N
neha3
Member
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).
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).
