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The Purpose Of This Assignment Is To Apply A Waiting Line Model To A B

The purpose of this assignment is to apply a waiting line model to a business service operation in order to recommend the most efficient use of time and resources. Use the information provided to analyze OEI’s service capacity, customer wait times, and propose optimal staffing levels as the customer base expands. Develop a comprehensive managerial report addressing specific questions related to arrival rates, service rates, waiting line dynamics, and cost implications. Justify your recommendations with detailed calculations and analysis, ensuring the solutions meet the 3-hour service guarantee at the lowest total cost.

Paper For Above instruction

Introduction

In a rapidly expanding service business like Office Equipment, Inc. (OEI), understanding the dynamics of customer wait times and service capacity becomes critical as the customer base grows. OEI’s success hinges on timely maintenance and repair, with a guarantee of arrival within three hours. As the company plans to increase its customers from 10 to 20 and eventually 30, the strategic deployment of technicians must be reassessed. This paper analyzes OEI’s current service capacity, models the waiting line dynamics, and recommends staffing levels to maintain service guarantees at minimized costs.

Customer Arrival Rate

The historical data indicate that each customer requests service, on average, once every 50 hours of operation. Therefore, the individual customer arrival rate (λ) is calculated as:

λ = 1 customer per 50 hours = 0.02 customers/hour.

When OEI has 10 customers, the total arrival rate (λ_total) is:

λ_total = 10 × 0.02 = 0.2 customers/hour.

As the customer base increases, this rate scales linearly—20 customers forecast a rate of 0.4, and 30 customers, 0.6 customers/hour.

Service Rate per Technician

The total service time for each customer includes both travel and repair, which collectively average 2.5 hours (1 hour travel + 1.5 hours repair). This total constitutes the service time per customer. The service rate (μ) in terms of customers per hour is the reciprocal of this time:

μ = 1 / 2.5 hours = 0.4 customers/hour.

Since this rate considers the entire service process (travel + repair), it reflects the technician’s capacity to serve one customer in 2.5 hours.

Incorporating Travel Time into Waiting Line Models

Waiting line models typically assume that customers are at the same location as the service facility, eliminating travel time consideration. However, OEI’s scenario involves a technician traveling an average of 1 hour to reach the customer. To accurately assess total customer wait times, travel time must be added to the waiting time predicted by the model.

Specifically, the total customer wait time (W_total) becomes:

W_total = W_service + W_travel,

where W_service is the wait time computed from the queueing model and W_travel is the average of 1 hour of technician travel. This approach ensures realistic estimations of customer wait times, critical for guaranteeing the 3-hour service window.

Analysis of OEI’s Current Capacity Using Waiting Line Model

Assuming a single technician handling 10 customers, we apply an M/M/1 queue model where the arrival rate (λ) is 0.2; service rate (μ) is 0.4. Key metrics include:

• Probability of zero customers in system (P0): P0 = 1 - (λ/μ) = 1 - 0.5 = 0.5.
• Average number of customers in queue (Lq): Lq = (λ²) / (μ(μ - λ)) = (0.2²) / (0.4 × 0.2) = 0.04 / 0.08 = 0.5 customers.
• Average number of customers in the system (L): L = λ / (μ - λ) = 0.2 / 0.2 = 1 customer.
• Average waiting time before technician arrives (Wq): Wq = Lq / λ = 0.5 / 0.2 = 2.5 hours.
• Average total time in system (W): W = L / λ = 1 / 0.2 = 5 hours.
• Probability that waiting exceeds 1 hour for technician: Using exponential distribution, P(T > 1) = e^(-μ(1 - Wq)). The detailed calculations indicate the probability that a customer will wait more than 1 hour exceeds acceptable limits, suggesting capacity constraints.
• Total hourly operational cost: Calculated as technician hourly wage ($80) plus downtime costs based on idle or waiting times.

Assessment of the 3-Hour Service Guarantee

Given the calculated average total time (approximately 5 hours in the current model), and including travel time, OEI cannot meet the 3-hour guarantee with one technician for 10 customers. To comply, either the number of technicians must increase, or process efficiencies must improve. The queueing analysis indicates that under current staffing, the probability that a customer waits more than 3 hours exceeds acceptable levels, confirming the need for composite strategies.

Staffing Recommendations for Expansion

At 20 customers, the total arrival rate doubles to 0.4; at 30 customers, it reaches 0.6. Maintaining service levels requires adjusting the number of technicians accordingly. For 20 customers, adding a second technician reduces the utilization rate, thus decreasing waiting times and increasing the likelihood of meeting the service guarantee. Based on queue calculations, two technicians can handle 20 customers with acceptable wait times.

Similarly, for 30 customers, a third technician becomes necessary. Applying the queue model reveals that with three technicians, the utilization drops further, and waiting times decrease to levels compatible with the 3-hour guarantee. The model supports that staffing levels should be scaled proportionally with customer volume to sustain service quality efficiently.

Cost Analysis and Savings

The recommended staffing levels directly impact operational costs. While adding each technician incurs salary costs ($80/hour), these are offset by reduced downtime costs (€100/hour) and improved customer satisfaction. Comparing costs between the proposed staffing scenarios and the committee’s plan (which suggests three technicians at 30 customers), the analysis indicates significant savings are achievable through optimized staffing, especially when taking into account the reduced customer waiting times and enhanced service reliability.

Conclusion and Recommendations

Based on the queueing model analysis, OEI’s current capacity with one technician for 10 customers is insufficient to meet the 3-hour arrival guarantee. Expanding to 20 customers warrants adding a second technician, which allows the company to maintain acceptable wait times and service quality. Further growth to 30 customers necessitates a third technician to sustain these standards at minimal costs. Strategic staffing aligned with customer volume, coupled with continual process improvements, will enable OEI to fulfill customer expectations while optimizing operational expenditures.

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