosted engines with less fuel to achieve the same fuel range and speed while taking less space. On the other hand, if your engine:fuel ratio is much larger than 3:1, it means you can use smaller and more boosted engines and more fuel to save space.

Going with max boosted engine is almost never the optimal choice except for extremely fast and short-ranged ships.

Interesting! Do you happen to have a useful link that lays out the math behind this? I don't doubt you at all, but I'm curious how optimizing two different but related quantities in speed and range would consistently give a nearly 3:1 ratio regardless of the desired values.

Not sure if it is posted on the forum anywhere, but here is the original post on Reddit:
https://www.reddit.com/r/aurora/comments/g94nko/ship_design_math_or_the_formula_behind_the/

Based on the derivation it is actually possible to solve for the optimal engine size, boost, and fuel amount for a given ship tonnage, fuel range, and speed. Then you can choose a design parameter close to this optimal point in-game to build your ship. Of course, there are calculators to find that optimal point for you

[nuclearslurpee]

Not sure if it is posted on the forum anywhere, but here is the original post on Reddit:
https://www.reddit.com/r/aurora/comments/g94nko/ship_design_math_or_the_formula_behind_the/

Based on the derivation it is actually possible to solve for the optimal engine size, boost, and fuel amount for a given ship tonnage, fuel range, and speed. Then you can choose a design parameter close to this optimal point in-game to build your ship. Of course, there are calculators to find that optimal point for you

Thanks! Makes sense looking at the calculation, though I'm going to have to take some time to get my head around how it works out into practical designs.

It does look like this calculation doesn't account for additional crew quarters required for the engines vs the fuel. I'm not sure yet if it would substantially change the result but it's probably worth a look, albeit crew quarters *really* suffer from the whole discrete-sizes thing which would add some more distance between the optimal and the possible.

[Iceranger]

Not sure if it is posted on the forum anywhere, but here is the original post on Reddit:
https://www.reddit.com/r/aurora/comments/g94nko/ship_design_math_or_the_formula_behind_the/

Based on the derivation it is actually possible to solve for the optimal engine size, boost, and fuel amount for a given ship tonnage, fuel range, and speed. Then you can choose a design parameter close to this optimal point in-game to build your ship. Of course, there are calculators to find that optimal point for you

Thanks! Makes sense looking at the calculation, though I'm going to have to take some time to get my head around how it works out into practical designs.

It does look like this calculation doesn't account for additional crew quarters required for the engines vs the fuel. I'm not sure yet if it would substantially change the result but it's probably worth a look, albeit crew quarters *really* suffer from the whole discrete-sizes thing which would add some more distance between the optimal and the possible.

I imagine it won't be too far off practically, and it should not affect the theoretical optimal point at all.

Note that an engine's crew requirement is engine size x engine power boost, which is proportional to the engine power. Since the optimality is established based on given fuel range, tonnage and speed, under these conditions the total engine power is fixed. Thus in theory the total crew required by those engines is fixed. Of course practically the crew count for each engine needs to be an integer, so crew number for each engine is rounded up or down, which may result in certain engine compositions having slightly more or less crew, but the difference should be very small.

[nuclearslurpee]

Okay, so I followed up and got my head around the math, and I wanted to share my findings as this all ties back to the initial question of this thread i.e. what should a "standard speed" look like?

To sum up, the Reddit post shared by Iceranger is completely right as expected. The optimal ratio of engine to fuel component size is 3 HS of engine to 1 HS of fuel barring small discretization errors and running up against tech limits. Interestingly, this ratio is completely independent of the engine power modifier and so by itself this ratio does not tell us what modifier is optimal (although it does invalidate my argument about engine vs fuel size efficiency), however the math is there to figure out the answer and it turns out that the answer depends strongly on the design doctrine for your ships.

So to start off I'll sum up the math from that Reddit post, we have expressions for ship speed and range in terms of various tech level values, design parameters, and target values. Anyone else who wants to please feel free to double-check this of course.

