Editing Network Time Protocol (section)

== Clock synchronization algorithm ==
[[File:NTP-Algorithm.svg|thumb|Round-trip delay time δ]]

A typical NTP client regularly [[Polling (computer science)|polls]] one or more NTP servers. The client must compute its time offset and [[round-trip delay]]. Time offset ''θ'' is positive or negative (client time > server time) difference in absolute time between the two clocks. It is defined by

<math display="block">\theta = \frac{(t_1 - t_0) + (t_2 - t_3 )}{2} ,</math>
and the round-trip delay ''δ'' by
<math display="block">\delta = {(t_3 - t_0 ) - ( t_2- t_1 )} ,</math>
where
*''t''<sub>0</sub> is the client's timestamp of the request packet transmission,
*''t''<sub>1</sub> is the server's timestamp of the request packet reception,
*''t''<sub>2</sub> is the server's timestamp of the response packet transmission and
*''t''<sub>3</sub> is the client's timestamp of the response packet reception.<ref name="Mills2010" />{{rp|19}}

To derive the expression for the offset, note that for the request packet,
<math display="block">t_0 + \theta + \delta/2 = t_1</math>
and for the response packet,
<math display="block">t_3 + \theta - \delta/2 = t_2</math>
Solving for ''θ'' yields the definition of the time offset.

The values for ''θ'' and ''δ'' are passed through filters and subjected to statistical analysis ("mitigation"). [[Outlier]]s are discarded and an estimate of time offset is derived from the best three remaining candidates. The clock frequency is then adjusted to reduce the offset gradually ("discipline"), creating a [[feedback loop]].<ref name="Mills2010">{{cite book|author=David L. Mills|title=Computer Network Time Synchronization: The Network Time Protocol|url=https://books.google.com/books?id=pdTcJBfnbq8C&pg=PA12|date=12 December 2010|publisher=Taylor & Francis|isbn=978-0-8493-5805-0|pages=12–|access-date=16 October 2016|archive-url=https://web.archive.org/web/20140718092324/http://books.google.com/books?id=pdTcJBfnbq8C&pg=PA12|archive-date=18 July 2014|url-status=live}}</ref>{{rp|20}}

Accurate synchronization is achieved when both the incoming and outgoing routes between the client and the server have symmetrical nominal delay. If the routes do not have a common nominal delay, a [[systematic bias]] exists of half the difference between the forward and backward travel times. A number of approaches have been proposed to measure asymmetry,<ref name="iL6pp">{{cite conference |last1=Gotoh |first1=T. |last2=Imamura |first2=K. |last3=Kaneko |first3=A. |title=Conference Digest Conference on Precision Electromagnetic Measurements |chapter=Improvement of NTP time offset under the asymmetric network with double packets method |conference=Conference on Precision Electromagnetic Measurements |pages=448–449 |year=2002 |doi=10.1109/CPEM.2002.1034915 |isbn=0-7803-7242-5}}</ref> but among practical implementations only chrony seems to have one included.<ref name="Ocilw"/><ref>{{cite web |title=sourcestats.c, function estimate_asymmetry() |url=https://git.tuxfamily.org/chrony/chrony.git/tree/sourcestats.c?h=4.3 |website=git.tuxfamily.org (chrony)}}</ref>