Question

3. Assume that the lifetimes (measured from the beginning of use) of light bulbs are i.i.d. random variables with distribution P(T> k) = (k +1)-, k = 0,1,2, ..., for some B > 0. (Note that time is measured in discrete units.) In a lightbulb socket in a factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (Xn)n>o be the irreducible Markov chain which records the age of the bulb currently in use in the socket (Xn = 0 at times when a bulb is replaced, corresponding to a new bulb). (a) Derive the transition probabilities of the chain. (b) For each value of B, determine if the chain is positive recurrent, null recurrent, or transient.

Answer 1

Pry PIxnty lXn> J Pry 0,4 1 , xt1 PIXnti= t [Xn= PET7131 T2 PIT22t 11 -B t PETZ Pa,s> PLT-1 T2 PIT2t PITZ 1- PIT2 ( + 1 -th -B Now, an t Bmoreanes, tt 2-t1 The ehain han finite irst onder morents and hence t posttive recvrrent ECCHD-LJES2,. - ane T