An Engineering Room Retains A Technical Specialist To Assist

An Engineering I Rm Retains A Technical Specialist To Assist Four Desi

An engineering firm has retained a technical specialist to assist four design engineers working on a project. The help provided by the specialist varies in time consumption, with some requests requiring memory recall, others necessitating computation, and some demanding significant search time. On average, each request for assistance takes approximately one hour. The engineers make help requests at an average rate of once per day. Since each assistance takes about one hour, each engineer can work independently for roughly seven hours before needing help. Also, engineers do not interrupt the specialist if he is already engaged with another problem, indicating a queueing situation with potentially infinite capacity. This scenario models a typical M/M/1 queue where requests arrive randomly, and service times follow an exponential distribution. The following analysis addresses the key queueing metrics: the average number of engineers waiting for assistance, the average waiting time for an engineer, and the probability that an engineer will need to wait in line for the specialist.

Paper For Above instruction

In this paper, we analyze the operational dynamics of a technical specialist assisting four engineers in a project environment, employing queueing theory principles to evaluate system performance. The problem at hand models a classical M/M/1 queue, in which the arrival process of requests follows a Poisson distribution, and service times are exponentially distributed. This model is appropriate given the random, independent nature of requests and the variability in assistance times, averaging one hour per request.

Introduction

Engineering projects often rely heavily on the support services of specialists, whose availability and workload significantly impact project timelines and productivity. The present scenario involves a single specialist assisting four engineers, with requests arriving at a steady rate, and each assistance session consuming approximately an hour. In queueing theory, such a system can be analyzed to determine key performance metrics like the average queue length, waiting times, and the likelihood of delays. Understanding these metrics can guide resource allocation and improve operational efficiency.

Model Assumptions and Parameters

The analysis assumes the following parameters:

• Arrival rate (λ): Requests per day = 1 per engineer per day × 4 engineers = 4 requests per day. Converting to per hour: λ = 4 requests / 8 working hours ≈ 0.5 requests per hour.
• Service rate (μ): Each request takes 1 hour, thus μ = 1 request per hour.

Since requests from engineers do not interrupt if the specialist is busy, the system follows an M/M/1 queue model with an infinite buffer capacity, meaning no request is lost or rejected but must wait in the queue if the specialist is occupied.

Calculations

Given λ = 0.5 requests/hour and μ = 1 request/hour, the system's utilization rate (ρ) is:

ρ = λ / μ = 0.5 / 1 = 0.5

This indicates the specialist is busy 50% of the time, and the remaining 50% of the time, requests are waiting.

a. Average number of engineers waiting for help

The average number of requests in the queue (Lq) in an M/M/1 system is given by:

Lq = ρ² / (1 - ρ)

Substituting ρ = 0.5:

Lq = (0.5)² / (1 - 0.5) = 0.25 / 0.5 = 0.5

Therefore, on average, there are 0.5 engineers waiting in line for assistance at any given time.

b. Average waiting time for an engineer

The average time an engineer waits in the queue (Wq) is given by:

Wq = Lq / λ

Using Lq = 0.5 and λ = 0.5 requests/hour:

Wq = 0.5 / 0.5 = 1 hour

Thus, an engineer will typically wait about one hour before receiving assistance.

c. Probability that an engineer has to wait in line

The probability that an engineer must wait (the queue is non-empty) is simply the system's utilization rate (ρ).

P(wait) = ρ = 0.5

This means there is a 50% chance that an engineer will find the specialist busy and will need to wait in line.

Discussion

The analysis reveals that despite the variable nature of assistance times, the queueing system maintains a moderate workload, with the specialist being busy only half of the time. The average wait time of one hour can impact project schedules, especially if multiple engineers need assistance concurrently. However, the probability of delay being only 50% suggests the current resource allocation is reasonable, yet there may be room for efficiency improvement or resource expansion if reduced wait times are desired.

Implications and Recommendations

Understanding the performance metrics enables better planning in engineering support environments. To decrease the average wait time, the firm could consider increasing the specialist’s capacity—either through additional specialists or optimizing existing processes to reduce assistance durations. Alternatively, scheduling assistance during periods of peak requests might also be optimized based on the queueing model. Ultimately, this analysis demonstrates the utility of queueing theory in operational decision-making in engineering project management.

Conclusion

Applying the M/M/1 queue model to the engineering environment with a single technical specialist effectively quantifies the expected system performance. The key metrics indicate a balanced workload with moderate delays, providing insight into resource utilization and support efficiency. Continuous monitoring and potential resource adjustments can further enhance productivity and reduce waiting times, contributing to the success of engineering projects.

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