1. A weight of L tons is held up by K cables that share the load equally. When one of the cables breaks, the remaining unbroken cables share the entire load equally. When the last cable breaks, we have a failure. The failure rate of a cable subject to M tons of load isM per year. The lifetimes of the K cables are independent of each other. Let X(t ) be the number of cables that are still unbroken at time t. Show that {X (t ), t > 0} is a CTMC, and find its rate matrix

2. A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers. What fraction of potential customers enter the system?

3. Johnson Medical Associates has two physicians on call, Drs. Dawson and Baick. Dr. Dawson is available to answer patients' calls for time periods that are exponentially distributed with mean 2 hours. Between those periods, he takes breaks, each of which being an exponential amount of time with mean 30 minutes. Dr. Baick works independently from Dr. Dawson, but with similar work patterns. The time periods she is available to take patients' calls and the times she is on break are exponential random variables with means 90 and 40 minutes, respectively. In the long run, what is the proportion of time in which neither of the two doctors is available to take patients' calls?