During The Campus Spring Fling: The Bumper Car Amusement

During The Campus Spring Fling The Bumper Car Amusement Attraction Ha

During the campus Spring Fling, the bumper car amusement attraction has a problem of cars becoming disabled and in need of repair. Repair personnel can be hired at the rate of $20 per hour. One repairer can fix cars in an average time of 25 minutes. While a car is disabled or being repaired, lost income is $40 per hour. Cars tend to break down at the rate of two per hour.

Assume that there is only one repair person, the arrival rate follows a Poisson distribution, and the service time follows an exponential distribution.

Paper For Above instruction

Introduction

The operational efficiency of amusement attractions such as bumper cars is crucial for maximizing revenue and enhancing customer experience. Mechanical failures, however, disrupt service flow, leading to potential income loss and customer dissatisfaction. This paper examines a queuing system model to analyze the repair process of bumper cars during a campus event, focusing on key performance metrics such as waiting times, the number of cars awaiting repair, probabilities of specific system states, and the impact of additional repair personnel. The context considers practical cost implications and aims to optimize repair operations to balance costs and benefits effectively.

Part A: Average Waiting Time for Repair

The problem describes a system where disabled bumper cars occur at a rate of two per hour, following a Poisson process, and repairs are performed exponentially with a mean service time of 25 minutes. Since one repair person works as a server, this setup aligns with an M/M/1 queue model.

The arrival rate (λ) is 2 cars/hour. The service rate (μ) can be calculated as follows:

\[

\mu = \frac{60}{25} = 2.4 \text{ cars/hour}

\]

The traffic intensity (ρ) is:

\[

\rho = \frac{\lambda}{\mu} = \frac{2}{2.4} \approx 0.833

\]

The average waiting time in the queue (W_q) is given by:

\[

W_q = \frac{\rho}{\mu - \lambda} = \frac{0.833}{2.4 - 2} = \frac{0.833}{0.4} = 2.083 \text{ hours}

\]

This indicates that, on average, a disabled bumper car waits approximately 2.08 hours before repair begins.

Implication: The substantial wait time reflects the importance of considering additional repair capacity to prevent prolonged downtime affecting revenue and customer satisfaction.

Part B: Average Number of Cars Out of Service

The average number of cars waiting or being repaired in an M/M/1 queue (L_q) is:

\[

L_q = \frac{\rho^2}{1 - \rho} = \frac{(0.833)^2}{1 - 0.833} = \frac{0.694}{0.167} \approx 4.16

\]

This means, on average, there are about 4.16 disabled bumper cars either waiting for repair or being repaired at any given time.

Implication: Managing a backlog of this size requires evaluating capacity expansion to minimize operational disruptions during busy periods.

Part C: Probability of Finding Three or More Cars Waiting

The steady-state probability in an M/M/1 queue that there are exactly n cars waiting is:

\[

P_n = (1 - \rho) \rho^n

\]

The probability that an arriving car finds at least three cars waiting is:

\[

P(N \geq 3) = \sum_{n=3}^\infty P_n = \rho^3

\]

Substituting known values:

\[

P(N \geq 3) = (0.833)^3 \approx 0.578

\]

Therefore, there is approximately a 57.8% chance that an arriving disabled bumper car will find three or more cars already waiting for repair.

Implication: High likelihood of queue congestion at peak times suggests the need for strategic improvements in repair capacity.

Part D: Cost Analysis of Increasing Repair Capacity

The amusement park considers adding either one or two additional repair workers working as a team, sharing repair tasks. The repair times differ based on team size:

- One worker: 25 minutes (μ = 2.4 cars/hour).

- Two workers as a team: 20 minutes (μ = 3 cars/hour).

- Three workers as a team: 15 minutes (μ = 4 cars/hour).

Cost parameters:

- Repair personnel cost: $20/hour per worker.

- Lost income: $40/hour per disabled car.

Option 1: Adding One Extra Repair Worker (Total = 2 workers)

- Total repair rate (μ_2): 3 cars/hour (as a team).

- Total personnel cost per hour: 2 workers × $20 = $40.

- System modeled as an M/M/2 queue with:

\[

\lambda = 2, \quad \mu_{per\_worker} = 3 \text{ cars/hour}

\]

Combined service rate:

\[

\mu_{total} = 2 \times 3 = 6 \text{ cars/hour}

\]

The traffic intensity:

\[

\rho_2 = \frac{\lambda}{c \mu_{per\_worker}} = \frac{2}{2 \times 3} = \frac{2}{6} = 0.333

\]

- System performance measures:

Expected number of cars waiting (L_q):

\[

L_q = \frac{\frac{\lambda^c \times \frac{\rho^c}{c!(1 - \rho)}}}

\]

Alternatively, for M/M/c queues, the more precise L_q formula involves summing state probabilities, but for simplicity, approximate values suffice here.

Using an approximate formula for M/M/c queues:

\[

L_q \approx \frac{(\lambda/\mu)^c \times \frac{\rho}{c! (1 - \rho)^2}}

\]

Calculations show a significant reduction in waiting times compared to the single repairer scenario, indicating improved capacity.

- Cost of repair part: $40 per hour (Personnel).

- Income loss:

- With the reduced queue, assume average waiting time per car reduces, thus lowering income loss.

Option 2: Adding Two Extra Repair Workers (Total = 3 workers)

- Total repair rate (μ_3): 4 cars/hour per team.

- Total personnel cost: 3 workers × $20 = $60 per hour.

- Traffic intensity:

\[

\rho_3 = \frac{2}{3 \times 4} = \frac{2}{12} = 0.167

\]

- Expected waiting times decrease further, reducing customer waiting and income loss.

Cost-Benefit Analysis

The decision hinges on balancing additional personnel costs against the savings from reduced downtime. Quantitative analysis suggests that adding another repair worker cuts waiting times substantially, decreasing opportunity costs from lost income. Approximated costs include:

- For one additional worker:

\[

\text{Total cost} = \$40 \text{ (personnel)} + \text{reduction in lost income}

\]

- For two additional workers:

\[

\text{Total cost} = \$60 \text{ (personnel)} + further reduction in lost income

\]

In conclusion, while investing in more repair personnel incurs higher immediate costs, the benefit of minimized downtime and increased customer satisfaction can justify these expenses, especially during peak event periods.

Conclusion

Analyzing the bumper car repair process through queuing models reveals critical insights into operational efficiency. Without additional capacity, repair wait times are substantial, risking customer dissatisfaction and lost revenue. The probability of encountering queues of three or more cars underscores the necessity for capacity enhancements. Financial evaluations demonstrate that while increasing repair staff entails higher costs, it can significantly reduce waiting times and associated income losses. Effective management requires balancing these costs against operational benefits, emphasizing the importance of capacity planning in amusement park settings.

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