Takes each passenger 30 seconds to buy a ticket and borad the ferry, 1 passenger arrives every 2 mins. how long until the ferry leaves?

Question

The passenger ferry doesn’t leave until all passengers have boarded and there are none waiting. It takes each passenger 30 seconds to buy a ticket and board the ferry. When boarding started there were 25 passengers waiting and more passengers were arriving at the rate of 1 every 2 minutes. How long is it before the ferry leaves?

Answer 1

This problem deals with “waiting lines,” which are called “queues” in England.  In Computer Science, we call this “queuing theory” and it is extremely important because lots and lots of things wait on each other.  In this problem, it takes the “server” 30 seconds to process (sell ticket and get person on board the ferry) each person.  Also, the ferry may leave when there are no people “queued” (in line), remaining to be processed.  [Notice how I reworded this problem in order to take the perspective of the server.]

 

So, the server begins processing.  As serving (selling tickets and getting people on board the ferry) proceeded, more people arrived.  How many?  Well, to serve the 25 initial people, it took 12.5 minutes; during this time, 6 more people (truncate 12.5/2) arrived.  Processing them took 3 minutes and 1 more person ( (30 second remainder + 3 minutes); truncate 3.5/2) arrived.  Processing that person took 30 seconds ( (1.5 minute remainder + 30 seconds); truncate 4.0/2 minutes).  This allows 1 last person to arrive just as the server is getting ready to close operation.  That person is processed in 30 more seconds.  That is not enough time for anyone else to arrive.  It is important to realize that at this point, the line (queue) is empty for the first time in this problem (otherwise, the ferry would have left already; this is why the server's perspective is so important).

 

The elapsed time is:   12.5 + 3 + 0.5 + 0.5  =  16 minutes 30 seconds(note: the server worked continuously through 33 events, each taking 30 seconds).  Because there is no person in line, the ferry can leave.

 

Checking the process in really pretty easy, but very, very important.

   In 16 minutes 30 seconds, how many people can be “processed?”   33

   Starting with 25 people in the queue and gaining 1 every 2 minutes, how many people must be processed in 16 minutes 30 seconds? 

              25 + truncate(16.5/2)

              = 25 + 8

              = 33        check!

 

Final note:  I assumed what is called a "single-server queue;" With a multi-server queue, the 25 people could have been 'processed' much quicker and not have to wait on the stragglers.

Answer 2

Hello Sarah,

 

For the 25 passengers that are in line at the start, it takes 12.5 minutes to accommodate this group (25*1/2min). In this time period, you would have gained an additional 6.25 people (12.5min/2ppl per minute), which in turn takes 3.125 minutes to accommodate.  

 

In the 3.125 minutes, you acquired an additional 1 9/16 of a person (25/16 divided by 2), which takes 47 seconds to accommodate this person (of course we know we can't have 9/16 of a person, but it gives us an idea of frequency).

 

We would not have gained any additional people during this time period (rate is 1 person every two minutes), thus we would just add the times for all the accommodations: 

 

12.5 + 3.125 + 0.783 (this is the 47 seconds)= 16.4 minutes 

 

Hope this helps!