Suppose a process has 32 stages and the time required to complete each stage is

Question

Suppose a process has 32 stages and the time required to complete each stage is an exponential random variable with a mean of 1 minute and 45 seconds a) Assume all the stages will be complete and each stage's time is independent of the others. Identify the distribution of the total time taken (in minutes). b) Identify the distribution of the number of stages completed in one hour. c) Using the distribution in part (a) or part (b), compute (or approximate) the probability of completing all 32 stages within one hour.

Explanation / Answer

Let Xi = time required to complete the ith stage, then the given statement means

Xi ~ Exp(1.75) and X1, X2, X3,….., X32 are iid Exp(1.75). Hence,

X = (X1 + X2 + X3 +….. + X32) ~ Exp(32 x 1.75). Evidently, X = total time taken.

Thus, the total time taken has an Exponential distribution with mean of 56 minutes.

This answers Part (a)

If inter-event time is Exponential with parameter (average time per event), then number of events per unit time period is Poisson with parameter = 1/.

Average time per stage is 1.75 minutes => number of stages completed in 1.75 minutes = 1 => number of stages completed in 1 hour = 60/1.75 = 32.29.

Hence, the distribution of Number of stages completed in 1 hour has a Poisson Distribution with parameter 32.29. This answers Part (b)

To answer Part (c), we may use (a) or (b).

Using (a), we want P(32 stages are completed in 1 hour/on an average 32 stages take 56 minutes or 0.9333 hour to get completed). This is same as finding P(X 1/given = 0.9333), where X is the Exponential variable with parameter 0.9333.

This probability = 1 – e – 1/0.9333 [by the cumulative distribution of Exponential]

= 1 – 0.3425 = 0.6565 ANSWER

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