1. Mr. Smith replaces the battery in his car as soon as it dies. Suppose the battery lifetimes are iid Erlang with parameters k = 3 and=1 (per year). Compute the long-run replacement rate if Mr. Smith replaces the battery upon failure

2. The lifetime of a machine is an Erlang random variable with parameters k = 2 and=0.2 (per day). When the machine fails, it is repaired. The repair times are Exp () with mean 1 day. Suppose the repair costs $10 per hour. The machine produces revenues at a rate of $200 per day when it is working. Compute the long run net revenue per day.

3. A machine is classified into one of the kodisi, namely: good, normal, or damaged condition. Suppose the machine is in good condition it will remain like this for a time of 1 and will then transition to one of the reasonable or damaged conditions with probabilities of 0.6 and 0.4 respectively. An engine under normal conditions will remain that way for 2 and will then break down. A damaged machine will be repaired, which takes 3, and when repaired will be in good condition with a probability of 0.75 and normal conditions with a probability of 0.25. (a) Determine the proportion of the machine in each condition. (b) From historical data, 1=4, 2=2, and 3=1, calculate the proportion of time for machines in each condition.