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 large system is controlled by n identical computers. Each computer independently alternates…
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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 Qi(t) be the probability that the ith computer is operational at time t and find limt→∞ Qi(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).
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