Customers arrive randomly, and Rich serves them one at a time. The arrival rate exceeds the service rate slightly (4 per hour vs. 8 per hour). This imbalance leads to a small queue, with an average of 0.5 customers waiting. This means Rich spends a little time idle waiting for customers, but not enough to significantly impact his overall productivity.
To find the average number of customers waiting for haircuts, we need to consider the balance between arrival and service rates.
1. Arrival Rate:
- Customers arrive at a rate of λ = 4 per hour. This means, on average, a customer arrives every 1/4 hour.
2. Service Rate:
- Rich can perform haircuts at a rate of μ = 8 per hour. This means, on average, he finishes a haircut every 1/8 hour.
3. Average Number Waiting:
- We can use Little's Law to find the average number waiting (L):
L = λ * W
where:
- L: Average number waiting
- λ: Arrival rate
- W: Average time spent in the system (waiting + service)
4. Solving for W:
- We need to find W first. Since arrivals and services follow Poisson and negative exponential distributions, the system is in steady-state, and W is simply the reciprocal of the service rate:
W = 1/μ = 1/8 hour
5. Calculating L:
- Now, plug the values into Little's Law:
L = λ * W = 4 * 1/8 = 0.5 customers
Therefore, on average, 0.5 customers are waiting for haircuts at Rich Dunn's Styling Shop.