Simultaneous Success-Run Chains

If every element from a source of identical particles has to perform a certain fixed success-run Markov chain with n + 2 states, and if the particles are put into the initial state one at a time, they act independently and reach the absorbing state n + 2 after n + 2 steps or they return to the source. This sequence of simultaneous success-run chains can be analysed by using a “basic” Markov chain with 2ⁿ different states that is studied in great detail. Among others, we derive the transition matrix and the stationary distribution. It further turns out that, for m ≧ n the basic Markov chain already behaves in a stationary way , so that it is easy to find the distribution of the number of absorbed particles at a certain instant, as well as the number of particles that return to the source. The proofs of the main theorems are based on mathematical induction and matrix methods.