Ship speed/velocity:

Code: [Select]

V = Cv * Me * Fp
    Cv = 750 * EP

    EP = base engine tech level in EP/HS
    Me = engine power modifier
    Fp = Fraction of ship mass dedicated to propulsion components, i.e. engines and fuel storage

Ship range:

Code: [Select]

R = Cr * Fp^1.5 / (Me^2.5 * Ne^0.5)
    Cr = 6.75E+4.5 * Cf * HS^0.5 / eta

    Ne = number of engines
    Cf = fuel storage size in L/HS
    HS = total ship size in HS
    eta = fuel efficiency tech level

The design parameters are Me, Fp, and Ne, as everything else is either a tech that we always want to use the latest level of or a parameter we set earlier in the ship design (ship size, in this case). Out of these parameters: Fp is usually determined by our ship design doctrine based on what we think will give us a good balance of weapons and speed; Ne should be as small as possible for efficiency, but since warships need redundancy in their important components we usually don't want Ne = 1 except on fighters/FACs, and additionally if we are reusing the same engines for multiple sizes of ship classes we have to compromise on this parameter; this leaves Me as the design parameter to optimize and we can do this in several different ways.

(1) Select speed, get Me, get range.

Code: [Select]

Me = v / Cv / Fp

R = Cr * Cv^2.5 * Fp^3.5 / (v^2.5 * Ne^0.5)
Very straightforward and simple. Good for fighters/FACs and if you're not worried about having a long range as long as your fleet has a uniform speed in combat. Another approach is to set a minimum speed (e.g. for survey ships or transports) and see how much range you can get out of it.

(2) Select range, get Me, get speed.

Code: [Select]

Me = Cr^0.4 * Fp^0.6 / (R^0.4 * Ne^0.2)

v = Cv * Cr^0.4 * Fp^1.6 / (R^0.4 * Ne^0.2)
Only slightly more complicated in terms of how much work you have to do to find Me. You can use this approach to ensure that all of your fleet units have a specific range, but this is often not really necessary if you have a functioning fleet logistics arm unless you expect to be sailing from X to Y at top speed without any resupply stations along the way. Probably a better use here is for commercial and survey ships that need long ranges. Another idea is to set a minimum operating range and see what kinds of speed you can get out of it.

(3) Optimize some relation between range and speed. If you're not able to precisely define the mission parameters of your fleet, this could be a useful approach to get a "balanced" fleet that can perform a variety of missions adequately. This is where calculators may not be as much help, the above two cases are easy but if you want to optimize a particular function of R and v you may have to DIY.

Example: Optimize the product R*v:

Code: [Select]

R*v = Cv * Cr * Fp^2.5 / (Me^1.5 * Ne^0.5)

Optimal Me is the minimum value available from your tech.
This is a rather simplistic example that shows an important point: if we optimize some product of R and v, to whatever powers we might choose to emphasize one over the other, we always end up with either the minimum or maximum engine power modifier as the optimized result (unless it drops out entirely). So products are not a useful optimization function.

What about addition? We can't add R and v directly because they have different units, but we can add scaled versions by dividing by target values Rt and vt:

Code: [Select]

Let Qt = R / Rt + v / vt

Qt = Cr * Fp^1.5 / (Rt * Me^2.5 * Ne^0.5) + Cv * Fp * Me / vt

dQt / dMe = 0 = Cv * Fp / vt - 5 * Cr * Fp^1.5 / 2 / Rt / Ne^0.5 / Me^3.5

Me = [(Cr / Rt) / (Cv / vt)]^(2/7) * (25 * Fp / 4 / Ne)^(1/7)
Obviously, this is the most overcomplicated approach so far, and you could certainly get even more complicated if you wanted (root mean squares, anyone?). Probably for most people it will suffice to optimize for speed or range and tweak the target value a few times until they're happy with both values.

Okay, so to tie back to the original question: what is a good "standard speed" for a given tech level? To answer this question I pulled the tech values from the Aurora DB for base engine tech, min/max engine power modifiers, and fuel efficiency tech and just set them up as somewhat arbitrary tech levels. Since a standard TN start gives the player NTE engine tech but only the baseline/conventional modifier and fuel techs (0.5, 1.0, and 1.0 respectively) I set this as TL1, and each successive tech level increments each tech if possible. Additionally, I set Fp = 0.4, Ne = 4, Cf = 50,000 L/HS, and the ship size 200 HS (10,000 tons). 40% of space dedicated to propulsion is a reasonable estimate (perhaps a bit low?), and four engines for a 10,000-ton destroyer makes sense if you want to re-use the engines for 5,000-ton frigates and 15,000-ton cruisers or something similar.

