There is a mathematical theory called queuing theory that studies ways in which computer jobs are fed in CPUs and researches on how these can be reduced to a minimum. Show how can a computer estimate the average number of jobs waiting at a queue?
Suppose that in a 5 sec interval jobs arrive as indicated in the following table ( Arrival time is assumed to be at the beginning of each second) In the first second jobs A and B arrive. During the second second B moves to the head of the line ( A job is completed as it took 1 sec to be served), and C and D jobs arrive and so on.
Find:
a. The mean number of jobs in line
b. The mode of the number of jobs in line
Time in seconds Jobs
1 A,B
2 C,D
3 -
4 E,F
5 -
Please everyone I need step by step so I will understand how to solve this kind of problem next time. Thanks in advance.
To estimate the average number of jobs waiting in the queue, you can use a few key formulas from queuing theory. Let's go step by step through the process:
Step 1: Calculate the number of jobs in the queue at the end of each time interval.
Using the given table, let's track the number of jobs in the queue at the end of each second:
Time: 1 - Jobs: A,B - Jobs in queue: 2
Time: 2 - Jobs: C,D - Jobs in queue: 4
Time: 3 - No new jobs arrived, so the number of jobs in the queue remains the same: 4
Time: 4 - Jobs: E,F - Jobs in queue: 6
Time: 5 - No new jobs arrived, so the number of jobs in the queue remains the same: 6
Step 2: Calculate the total number of jobs in the queue over the time interval.
Now, sum up the number of jobs in the queue at the end of each second:
Total jobs in the queue = 2 + 4 + 4 + 6 + 6 = 22
Step 3: Calculate the average number of jobs in the queue.
To find the average number of jobs in the queue, divide the total number of jobs by the total time:
Average number of jobs in the queue = Total jobs in the queue / Total time
In this case, the total time is 5 seconds, and we already found that the total number of jobs is 22. Therefore:
Average number of jobs in the queue = 22 / 5 = 4.4
So, the estimated mean number of jobs in the queue is 4.4.
Step 4: Calculate the mode of the number of jobs in the queue.
The mode represents the most frequent number of jobs in the queue. To find the mode, identify the number that appears most often in the list of jobs in the queue:
Number of jobs in the queue: 2 4 4 6 6
In this case, both 4 and 6 appear twice. So, the mode of the number of jobs in the queue is 4 and 6.
That's it! You have now calculated both the mean and mode of the number of jobs in the queue using queuing theory.