Question

3. The Solomon, Smith, and Samson law firm produces many legal documents that must be word processed for clients and the firm. Requests average eight pages of documents per hour, and they arrive according to a Poisson distribution. The secretary can word process 10 pages per hour on average according to an exponential distribution. a. What is the average utilization rate of the secretary? b. What is the probability that more than four pages are waiting or being word processed? c. What is the average number of pages waiting to be word processed?

Answer 1

a. What is the average utilization rate of the secretary?

Utilization rate (?): Arrival rate (?)/ Service rate (?)

= 8/10

= 0.8 or 80% utilization

b. What is the probability that more than four pages are waiting or being word processed?

P_n = (1- ?) (?)ⁿ

P₄ = (1-0.8)*(0.8)⁴=0.0819

P₃ = (1-0.8)*(0.8)³=0.1024

P₂ = (1-0.8)*(0.8)²=0.1280

P₁ = (1-0.8)*(0.8)¹=0.1600

P₀ = (1-0.8)*(0.8)⁰=0.2000

Total probability = (P₀) + (P₁) + (P₂) + (P₃) + (P₄)

= 0.0819 + 0.1024 + 0.1280 + 0.1600 + 0.2000

= 0.6723

Therefore, the probability of more than four pages in the system is 1 – 0.6723 = 0.3277

c. What is the average number of pages waiting to be word processed?

L_q= ?*L

L= (?/µ-?)

?= ?/ ? = 8/10

L_q= ?*L

= (8/10) * (8/10-8)

= 0.8 * 4

= 3.2 Pages