Each of the above optimization methods is compared (for case 3, the addition method is compared as the product method is overly simplistic and useless). For the target speed I used a prescribed EP modifier linearly interpolated from 1.0 to 3.0 across the tech levels (i.e. 1.0 at TL1, 3.0 at TL14), and the target range is 20 billion km times SQRT(tech level / 5) (this is scaled to give 20b km range at ion drives which was arbitrarily chosen based on the starting ships in Steve's Imperium of Man fiction). The results are below in the table, noting that the EP modifiers are locked to the allowed range at each tech level - speeds in km/s, ranges in billion km.

Tech Level	Engine Tech	Base EP	Min EM mod	Max EP mod	Fuel eff	Target speed	Case 1 EP mod	Case 1 range	Target range	Case 2 EP mod	Case 2 speed	Case 3 EP mod	Case 3 speed	Case 3 range
1	NTE	5.0	0.5	1.0	1.0	1500	1.00	19.1	8.9	1.00	1500	1.00	1500	19.1
2	INTE	6.4	0.4	1.25	0.9	2215	1.15	14.8	12.6	1.23	2361	1.25	2400	12.1
3	NPE	8.0	0.3	1.5	0.8	3138	1.31	12.2	15.5	1.19	2853	1.5	3600	8.7
4	INPE	10.0	0.25	1.75	0.7	4385	1.46	10.6	17.9	1.18	3551	1.63	4901	8.0
5	Ion Drive	12.5	0.2	2.0	0.6	6057	1.62	9.6	20	1.20	4515	1.70	6381	8.4
6	MP Drive	16.0	0.15	2.5	0.5	8492	1.77	9.2	21.9	1.25	5994	1.79	8603	8.9
7	Int CF	20	0.1	3.0	0.4	11538	1.92	9.3	23.7	1.32	7944	1.91	11483	9.4
8	Mag CF	25	0.1	3.0	0.3	15577	2.08	10.2	25.3	1.45	10847	2.08	15629	10.2
9	Ine CF	32	0.1	3.0	0.25	21415	2.23	10.3	26.8	1.52	14587	2.20	21150	10.6
10	Solid AM	40	0.1	3.0	0.2	28615	2.38	10.9	28.3	1.63	19520	2.36	28291	11.2
11	Gas AM	50	0.1	3.0	0.16	38077	2.54	11.6	29.7	1.74	26175	2.52	37852	11.8
12	Plasma AM	64	0.1	3.0	0.125	51691	2.69	12.84	31.0	1.89	36343	2.72	52220	12.5
13	Beam AM	80	0.1	3.0	0.1	68308	2.85	14.0	32.2	2.04	48881	2.91	69882	13.2
14	Photonic	100	0.1	3.0	0.1	90000	3.00	12.2	33.5	2.01	60203	2.92	87742	13.1

I'm not going to claim that this is a terribly informative data set, my aim is just to show what you can expect to see when you optimize for different quantities and how one might go about using this information to determine standard speeds for their fleets. In most cases, it's probably best to just optimize for a fixed speed or range (cases 1 and 2), and in this case using a calculator is probably the way to go if you want to get the best designs. However, if you really want to get into it you can devise various kinds of metrics that probably will work better than my examples in case 3, including messing with your choice of target values.

Anyways, hope this is at least mildly interesting to one of you nerds out there.

I imagine it won't be too far off practically, and it should not affect the theoretical optimal point at all.

Note that an engine's crew requirement is engine size x engine power boost, which is proportional to the engine power. Since the optimality is established based on given fuel range, tonnage and speed, under these conditions the total engine power is fixed. Thus in theory the total crew required by those engines is fixed. Of course practically the crew count for each engine needs to be an integer, so crew number for each engine is rounded up or down, which may result in certain engine compositions having slightly more or less crew, but the difference should be very small.

