A large system is controlled by n identical computers. Each computer independently alternates between an operational state and a repair state. The duration of the operational state, from completion of...

A large system is controlled by n identical computers. Each computer independently alternates between an operational state and a repair state. The duration of the operational state, from completion of one repair until the next need for repair, is a random variable X with finite expected duration E [X]. The time required to repair a computer is an exponentially distributed random variable with density λe^-λt. All operating durations and repair durations are independent. Assume that all computers are in the repair state at time 0.

a) For a single computer, say the ith, do the epochs at which the computer enters the repair state form a renewal process? If so, find the expected inter-renewal interval.

b) Do the epochs at which it enters the operational state form a renewal process?

c) Find the fraction of time over which the ith computer is operational and explain what you mean by fraction of time.

d) Let Q_i(t) be the probability that the ith computer is operational at time t and find lim_t→∞
Q_i(t).

e) The system is in failure mode at a given time if all computers are in the repair state at that time. Do the epochs at which system failure modes begin form a renewal process?

f) Let Pr{t} be the probability that the the system is in failure mode at time t. Find limt!1 Pr{t}. Hint: look at part d).

g) For small, find the probability that the system enters failure mode in the interval (t,t + ] in the limit as t → ∞.

h) Find the expected time between successive entries into failure mode.

i) Next assume that the repair time of each computer has an arbitrary density rather than exponential, but has a mean repair time of 1/λ. Do the epochs at which system failure modes begin form a renewal process?

j) Repeat part f) for the assumption in (i).