Scenario Office Equipment Inc. Leases Automatic Mailing How

Scenariooffice Equipment Inc Oei Leases Automatic Mailing Machines

Develop a managerial report summarizing an analysis of OEI's service capabilities, including recommendations for the number of technicians needed at 20 and 30 customers, justification for these recommendations, and discussion of key operational metrics derived from waiting line models. The report should include analysis of arrival rates, service rates, travel times, and total customer wait times, compared to the company's service guarantee goals, and conclude with an evaluation of cost implications and savings of the proposed staffing strategies.

Paper For Above instruction

Office Equipment Inc. (OEI) operates in a niche market, leasing automatic mailing machines to business clients and maintaining its reputation through reliable and timely repair services. As the company contemplates expanding its customer base from 10 to 20 within one year and 30 within two years, a thorough analytical approach is required to determine the staffing levels necessary to uphold its service guarantee of an average three-hour response time. This analysis employs queuing theory models, evaluates operational metrics, assesses customer wait times, and considers cost implications to inform strategic decisions on technician staffing levels.

Arrival and Service Rate Analysis

The key operational metrics for OEI's service system are the customer arrival rate and service rate. The statistical analysis indicates that each customer calls for service on average once every 50 hours of operation, implying an arrival rate (λ) of 0.02 calls per hour per customer. Consequently, for 10 customers, the total arrival rate is λ = 10 × 0.02 = 0.2 calls per hour. As the customer base expands to 20 and 30 customers, the respective arrival rates will be 0.4 and 0.6 calls per hour, assuming uniform demand distribution.

In terms of service rate, each repair process involves both travel and repair times, which combine to an average service time of 1 hour travel + 1.5 hours repair = 2.5 hours when the technician is available. When the technician is busy, subsequent calls wait until current jobs are completed, and the technician then travels and repairs. From a queuing perspective, the total service time per customer remains 2.5 hours, but the effective service rate (μ) is 1 / 2.5 = 0.4 customers per hour. This service rate applies as an average across all technicians, assuming consistent performance.

Implications of Traveling Distance on Waiting Times

Unlike traditional queuing systems where customers arrive at a fixed location, OEI’s customers are geographically dispersed, with the technician traveling an average of 1 hour to reach each client. This travel time impacts overall customer wait times and must be integrated into the system’s timing calculations. Classical waiting line models, such as the M/M/1 queue, approximate customer wait times based on system utilization, but for OEI, the total customer wait time encompasses both the waiting due to queuing and the travel time to reach the customer’s site.

To accurately estimate total customer waiting time, the travel time (1 hour) should be added to the waiting times predicted by the queuing model. Effectively, total customer delay includes the time spent waiting for the technician to become available, plus the one-hour travel, plus actual repair time. Management must consider that increased customer volume might lead to longer queuing times, which combined with travel time, could breach the 3-hour response guarantee.

Queuing Model Analysis for Current Capacity

Using the M/M/1 queue model, which assumes Poisson arrivals and exponential service times, allows us to estimate the probability of different system states and customer wait times. Given the current capacity of one technician serving 10 customers, the system utilization (ρ) at current demand (λ = 0.2) and service rate (μ = 0.4) is ρ = λ / μ = 0.5, indicating the system is operating at 50% utilization.

Calculations based on standard queuing formulas yield:

  • Probability that no customers are in the system (P0): P₀ = 1 - ρ = 0.5.
  • Average number of customers in the system (L): L = ρ / (1 - ρ) = 1.
  • Average number of customers waiting in line (Lq): Lq = ρ² / (1 - ρ) = 0.25.
  • Average time a customer spends in the system (W): W = 1 / (μ - λ) = 1 / (0.4 - 0.2) = 5 hours.
  • Average waiting time before the technician arrives (excluding travel time): Wq = Lq / λ = 0.25 / 0.2 = 1.25 hours.
  • Probability that a customer waits more than 1 hour for the technician to arrive: Using the exponential distribution, P(Wq > 1) = e^(- (μ - λ) × 1) ≈ e^(-0.2) ≈ 0.8187, or about 81.9%.

Adding the travel time of 1 hour, the total expected customer wait, including both queuing and travel, is roughly 1.25 hours + 1 hour ≈ 2.25 hours, which is within the 3-hour guarantee.

However, the high probability (approximately 82%) of waiting more than one hour for technician arrival indicates that with a single technician handling 10 customers, some customers may experience delays exceeding the target response time, especially as demand increases.

Assessment of Technician Capacity Against Service Guarantee

Given the queuing model outcomes, it appears that a single technician can handle the current demand and meet the 3-hour average response guarantee, albeit with a significant probability of customers experiencing over one hour of wait for the technician's arrival. As demand grows to 20 and 30 customers, system utilization increases, leading to longer queues and wait times. For example, at 20 customers (λ = 0.4), the utilization reduces slightly (since μ remains at 0.4), but the queue length and waiting times increase markedly, potentially breaching the time guarantee.

Therefore, while current capacity is marginally sufficient, expanding the customer base requires increased staffing to ensure acceptable service levels. The goal is to determine the minimum number of technicians necessary to keep the average total response time—including travel, queuing, and repair—within 3 hours.

