Question

Consider a single machine workstation to which jobs arrive according to a Poisson distribution with a mean arrival rate of 12 jobs/hour. The process time of the workstation is exponentially distributed with a mean of 4 minutes. The expected number of jobs at the workstation at any given point of time is ________\ \text{(round off to the nearest integer).}

Hint:

In queueing theory, the expected number of jobs in the system is determined by the ratio of arrival and service rates. Ensure to use the correct units for consistency.

Answer 1

Correct Answer: 4

In a queueing system, the expected number of jobs in the system (denoted as \( L \)) can be computed using the formula for the M/M/1 queue: \[ L = \frac{\lambda}{\mu - \lambda} \] Where:
- \( \lambda = 12\ \text{jobs/hour} \) is the arrival rate,
- \( \mu = \frac{60}{4} = 15\ \text{jobs/hour} \) is the service rate (since the mean service time is 4 minutes).
Substitute the values into the formula: \[ L = \frac{12}{15 - 12} = \frac{12}{3} = 4 \] Thus, the expected number of jobs at the workstation is: \[ \boxed{4} \]