# Comparison of Serial vs Parallel Loan Application Processing This analysis compares **serial (specialized)** and **parallel (generalized)** processing models for a bank's loan application system. There are 4 workers and 4 tasks: 1. Credit check 2. Preparing the loan covenant 3. Pricing the loan 4. Disbursement of funds The performance measure is **mean response time per customer**. Simulations were run using: - **Model 1:** Serial processing - **Model 2:** Parallel processing - **IRN:** Independent Random Numbers - **CRN (w/ Sync):** Common Random Numbers, with synchronization - **CRN (w/o Sync):** Common Random Numbers, without synchronization The table below summarizes 10 replications of average response times (minutes): | Replication | Model 1 | Model 2 IRN | Model 2 CRN w/Sync | Model 2 CRN w/o Sync | |-------------|---------|-------------|--------------------|----------------------| | 1 | 24.82 | 17.13 | 18.63 | 22.79 | | 2 | 44.28 | 15.67 | 9.86 | 14.08 | | 3 | 24.23 | 27.3 | 16.41 | 21.24 | | 4 | 19.42 | 11.36 | 8.90 | 15.72 | | 5 | 34.94 | 11.71 | 19.63 | 22.54 | | 6 | 46.71 | 11.66 | 17.57 | 17.78 | | 7 | 45.89 | 17.42 | 18.32 | 20.69 | | 8 | 31.3 | 12.58 | 13.13 | 13.77 | | 9 | 26.77 | 9.8 | 10.46 | 23.77 | | 10 | 22.79 | 17.15 | 13.62 | 23.85 | --- ## a. Comparison and Comments ### Mean Response Times Let's calculate the mean response times for each model: ```markdown Mean(Model 1) = (24.82 + 44.28 + 24.23 + 19.42 + 34.94 + 46.71 + 45.89 + 31.3 + 26.77 + 22.79) / 10 ≈ 32.215 min Mean(Model 2 IRN) = (17.13 + 15.67 + 27.3 + 11.36 + 11.71 + 11.66 + 17.42 + 12.58 + 9.8 + 17.15) / 10 ≈ 15.978 min Mean(Model 2 CRN w/Sync) = (18.63 + 9.86 + 16.41 + 8.90 + 19.63 + 17.57 + 18.32 + 13.13 + 10.46 + 13.62) / 10 ≈ 14.753 min Mean(Model 2 CRN w/o Sync) = (22.79 + 14.08 + 21.24 + 15.72 + 22.54 + 17.78 + 20.69 + 13.77 + 23.77 + 23.85) / 10 ≈ 19.923 min ``` ### Observations - **Parallel (Model 2) is consistently faster** than Serial (Model 1) under all random number schemes. - Using **CRN with synchronization** slightly reduces variability and mean response time, compared to independent random numbers. - **CRN without synchronization** results in higher mean response time and more variability compared to CRN with sync, but still outperforms serial processing. - **Serial processing (Model 1)** has a much higher mean response time and greater variability, indicating a bottleneck effect typical of serial queues. ### Statistical Efficiency - Using **common random numbers (CRN)** allows for paired comparisons and reduces variance of the difference between models, making statistical tests more powerful. - **Synchronization** of random numbers further enhances this effect. --- ## b. Required Number of Replications Suppose the goal is to **detect a difference of 4 minutes in mean response time per customer**. ### Steps 1. **Estimate the standard deviation (s) of the paired differences** between serial and parallel models (using CRN with sync for highest efficiency). 2. Use the formula for the required number of replications for a two-sided confidence interval: \[ n = \left( \frac{z_{\alpha/2} \cdot s}{\delta} \right)^2 \] where: - \(z_{\alpha/2}\) = 1.96 for 95% confidence - \(s\) = sample standard deviation of the paired differences - \(\delta\) = minimum detectable difference (4 minutes) #### Calculate Paired Differences (Serial - Parallel, CRN with Sync) | Replication | Serial (M1) | Parallel (M2 CRN w/Sync) | Difference | |-------------|-------------|--------------------------|------------| | 1 | 24.82 | 18.63 | 6.19 | | 2 | 44.28 | 9.86 | 34.42 | | 3 | 24.23 | 16.41 | 7.82 | | 4 | 19.42 | 8.90 | 10.52 | | 5 | 34.94 | 19.63 | 15.31 | | 6 | 46.71 | 17.57 | 29.14 | | 7 | 45.89 | 18.32 | 27.57 | | 8 | 31.3 | 13.13 | 18.17 | | 9 | 26.77 | 10.46 | 16.31 | | 10 | 22.79 | 13.62 | 9.17 | Calculate the sample mean and standard deviation \(s\): \[ \text{Mean} = \frac{6.19 + 34.42 + 7.82 + 10.52 + 15.31 + 29.14 + 27.57 + 18.17 + 16.31 + 9.17}{10} = 17.962 \] \[ s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (d_i - \bar{d})^2 } \] Calculating each squared difference and the sum: - (6.19 - 17.962)^2 = 139.07 - (34.42 - 17.962)^2 = 269.17 - (7.82 - 17.962)^2 = 103.23 - (10.52 - 17.962)^2 = 55.50 - (15.31 - 17.962)^2 = 7.02 - (29.14 - 17.962)^2 = 124.79 - (27.57 - 17.962)^2 = 92.47 - (18.17 - 17.962)^2 = .04 - (16.31 - 17.962)^2 = 2.73 - (9.17 - 17.962)^2 = 77.25 Sum = 871.27 \[ s = \sqrt{ \frac{871.27}{9} } = \sqrt{96.81} \approx 9.84 \] #### Plug into the formula \[ n = \left( \frac{1.96 \cdot 9.84}{4} \right)^2 = \left( \frac{19.27}{4} \right)^2 = (4.8175)^2 \approx 23.23 \] **Thus, about 24 replications** are needed to detect a 4-minute difference at 95% confidence. --- # Final Summary - **Parallel (generalized) processing significantly outperforms serial (specialized) processing** in terms of mean response time for the loan application system, regardless of the random number scheme used. - **Using common random numbers (CRN), especially with synchronization, reduces variance and allows for more precise comparison** between models. - To reliably detect a 4-minute difference in mean response time per customer at the 95% confidence level, **approximately 24 replications** are required under the current observed variability. - The data strongly supports using a parallel model with CRN synchronization for both operational efficiency and experimental analysis. --- **Note:** - For real-world implementation, the bank manager should consider not just mean performance but also variability, process robustness, and other operational constraints. - The analysis assumes the validity of the simulation model and independence between customers. - Further analysis could examine resource utilization and cost implications.

# A. Comparison and Comments The analysis reveals that **parallel processing (Model 2)** consistently yields lower mean response times than **serial processing (Model 1)** across all sampling methods. This indicates that a parallel approach is more efficient, as it minimizes bottlenecks and enhances operational throughput. ### Summary for A - Parallel processing outperforms serial processing in average response times, emphasizing the efficiency of simultaneous task execution. - The use of common random numbers, especially with synchronization, enhances the reliability of the results and reduces variability in response times. --- # B. Required Number of Replications To detect a difference of 4 minutes in mean response time with 95% confidence, approximately **24 replications** are required based on the observed data variability. This ensures a statistically valid comparison, allowing for an effective assessment of the differences between the processing models. ### Summary for B - The calculated number of replications needed to identify a significant difference provides a quantitative basis for experimental design. - Ensuring this number of replications will enhance the robustness of the conclusions drawn from the loan processing performance analysis.