Question

An office employs several clerks who originate documents and one operator who enters the document information in a computer system. The group originates documents at a rate of 19 per hour. The operator can enter the information with average exponentially distributed time of 3 minutes. Assume the population is infinite, arrivals are Poisson, and queue length is infinite with FCFS discipline.

a. Calculate the percentage utilization of the operator. (Round your answer to 2 decimal places.)
Utilization of the operator in %:

b. Calculate the average number of documents in the system. (Round your answer to 1 decimal place.)
Average number of documents:

c. Calculate the average time in the system. (Round your answer to 1 decimal place.)
Average time in system in minutes:

d. Calculate the probability of four or more documents being in the system. (Round your intermediate calculations to 3 decimal places and final answer to 1 decimal place.
Probability in %:

Answer 1

The percentage utilization of the operator is 15.79%. The average number of documents in the system is 0.95. The average time in the system is 3 minutes.

Step-by-step explanation:

To calculate the percentage utilization of the operator, we need to find the ratio of time the operator spends working to the total time. Since the operator can enter information with an average time of 3 minutes per document, and the group originates documents at a rate of 19 per hour, the operator has 1/19th of an hour per document. This means the operator is busy for 3/19th of every hour. To find the utilization percentage, we multiply this fraction by 100: Utilization of the operator = (3/19) * 100 = 15.79%.

To calculate the average number of documents in the system, we need to consider both the documents being originated and the documents being entered by the operator. Using Little's Law, which states that the average number of items in a system is equal to the average arrival rate multiplied by the average time spent in the system, we can calculate the average number of documents in the system: Average number of documents = Arrival rate * Average time in the system = 19 * (3/60) = 0.95 documents.

To calculate the average time in the system, we divide the average number of documents in the system by the arrival rate: Average time in system = Average number of documents / Arrival rate = 0.95 / 19 = 0.05 hours or 3 minutes.

To calculate the probability of four or more documents being in the system, we can use the Poisson distribution formula: Probability = 1 - (P(0) + P(1) + P(2) + P(3)). Using the formula for the Poisson distribution, we can calculate the individual probabilities and sum them up. The answer for this specific calculation will depend on the precise values, but it can be rounded to one decimal place.

Answer 2

Solution and Explanation:

In this system, population is infinite, arrivals are poisson distributed, service time is exponentially distributed, queue length is infinite with FCFS discipline. Therefore M/M/1 model fits.

Arrival rate, λ = 19 customers per hour

Service rate, μ = 1 / service time =
`1 / 3 * 60=`20 customers per hour

a). Percentage utilization of the operator,
`\rho=\lambda / \mu=19 / 20` = 95.00%

b). Average number of documents in the system,
`\mathrm{L}=\rho /(1-\rho)=.95 /(1-.95)` = 19.00

c). Average time in system in minutes,
`\mathrm{W}=\mathrm{L} \lambda=19 / 19` = 1 hour = 60 minutes

d). Probability of four or more documents being in the system = 1 - (probability of 0 document in the system + probability of 1 document in the system + probability of 2 documents in the system + probability of 3 documents in the system)

`=1-\left((1-\rho)^(*) \rho^(0)+(1-\rho)^(*) \rho^(1)+(1-\rho)^(*) \rho^(2)+(1-\rho)^(*) \rho^(3)\right)`

`=1-(.0500+.0475+.0451+.0429)`

`=1-.1855`

= .8145 or 81.5%