The Solomon, Smith, and Samson law firm produces many legal documents that must

Question

The Solomon, Smith, and Samson law firm produces many legal documents that must be word processed for clients and the firm. Requests average 15 pages of documents per hour, and they arrive according to a Poisson distribution. The secretary can word process 1616 pages per hour on average according to an exponential distribution. a. The average utilization rate of the secretary is _____%utilization. (Enter your response as a percentage rounded to the nearest whole number.) b. The probability that more than four pages are waiting or being word processed is ____ (Enter your response rounded to three decimal places.) c. The average number of pages waiting to be word processed is ____pages. (Enter your response rounded to two decimal

Explanation / Answer

Note: The question is being solved using the assumption that 1616 pages is a typing error and it is 16 pages instead.

To be calculated:

(a) Average utilization rate

(b) Probability that more than four pages are waiting

(c) Average number of pages waiting

Given values:

Number of servers, m = 1

Processing/Service rate, p = 16 pages per hour

Request arrival rate, a = 15 pages per hour

Solution:

(a) The average utilization of the secretary is calculated as;

Utilization, u = a / p

where,

a = request arrival rate

p = service or processing rate

Putting the given values in the above formula, we get;

Utilization, u = 15 / 16 = 0.94

In percentage terms,

u = 0.94 x 100 = 94%

The average utilization rate of the secretary is 94%

(b) The Probability that more than four pages are waiting is calculated as;

P (n > 4) = (a / p) ^ (m+1)

where

a = request arrival rate

p = service or processing rate

m = 4

Putting the given values in the above formula, we get;

P (n > 4) = (15 / 16) ^ (4+1)

P (n > 4) = 0.938 ^ 5

P (n > 4) = 0.726

The probability that more than four pages are waiting or being word processed is 0.726

(c) The average number of pages waiting to be word processed is calculated using the below formula:

Number of pages waiting = [a^2 / p*(p - a)]

where,

a = request arrival rate

p = service or processing rate

Putting the given values in the above formula, we get;

Number of pages waiting = [(15^2) / 16*(16 - 15)]

Number of pages waiting = 225 / 16

Number of pages waiting = 14.06

The average number of pages waiting to be word processed is 14.06 pages

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