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If, by whatever means,
lim
k
P
k
{\textstyle \lim _{k\to \infty }\mathbf {P} ^{k}}
is found, then the stationary distribution of the Markov chain in question can be easily determined for any starting distribution, as will be explained below. If all states in an irreducible Markov chain are ergodic, then the chain is said to be ergodic. . Each element of the one-step transition probability matrix of the EMC, S, is denoted by sij, and represents the conditional probability of transitioning from state i into state j.
To find the stationary probability distribution vector, we must next find
{\displaystyle \varphi }
such that
with
{\displaystyle \varphi }
being a row vector, such that all elements in
{\displaystyle \varphi }
are greater than 0 and
1
{\displaystyle \|\varphi \|_{1}}
= 1. A Markov-chain has the property that future states are dependent only on present states (this is known as the Markov property).
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Markovian systems appear extensively in thermodynamics and statistical mechanics, whenever probabilities are used to represent unknown or unmodelled details of the system, if it can be assumed that the dynamics are time-invariant, and that no relevant history need be considered which is not already included in the state description. For example, an M/M/1 queue is a CTMC on the non-negative integers where upward transitions from i to i+1 see here now at rate λ according to a Poisson process and describe job arrivals, while transitions from i to i–1 (for i1) occur at rate μ (job service times are exponentially distributed) and describe completed services (departures) from the queue. .