Andrew S. Tanenbaum (OS) Edition 4 Exercise 2 Question 8 (Page No. 174)

Question

Consider a multiprogrammed system with degree of $6$ (i.e., six programs in memory at the same time). Assume that each process spends $40\%$ of its time waiting for I/O. What will be the CPU utilization?

Answer 1

The probability of all processes which are waiting for I/O $=(0.4)^6 = 0.004096$

$\therefore$ CPU Utilization $=1-0.004096 = 0.995904$

Comments

can you tell me what is wrong in my approach …

lets assume the total time taken to execute all the 6 processes is x

then the question says 40% of time is spent on IO ie 40% of x is on IO .. ie 0.4x

so the amount of time CPU is use is  is x – 0.4x = 0.6x

so the CPU utilization is useful time / total time = 0.6x / x  ie 0.6 but that not the answer can you tell me where i am going wrong …

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@sukesh_reddy, Your assumption is that if one process is spending 40% of time in IO, then the CPU handling 6 processes will also spend 40% of it’s time in IO.

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I have an alternate view, which I'm not able to understand why it gives the wrong answer.
Suppose all the processes take 1unit of time, hence any one of the 6 will take 0.4 time in IO and 0.6 time in Execution.
Considering all 6 processes are pipelined in the CPU, then the total utilization should be (useful time) / (total time) = $0.6*6/(0.6*6+0.4) = 0.9$

Answer 2

In a multiprogrammed system, multiple programs are kept in memory and are executed by the CPU in a way that maximizes the use of the CPU and other resources. In a system with a degree of 6, there are six programs in memory at any given time.

If each process spends 40% of its time waiting for I/O, this means that the CPU is not being used for 40% of the time for each process. Since there are six programs, this means that the CPU is not being used for a total of 40% * 6 = 240% of the time.

Since the total time is 100%, this means that the CPU utilization in this system will be 100% - 240% = -140%. This is not possible, since utilization must be a non-negative value.

One possible explanation for this result is that the assumptions made in the question are not accurate. It is not possible for a process to spend more time waiting for I/O than the total amount of time that it has to execute. This means that the assumption that each process spends 40% of its time waiting for I/O is incorrect, and the correct value is likely lower than 40%.

Alternatively, the question might be asking for the percentage of time that the CPU is being used by at least one of the processes. In this case, the answer would be 100% - 40% = 60%. This means that the CPU is being used by at least one process for 60% of the time, and is idle for the remaining 40% of the time.