1. **Problem Statement:** Charles, a barber, can shave 4 customers per hour (service rate $\mu=4$), and customers arrive at a rate of 3 per hour (arrival rate $\lambda=3$). We need to find: i. Proportion of time Charles is idle. ii. Probability a customer receives immediate service. iii. Average number of customers in the system. iv. Average time a customer spends in the system. 2. **Formulas and Concepts:** This is an M/M/1 queue (Poisson arrivals, exponential service, single server). - Utilization factor $\rho = \frac{\lambda}{\mu}$. - Proportion of time idle = $1 - \rho$. - Probability of immediate service = $1 - \rho$ (since immediate service means no queue). - Average number in system $L = \frac{\rho}{1-\rho}$. - Average time in system $W = \frac{1}{\mu - \lambda}$. 3. **Calculations:** - Calculate $\rho$: $$\rho = \frac{3}{4} = 0.75$$ - i. Proportion of time idle: $$1 - \rho = 1 - 0.75 = 0.25$$ - ii. Probability of immediate service: $$1 - \rho = 0.25$$ - iii. Average number in system: $$L = \frac{0.75}{1 - 0.75} = \frac{0.75}{0.25} = 3$$ - iv. Average time in system: $$W = \frac{1}{4 - 3} = 1 \text{ hour}$$ 4. **Explanation:** - Utilization $\rho$ tells us how busy Charles is; 0.75 means he is busy 75% of the time. - The idle time is the remaining 25% when no customers are being served. - Immediate service probability equals idle time because if Charles is free, the arriving customer is served immediately. - Average number in system $L$ includes customers being served and waiting. - Average time $W$ is how long a customer spends from arrival to departure. **Final answers:** - Proportion idle = 0.25 - Probability immediate service = 0.25 - Average number in system = 3 - Average time in system = 1 hour