We consider time-inhomogeneous Markov chains on a finite state-space, whose transition probabilitiesp,y(/) = Cjjt(t) Vi J are proportional to powers of a vanishing small parameter e(t). We determine the precise relationship between this • chain and the corresponding time-homogeneous chains p,y = c^e 1^, as e ^ 0.Let | vj) be the steady-state distribution of this time-homogeneous chain. We characterize the orders ( JJ,) in v-= Q(e m ). We show that if e(t) \ 0 slowly enough, then the timewise occupation measures /3,-:= sup( 0| SJ1, «(/)* Prob(x(O = /') = +oo J, called the recurrence orders, satisfy ft -fy = ij y -TJ,. Moreover, if S : = [vilvi = min jVj] is the set of ground states of the timehomogeneous chain, thenx(t) -* Q, in an appropriate sense, whenever i/(f) is "cooled" slowly. We also show that there exists a critical p* such that x(t) -> 9 if and only if £,°L, e(t) p * = +oo. We characterize this critical rate as p* = max, e s min,« e S min p = ( , = , 0 , Vv= ,-) maxo s * s/ v-i(K>/ t+1 + Vt k -V,)-Finally, we provide a graph algorithm for determining the orders (%) and (/?,-] and the critical rate p*.

Hidden Markov Model (HMM) is a stochastic model where all the states are hidden and emit outputs that are observable. HMM is intertwined with artificial intelligence that allows it to be applied to latest technologies like speech and handwriting recognition. It has proved to be an extremely valuable tool in the field of bioinformatics and has been extensively used in sequencing, alignment, homology prediction and protein secondary structure prediction. In order to understand HMM, we have used a simple problem based on land used for pigeon pea cultivation throughout years which has been illustrated using MATLAB and studied using matrix multiplication and probability concepts. This paper also outlines the technique used for studying gene prediction using HMM. It heavily draws content from the forward and backward recursions which were given by Ruslan L Stratonovichlt in 1960. It is a nexus of complicated ideas and calculations, including various algorithms like Viterbi and Forward algorithm which can be implemented using various programming languages such as C, C++ and tools like Matlab. Even though the concept is about 60 years old,it is one of the most detailed algorithms available to date. Due to ever increasing applications, it has become a constant in the curriculum for students of different backgrounds like computer science, electronics and bioinformatics.