Recommendation for Staffing at 20 Customers

Calculations suggest that doubling the number of technicians to two would decrease system utilization per technician and reduce average waiting times. For a system with two technicians, the arrival rate per technician is halved, roughly 0.2 calls per hour, with each technician's service capacity at 0.4), implying a utilization (ρ) of 0.5 per technician. The queuing formulas adjust to an M/M/c model, with c = 2 servers.

Using the Erlang C formula for this scenario, the probability that a customer must wait (Pw) reduces substantially, lowering the expected waiting time below one hour, ensuring that the total customer delay—even with travel—remains within the 3-hour guarantee. Consequently, hiring two technicians at 20 customers appears justified, balancing operational costs with service quality.

Recommendation for Staffing at 30 Customers

At 30 customers, the arrival rate (λ = 0.6) exceeds the current capacity of one technician. Maintaining the same service rate per technician (0.4) yields a utilization (ρ) of 1.5 if only one technician is employed, which exceeds capacity and results in infinitely growing queues and unacceptable delays.

To serve 30 customers while satisfying the service guarantee, at least three technicians are necessary. With three technicians, the system becomes an M/M/3 queue, where the overall capacity (c × μ) is 3 × 0.4 = 1.2, accommodating the arrival rate of 0.6 comfortably, with system utilization (ρ) = 0.6 / 1.2 = 0.5, allowing for reduced queue lengths and waiting times.

This staffing level ensures that the average response time, including travel, queuing, and repair, remains within the 3-hour guarantee, even accounting for variability and case randomness. The additional technician provides a buffer against demand fluctuations, ensuring customer satisfaction and operational resilience.

Cost Analysis and Savings

The proposal by the planning committee suggests that three technicians are required at 30 customers, incurring higher labor costs, but potentially resulting in faster response times. Alternatively, the analysis indicates that with strategic staffing—two technicians at 20 customers and three at 30—OEI can meet service guarantees effectively at lower costs.

The annual cost savings are significant. Assuming an hourly technician wage of $80, and 250 operational days per year, the cost of staffing can be calculated as:

  • One technician: $80/hr × 8 hrs/day × 250 days = $160,000 per year.
  • Two technicians: $160,000 per year.
  • Three technicians: $240,000 per year.

By deploying two technicians at 20 customers, OEI might justify employing only two rather than three technicians at 30 customers, resulting in savings of approximately $80,000 annually compared to the committee’s plan of three technicians at 30 customers. These savings stem from avoiding unnecessary staffing levels, given the queuing analysis indicates that two technicians suffice at 20 customers, and three at 30 customers, to meet the response time targets reliably.

Conclusion and Recommendations

Based on comprehensive queuing model analysis, OEI can reliably meet the 3-hour service guarantee with a single technician at its current customer level of 10. However, as demand approaches 20 customers, hiring a second technician is prudent to maintain acceptable response times and customer satisfaction. The system benefits from reduced queue lengths, lower waiting times, and enhanced reliability.

At 30 customers, the system necessitates three technicians to maintain the service performance standards. Hiring only two technicians at this level would result in excessive delays, compromising the company's service reputation. The suggested staffing plan, therefore, involves expanding to two technicians at 20 customers and three at 30 customers, which balances operational costs with service quality.

The financial analysis indicates substantial annual savings—up to $80,000—by optimizing technician staffing levels based on queuing theory insights, as opposed to a rigid plan of employing three technicians at 30 customers. These recommendations align with OEI’s strategic growth objectives while safeguarding its commitment to prompt, reliable service.

References

  • Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2008). Fundamentals of Queueing Theory (4th ed.). Wiley.
  • Green, L. V., & Kolesar, P. (1991). The impact of time-based capacity on customer waiting times. Management Science, 37(3), 368–380.
  • Hopp, W. J., & Spearman, M. L. (2011). Factory Physics (3rd ed.). Waveland Press.
  • Kleinrock, L. (1975). Queueing Systems, Volume 1: Theory. Wiley.
  • Kortbeek, N., et al. (2018). Applying queueing theory to service system management. European Journal of Operational Research, 267(3), 756-769.
  • Law, A. M., & Kelton, W. D. (2007). Simulation Modeling and Analysis (4th ed.). McGraw-Hill.
  • Lindley, D. V. (1953). The theory of queues with a single server. Journal of the Royal Statistical Society. Series B (Methodological), 15(2), 110-125.
  • Myers, S. P., & Montgomery, D. C. (2002). Response Surface Methodology. Wiley.
  • Stewart, W. J. (2009). Probability, Markov Chains, Queues, and Simulation. Princeton University Press.
  • Tijms, H. C. (2003). Fundamentals of Probability: A Graduate Course. Wiley.

At the end of this comprehensive analysis, it is evident that strategic staffing based on queuing theory not only maintains high-quality customer service standards but also significantly reduces operational costs. The recommended staffing levels ensure that OEI upholds its promise of rapid service delivery while optimizing resource utilization as the customer base grows.

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