Let's break down and **solve** the queueing problem step by step: --- ### **Step 1: Understand the Problem** - **Customer Arrivals:** Every 40 ± 35 seconds (mean = 40 sec, SD = 35 sec, assume exponential for simplicity). - **Tellers:** 2 tellers. - **Service Time:** 75 ± 25 seconds (mean = 75 sec, SD = 25 sec, assume exponential). - **Current System:** 2 separate lines, customers choose a teller at random. - **Proposed System:** Single line for both tellers (like Lotta Trust Bank). - **Simulation Periods:** Ignore first 15 min for warm-up, analyze statistics for the next 2 hours. --- ### **Step 2: Convert to Queueing Models** #### **A. Two Separate Lines (Current)** - 2 independent M/M/1 queues. - **Arrival Rate per Teller:** Each teller gets half the arrivals. - **λ (per teller):** \( \frac{1}{40} × 0.5 = 0.0125 \) per sec ≈ 45 per hour ÷ 2 = 22.5/hr - **μ (per teller):** \( \frac{1}{75} = 0.0133 \) per sec ≈ 48/hr #### **B. Single Line (Proposed)** - 1 M/M/2 queue. - **λ (total):** \( \frac{1}{40} = 0.025 \) per sec ≈ 90/hr - **μ (per teller):** \( \frac{1}{75} = 0.0133 \) per sec ≈ 48/hr --- ### **Step 3: Calculate Performance Measures** #### **A. Two Independent M/M/1 Queues** - Per teller: - **λ = 22.5/hr** - **μ = 48/hr** - **ρ (utilization):** \( \rho = \frac{\lambda}{\mu} = \frac{22.5}{48} = 0.469 \) - **Mean number in system (L):** \( L = \frac{\rho}{1-\rho} = \frac{0.469}{1-0.469} = 0.883 \) - **Mean time in system (W):** \( W = \frac{1}{\mu - \lambda} = \frac{1}{48-22.5} = 0.04 \) hr = 2.4 min - **Mean waiting time in queue (Wq):** \( W_q = \frac{\rho}{\mu-\lambda} = \frac{0.469}{25.5} = 0.0184 \) hr = 1.1 min #### **B. Single M/M/2 Queue** - **λ = 90/hr** - **μ = 48/hr** - **c = 2 (servers)** - **ρ (overall):** \( \rho = \frac{\lambda}{c \mu} = \frac{90}{96} = 0.938 \) - **Use M/M/c formulas:** - **P0 (prob. system empty):** \[ P_0 = \left[ \sum_{n=0}^{c-1} \frac{(c \rho)^n}{n!} + \frac{(c \rho)^c}{c!} \frac{1}{1-\rho} \right]^{-1} \] - For c=2: \[ P_0 = \left[ \frac{(2 \times 0.938)^0}{0!} + \frac{(2 \times 0.938)^1}{1!} + \frac{(2 \times 0.938)^2}{2!} \frac{1}{1-0.938} \right]^{-1} \] - Compute: - \( (2 \times 0.938) = 1.876 \) - Terms: \( 1 + 1.876 + \frac{1.876^2}{2} \frac{1}{0.062} \) - \( 1.876^2 = 3.519 \) - \( \frac{3.519}{2} = 1.759 \) - \( \frac{1.759}{0.062} = 28.37 \) - Total inside sum: \( 1 + 1.876 + 28.37 = 31.246 \) - \( P_0 = \frac{1}{31.246} = 0.032 \) - **Mean number in queue (Lq):** \[ L_q = P_0 \frac{(c \rho)^c \rho}{c! (1-\rho)^2} \] - \( (1.876)^2 = 3.519 \) - \( \frac{3.519 \times 0.938}{2 \times (0.062)^2} \) - \( 2 \times 0.0038 = 0.0076 \) - \( 3.519 \times 0.938 = 3.303 \) - \( \frac{3.303}{0.0076} = 434.6 \) - \( L_q = 0.032 \times 434.6 = 13.9 \) This seems high. Let's double-check using standard formula for M/M/c: \[ L_q = \frac{P_0 (c \rho)^c \rho}{c! (1-\rho)^2} \] Alternatively, let's use an online calculator for M/M/2 queue: - λ = 90/hr, μ = 48/hr, c = 2 - **Utilization per teller:** \( 90/96 = 0.938 \) - **Mean delay in queue (Wq):** For high utilization, the queue grows rapidly. --- ### **Step 4: Compare Results** #### **A. Teller Utilization** - **Separate lines:** 0.469 (46.9%) - **Single line:** 0.938 (93.8%) (across both tellers; each teller busy 93.8% of the time) #### **B. Mean Delay in Queue** - **Separate lines:** 1.1 min - **Single line:** Much higher due to high utilization (estimate > 10 min) #### **C. Proportion of Customers Waiting > 1 min, > 3 min** - **Separate lines:** Use exponential waiting time distribution: - \( P(\text{wait} > t) = e^{-\mu (1-\rho) t} \) - For 1 min (t = 1/60 hr): - \( \mu(1-\rho) = 48 \times (1-0.469) = 25.5 \) - \( P(\text{wait} > 1) = e^{-25.5 \times (1/60)} = e^{-0.425} = 0.654 \) - For 3 min: - \( P(\text{wait} > 3) = e^{-25.5 \times (3/60)} = e^{-1.275} = 0.279 \) - **Single line:** Because utilization is so high, the probability is much higher that customers wait long times. --- ### **Step 5: Final Answer and Recommendation** #### **Summary Table** | Discipline | Teller Utilization | Mean Delay (min) | P(wait > 1 min) | P(wait > 3 min) | |--------------------|-------------------|------------------|-----------------|-----------------| | Separate lines | 46.9% | 1.1 | 65.4% | 27.9% | | Single line | 93.8% | >10 | Much higher | Much higher | #### **Conclusion** - **The current system (separate lines) is significantly faster for customers** under the given arrival and service rates, with much lower teller utilization but also much less customer delay. - **Switching to a single line with two tellers increases employee utilization but drastically increases customer waiting times** due to the high utilization (ρ > 0.9). - **Recommendation:** **Do NOT switch to a single line unless you can increase the number of tellers or decrease service time.** --- ### **If you need simulation code (in Python or Excel), let me know!**