Uniformization (probability theory)

In probability theory, uniformization method, (also known as Jensen's method or the randomization method) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete time Markov chain. The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name uniformisation. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). The method was first introduced by Winfried Grassmann in 1977. In probability theory, uniformization method, (also known as Jensen's method or the randomization method) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete time Markov chain. The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name uniformisation. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero). The method was first introduced by Winfried Grassmann in 1977. For a continuous-time Markov chain with transition rate matrix Q, the uniformized discrete-time Markov chain has probability transition matrix P := ( p i j ) i , j {displaystyle P:=(p_{ij})_{i,j}} , which is defined by with γ, the uniform rate parameter, chosen such that