Consider a system that is always in one of states, numbered 1 through . Every time a clock ticks, the system updates itself according to an matrix of transition probabilities, the entry of which gives the probability that the system moves from state to state at any clock tick. A Markov chain is a system like this, in which the next state depends only on the current state and not on previous states. Powers of the transition matrix approach a matrix with constant columns as the power increases. The number to which the entries in the column converge is the asymptotic fraction of time the system spends in state .