ABSTRACT
We shall now continue our investigation of further random processes in continuous time. In the Poisson process in Chapter 5, the probability of a further event was independent of the current number of events or readings in the Geiger counter analogy: this is a specific assumption in the definition of the Poisson process (Section 5.7). On the other hand, in birth and death processes, the probability of a birth or death will depend on the population size at time t. The more individuals in the population, the greater the possibility of a death, for example. As for Markov chains, birth and death processes are further examples of Markov processes. Additionally they include queueing processes, epidemics, predator–prey competition, and others. Markov processes are characterized by the condition that future development of these processes depends only on their current states and not their history up to that time. Generally Markov processes are easier to model and analyse, and they do include many interesting applications. Non-Markov processes in which the future state of a process depends on its whole history are generally harder to analyse mathematically.