That makes sense. I'm still curious how it would shake out from the math and if there's any implications at all hidden away in there, but that's an adventure for another day I think.

[misanthropope]

sure, now include the cost of tankers and fuel infrastructure, and see if that "optimality" doesn't change a wee bit.

It doesn't change anything.
Warships should be about only two things: payload size, and ship speed. Anything that increases one of those, or the two, is good. Tankers and fuel don't factor into this.

yeah because the resources to build those things grow on trees, unlike the resources for building ships, which you have to budget.

[Zincat]

I find it entirely fascinating that I've used something very similar to this 3HS engine / 1HS fuel proportion myself, without actually doing the math. That is because I am mostly a roleplayer and so I don't try to strictly optimize towards some sort of "optimal result", but rather towards what I want for my roleplay.

I will preface that I mostly use beam warships, missiles are either for bomber squadrons (carrier based) or for cleaning up/ testing defenses/killing civilians/harassment in general. And for missile defense bases in orbit around important planets.

My general design policy is to set 10% of the warship design for fuel and 30-35% for the engine, so very close to that proposed 3:1 ratio. I generally set my desired range, and then push up the engine multiplier as much as possible while still keeping that range.

I feel that with beam warships, range is really the most "important" value, as a ship that cannot quickly defend my territory is not useful. Of course tankers can help with transfers to different parts of my territory, or to get close to the enemy's territory. But once the action starts, a beam fleet NEEDS enough range to fight. After that, it's speed speed speed.

As such, I don't have any "standard" range value, because it depends entirely of the geography of the galaxy, the size of the systems involved, and of course the technology available.


Of course I also do build some different ships from time to time, like "system patrol boats" that are very fast but with short range or similar.

[nuclearslurpee]

yeah because the resources to build those things grow on trees, unlike the resources for building ships, which you have to budget.

Tankers are usually pretty cheap - a couple commercial engines, a fuel pump, and a few dozen fuel storage modules - and once you have them you can keep them for a long time without needing to replace them, they might not keep up to speed with your main fleet but smart pre-positioning makes up for this.

[TheTalkingMeowth]

Tankers are cheap, but the fuel to put in them is not.

I find the tempo of operations to be limited by (in order of increasing importance) officer/crew availability, warship building, missile manufacturing, and fuel.

That is, fuel is a bigger limit than anything else. This is partially because I play with inexperienced fleets and so burn lots and lots of fuel training my ships, but the fact is, fuel harvesters are really expensive. And boosted engines suck huge amounts of fuel, for really pretty negligible reductions in propulsion fraction.

In a 15000 ton ship with INPE engines with 1.75 max boost, I found I could go for 0 boost and 500,000 liters of fuel, or save like 500 tons by going for a boosted engine...at the price of needing 1,000,000 liters of fuel (rough numbers). Target speed was 4000km/s, target range 24 million kilometers.

The "optimal" choice is the boosted engine, but I went with the unboosted ones. I have in the vicinity of 10 million tons of fuel harvesters, several fuel production upgrades, stacked mining command bonuses to 35% (plus ship COs), and fuel is STILL an issue.

[Steve Walmsley]

I also tend to go for non-boosted engines for warships. The strategic penalties in fuel generation and distribution usually outweigh the tactical advantages of faster ships.

[misanthropope]

and the lots-of-boosted-engines fetish runs you out of gallicite uber-fast. if duranium and uridium are just piling up because your use-fraction of gallicite is greatly higher than the incidence rate of gallicite, then you've got a problem parallel to and even more expensive than an extravagant fuel pipeline:  you've bloated your requirement for mines.  it's a way bigger budget buster than needing to expand missile production.

the zippy sexy fleet paradigm isn't just "six parts support systems for one part weapons", it's also "three parts logistical tail for one part fleet"

[xenoscepter]

 - My 2 Cents: I put boosted engines on many of my combat Fighters, but also many of my Corvettes and a good many of my Tugs. I tend to use them a lot on warships that are intended for in-system defense. They need to be supported by a fueling & maintenance facility, but they work really well for that purpose. 1-3 months of deployment (3 preferred) and 1 Engineering Space per 1,000 tons (20 HS) are typically fine for such a ship.

