1. **Problem statement:** We have a duplicating machine with a Poisson service rate of mean $\mu = 10$ jobs/hour and arrivals at rate $\lambda = 5$ jobs/hour over an 8-hour workday. We want to analyze: i) equipment utilization ii) percent time an arrival waits iii) average system time iv) average cost due to waiting and operating the machine 2. **Calculate equipment utilization ($\rho$):** $$\rho = \frac{\lambda}{\mu} = \frac{5}{10} = 0.5$$ This means the machine is busy 50% of the time. 3. **Percent time an arrival has to wait (probability queue forms):** For an M/M/1 queue, the probability an arrival waits is the utilization: $$P_{wait} = \rho = 0.5 = 50\%$$ 4. **Average system time ($W$):** The average time a job spends in the system (waiting + service) is: $$W = \frac{1}{\mu - \lambda} = \frac{1}{10 - 5} = \frac{1}{5} = 0.2 \text{ hours} = 12 \text{ minutes}$$ 5. **Average cost due to waiting and operating the machine:** - Cost of secretary time: 3.5 per hour - Machine operates 8 hours, cost per hour unknown, assume cost $C_m$ per hour (not given, so we consider only secretary cost) Calculate average number of jobs in system ($L$): $$L = \lambda W = 5 \times 0.2 = 1$$ Average waiting time in queue ($W_q$): $$W_q = W - \frac{1}{\mu} = 0.2 - 0.1 = 0.1 \text{ hours} = 6 \text{ minutes}$$ Average number of jobs waiting ($L_q$): $$L_q = \lambda W_q = 5 \times 0.1 = 0.5$$ Cost due to waiting secretary time: $$\text{Cost}_{wait} = L_q \times 3.5 = 0.5 \times 3.5 = 1.75 \text{ per hour}$$ Cost of operating machine for 8 hours (assuming cost per hour $C_m$ unknown, so omitted). Total average cost per hour due to waiting and operation (excluding machine cost): $$1.75$$ **Final answers:** - i) Equipment utilization $\rho = 0.5$ - ii) Percent time arrival waits = 50% - iii) Average system time $W = 0.2$ hours (12 minutes) - iv) Average cost due to waiting = 1.75 per hour (secretary time only)