Problem 4 25 marks A new light fast internet service is on t

Problem 4 (25 marks). A new light fast internet service is on testing. It starts working from unstable speed (U). As time goes its speed changes to slow (S) or normal (N) or light fast (F). The status of the internet speed follows a Markov chain having transition matrix S U N F S 1.0 0.0 0.0 0.0 p= 0.1 0.2 0.6 0.1 N 0.0 0.0 0.7 0.3 F 0.0 0.0 0.4 0.6 (a) [4 marks] Determine class (transient or recurrent) of each state. (b) 7 marks] Find the expected time to exit from status U. (c) [7 marks Compute the probability the internet speed drops down to slow eventually (d) [7 marks] Compute the limit transition probability limn opna,x) for each x = S, U, N, F.

Solution

U is transient as probability that the state will ever return to U if it is U is not 1. since U will not come again if goes to S, N or F.

U is recurrent as probability that the state will ever return to S if it is S is equal to 1 Since S is an absorbing state

N is recurrent as probability that the state will ever return to N if it is N is equal to 1. Since N can move to F or stay at N and F is moving to N with some positive probability.

F is recurrent as probability that the state will ever return to F if it is F is equal to 1. Since F can move to N or stay at F and N is moving to F with some positive probability.

b) Expected Time to exit from status U

probabity that U will remain U after one time step is 0.2

probabity that U will not remain U after one time step is 0.8

every time step is independent . Will follow negative binomial

So Expected time = 1/0.8 = 1.25

c) Once its slow its slow forever.

Once its N or F it will never be S again.

So the required probability is the probability that U changes to S

which is equal to 0.1

d)limit transition probability

U remains U = (0.2)^n goes to zero

U goes to S = 0.1