Let's solve each part step by step. --- ### **Given Data** - Arrival rate (\(\lambda\)) = 40 per hour - Service rate (\(\mu\)) = 60 per hour - System works for 8 hours a day This is a classic **M/M/1 queue** (single server, Poisson arrivals, exponential service time). --- ## **1. The average time in the system (\(W\))** For an M/M/1 system: \[ W = \frac{1}{\mu - \lambda} \] Substitute values: \[ \mu = 60, \quad \lambda = 40 \] \[ W = \frac{1}{60 - 40} = \frac{1}{20} \text{ hours} = 3 \text{ minutes} \] **Answer:** **Average time in the system = 3 minutes** --- ## **2. The average length of the queue (\(L_q\))** \[ L_q = \frac{\lambda^2}{\mu(\mu - \lambda)} \] Substitute values: \[ L_q = \frac{40^2}{60 \times (60-40)} = \frac{1600}{60 \times 20} = \frac{1600}{1200} = 1.33 \] **Answer:** **Average length of the queue = 1.33 machines** --- ## **3. Implied cost of machine's time (Cost analysis if faster service is rejected)** If owner rejects faster service (from 60/hr to 80/hr) which would cost Rs. 800 more per day, what is the implied cost per machine per hour? ### **Step 1: Calculate Average Time Saved per Machine with Faster Service** - New service rate, \(\mu' = 80\) per hour \[ W' = \frac{1}{\mu' - \lambda} = \frac{1}{80-40} = \frac{1}{40} \text{ hours} = 1.5 \text{ minutes} \] - Old average time in system = 3 minutes (from above) - New average time in system = 1.5 minutes (from above) - **Time saved per machine** = \(3 - 1.5 = 1.5\) minutes = \(0.025\) hours ### **Step 2: Find Total Time Saved in a Day** - Number of arrivals per day = \(40 \times 8 = 320\) machines - **Total time saved per day:** \[ 320 \times 0.025 = 8 \text{ hours} \] ### **Step 3: Implied Cost per Hour** - Cost to save 8 hours = Rs. 800 - **Implied cost per hour:** \[ \text{Implied cost per hour} = \frac{800}{8} = Rs. 100 \text{ per hour} \] --- ## **Final Answers (Summary)** 1. **Average time in the system:** \(3\) minutes 2. **Average length of the queue:** \(1.33\) machines 3. **Implied cost of machine's time:** **Rs. 100 per hour** --- **Let me know if you want detailed formulas or further explanations!**