(a) To find the probability that an arriving job will require more than 20 minutes to be processed, we can use the exponential distribution. The average processing time per job is given as 3 minutes, so we can calculate the rate parameter (λ) as 1/3 (since the average is the reciprocal of the rate parameter).

The probability that a job takes more than 20 minutes to be processed can be calculated using the cumulative distribution function (CDF) of the exponential distribution. The CDF of the exponential distribution is given by P(X x) = 1 - e^(-λx), where X is the random variable representing the processing time and x is the desired time threshold.

Substituting the values, we have P(X 20) = 1 - e^(-(1/3)*20) = 1 - e^(-20/3) ≈ 0.1813.

Therefore, the probability that an arriving job will require more than 20 minutes to be processed is approximately 0.1813.

(b) To find the average number of jobs waiting in the queue, we can use the concept of the arrival rate and the service rate. The arrival rate is given as one job every 4 minutes, and the processing time per job is exponentially distributed with an average time of 3 minutes.

The average number of jobs waiting in the queue can be calculated using the formula: Lq = λ^2 / (μ(μ - λ)), where λ is the arrival rate and μ is the service rate.

Substituting the values, we have Lq = (1/4)^2 / (1/3 * (1/3 - 1/4)) = 1/16 / (1/3 * (1/12)) = 3/4.

Therefore, the average number of jobs waiting in the queue is 3/4.

Question

(a) To find the probability that an arriving job will require more than 20 minutes to be processed, we can use the exponential distribution. The average processing time per job is given as 3 minutes, so we can calculate the rate parameter (λ) as 1/3 (since the average is the reciprocal of the rate parameter).

The probability that a job takes more than 20 minutes to be processed can be calculated using the cumulative distribution function (CDF) of the exponential distribution. The CDF of the exponential distribution is given by P(X > x) = 1 - e^(-λx), where X is the random variable representing the processing time and x is the desired time threshold.

Substituting the values, we have P(X > 20) = 1 - e^(-(1/3)*20) = 1 - e^(-20/3) ≈ 0.1813.

Therefore, the probability that an arriving job will require more than 20 minutes to be processed is approximately 0.1813.

(b) To find the average number of jobs waiting in the queue, we can use the concept of the arrival rate and the service rate. The arrival rate is given as one job every 4 minutes, and the processing time per job is exponentially distributed with an average time of 3 minutes.

The average number of jobs waiting in the queue can be calculated using the formula: Lq = λ^2 / (μ(μ - λ)), where λ is the arrival rate and μ is the service rate.

Substituting the values, we have Lq = (1/4)^2 / (1/3 * (1/3 - 1/4)) = 1/16 / (1/3 * (1/12)) = 3/4.

Therefore, the average number of jobs waiting in the queue is 3/4.

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