ct4 october 2010 Q11

Q 11) At a certain airport, taxis for the city centre depart from a single terminus. The taxis
are all of the same make and model, and each can seat four passengers (not including
the driver). The terminus is arranged so that empty taxis queue in a single line, and
passengers must join the front taxi in the line. As soon as it is full, each taxi departs.
A strict environmental law forbids any taxi from departing unless it is full. Taxis are
so numerous that there is always at least one taxi waiting in line.
Customers arrive at the terminus according to a Poisson process with a rate β per minute.

(i) Explain how that the number of passengers waiting in the front taxi can be
modelled as a Markov jump process.
(ii) Write down, for this process:
(a) the generator matrix
(b) Kolmogorov’s forward equations in component form
[4]
(iii) Calculate the expected time a passenger arriving at the terminus will have to wait until his or her taxi departs.

The four-passenger taxis were highly polluting, and the government instituted a
“scrappage” scheme whereby taxi drivers were given a subsidy to replace their old
four-passenger taxis with new “greener” models. Two such models were on the
market, one of which had a capacity of three passengers and the other of which had a
capacity of five passengers (again, not including the driver in each case). Half the
taxis were replaced with three-passenger models, and half with five-passenger
models.
Assume that, after the replacement, three-passenger and five-passenger models arrive
randomly at the terminus.
(iv) Write down the transition matrix of the Markov jump chain describing the
number of passengers in the front taxi after the vehicle replacement. [2]
(v) Calculate the expected waiting time for a passenger arriving at the terminus
after the vehicle scrappage scheme and compare this with your answer to part (iii).

 

Hi,

An urgent helpis required on this as well...

 

Hi,

I have get thru with the generator matrix & Komogrov equations but not able to get thru the rest of the part. Can anyone help me with this.

 

Hi Neha,

I see u already have parts (i) and (ii) so i'll give my thoughts on (iii).

When the passenger arrives the front taxi has either 0,1,2,3 passengers (4 possible states) waiting already so it's best to split it into cases, with an assumed 1/4 prob of each being the case.

If 3, waiting time is 0 as the taxi leaves immediately (and process jumps to state0).

If 2, the chain enters state 3 and the waiting time is then the holding time in state 3... ie. 1/beta

If 1, chain enters state 2 and waiting time is the sum of the holdings times in 2 and 3. Similarly for 0.

For part (iv) I would again split into equally probably cases of having a 3-seater and having a 5-seater.

Hope this helps!

 

Thanks.

Please can you also explain the matrix in part (iv) and rest is the same like (iii)

 

okay well ggrzz was sort of on the right track but is still incorrect so I will explain it again

For iii)
Theres sort of 2 parts here
*Lets say you are the first passenger to arrive: this means you still need to wait for 3 more passengers to arrive, this is expected to take 3*Beta
*If you are the 2nd passenger to arrive: this means you still need to wait for 2 more passengers to arrive, this is expected to take 2*Beta
*If you are the 3rd passenger to arrive: this means you still need to wait for 1 more passenger to arrive, this is expected to take Beta
*If you are the 4th passenger to arrive, this means you leave immediately and the expected time is 0

Now we need to consider the probability of arriving 1st 2nd 3rd or 4th -> here we assume that each is equal by the way the question is worded
so each case occurs with probability 1/4

therefore the EXPECTED waiting time is 1/4(3*beta + 2*beta +beta) = 3/2*Beta

 

For Part iv) note that it is asking for a transition matrix and NOT a generator matrix as in part i)
-things to note here: assumptions are that once a passenger is waiting they will never leave.
-What this means is that: we go from 0 to 1 passenger with probability 1;
-we go from 1 to 2 passengers with probability 1;
-now the "catch" of the question: if it is a 3 seater taxi and another person arrives we move back to 0 passengers waiting; if it is a 5 seater passenger we move to 4 ppl waiting
-> each of this occur with equal probability of 0.5.
-if it were a 5 seater, and there are 4 ppl waiting we go back to 0 ppl waiting with probability 1

-i cant draw the matrix here - but the examiners solution should suffice.

 

for part v)
-note that if it is a 3 passenger taxi, using the method in iii) we get the EXPECTED waiting time to be 1/3(2*beta + 1*beta) = beta
-if it was a 5 person taxe using the method in iii) we get that the EXPECTED waiting time to be 1/5(4*beta+3*beta+2*beta+1*beta) = 2*beta
-this is where the "catch" of the question comes in:
since the 5 seater passenger taxi can take 5 ppl, it will be there for longer on average than the 3 seater taxi
-this means that the overall expected waitingtime becomes
5/8*2beta+3/8*beta = 13/8*beta
-Note this is NOT 1/2*beta+1/2*2beta = 1.5beta (which would equal the answer from iii)
-BECAUSE of this conditioning the expected waiting time is now longer than the expected waiting time from iii) (however the variance in waiting times is likely to have increased - but is not addressed in this question)
-based purely on the expected waiting times - it _appears_ as if the service has worsened (since each person expects to wait longer)
- in reality people that happen to get a 3 seater will experience better service
and people who happen to get a 5 seater will experience worse service than before
-and overall on average everyone receives worse service

 

Are you writing CT4 next week Friday (15th of April)?
If so, are you aware that there are memo's for these questions available (granted I included additional explanation) but if with the memo's at this stage you could not figure out the answers then I think you need to do some serious studying between now and then

 

Hi Rowan,

Thanks for the explaination.

I get thru the answer what was mentioned in the examiner's report. However just wanted to check whether my strategy was correct or not.

Also, you wrote abt some memo's available for the same ???

 

When i said memo's I was refering to the examiners report