Abstract When it is required to study the behaviour of any system after start-up or repair, transient solutions are particularly important. Also, these solutions become necessary when the behaviour need to be studied over a fixed period of time during which the system is required to perform a given mission. There are a number of methods available to solve the first order differential equations of any system, including: the Laplace Transform method, and the Matrix Multiplication method. Both methods are used in this paper to calculate the state-probabilities for four-state and five-state models, respectively, and the results are compared. A comparison is also made between the system “availability” and “unavailability” for both four-state and five-state models using the Laplace Transform and the Matrix Multiplication methods.