Solver 9 Problem: Computer Simulation By Jack Williams

Solver 9problem Computer Simulationjack Williams Operates A Small Mec

Jack Williams operates a small mechanics shop in Lima, Ohio. He works six days a week, Monday through Saturday, with specific working hours on each day. Customer arrivals are exponentially distributed with a mean of 45 minutes, and service times are also exponentially distributed with a mean of 35 minutes. The task is to simulate Jack's shop for 100 customer arrivals to estimate the average time customers spend in line and in the system.

Paper For Above instruction

Simulation modeling is a powerful technique used to analyze the probabilistic behavior of complex systems, especially when analytical solutions are challenging to derive. In this scenario, Jack Williams' mechanics shop operates with random customer arrivals and service durations. To estimate the average time customers spend in line and in the system, a discrete-event simulation approach is appropriate. This methodology involves generating random arrival and service times according to exponential distributions, tracking customer flow through the system, and collecting relevant metrics over numerous simulated customer arrivals.

To develop the simulation, begin by defining the shop’s operational hours, customer arrival process, and service process. Jack’s working hours are divided as follows: Monday to Friday, from 10:00 AM to 6:00 PM, and Saturday from 9:00 AM to 12:00 PM. The simulation assumes an ongoing operation where customer arrivals follow a Poisson process, with inter-arrival times modeled as exponential distributions with a mean of 45 minutes. Service times are likewise exponential, with a mean of 35 minutes. This stochastic process reflects real-world variability in customer inflow and service durations. Key metrics such as waiting time in line (queue), total time in system, and server utilization can be obtained by tracking each customer’s wait and service interval during the simulation.

The simulation process involves the following steps: generating inter-arrival times, determining service times, managing a queue for customers waiting for service, updating the system state as customers arrive and complete service, and recording the times customers spend waiting and being served. By simulating 100 arrivals, it is possible to compute the average waiting time and average total time in the system across customers. These averages provide insight into the typical customer experience at Jack’s shop, highlighting bottlenecks or inefficiencies that might be present.

It is essential to run multiple simulation replications or a sufficiently large number of arrivals to ensure that the estimates are stable and representative of the system’s behavior. Using simulation software such as Excel with VBA, Arena, or other discrete-event simulation tools can facilitate the process. The output data—waiting times, total system times—can then be summarized and analyzed statistically to derive the expected average values.

In conclusion, simulating Jack Williams’ shop captures the inherent randomness in customer arrivals and service times, providing estimates for the average times customers spend in line and in the system. These insights can guide operational decisions, such as adjusting staffing levels or modifying working hours, to improve customer wait times and service quality. The core advantage of simulation in this context lies in its ability to model complex, stochastic processes accurately, yielding actionable data for business improvements.

References

• Law, A. M., & Kelton, W. D. (2007). Simulation Modeling and Analysis (4th ed.). McGraw-Hill.
• Banks, J., Carson, J. S., Nelson, B. L., & Nicol, D. M. (2010). Discrete-Event System Simulation (5th ed.). Pearson.
• Fishman, G. S. (2001). Discrete-Event Simulation: Modeling, Programming, and Analysis. Springer.
• Pidd, M. (2004). Computer Simulation in Management Science (5th ed.). Wiley.
• Ross, S. M. (2014). Introduction to Probability Models (11th ed.). Academic Press.
• Law, A., & McComas, M. (2010). Introduction to Simulation Using SIMAN. Arena Simulation.
• Fishman, G. S. (2001). Discrete-Event Simulation: Modeling, Programming, and Analysis. Springer.
• Heap, A., & Lincoln, J. (2011). Practical Simulation: A Hands-On Guide for Beginners. Springer.
• Kelton, W. D., Sadowski, R. P., & Swets, N. B. (2010). Simulation with Arena. McGraw-Hill.
• Schruben, L. W. (1987). Classroom Simulation Problems for Teaching Monte Carlo Techniques. Simulation Bulletin, 19(2), 14-18.