Question

1. suppose jobs arrive at a single - machine workstation at a rate of 20 per hour and the average process time is 2.5 minutes.

(a) what is the utilization of the machine?
(b) suppose that interarrival and process times are exponential.
i) what is the average time a job spends at the station?
ii) what is the average number of jobs at the station?

Explanation:

Step1: Convert process time to hourly rate

The average process time is 2.5 minutes per job, so the machine's processing rate $\mu$ is $\frac{60}{2.5} = 24$ jobs per hour.

Step2: Calculate machine utilization (a)

Utilization $
ho$ is arrival rate $\lambda$ over processing rate $\mu$.
$
ho = \frac{\lambda}{\mu} = \frac{20}{24} = \frac{5}{6} \approx 0.833$

Step3: Find average time at station (b-i)

Use M/M/1 queue formula: $W = \frac{1}{\mu - \lambda}$
$W = \frac{1}{24 - 20} = 0.25$ hours, convert to minutes: $0.25 \times 60 = 15$ minutes

Step4: Find average number of jobs (b-ii)

Use M/M/1 queue formula: $L = \frac{\lambda}{\mu - \lambda}$
$L = \frac{20}{24 - 20} = 5$

Answer:

(a) $\frac{5}{6}$ or approximately 0.833
(b) i) 15 minutes (or 0.25 hours)
ii) 5