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Chapter 8: Problem 5

It follows from Exercise 4 that if, in the \(M / M / 1\) model, \(W_{Q}^{*}\) is the amount of time that a customer spends waiting in queue, then $$ W_{Q}^{*}=\left\\{\begin{array}{ll} 0, & \text { with probability } 1-\lambda / \mu \\ \operatorname{Exp}(\mu-\lambda), & \text { with probability } \lambda / \mu \end{array}\right. $$ where \(\operatorname{Exp}(\mu-\lambda)\) is an exponential random variable with rate \(\mu-\lambda\). Using this, find \(\operatorname{Var}\left(W_{Q}^{*}\right)\).

Short Answer

Expert verified
The variance of the waiting time in the \(M/M/1\) model is: $$ \operatorname{Var}(W_Q^*) = \frac{\lambda}{\mu^2(\mu - \lambda)^2} $$

Step by step solution

01

Find the Expected Value of Waiting Time

To calculate the expected value of \(W_Q^*\), we will use the definition of expected value in the context of probability distribution: $$ E(W_Q^*) = \sum_i x_i P(x_i) $$ where \(x_i\) represents the possible waiting times and \(P(x_i)\) represents their corresponding probabilities. According to the given distribution: $$ W_Q^* = \begin{cases} 0, & \text{ with probability } 1 - \frac{\lambda}{\mu} \\ \operatorname{Exp}(\mu - \lambda), & \text{ with probability } \frac{\lambda}{\mu} \end{cases} $$ Thus, the expected value of waiting time is: $$ E(W_Q^*) = 0 \times (1 - \frac{\lambda}{\mu}) + \frac{1}{\mu - \lambda} \times \frac{\lambda}{\mu} $$ $$ E(W_Q^*) = \frac{\lambda}{\mu(\mu - \lambda)} $$
02

Find the Expected Value of the Squared Waiting Time

Now, we need to find the expected value of the squared waiting time \(E\left((W_Q^*)^2\right)\). We will again use the definition of expected value: $$ E\left((W_Q^*)^2\right) = \sum_i x_i^2 P(x_i) $$ Using the given distribution: $$ E\left((W_Q^*)^2\right) = 0^2 \times (1 - \frac{\lambda}{\mu}) + \left(\frac{2}{(\mu - \lambda)^2}\right) \times \frac{\lambda}{\mu} $$ $$ E\left((W_Q^*)^2\right) = \frac{2\lambda}{\mu(\mu - \lambda)^2} $$
03

Calculate the Variance of Waiting Time

Now that we have both the expected value of waiting time and the expected value of the squared waiting time, we can calculate the variance using the formula: $$ \operatorname{Var}(W_Q^*) = E\left((W_Q^*)^2\right) - \left(E(W_Q^*)\right)^2 $$ Substitute the values calculated in steps 1 and 2: $$ \operatorname{Var}(W_Q^*) = \frac{2\lambda}{\mu(\mu - \lambda)^2} - \left(\frac{\lambda}{\mu(\mu - \lambda)}\right)^2 $$ After simplification, the variance of waiting time in the \(M/M/1\) model is: $$ \operatorname{Var}(W_Q^*) = \frac{\lambda}{\mu^2(\mu - \lambda)^2} $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Waiting Time in Queue Systems
In the realm of queue models, such as the M/M/1 queue, waiting time refers to the duration a customer spends waiting in line before receiving service. Let's break down this concept in the context of a single-service queue system. In the given scenario, not every customer will experience a waiting time. Some might be served immediately if the system is free. Here, there's a certain probability that a customer will wait for some time, described by specific probability rules.
When a customer enters a busy system, their waiting time can be described as a stochastic process, meaning it's influenced by randomness and probability. For complex systems, understanding how long a customer might wait is crucial for efficiency, planning, and customer satisfaction. Knowing the expected waiting time allows businesses to forecast system behavior and optimize service processes.
The Role of Exponential Distribution
Exponential distribution is a fundamental concept in queueing theory and is often used to model the time between arrivals in a continuous-time waiting line model. It is characterized by its memoryless property, which implies the likelihood of a future event occurring depends only on the present and not any past events.
In the scenario we're examining, the waiting time for customers who enter a busy system is represented as an exponentially distributed random variable with a rate \(\mu - \lambda\). This suggests that the time until the next service completion is consistent and random, with a specific average rate of occurrence.
The exponential distribution simplifies the analysis of queue systems by providing a manageable mathematical model. Understanding this helps in accurately predicting system behavior and waiting times.
How Variance Calculation is Used
Variance is a statistical measure that gives us an understanding of how much the waiting time in the M/M/1 queue can vary for different customers. Essentially, it quantifies the spread or dispersion of the waiting time around the mean or expected value. To calculate variance for the waiting time in this queuing model, we use the formula \(\text{Var}(W_Q^*) = E((W_Q^*)^2) - (E(W_Q^*))^2\), where \( E((W_Q^*)^2) \) is the expected value of the squared waiting time.
The variance provides insights into the level of uncertainty and inconsistency in waiting times, revealing how predictable or unpredictable the service process can be. A low variance indicates uniformity where all customers have similar waiting times, whereas a high variance suggests more inconsistency.
Exploring the M/M/1 Queue Model
The M/M/1 queue model is a classic representation in queueing theory, used to depict a service system with a **single server**, where arrivals and service times are modeled as memoryless or exponentially distributed processes. The notation "M/M/1" signifies:
  • M: Markovian (exponential) arrival process
  • M: Markovian (exponential) service process
  • 1: One server handling the tasks
Such a model provides an analytical framework to assess the behavior of a queue, calculating metrics like average waiting time, server utilization, and probability of queue length.
This model assumes customer arrivals are random and independent, with each experiencing a potentially different waiting time based on variables of the distribution. It finds applications in various fields, from production lines to computer networks, aiding in optimizing service efficiency.
Implications of Expected Value in Queue Systems
The concept of expected value is integral in queue analysis. It represents the average or mean waiting time a customer can expect when joining the queue. Calculating the expected value involves averaging all possible waiting times, weighted by their respective probabilities. In the given solution, the expected value for the waiting time is calculated using the formula: \( E(W_Q^*) = \frac{\lambda}{\mu(\mu - \lambda)} \). This value indicates the mean time a customer waits before receiving service in an efficient system. Having an expected value helps businesses anticipate average queue lengths, allowing them to manage customer expectations and optimize resource allocation effectively. It's a valuable measure to ensure a balance between service capacity and demand, maintaining quality and saving costs.

