Question

The average amount of time that a customer waits in line for service is given by \(W(x, y)=\frac{1}{x-y}, \quad y

Short answer

The average waiting time of the customers for each point is: (a) \(\frac{1}{5}\) hours, (b) \(\frac{1}{3}\) hours, (c) \(\frac{1}{6}\) hours, (d) \(\frac{1}{2}\) hours

Step-by-step solution

Calculate W for (15,10)

Here, x=15 and y=10. The waiting time is calculated using the formula \(W(15, 10) = \frac{1}{15-10}\)

Calculate W for (12,9)

Here, x=12 and y=9. The waiting time is calculated using the formula \(W(12, 9) = \frac{1}{12-9}\)

Calculate W for (12,6)

Here, x=12 and y=6. The waiting time is calculated using the formula \(W(12, 6) = \frac{1}{12-6}\)

Calculate W for (4,2)

Here, x=4 and y=2. The waiting time is calculated using the formula \(W(4, 2) = \frac{1}{4-2}\)

Key concepts

Queueing Theory

Queueing theory explores the dynamics of queues in various scenarios, such as customers waiting for service or data packets waiting to be transmitted over a network. The theory primarily aims to predict queue lengths and waiting times, ensuring efficiency in systems like customer service centers, computer networks, and manufacturing.

At the heart of queueing theory lies the balance between demand (arrival rate) and the ability to meet that demand (service rate). Calculations often involve determining the average waiting time a member (customer or item) spends in a queue. The formula used in the textbook exercise, \(W(x, y) = \frac{1}{x - y}\), showcases a direct application of these principles. Here, \(y\) is the rate at which customers arrive, and \(x\) is the rate at which they are served. When service rate exceeds the arrival rate (\(x > y\)), it indicates that the system can manage the incoming demand without causing excessive delays. Conversely, if the service and arrival rates are close or the arrival rate is higher, queues can become longer and waiting times increase dramatically.

Understanding and applying queueing theory allows organizations to design more efficient systems. For example, a bank may increase the number of tellers during peak hours to reduce customer wait times, or an IT company may enhance bandwidth to prevent data packet bottlenecks.

Service Rate

The service rate, denoted by \(x\) in our exercise, refers to the average capacity of a service system to handle requests or customers per unit of time. In the context of queueing theory, it's a vital part of the equation for calculating the average waiting time.

Service rate is not just about speed. It reflects effectiveness, efficiency, and the quality of the service rendered. An optimal service rate should match or exceed the arrival rate to prevent backlog and ensure smooth operation. In the provided formulas, a higher service rate compared to the arrival rate (\(x > y\)) means shorter waiting times, as seen in the exercise solutions.

Improving Service Rate

Businesses or systems might improve their service rate by training staff, adopting new technology, or restructuring processes to be more efficient. In the scenarios given in the textbook problem, service rates are varied to demonstrate different waiting times. A high service rate is indicative of a proficient system, and its importance cannot be overstressed in service design.

Arrival Rate

The arrival rate, denoted by \(y\) in our exercise, is a measure of how frequently customers or requests arrive to a system requiring service, within a given time frame. It's a critical factor in queueing theory, as it influences the size of the queue and, consequently, the average waiting time.

The arrival rate can be variable or stable and is often influenced by external factors such as marketing campaigns, seasonal demand, or economic conditions. In the exercise, you can see how different rates affect waiting times. Understanding these variations helps in planning and adjusting service capabilities accordingly.

Managing Arrival Rate

To manage an increased arrival rate, a system could introduce appointment scheduling, or a virtual queue system. On the other hand, if the system experiences low arrival rates, it might need to consider strategies to attract more customers or optimize periods of low activity. The fundamental aim is always to maintain a balance between arrival and service rates, ensuring efficient operation without overextending resources or causing customer dissatisfaction due to long wait times.