Arrivals At A Telephone Booth Are Considered To Be Poisson With An Average Time Of 10 Minutes Between One Arrival And The Next. The Length Of A Phone Call Is Assumed To Be Distributed Exponentially With A Mean Of 3 Minutes.(a) What Is The

Apr 8, 2025 by ADMIN 239 views

**Understanding the Arrival and Service Process at a Telephone Booth**

Introduction

In this article, we will delve into the world of queueing theory and explore the arrival and service process at a telephone booth. We will assume that the arrivals at the telephone booth follow a Poisson distribution, and the length of a phone call is exponentially distributed. Our goal is to understand the underlying processes and derive key metrics that describe the system's behavior.

Arrival Process

The arrival process at the telephone booth is considered to be Poisson with an average time of 10 minutes between one arrival and the next. This means that the number of arrivals in a given time interval follows a Poisson distribution. The Poisson distribution is characterized by a single parameter, λ (lambda), which represents the average rate of arrivals per unit time.

In this case, λ = 1/10, since there is an average of one arrival every 10 minutes. The Poisson distribution is given by the formula:

P(X = k) = (e^(-λ) * (λ^k)) / k!

where k is the number of arrivals, e is the base of the natural logarithm, and ! denotes the factorial function.

Service Process

The length of a phone call is assumed to be exponentially distributed with a mean of 3 minutes. This means that the time between phone calls follows an exponential distribution. The exponential distribution is characterized by a single parameter, μ (mu), which represents the average rate of service per unit time.

In this case, μ = 1/3, since there is an average of one phone call every 3 minutes. The exponential distribution is given by the formula:

f(x) = (1/μ) * e^(-x/μ)

where x is the time between phone calls, and e is the base of the natural logarithm.

Queueing System

The telephone booth can be modeled as a queueing system, where customers (phone calls) arrive and wait in a queue for service. The queueing system is characterized by the following parameters:

Arrival rate: λ = 1/10
Service rate: μ = 1/3
Number of servers: 1 (since there is only one telephone booth)
Queue capacity: infinite (since there is no limit to the number of phone calls that can wait in the queue)

Key Metrics

To understand the behavior of the queueing system, we need to derive key metrics such as the average number of customers in the system, the average waiting time in the queue, and the probability of having a customer in the system.

Average Number of Customers in the System

The average number of customers in the system (L) can be calculated using the formula:

L = ρ / (1 - ρ)

where ρ is the utilization factor, given by:

ρ = λ / μ

In this case, ρ = (1/10) / (1/3) = 3/10.

Substituting the value of ρ into the formula for L, we get:

L = (3/10) / (1 - 3/10) = 3/7

So, the average number of customers in the system is 3/7.

Average Waiting Time in the Queue

The average waiting time in the queue (Wq) can be calculated using the formulaWq = L / (λ * (1 - ρ))

Substituting the values of L and ρ, we get:

Wq = (3/7) / ((1/10) * (1 - 3/10)) = 30/7

So, the average waiting time in the queue is 30/7 minutes.

Probability of Having a Customer in the System

The probability of having a customer in the system (P) can be calculated using the formula:

P = ρ

In this case, P = 3/10.

So, the probability of having a customer in the system is 3/10.

Conclusion

In this article, we have explored the arrival and service process at a telephone booth, assuming that the arrivals follow a Poisson distribution and the length of a phone call is exponentially distributed. We have derived key metrics such as the average number of customers in the system, the average waiting time in the queue, and the probability of having a customer in the system. These metrics provide valuable insights into the behavior of the queueing system and can be used to optimize the system's performance.

References

[1] Cox, D. R., & Smith, W. L. (1961). Queues. Chapman and Hall.
[2] Kendall, D. G. (1953). Stochastic processes. Academic Press.
[3] Kleinrock, L. (1975). Queueing systems, volume 1: Theory. Wiley.

Note: The references provided are a selection of classic texts in the field of queueing theory. They are not exhaustive, and readers are encouraged to explore further references for a more comprehensive understanding of the subject.

Introduction

In our previous article, we explored the arrival and service process at a telephone booth, assuming that the arrivals follow a Poisson distribution and the length of a phone call is exponentially distributed. We derived key metrics such as the average number of customers in the system, the average waiting time in the queue, and the probability of having a customer in the system. In this article, we will address some frequently asked questions (FAQs) related to the arrival and service process at a telephone booth.

Q: What is the difference between a Poisson distribution and an exponential distribution?

A: A Poisson distribution is a discrete distribution that models the number of arrivals in a given time interval, while an exponential distribution is a continuous distribution that models the time between arrivals. In the context of the telephone booth, the Poisson distribution models the number of phone calls arriving in a given time interval, while the exponential distribution models the time between phone calls.

Q: Why is the exponential distribution used to model the time between phone calls?

A: The exponential distribution is used to model the time between phone calls because it is a memoryless distribution, meaning that the time between phone calls is independent of the time since the last phone call. This is a reasonable assumption in the context of the telephone booth, where the time between phone calls is not influenced by the time since the last phone call.

Q: What is the utilization factor (ρ) and how is it calculated?

A: The utilization factor (ρ) is a measure of the proportion of time that the telephone booth is busy. It is calculated as the ratio of the arrival rate (λ) to the service rate (μ). In the context of the telephone booth, ρ = λ / μ = (1/10) / (1/3) = 3/10.

Q: What is the average number of customers in the system (L) and how is it calculated?

A: The average number of customers in the system (L) is a measure of the average number of phone calls in the system, including those in the queue and those being served. It is calculated using the formula L = ρ / (1 - ρ). In the context of the telephone booth, L = (3/10) / (1 - 3/10) = 3/7.

Q: What is the average waiting time in the queue (Wq) and how is it calculated?

A: The average waiting time in the queue (Wq) is a measure of the average time that a phone call spends waiting in the queue. It is calculated using the formula Wq = L / (λ * (1 - ρ)). In the context of the telephone booth, Wq = (3/7) / ((1/10) * (1 - 3/10)) = 30/7.

Q: What is the probability of having a customer in the system (P) and how is it calculated?

A: The probability of having a customer in the system (P) is a measure of the probability that there is at least one phone call in the system. It is calculated using the formula P = ρ. In the context of the telephone booth, P = 310.

Q: How can the arrival and service process at a telephone booth be optimized?

A: The arrival and service process at a telephone booth can be optimized by adjusting the arrival rate (λ) and the service rate (μ) to minimize the average waiting time in the queue (Wq) and the probability of having a customer in the system (P). This can be achieved by increasing the service rate (μ) or decreasing the arrival rate (λ).

Conclusion

In this article, we have addressed some frequently asked questions (FAQs) related to the arrival and service process at a telephone booth. We have provided explanations and calculations for key metrics such as the utilization factor (ρ), the average number of customers in the system (L), the average waiting time in the queue (Wq), and the probability of having a customer in the system (P). We have also discussed ways to optimize the arrival and service process at a telephone booth.

References

[1] Cox, D. R., & Smith, W. L. (1961). Queues. Chapman and Hall.
[2] Kendall, D. G. (1953). Stochastic processes. Academic Press.
[3] Kleinrock, L. (1975). Queueing systems, volume 1: Theory. Wiley.