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Suppose that a car rental agency offers insurance for a week that will cost $10 per day. A minor fender bender will cost $1,500, while a major accident might cost $15,000 in repairs. Without the insurance, you would be personally liable for any damages. What should you do? Clearly, there are two decision alternatives: take the insurance or do not take the insurance.

The uncertain consequences, or events that might occur, are that you would not be involved in an accident, that you will be involved in a fender bender, or that you would be involved in a major accident. Assume that researched insurance industry statistics and found out that the probability of a major accident is 0.05% and the probability of a fender bender is 0.16%. What is the expected value decision? Would you choose this? Why or why not?

What would be some alternate ways to evaluate risk? (2) Suppose that the service rate to a waiting line system is 10 customers per hour (exponentially distributed). Analyze how the average waiting time is expected to change as the arrival rate varies from two to ten customers per hour (exponentially distributed). Please be sure your work is organized, legible, and your responses are substantive. You need to submit all details of your work including Excel sheets used to arrive at the solution. It is not enough to attach your Excel sheet. You MUST provide interpretation of results and describe conclusions.

Paper For Above instruction

Introduction

The decision to purchase insurance when renting a vehicle involves evaluating the potential costs associated with possible accidents against the cost of the insurance premium. This analysis hinges on understanding probabilities of different accident types, associated costs, and calculating the expected value of each option—insuring or not insuring. Furthermore, assessing queue performance in a service system with varying customer arrival rates provides insights into how delay times change with system load, essential for efficient resource allocation. This paper explores these two scenarios comprehensively.

Part 1: Expected Value Analysis for Car Rental Insurance

In assessing insurance choices, expected value (EV) serves as a quantitative tool combining probabilities and costs to inform decision-making. The fundamental question is whether the EV of purchasing insurance outweighs the potential costs of accidents not covered without insurance.

The insurance fee involves a daily charge of $10, totaling $70 for a week. The related accident costs are $1,500 for a minor fender bender and $15,000 for a major accident. Using industry statistics, the probabilities are 0.16% (or 0.0016) for a fender bender and 0.05% (or 0.0005) for a major accident, with the remaining probability of no accident as 1 minus the sum of these probabilities.

Calculating the expected costs without insurance involves multiplying the probability of each event by its cost:

• No accident: Probability = 1 - 0.0016 - 0.0005 = 0.9979
• Fender bender: Probability = 0.0016; cost = $1,500
• Major accident: Probability = 0.0005; cost = $15,000

Expected cost without insurance:

EV = (0.9979 × $0) + (0.0016 × $1,500) + (0.0005 × $15,000) = $0 + $2.4 + $7.5 = $9.9

Since the insurance costs $70, the overall expected cost if insuring is $70 plus the expected accident costs, which are avoided if incidents occur but paid out if accidents happen. Therefore, the EV of purchasing insurance, considering the premium and the avoided costs, is:

EV with insurance = $70 (premium) + probability of accident × cost if uninsured (since the insured's liability is covered), but since insurance transfers the losses, the EV simplifies to the premium, assuming the insurance covers all damages.

Comparing the two options, the expected cost without insurance ($9.90) is significantly lower than the insurance premium ($70). Given this, the rational economic decision is to not purchase the insurance based solely on expected costs. However, this ignores risk aversion; some renters might prefer peace of mind despite the higher expected costs.

Alternative risk evaluation methods

Besides expected value calculations, other approaches include:

• Risk Acceptability: Assessing whether the potential financial loss, especially in the case of a major accident, is tolerable.
• Variance and Standard Deviation: Measuring the variability of potential costs to account for risk heterogeneity.
• Decision Trees: Visualizing different outcomes and their probabilities to facilitate complex decision-making.
• Utility Theory: Incorporating individual risk preferences by assigning utility values to outcomes rather than monetary value alone.

Part 2: Queue Analysis - Impact of Arrival Rate on Waiting Time

The queue system under consideration operates with a service rate (µ) of 10 customers per hour, with arrivals following a Poisson process (exponentially distributed inter-arrival times). The relationship between the arrival rate (λ) and waiting time is fundamental in operations management, often analyzed via the M/M/1 queue model.

The average waiting time in the queue (Wq) for an M/M/1 system is given by:

Wq = λ / [μ(μ - λ)]

As λ varies from 2 to 10 customers per hour, we can analyze how Wq changes. When λ approaches μ (10), the waiting time increases dramatically due to system congestion, stalling the queue. When λ is low, waiting times are minimal.

Calculations for selected λ values:

This analysis demonstrates that as arrival rate approaches service capacity, waiting time increases exponentially, indicating system instability at λ = 10. Operationally, to prevent excessive delays, the agency should maintain arrival rates well below the service rate.

Conclusion

The first scenario revealed that, from an expected-value perspective, purchasing insurance for the rental car is not cost-effective, considering the low probability of severe accidents combined with the relatively low expected losses. However, individual risk preferences could influence decision-making beyond pure economic rationale.

In the queue analysis, the results clearly indicate that as customer arrivals increase, the average waiting time escalates dramatically near system capacity, underscoring the importance of capacity planning and load management in service systems. Maintaining arrival rates below the service capacity ensures minimal delays and efficient operations.

Overall, systematic analysis using probabilistic and queuing models provides valuable insights into risk management and operational efficiency, essential for informed decision-making in both insurance decisions and service system design.

References

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