star trek - Why is the Federation so poorly patrolled?

After reading this answer to the question Roughly how many starships are in Starfleet? I was left with the question of: why is the Enterpise so frequently "the only ship in range" whenever something awful happens? This would seem to indicate that the Federation is not very good at managing its resources.

Answer

Here's one way to think about it: based on the maps shown in Can anyone explain the Star Trek astrography (i.e. quadrants)?, you can estimate the size of the region of space explored by the Federation and its allies. The map shows that they appear to have a reach of about 15000 LY across, end-to-end, but that's not a circular region, so it's probably more realistic to approximate the cross-sectional area of explored space shown in the map at about 100 million square light years (10000x10000). Now, the galactic disk is a few thousand light years thick, depending on which definition you use. I would estimate that the Federation has probably explored the central 2500 LY or so.¹ So the volume of explored space is probably roughly approximated by a cylinder 10000 LY in diameter and 2500 LY high, which would have a volume of about 200 billion cubic light years.

Now, in Roughly how many starships are in Starfleet?, it was estimated that there are about 30000 Federation-aligned ships among Starfleet and its allies. But presumably Starfleet can only send its own ships to investigate things, so the number of ships that can meaningfully be considered in range for something is lower. Even with an optimistic estimate of 10000 actual Starfleet ships, if you spread those 10000 starships over the supposed volume of explored space, that's one ship per 20 million cubic light years. Anything in that 20 million cubic light year space has to be handled by the one ship. And realistically, even if the Enterprise isn't the closest ship, there's a decent chance that the actual closest ship is already occupied doing something else.

You can get a rough estimate of the average distance between ships by taking the cube root of that 20 million LY^3 volume: it works out to about 270 light years. According to HorusKol's comment on this answer, warp 9.6 is about 10 LY per day, which means it would take about a month at even that high warp factor for one Starfleet vessel to rendevouz with another, on average.

Something else you might consider, though, is that Starfleet ships probably spend more of their time on the borders of Federation space than zipping around through the middle of it, because interesting things in the middle of the Federation have probably already been discovered. By the same token, the top and bottom of the cylinder are not likely to be that interesting because the density of stars gets so low. So you could do an alternate calculation of how Starfleet ships would distributed across the border area of Federation space. Considering only the "side" of the cylinder I mentioned before, it has an area of 80 million square light years. Let's say half of Starfleet is assigned to border patrol and exploration, on average, so you have 5000 ships covering that area. That corresponds to one ship per ~16000 square light years, for an average distance between ships on the border of 125 LY.

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¹The galactic disk doesn't have a definite thickness because the star density gradually decreases as you move away from the galactic plane. Carroll and Ostlie's Modern Astrophysics gives a number density of

(p.919, first edition, 1996). This is a superposition of two profiles, the thin disk and the thick disk, with the characteristic length scales z_thin = 325pc = 1059 LY and z_thick = 1.4kpc = 4500LY. My estimate of 2500 LY comes from guessing that the Federation would consider it worth exploring out to the point where the stellar density drops to roughly 10% of its value in the galactic plane.