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June 4th, 2013, 02:22 PM   #1
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The coming of convoy

(i) Merchant ships sailing independently take 75% of the time to complete voyages compared to ships sailing in a convoy but lose 14% of their number to submarines on each voyage, whilst convoyed ships lose 5% per voyage.
We start with a given fleet of merchant ships and must decide whether to use convoy for all of them or let them all sail independently. We can produce ships quickly enough to replace all those lost in convoy. Show that in the time it takes to make three convoy voyages, an independently sailing fleet will have made more voyages than a convoyed one but the position will be reversed for the time it takes to make six. What will happen over a long time?

(ii) Suppose we can produce merchant ships at a rate ? (ships per unit time) and that we lose merchant ships at a rate ? (ships per ship afloat per unit time). Explain briefly why, with this model, the size x(t) of our fleet is governed by the differential equation ?= -?x + ? ,

and deduce that

x(t)= (?/?)+ (X - ?/?)e^(-?t), where X is the size of our fleet when t=0. What happens to the size of our fleet when t is large?



Any help on the above questions would be much appreciated. The question comes from T Korner's 'Pleasures of Counting' (exercise 2.2.1).
calvinnesbitt is offline  
 
June 4th, 2013, 02:56 PM   #2
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Re: The coming of convoy

Quote:
Originally Posted by calvinnesbitt
(i) Merchant ships sailing independently take 75% of the time to complete voyages compared to ships sailing in a convoy but lose 14% of their number to submarines on each voyage, whilst convoyed ships lose 5% per voyage.
We start with a given fleet of merchant ships and must decide whether to use convoy for all of them or let them all sail independently. We can produce ships quickly enough to replace all those lost in convoy. Show that in the time it takes to make three convoy voyages, an independently sailing fleet will have made more voyages than a convoyed one but the position will be reversed for the time it takes to make six.
I don't understand this question. It appears to ask nothing about the number of ships that are lost. Obviously, if it take ships sailing independently only 75% of the time of convoys, in the time it takes the convoys to make three trips, the independently sailing ships will make (4/3)(3)= 4 trips.. But that is certainly NOT "reversed for the time it takes to make six". In the time the convoys make 6 trips, the independently sailing ships will make (4/3)(6)= 8 trips.
Quote:
What will happen over a long time?

(ii) Suppose we can produce merchant ships at a rate ? (ships per unit time) and that we lose merchant ships at a rate ? (ships per ship afloat per unit time). Explain briefly why, with this model, the size x(t) of our fleet is governed by the differential equation ?= -?x + ? ,
? is the rate of change of x. During "unit time", x changes in two ways: it can decrease because of losing ships. The rate is ? "per ship afloat per unit time" so since there are x ships afloat, it will decrease by ?x. it can increase because of new ships. The rate is ? new "ships per unit time". Putting those together, we have ?= -?x + ?.

Quote:
and deduce that

x(t)= (?/?)+ (X - ?/?)e^(-?t), where X is the size of our fleet when t=0.
The given equation is which we can write as .
Integrate that.

Quote:
What happens to the size of our fleet when t is large?
Once you have integrated, take the limit as t goes to infinity.

Quote:

Any help on the above questions would be much appreciated. The question comes from T Korner's 'Pleasures of Counting' (exercise 2.2.1).
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August 17th, 2014, 01:35 AM   #3
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I've also found this question difficult to understand, (part (i) at least), and, as this is the only place where it is discussed, excuse me for answering an old thread.

I think the key to understanding part (i) comes from the sentence "We can produce ships quickly enough to replace all those lost in convoy.". It is never claimed that ships lost sailing independently can be replaced (completely).
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