Question

Students arrive at the Administrative Services Office at an average of one every 20 minutes, and their requests take on average 16 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times.

a. What percentage of time is Judy idle? (Round your answer to 1 decimal place.)

b. How much time, on average, does a student spend waiting in line? (Do not round intermediate calculations. Round your answer to 1 decimal place.)

c. How long is the (waiting) line on average? (Round your answer to 2 decimal places.)

d. What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line? (Do not round intermediate calculations. Round your answer to 4 decimal places.)

Answer 1

Solution-

In queuing theory we learn about the waiting-line frameworks.

three pieces of a waiting-line framework are

(1) appearances or inputs to the framework,

(2) queue discipline, and

(3) administration office. Waiting-line models with Poisson dispersed appearances, FIFO discipline, and a single-administration stage are probably the most generally utilized waiting line models.

(a)

According to given issue, the appearance rate λ=60/20=3per hour and the administration rate μ=60/16=3.75 every hour.

Idle probability = 1−λ/μ=1−3/3.75

=0.20 or 20% of the time.

(b)

Average waiting time in the queue = λ/μ(μ−λ)=3/[3.75(3.75−3)]

= 1.06666667 hour

=1.1 hour

(c)

Average queue length = Average waiting time in the queue ×λ= 1.06666667 ×3= 3.2

(d)

Probability of in any event one other student in line = Probability of at any rate two students in the framework = (λ/μ)3=0.512 =0.51

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