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Most popular questions from this chapter

In a queue with unlimited waiting space, arrivals are Poisson (parameter \(\lambda\) ) and service times are exponentially distributed (parameter \(\mu\) ). However, the server waits until \(K\) people are present before beginning service on the first customer; thereafter, he services one at a time until all \(K\) units, and all subsequent arrivals, are serviced. The server is thea "idle" until \(K\) new arrivals have occumed. (a) Define an appropriate state space, draw the transition diagram, and set up the balance equations. (b) In terms of the limiting probabilities, what is the average time a customer spends in queue? (c) What conditions on \(\lambda\) and \(\mu\) are necessary?

A group of \(m\) customers frequents a single-server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon completing service the customer departs the system, but then returns after an exponential time with rate \(\theta\). All service times are exponentially distributed with rate \(\mu\). (a) Define states and set up the balance equations. In terms of the solution of the balance equations, find (b) the average rate at which customers enter the station. (c) the average time that a customer spends in the station per visit.

Customers arrive at a two-server system at a Poisson rate \(\lambda\). An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding tw? others in the system will wait in line for the first free server. An arrival finding three in the system will not enter. All service times are exponential with rate \(\mu\), and once a customer is served (by either server), he departs the system. (a) Define the states. (b) Find the long-run probabilities. (c) Suppose a customer arrives and finds two others in the system. What is the expected times he spends in the system? (d) What proportion of customers enter the system? (e) What is the average time an cntering customer spends in the system?

Two customers move about among three servers. Upon completion of service at server \(i\), the customer leaves that server and enters service at whichever of the other two servers is free. (Therefore, there are always two busy servers.) If the service times at server \(i\) are exponential with rate \(\mu_{\ell}, i=1,2,3\), what proportion of time is server \(i\) idle?

Consider a single-server queue with Poisson arrivals and exponential service times having the following variation: Whenever a service is completed a departure occurs only with probability \(\alpha\). With probability \(1-\alpha\) the customer, instead of leaving, joins the end of the queue. Note that a customer may be serviced more than once. (a) Set up the balance equations and solve for the steady-state probabilities, stating conditions for it to exist. (b) Find the expected waiting time of a customer from the time he arrives until he enters service for the first time. (c) What is the probability that a customer enters service exactly \(n\) times, \(n=\) \(1,2, \ldots ?\) (d) What is the expected amount of time that a customer spends in service (which does not include the time he spends waiting in line)? Use part (c). (c) What is the distribution of the total length of time a customer spends being served? 7 Is it memoryless?
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