Question

In the two-class priority queueing model of Section \(8.6 .2\), what is \(W_{Q}\) ? Show that \(W_{Q}\) is less than it would be under FIFO if \(E\left[S_{1}\right]E\left[S_{2}\right]\)

Short answer

In the two-class priority queueing model, the overall waiting time in the queue \(W_Q\) is given by: \(W_Q = P_2 \times W_{Q2}\), where \(W_{Q2} = P_1 \times E[S_1]\). When \(E[S_1] < E[S_2]\), \(W_Q\) is less than it would be under FIFO, and when \(E[S_1] > E[S_2]\), \(W_Q\) is greater than it would be under FIFO.

Step-by-step solution

Understand the two-class priority queueing model

Two-class priority queueing model consists of two classes of customers, Class 1 and Class 2. Class 1 customers have a higher priority, and their service times are defined by \(S_1\). Class 2 customers have a lower priority, with service times defined by \(S_2\). We assume that customers will be served based on their priorities, where Class 1 customers will always be served first.

Calculate the waiting time in the queue for Class 1 customers

According to the model, there will be no waiting time in the queue for Class 1 customers since they always have a higher priority and will be served immediately. So, the waiting time in the queue for Class 1 customers, denoted by \(W_{Q1}\), is 0.

Calculate the waiting time in the queue for Class 2 customers

We can define the waiting time in the queue for Class 2 customers \(W_{Q2}\) as the expected service time for Class 1 customers \((E[S_1])\) multiplied by the probability that a Class 1 customer arrives, denoted by \(P_1\). So, we write the equation for \(W_{Q2}\) as: \[W_{Q2} = P_1 \times E[S_1]\]

Calculate the overall waiting time in the queue, W_Q

Now, to calculate the overall waiting time in the queue (\(W_Q\)), we need to consider both Class 1 and Class 2 customers. We can define \(W_Q\) as a weighted average of the waiting times for Class 1 and Class 2 customers, given by: \[W_Q = P_1 \times W_{Q1} + P_2 \times W_{Q2}\] Since \(W_{Q1} = 0\), the equation simplifies to: \[W_Q = P_2 \times W_{Q2}\]

Analyze the impact of E[S_1] on W_Q

We are given two cases to analyze: when \(E[S_1] < E[S_2]\) and when \(E[S_1] > E[S_2]\). Case 1: \(E[S_1] < E[S_2]\) In this case, we can say that the service time of Class 1 customers is smaller than that of Class 2 customers, which means that there will be less idle time for the server. As a result, there will be a higher probability that Class 2 customers will have to wait in the queue since Class 1 customers are served faster. Thus, \(W_Q\) will be less than it would be under FIFO in this case. Case 2: \(E[S_1] > E[S_2]\) In this case, we can say that the service time of Class 1 customers is greater than that of Class 2 customers, leading to more idle time for the server. Consequently, there will be a lower probability that Class 2 customers will have to wait in the queue, as Class 1 customers take longer to be served. Therefore, \(W_Q\) will be greater than it would be under FIFO in this case. In conclusion, we have shown that when \(E[S_1] < E[S_2]\), the waiting time in the queue \(W_Q\) is less than it would be under FIFO, and when \(E[S_1] > E[S_2]\), the waiting time in the queue \(W_Q\) is greater than it would be under FIFO.

Key concepts

Waiting Time in Queue

Understanding the waiting time in a queue is crucial in many service industries as it helps to gauge the efficiency of service provision. In a priority queueing model, the waiting time in queue (\( W_Q \)) represents the time a customer spends in the queue before being served. In the provided exercise, a two-class priority system is examined to determine the impact of different average service times (\( E[S_1] \) and \( E[S_2] \)) on \( W_Q \).

For Class 1 customers with higher priority, the waiting time is zero because they are served immediately (\( W_{Q1} = 0 \)). For Class 2 customers, the waiting time (\( W_{Q2} \)) is the expected service time of Class 1 customers multiplied by the probability of Class 1 customer arrivals (\( P_1 \times E[S_1] \)). This introduces the dependency of Class 2 waiting times on Class 1's average service time. It illustrates that the shorter the service time for the higher priority customers, the less the impact on the lower priority's waiting time, and vice versa. This concept is fundamental as it demonstrates how service efficiency for one class influences the experience of another class in prioritized environments.

Service Times

Service times play a pivotal role in the functionality of queueing systems. Defined as the time required to serve a customer, the average service times (\( E[S_1] \) and \( E[S_2] \)) directly influence the waiting time in queue, as shown in the priority queueing model. When analyzing the differences in service times between two classes of customers, you can predict the subsequent effects on the queue's dynamics.

For instance, if the average service time for higher priority customers (\( E[S_1] \)) is less than that of lower priority customers (\( E[S_2] \)), the queue is less likely to be long for subsequent customers, as they are served more quickly. Conversely, if the high-priority customers take longer on average to serve (\( E[S_1] > E[S_2] \)), this can create delays in the system, increasing waiting times for others. Acknowledging these nuances helps to manage and design more efficient queueing systems.

FIFO (First In First Out)

FIFO, an acronym for First In, First Out, is a fundamental concept in queueing theory where customers are served in the exact order they arrive. This method assumes no priority differences between customers; hence it is 'fair' to all. The exercise compares the priority queueing model's waiting time (\( W_Q \)) to what it would be under a FIFO system when varying the average service times of the different customer classes. In scenarios where the service time of higher priority customers is shorter than that of lower priority customers (\( E[S_1] < E[S_2] \)), the waiting time under priority queueing would be less than FIFO. The inverse is true when high priority customers take longer to serve (\( E[S_1] > E[S_2] \)).

Incorporating FIFO systems is often straightforward, but it can lead to inefficiencies in cases where prioritization is necessary due to varying urgency or importance of service.

Queueing Theory

Queueing theory is a mathematical study of waiting lines or queues, which provides the tools for making predictions and optimizations in service-oriented systems. It encompasses the analysis of several key components such as arrival rates, service rates, the number of servers, and the priority rules, which collectively influence the system's performance. The exercise at hand uses queueing theory principles to assess the priority queueing model's effectiveness versus a FIFO system, given different service times. By using these principles, we can predict the waiting times and thereby devise ways to manage queues more effectively.

An understanding of queueing theory not only helps in predicting the waiting time in queues (\( W_Q \)) but also empowers businesses and service providers to enhance customer satisfaction by minimizing wait times through the optimization of service processes and queue management.