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Chapter 8: Problem 37

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 (i) go to station 2, (ii) go to station 3, or (iii) 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. (a) What is the average number of customers in the system (consisting of all three stations)? (b) What is the average time a customer spends in the system?

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