Editing Markov chain (section)

===Queueing theory===
{{Main|Queueing theory}}Markov chains are the basis for the analytical treatment of queues ([[queueing theory]]). [[Agner Krarup Erlang]] initiated the subject in 1917.<ref name="MacTutor|id=Erlang">{{MacTutor|id=Erlang}}</ref> This makes them critical for optimizing the performance of telecommunications networks, where messages must often compete for limited resources (such as bandwidth).<ref name="CTCN">S. P. Meyn, 2007. [http://www.meyn.ece.ufl.edu/archive/spm_files/CTCN/MonographTocBib.pdf Control Techniques for Complex Networks] {{webarchive|url=https://web.archive.org/web/20150513155013/http://www.meyn.ece.ufl.edu/archive/spm_files/CTCN/MonographTocBib.pdf |date=2015-05-13}}, Cambridge University Press, 2007.</ref>

Numerous queueing models use continuous-time Markov chains. For example, an [[M/M/1 queue]] is a CTMC on the non-negative integers where upward transitions from ''i'' to ''i''&nbsp;+&nbsp;1 occur at rate ''λ'' according to a [[Poisson process]] and describe job arrivals, while transitions from ''i'' to ''i''&nbsp;–&nbsp;1 (for ''i''&nbsp;>&nbsp;1) occur at rate ''μ'' (job service times are exponentially distributed) and describe completed services (departures) from the queue.