Final answer:
The probability that a given disk drive is busy is 53%, calculated using the utilization factor. The probability that no disk drives are busy can be calculated with (1-0.53)^3. To find the average number of requests in the system, use Little's Law with the average service time and arrival rate.
Step-by-step explanation:
To solve the storage system problem involving disk drives, we first need to understand the properties of Poisson and exponential distributions. The average arrival rate of 53 storage requests per second and an average service time of 0.03 seconds per request provide the necessary parameters.
a. To determine the probability that a given disk drive is busy, we calculate the utilization factor (ρ). The utilization factor is the average service rate (μ) multiplied by the average time to service a request. With three disk drives, the service rate is 3/0.03 or 100 requests per second. Thus ρ = 53/100 = 0.53. This means each disk drive is busy 53% of the time.
b. The probability that no disk drives are busy can be calculated using the formula P(0) = (1-ρ)^(number of servers), which in this case is (1-0.53)^3.
c. The probability that a storage request will have to wait is equal to 1 minus the probability that all servers are free. We can use the Erlang B formula or the formula for the probability of zero customers in the system to find this.
d. To find the average number of storage requests in the system, we use Little's Law: L = λ * W, where λ is the arrival rate and W is the average time a storage request spends in the system (including both waiting and service time).