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Chapter 8: Problem 33

In an \(M / G / 1\) queue, (a) what proportion of departures leave behind 0 work? (b) what is the average work in the system as seen by a departure?

Short Answer

Expert verified
(a) The proportion of departures that leave behind 0 work is equal to the probability of the system being idle, which can be calculated as: \[P_0 = 1 - \rho\] (b) The average work in the system as seen by a departure can be found by dividing the average number of customers in the system (L) by the arrival rate (\(\lambda\)): \[W = \frac{L}{\lambda}\]

Step by step solution

01

Computing the Utilization Factor

First, calculate the utilization factor, denoted as \(\rho\). The utilization factor represents the proportion of time the server is busy. It can be calculated as: \[\rho = \frac{\lambda}{\mu}\]
02

Applying the Pollaczek-Khintchine Formula

Using the Pollaczek-Khintchine formula, we can find the average number of customers in the system by substituting the values of \(\lambda\), \(\mu\), and \(\sigma^2\) as well as the utilization factor \(\rho\) into the formula: \[L = \rho + \frac{\lambda^2 \sigma^2 + \lambda^2 \mu^2}{2(1-\rho)}\]
03

Answering Question (a) - Proportion of Departures

To find the proportion of departures that leave behind 0 work, we must first understand the concept of an idle period. An idle period occurs when there are no more customers in the system, and the server is idle. The proportion of departures that leave behind 0 work is equal to the probability of the system being idle, which can be calculated as: \[P_0 = 1 - \rho\]
04

Answering Question (b) - Average Work in System

To find the average work in the system as seen by a departure, we must find the average number of customers in the system (L) that we calculated in Step 2. Then, we need to divide the average number of customers (L) by the arrival rate (\(\lambda\)) to get the average work in the system: \[W = \frac{L}{\lambda}\] Now, we have broken down the exercise and explained how to compute the proportion of departures that leave behind 0 work and the average work in the system as seen by a departure for an M/G/1 queueing system.

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Most popular questions from this chapter

Consider a network of three stations. Customers arrive at stations 1 , 2, 3 in accordance with Poisson processes having respective rates \(5,10,15\). The service times at the three stations are exponential with respective rates \(10,50,100\). A customer completing service at station 1 is equally likely to (a) 8o to station 2, (b) go to station 3 , or \((\mathrm{c})\) leave the system. A customer departing service at station 2 always goes to station 3. A departure from service at station 3 is equally likely to either go to station 2 or leave the system. (i) What is the average number of customers in the system (consisting of all three stations)? (ii) What is the average time a customer spends in the system?

Suppose in Exercise 17 we want to find out the proportion of time there is a type 1 customer with server \(2 .\) In terms of the long-run probabilities given in Exercise 17 , what is (a) the rate at which a type 1 customer enters service with server \(2 ?\) (b) the rate at which a type 2 customer enters service with server \(2 ?\) (c) the fraction of server 2 's customers that are type \(1 ?\) (d) the proportion of time that a type 1 customer is with server \(2 ?\)

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 two 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 \(\mu_{\text {, }}\) 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 time he spends in the system? (d) What proportion of customers enter the system? (c) What is the average time an entering customer spends in the system?

For the \(M / G / 1\) queue, let \(X_{n}\) denote the number in the system left behind by the \(n\) th departure. (a) If $$ X_{n+1}=\left\\{\begin{array}{ll} X_{n}-1+Y_{n}, & \text { if } X_{n} \geq 1 \\ Y_{n}, & \text { if } X_{n}=0 \end{array}\right. $$ what does \(Y_{n}\) represent? (b) Rewrite the preceding as $$ X_{n+1}=X_{n}-1+Y_{n}+\delta_{n} $$ where $$ \delta_{n}=\left\\{\begin{array}{ll} 1, & \text { if } X_{n}=0 \\ 0, & \text { if } X_{n} \geq 1 \end{array}\right. $$ Take expectations and let \(n \rightarrow \infty\) in Equation \((8.58)\) to obtain $$ E\left[\delta_{d}\right]=1-\lambda E[S] $$ (c) Square both sides of Equation ( \(8.58\) ), takes expectations, and then let \(n \rightarrow \infty\) to obtain $$ E\left[X_{*}\right]=\frac{\lambda^{2} E\left[S^{2}\right]}{2(1-\lambda E[S])}+\lambda E[S] $$ (d) Argue that \(E\left[X_{w}\right]\), the average number as seen by a departure, is equal to \(L\).

Carloads of customers arrive at a single-server station in accordance to a Poisson process with rate 4 per hour. The service times are exponentially distributed with rate 20 per hour. If each carload contains cither 1,2 , or 3 customers with respective probabilities \(\frac{1}{8}, \frac{1}{2}, \frac{1}{4}\), compute the average customer delay in queue.
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