Question

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

The variance of the waiting time in the M/M/1 queuing model, \(Var(W_Q^*)\), is given by the formula: \[ \operatorname{Var}\left(W_{Q}^{*}\right) = \frac{\lambda(\mu^2 + \lambda^2 - 2\mu\lambda)}{\mu^2(\mu - \lambda)^2} \]

Step-by-step solution

Find the expectation \(\mathbb{E}(W_Q^{*})\)

To find the expectation of \(W_Q^*\), we will use the definition of expectation for a random variable that takes on different values with different probabilities. In this case, \(W_Q^*\) takes on the value 0 with probability \((1 - \frac{\lambda}{\mu})\) and follows an exponential distribution with rate \((\mu - \lambda)\) with probability \(\frac{\lambda}{\mu}\). Thus, the expectation can be written as: \[ \mathbb{E}(W_Q^{*}) = 0 \times (1 - \frac{\lambda}{\mu}) + \frac{1}{\mu - \lambda} \times \frac{\lambda}{\mu} \]

Simplify the expression and calculate \(\mathbb{E}(W_Q^{*})\)

Now, let's simplify the expression for \(\mathbb{E}(W_Q^{*})\): \[ \mathbb{E}(W_Q^{*}) = \frac{\lambda}{\mu(\mu - \lambda)} \]

Find the expectation \(\mathbb{E}(W_Q^{*2})\)

To find the expectation of the square of the waiting time, \(\mathbb{E}(W_Q^{*2})\), we can similarly use the definition of expectation. The square of the waiting time is 0 with probability \(1 - \frac{\lambda}{\mu}\) and follows an exponential distribution with rate \((\mu-\lambda)\) squared with probability \(\frac{\lambda}{\mu}\). Thus, the expectation can be written as: \[ \mathbb{E}(W_Q^{*2}) = 0 \times (1 - \frac{\lambda}{\mu}) + \frac{2}{(\mu - \lambda)^2} \times \frac{\lambda}{\mu} \]

Simplify the expression and calculate \(\mathbb{E}(W_Q^{*2})\)

Now, let's simplify the expression for \(\mathbb{E}(W_Q^{*2})\): \[ \mathbb{E}(W_Q^{*2}) = \frac{2\lambda}{\mu(\mu - \lambda)^2} \]

Compute the variance \(\operatorname{Var}\left(W_{Q}^{*}\right)\)

Now we are ready to compute the variance using the formula \(\operatorname{Var}\left(W_{Q}^{*}\right) = \mathbb{E}(W_Q^{*2}) - (\mathbb{E}(W_Q^{*})^2\). Plugging in the values from previous steps, we get: \[ \operatorname{Var}\left(W_{Q}^{*}\right) = \frac{2\lambda}{\mu(\mu - \lambda)^2} - \left(\frac{\lambda}{\mu(\mu - \lambda)}\right)^2 \]

Simplify the expression to obtain the final answer

Now, let's simplify the expression for the variance: \[ \operatorname{Var}\left(W_{Q}^{*}\right) = \frac{\lambda(\mu^2 + \lambda^2 - 2\mu\lambda)}{\mu^2(\mu - \lambda)^2} \] Now we have the answer for the variance of the waiting time in an M/M/1 queuing model: \[ \operatorname{Var}\left(W_{Q}^{*}\right) = \frac{\lambda(\mu^2 + \lambda^2 - 2\mu\lambda)}{\mu^2(\mu - \lambda)^2} \]

Key concepts

Queueing Theory

Queueing theory is the mathematical study of waiting lines, or queues. At its core, this discipline analyzes processes through which objects—like people, vehicles, or data packets—queue up to receive some type of service.

One of the simplest and most commonly used models in queueing theory is the M/M/1 queue model, which represents systems with a single service channel (1), Poisson arrival patterns (M for 'Markovian'), and exponential service times (the second M). This model is an invaluable tool for understanding and optimizing a variety of service and operational systems, from customer service desks to network routers.

Understanding the behavior of queues helps organizations to manage resources efficiently, reduce waiting times, and improve customer satisfaction. Queueing theory provides metrics such as average waiting time, queue length, and the probability of finding the system busy, which are essential in designing and managing service systems.

The calculation of such metrics often involves understanding the nature of the arrivals and service times, which leads us to an integral part of M/M/1 and queueing theory: the exponential distribution.

Exponential Distribution

The exponential distribution is crucial in queueing theory, particularly in the context of the M/M/1 queue model. It characterizes the time between events in a Poisson process, where events occur continuously and independently at a constant average rate.

In the context of queues, these 'events' can be interpreted as customer arrivals or service completions. The exponential distribution is memoryless, which means that the probability of an event occurring in the future is independent of any past events. This property is particularly suited to model random arrival and service patterns typical of many queuing systems, as it simplifies the analysis significantly.

For instance, in our given problem, the service times are explicitly modeled as an exponential random variable with a rate of \(\mu - \lambda\). This rate is the difference between the service rate \(\mu\) and the arrival rate \(\lambda\), indicating the system's capacity to handle the workload. The probability associated with the time a customer spends waiting in the queue, therefore, follows an exponential distribution with the specified rate, giving us insight into how long customers might expect to wait before being served.

Variance of Waiting Time

The variance of waiting time is a statistical measure of how much the length of time that customers wait in queue varies. It's important because it reflects the consistency of the waiting experience. A high variance indicates that waits can fluctuate wildly, while a low variance means that they're relatively stable and predictable.

In the M/M/1 queue model, the variance of the waiting time in the queue \(\operatorname{Var}(W_{Q}^{*})\) provides valuable insight into the variability that customers may experience. Variance is calculated using the expectation of the waiting time and the expectation of the waiting time squared. The step-by-step solution for calculating these expectations and the final variance gives us a clear picture of what customers might endure.

The final expression for variance can be somewhat complex, as seen in the exercise, but it encapsulates important factors like the rates of arrival and service. Understanding this variance helps businesses and service providers to identify strategies for managing queues better—through staffing adjustments, process improvements, or customer communication—in order to minimize variability in service and enhance the overall customer experience.