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Chapter 11: Problem 34

Suppose in Example \(11.19\) that no new customers are allowed in the system after time \(t_{0} .\) Give an efficient simulation estimator of the expected additional time after \(t_{0}\) until the system becomes empty.

Short Answer

Expert verified
The efficient simulation estimator of the expected additional time after \(t_0\) until the system becomes empty can be obtained by simulating the remaining system behavior for the customers still in the system at time \(t_0\). Calculate the remaining service time, \(T_i\), for each customer and find the minimum, \(T_{min}\), among them. Update the departure times by adding the remaining service time to \(t_0\) and get the additional time, \(E[D_i - t_0]\). Repeat the simulation for multiple independent runs and calculate the average additional time, \(\frac{1}{n}\sum_{j=1}^{n}E_{j}\), to get the estimator.

Step by step solution

01

Define the properties of the given queueing system

Start by understanding the properties of the given queueing system in Example 11.19. Assume that the queue is a single-server queue with the first-in, first-out (FIFO) discipline and that customers arrive according to a Poisson process with an arrival rate of \(λ\). The service time for each customer is exponentially distributed with a service rate of \(μ\). Note that no new customers are allowed to enter the system after time \(t_0\).
02

List the required variables for the simulation

In order to build a simulation estimator of the expected additional time until the system becomes empty, we need to define the following variables: 1. \(N_t\): The number of customers present in the system at time \(t\). 2. \(A_i\): The arrival time of the \(i\)th customer still in the system at time \(t_0\). 3. \(S_i\): The service time of the \(i\)th customer still in the system at time \(t_0\). 4. \(T_i\): The remaining service time of the \(i\)th customer at time \(t_0\). 5. \(D_i\): The departure time of the \(i\)th customer after time \(t_0\).
03

Simulate the remaining system behavior

For the simulation, we need to calculate the customers' departure times after \(t_0\), as they determine the additional time until the system becomes empty. Here's an approach to perform the simulation: 1. During the simulation, maintain a list of customers still in the system. 2. Calculate the remaining time, \(T_i\), for all customers in the system by subtracting the time they have already spent in the service from their total service time, i.e., \(T_i = S_i - (t_0 - A_i)\) for each customer who is currently being served, and \(T_i = S_i\) for each customer still in the queue. 3. Find the minimum remaining time, \(T_{min}\), among all customers still in the system. 4. Update the departure times: \(D_i = t_0 + T_i\) for all remaining customers in the system. 5. Find the additional time, \(E[D_i - t_0]\), for the system to become empty after \(t_0\).
04

Repeat the simulation and calculate the average additional time

To obtain an efficient simulation estimator of the expected additional time, repeat the simulation for a large number of independent runs (say, \(n\)) and calculate the average additional time: 1. For each simulation run \(j\), calculate the additional time, \(E_{j}\), as in step 3. 2. After all \(n\) simulation runs, calculate the average additional time over all runs: \[\frac{1}{n}\sum_{j=1}^{n}E_{j}\] This average additional time will be the estimator of the expected additional time after \(t_0\) until the system becomes empty.

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