Problem $1$ Define these terms:

$1.$ Markov Chain

$2.$ Jackson Network Queuing

$3.$ G$/$G$/1$ Queue

$4.$ M$/$M$/1$ Queue

$5.$ Exponential Arrivals

$6.$ Fault tree reliability

Problem $2$ Customers arrive at a rate of $5$ per hour. They are serviced by $1$ person at a rate of $7$ per hour. What is the average number of customers in the line? Assume exponential arrivals and service times.

Problem $3-$ Develop a markov chain problem to describe a credit card? Develop numbers and estimated the value of the typical credit card user? Use your Markov chain to discuss what factors make a good customer. Hint, use the balance as the state with monthly transitions $($payment period$).$

Problem $4-$ Write a Markov Chain that has a transient state but no absorbing state.

Problem $5-$ You have $5$ pumps and $1$ spare. Pumps fail at the rate of $1$ every $5$ years. When failed the pump is replaced instantly. A new pump takes $\mathrm{.}25$ years to order. All distrubtions are exponential. The system fails when pump is broken.

Write the transition diagram for the system. Try to solve the system of equations. Difficult problem. Consider the state of the system $(6$ working $/0$ broken$).$ Make an assumption about the spare $($does it age$).$