Case Analysis 31: Eugene Is A Business Analyst At Burger Kin ✓ Solved

Case Analysis 31 Eugene Is A Business Analyst At Burger King He Was

Analyze data on consumer arrivals and service rates to assess queueing system performance at Burger King. Calculate relevant parameters for different scenarios, interpret results, and explore the impact of changing arrival and service rates. Additionally, discuss the biblical perspectives on waiting, linking the concept to spiritual insights.

Sample Paper For Above instruction

Introduction

Effective management of customer flow in fast-food restaurants like Burger King relies heavily on understanding queueing dynamics. In this analysis, we examine a single-server queueing model using arrival and service rates to compute key performance metrics. The goal is to assess how changes in these rates influence system performance and to explore the philosophical implications of waiting from a biblical perspective.

Part A: Calculating Arrival and Service Rates per 10 Minutes

The initial data indicates that the average customer arrivals are 15 per hour, and the cashier can process 18 per hour. To convert these to a 10-minute period, we apply the following calculations:

• Arrival rate (λ): (15 customers/hour) / 6 = 2.5 customers per 10 minutes
• Service rate (μ): (18 customers/hour) / 6 = 3 customers per 10 minutes

Therefore, the arrival rate per 10-minute interval is 2.5, and the service rate per 10-minute interval is 3. These foundational parameters set the stage for subsequent queue performance analysis.

Part B: Computing Queue Metrics Using Excel Model (Baseline Scenario)

Using the calculated λ = 2.5 and μ = 3, we input these values into the Cell cells corresponding to arrival (λ) and service (μ) rates in the Excel model. From queue theory formulas for an M/M/1 model, we derive:

• System utilization (ρ): ρ = λ / μ = 2.5 / 3 ≈ 0.8333
• Probability of zero customers in the system (P₀): P₀ = 1 - ρ ≈ 0.1667
• Average number of customers in the queue (Lq): Lq = ρ² / (1 - ρ) ≈ 4.938
• Average number of customers in the system (L): L = ρ / (1 - ρ) ≈ 5.0
• Average waiting time in queue (Wq): Wq = Lq / λ ≈ 1.975 minutes
• Average time in system (W): W = L / λ ≈ 2.0 minutes
• Probability that an arriving customer has to wait (Pw): Pw = ρ ≈ 83.33%

Each value illustrates the system's performance, indicating a high likelihood of customers waiting and substantial congestion during peak times.

Part C: Impact of Increased Service Rate (Scenario 1)

Suppose the service rate increases from 3 to 4 customers per 10 minutes, with the arrival rate remaining at 2.5. Recomputing the metrics:

• Utilization (ρ): 2.5 / 4 = 0.625
• P₀: 1 - 0.625 = 0.375
• Lq: (0.625)² / (1 - 0.625) ≈ 0.833
• L: 0.625 / (1 - 0.625) ≈ 1.666
• Wq: 0.833 / 2.5 ≈ 0.333 minutes
• W: 1.666 / 2.5 ≈ 0.666 minutes
• Pw: 62.5%

The increased service capacity significantly reduces queue length, waiting times, and the probability of having to wait, thus improving overall efficiency.

Part D: Effects of Increased Arrival Rate (Scenario 2)

Now, if the arrival rate increases to 2.8 per 10 minutes while the service rate remains at 3, the recalculated metrics are:

• Utilization (ρ): 2.8 / 3 ≈ 0.9333
• P₀: 1 - 0.9333 ≈ 0.0667
• Lq: (0.9333)² / (1 - 0.9333) ≈ 13.07
• L: 0.9333 / (1 - 0.9333) ≈ 14
• Wq: 13.07 / 2.8 ≈ 4.67 minutes
• W: 14 / 2.8 ≈ 5 minutes
• Pw: 93.33%

The increased arrival rate causes a dramatic rise in queue length and waiting times, indicating congestion and potential service delays.

Part E: Comparative Analysis and Theoretical Implications

Comparing the three scenarios reveals that manipulating the service or arrival rate significantly affects queue performance metrics. Under baseline conditions, the system operates with a high probability of waiting, with about 83% of customers experiencing delays. Improving the service rate (Scenario 1) leads to a marked decrease in queue length and waiting times, enhancing customer satisfaction and operational efficiency. Conversely, increasing arrival rates (Scenario 2) results in congestion, longer waits, and decreased service quality.

Theoretically, this underscores queueing sensitivity to system utilization. When utilization exceeds 0.8 (80%), queues tend to grow rapidly. Managing arrival and service rates is crucial for maintaining equilibrium and ensuring customer convenience. These results align with operational strategies that emphasize balancing capacity with demand.

From a biblical perspective, the concept of waiting can be viewed as a spiritual and moral discipline. Scriptures like Isaiah 40:31 highlight the importance of patience and faith during periods of waiting—"But those who wait on the Lord shall renew their strength." Similarly, James 5:7 encourages believers to be patient in suffering, trusting God's timing. Waiting teaches humility, trust, and resilience—qualities vital both in queue management and spiritual growth. It reflects the understanding that good things often require patience and perseverance, reminding us of divine providence and the importance of faith in times of uncertainty.

Conclusion

This analysis demonstrates how queueing theory provides valuable insights into operational performance, highlighting the delicate balance between arrival and service rates. Adjustments in these parameters significantly influence waiting times and system efficiency. Additionally, integrating biblical principles enriches our understanding of waiting beyond logistics, emphasizing patience, faith, and resilience as virtues applicable in both everyday life and spiritual journeys.

References

• Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2008). Fundamentals of Queueing Theory (4th ed.). Wiley.
• Wolff, R. P., & Wicks, E. M. (2014). Queueing analysis for service systems. Operations Research Perspectives, 1, 73-78.
• Harris, C. M. (2002). The Theory of Queueing Systems. Mathematical Modeling, 5(2), 113-130.
• Allen, A. O. (1990). Probability, Statistics, and Queueing Theory. Academic Press.
• Isaiah 40:31 (NIV). Bible Gateway
• James 5:7 (NIV). Bible Gateway
• West, J. P. (2019). Waiting on the Lord: Biblical perspectives on patience. Journal of Theology and Ministry, 36(4), 45-57.
• Smith, L. A. (2017). Managing customer flows: Queue theory applications in retail. Operations Management Review, 9(3), 102-115.
• Lee, S., & Kim, H. (2016). Service rate optimization in fast-food restaurant queues. International Journal of Service Operations and Informatics, 11(2), 157-172.
• Phillips, R. T. (2020). Queueing theory and customer satisfaction. Service Science, 12(1), 25-40.