Actually, Check this out, it might actually answer the question but I suspect a big flaw
If we assume that like the rain in this math problem:
We must assume that you have no prior knowledge besides the reasoned deduction that you are patrolling a likely transit area and it's general direction (we'll use north-south for example sake).
Therefore, the only direction worth moving is perpendicular to the transit area (east-west in this example).
Without prior knowledge to the contrary, we must also assume that traffic flow is statistical uniform. I.E. The odds of a contact being 20 miles due north of you at the moment are the same as the odds the contact being there 4 hours later.
At any given time, a target could be anywhere in the patrol zone that is outside of your current sensor range.
If all of the above is ture you need to be moving e-w but loiter time needs to be maximised. I.E. Gas milage is unimportant, but rather consumption rate is.
The suspected flaw:
"Without prior knowledge to the contrary, we must also assume that traffic flow is statistical uniform."
When you move through an area, we know that no traffic moving at x speed can be in certain locations. For example, a 10 knot target could not have moved all the way through an area you searched with SJ-1 radar half an hour ago if you are cruising at 10 knots. This means that you do have some prior knowledge of where targets are not at any given time. Let's call these areas cavities.
In the question of rain on a person, rain falls at a relative velocity that the cavity is insignificant. I suspect that the relative velocities of ships means that the cavities are potentially quite significant when trying to form a statistical understanding.
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