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Chapter 11: Problem 31

Consider a queueing system in which each service time, independent of the past, has mean \(\mu\). Let \(W_{n}\) and \(D_{n}\) denote, respectively, the amounts of time customer \(n\) spends in the system and in queue. Hence, \(D_{n}=W_{n}-S_{n}\) where \(S_{n}\) is the service time of customer \(n\). Therefore, $$ E\left[D_{n}\right]=E\left[W_{n}\right]-\mu $$

Short Answer

Expert verified
The expected waiting time in the queue for customer \(n\) can be calculated using the formula: \(E[D_n] = E[W_n] - \mu\), where \(E[W_n]\) is the expected amount of time a customer spends in the system and \(\mu\) is the mean service time for each customer.

Step by step solution

01

Find the expected waiting time in the system, E[W_n]

To find \(E\left[D_{n}\right]\), we first need to find the expected amount of time a customer spends in the system, \(E\left[W_{n}\right]\). Since this information is not given in the exercise, we will assume that it is a known value. We will use the known value for \(E\left[W_{n}\right]\) in our calculations.
02

Identify the expected service time E[S_n]

The mean service time for each customer is given as \(\mu\). Since the service times are independent of the past, the expected service time for each customer is equal to the mean service time, that is: \(E[S_n] = \mu\)
03

Calculate the expected waiting time in the queue E[D_n]

Now that we know the expected waiting time in the system \(E[W_n]\) and the expected service time \(E[S_n] = \mu\), we can calculate the expected waiting time in the queue using the given formula: \( D_n = W_n - S_n \) Taking expectations of both sides of the equation: \(E[D_n] = E[W_n] - \mu\) Thus, we found the expected waiting time in the queue for customer n, which is given by: \(E[D_n] = E[W_n] - \mu\)

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