All perfect Praise is to Allah.
peace and blessings be upon the noble prophet and those who follow in his path till the end of time.

Firstly:
Al-Bukhaari (1131) and Muslim (1159)
narrated from ‘Abdullah ibn ‘Amr ibn
al-‘Aas (may Allah be pleased with him)
that the Messenger of Allah (blessings
and peace of Allah be upon him) said to
him: “the dearest of prayer to Allah is
the prayer of Dawood (peace be upon
him). He used to sleep for half the night,
get up and pray for one third of it, and
sleep for one sixth of it, and he used to
fast one day and not the next.”
What the hadith means is that the best
of night prayers is the prayer of the
Prophet of Allah Dawood (peace be upon
him) as he – firstly – used to sleep for
half the night. Whoever wants to put this
into practice today should work out the
time from after ‘Isha’ prayer until Fajr,
then sleep for half of that time.
Shaykh ‘Abd al-Muhsin al-‘Abbaad said:
Sleeping for half of the night means
after ‘Isha’ prayer; he should work out
the time from after ‘Isha’ prayer until
the break of dawn, spend half of it
sleeping, then after that wake up for
one third of it.
Sharh Sunan Abi Dawood (13/287)
Then after that time, Dawood ( peace be
upon him) would get up to pray, and he
would pray for one third of the night,
then he would sleep for one sixth, until
dawn (Fajr prayer).
The wisdom behind that is so that one
will not get tired or bored, and so that
one will get up to pray Fajr and recite
the morning adhkaar with energy, and
will not feel tired or lazy, and so that he
will start his daily activities in this
manner. Then he will be able to do what
is required of him of duties towards his
family, children and other people, and
towards his regular work, so he will not
go to work feeling sleepy. And there are
other benefits as well.
Al-Haafiz (may Allah have mercy on him)
said:
Al-Muhallab said: Dawood (peace be
upon him) used to rest by sleeping at
the beginning of the night, then he
would get up at that time when Allah
calls out: “Is there anyone who will ask,
so that I may give him?” Then he would
sleep for the remainder of the night in
order to rest after spending time
praying. This way is dearest to Allah
because it is easier for the individual
and helps him to avoid getting tired. The
Prophet (blessings and peace of Allah be
upon him) said: “Allah does not tire until
you tire.” And Allah loves to bestow His
grace constantly and grant one favour
after another. That is kinder to oneself
because sleeping after praying qiyaam
allows the body to rest, takes away any
harm that results from staying up at
night and protects one from feeling
weak, unlike staying up all night until
dawn. It also serves another interest,
which is that one will be able to pray
Fajr and recite the adhkaar with energy
and focus, and it is closer to avoiding
showing off, because the one who sleeps
for the last sixth of the night will wake
up with a regular colour (as opposed to
looking pale and tired), which is more
likely to help him conceal what he did at
night, if anyone sees him. This was
pointed out by Ibn Daqeeq al-‘Eid. End
quote.
Ibn al-Qayyim (may Allah have mercy on
him) said:
This clearly indicates that it is only
dearer to Allah for this reason, which is
that it gives the individual a break in
fasting and praying qiyaam, during which
he may rest and re-energise himself,
which will help him to fulfil his various
duties.
End quote from Tahdheeb Sunan Abi
Dawood (1/475)
Ibn ‘Uthaymeen (may Allah have mercy
on him) said:
Praying tahajjud at night is one of the
best acts of worship and is the best of
prayers after the obligatory prayers.
Prayer at night is better than prayer
during the day, especially in the last
third of the night. The best way to divide
the night is the prayer of Dawood; he
used to sleep for half of the night, get
up and pray for one third of it, and sleep
for one sixth. The Prophet (blessings and
peace of Allah be upon him) also used to
do that sometimes; in fact he used to do
that most of the time. Based on that, we
say: the best of night prayers are those
that are done after one half of the night
has passed, until one sixth of the night is
left. End quote.
Fataawa Noor ‘ala ad-Darb (1/161)

to be coontinued.

http://islamqa.info/en/194615