 - Here are some examples:
   - Fortune Class Frigate: http://aurora2.pentarch.org/index.php?topic=11551.msg135214#msg135214
   - Starlance & Starshield: http://aurora2.pentarch.org/index.php?topic=10971.msg126374#msg126374

[Bremen]

and the lots-of-boosted-engines fetish runs you out of gallicite uber-fast. if duranium and uridium are just piling up because your use-fraction of gallicite is greatly higher than the incidence rate of gallicite, then you've got a problem parallel to and even more expensive than an extravagant fuel pipeline:  you've bloated your requirement for mines.  it's a way bigger budget buster than needing to expand missile production.

the zippy sexy fleet paradigm isn't just "six parts support systems for one part weapons", it's also "three parts logistical tail for one part fleet"

Also they tend to explode, which I think people tend to underestimate. I've found that ships tend to be destroyed by engine explosions much earlier and more frequently than you'd think.

[nuclearslurpee]

Hey folks,

So earlier in the thread, it was stated that the optimal tonnage-efficient ratio of engine mass to fuel mass is 3:1 given a target speed and/or range. This all checks out mathematically and as I showed allows you to easily design engines given a target speed, range, or both if you define a suitable criterion to balance between the two.

However, this analysis was done by considering the "propulsion mass" of the ship to be the sum of the engine and fuel masses. The question was raised, but not answered conclusively: what if we include the crew quarters requirement for the engines (fuel modules require no crew) in the propulsion mass? Does the optimal ratio change, and if so is it significant? I remained curious...

So I hired a Power and Propulsion scientist and gave him a couple dozen labs, and 10,000 RP later this is what he came up with pending peer review:

Skipping over most of the derivations, which are almost identical to those given previously, we have

Code: [Select]

Speed: v = 1000 * Fe * Be * EP
    Fe: engine mass fraction, Me / Ms
        Me: total engine mass
        Ms: total ship mass
    Be: engine power modifier
    EP: engine power

Range: R = 3.6 * 10^5.5 * Ff * Cf / eta / Be^2.5 * SQRT(Fe * Ms / Ne)
    Ff: fuel mass fraction, Mf / Ms
       Mf: total fuel mass
    Cf: fuel capacity per HS, usually 50,000 L/HS
    eta: fuel efficiency tech level
    Ne: number of engines
These can be derived by copying equations from the wiki/changelogs and doing algebra. Substitution to eliminate Be gives the optimal ratio of Fe to Ff.

However, if we include the crew quarters mass, then the propulsion fraction changes with an additional term:

Code: [Select]

Propulsion fraction: Fp = Fe + Ff + Fc
    Fc: crew quarters mass fraction, Mc / Ms
   
    Mc = 0.02 * Me * Be * Td^(1/3)
        Td: deployment time in months

Thus, Fc = 0.02 * Fe * Be * Td^(1/3)

and Fp = Fe * [1 + 0.02 * Be * Td^(1/3)] + Ff

The optimal engine mass fraction is then found:

Code: [Select]

Maximize Ff * Fe^3 = {Fp - Fe * [1 + 0.02 * Be * Td^(1/3)]} * Fe^3

Result: Fe = 0.75 * Fp / [1 + 0.02 * Be * Td^(1/3)]
This means that the optimal engine mass fraction will be reduced compared to the case where we neglect the crew quarters. Interestingly, the optimal fuel mass fraction does not change and remains at 1/4 of the total propulsion mass fraction.

When is this actually significant? Let's look at a table and see:

	Td (m)
Be	1	3	6	9	12	18	24	36	48	60
0.1	74.9%	74.8%	74.7%	74.7%	74.7%	74.6%	74.6%	74.5%	74.5%	74.4%
0.25	74.6%	74.5%	74.3%	74.2%	74.2%	74.0%	73.9%	73.8%	73.7%	73.6%
0.5	74.3%	73.9%	73.7%	73.5%	73.3%	73.1%	72.9%	72.6%	72.4%	72.2%
0.75	73.9%	73.4%	73.0%	72.7%	72.5%	72.2%	71.9%	71.5%	71.1%	70.8%
1.0	73.5%	72.9%	72.4%	72.0%	71.7%	71.3%	70.9%	70.4%	69.9%	69.6%
1.25	73.2%	72.4%	71.7%	71.3%	70.9%	70.4%	70.0%	69.3%	68.8%	68.3%
1.5	72.8%	71.9%	71.1%	70.6%	70.2%	69.5%	69.0%	68.2%	67.6%	67.1%
2.0	72.1%	70.9%	69.9%	69.2%	68.7%	67.9%	67.2%	66.2%	65.5%	64.8%
2.5	71.4%	70.0%	68.8%	67.9%	67.3%	66.3%	65.5%	64.4%	63.5%	62.7%
3.0	70.8%	69.0%	67.6%	66.7%	65.9%	64.8%	63.9%	62.6%	61.6%	60.7%

Generally it's a fairly minor effect, certainly your commercial ship designs will not be affected very much. As expected, the effect is mostly visible at higher EP modifiers, which are mostly used for fighters and other small craft that don't need range to do their jobs. Otherwise, the main effect you'll see will be on long-range survey craft, as even with Be = 1.0 the optimal engine mass fraction drops below 70% for very long deployment times. Otherwise though, it doesn't matter a whole lot in terms of the engine mass, and the difference is more philosophical than practical

There is a similar impact on the EP modifier needed to reach a specified speed and/or range:

Code: [Select]

Speed
Without crew mass: Be = v / 750 / EP / Fp
With crew mass: Be = v / [750 * EP * Fp - 0.02 * v * Td^(1/3)]
For example, if you want to design for a fleet speed of v = 750 * EP * Fp, you would come up with an EP modifier of 1.0 if you neglected crew mass. If you include crew mass in the calculation, the required modifier increases to e.g. 1.05 for a 12-month deployment time.

Code: [Select]

Range
Without crew mass: Be = 607.5 * Fp^0.6 * Cf^0.4 * Ms^0.2 / Ne^0.2 / eta^0.4 / R^0.4
With crew mass: Be^5 * [1 + 0.02 * Be * Td^(1/3)] = 6.075E10 * Fp^3 * Cf^2 * Ms / Ne / eta^2 / R^2
This is the messy bit and you have to solve this iteratively, but as a simple example: if you want to design for a range of 6.075E10 * Fp^3 * Cf^2 * Ms / Ne / eta^2 you would come up with an EP modifier of 1.0 if you neglected crew mass. If you include crew mass in the calculation and break out your trusty Trans-Newtonian graphing calculator, the required modifier decreases slightly to e.g. 0.99 for a 12-month deployment. Thankfully the effect is minimal, because no one wants to do this calculation every time they design a new cruiser.

Again, just to emphasize, the difference here is largely one of philosophy, essentially whether or not you consider the necessary crew support to operate a system as part of that system. If you want to be very exact and say "I want exactly 40% of my ship mass to be related to propulsion, 40% to be payload, and 10% to be maintenance/DC (also 10% is armor, but shhh)", then this approach I've followed here lets you do that. On the other hand, if you don't care about that extra few% of mass, you can stick to counting only engine and fuel mass and chalk up the crew mass to overhead so you don't have to think about it. Both approaches give you "optimized" designs, the difference is just how you want to define "optimized".


ADDENDUM: For those who care about that 1% difference, you can estimate Be for a target range pretty closely:

Code: [Select]

Let B0 be the EP modifier you would calculate without the crew mass fraction:
  B0 = 607.5 * Fp^0.6 * Cf^0.4 * Ms^0.2 / Ne^0.2 / eta^0.4 / R^0.4

Then approximately (within 5% relative error at the smallest and largest Td values):
  Be = B0 * [1 - (Td/2.25E7)^0.328]
Note that since Td is in months we're talking about a very negligible difference, at most maybe 2% smaller for very long deployment times, but if you really care now you know how to get a good estimate.

[EvadingHostileFleets]

High-boost drives may be viable if paired with tugs to ferry ships around to actually get to places without burning tanker's worth of fuel per ship. But deployment speed suffers and you have to keep a substaintital fleet of tugs which is not optimal, unless you are into space trucks and like to tug around engineless payload - I saw such people.