91Ó°ÊÓ

Assume that a computer system is in one of three states: busy, idle, or undergoing repair, respectively denoted by states 0,1 , and 2 . Observing its state at 2 P.M. each day, we believe that the system approximately behaves like a homogeneous Markov chain with the transition probability matrix: $$ P=\left[\begin{array}{lll} 0.6 & 0.2 & 0.2 \\ 0.1 & 0.8 & 0.1 \\ 0.6 & 0.0 & 0.4 \end{array}\right] \text {. } $$ Prove that the chain is irreducible, and determine the steady-state probabilities.

Consider a computer system with a CPU and one disk drive. After a burst at the \(\mathrm{CPU}\) the job completes execution with probability \(0.1\) and requests a disk I/O with probability \(0.9\). The time of a single CPU burst is exponentially distributed with mean \(0.01 \mathrm{~s}\). The disk service time is broken up into three phases: exponentially distributed seek time with mean \(0.03 \mathrm{~s}\), uniformly distributed latency time with mean \(0.01 \mathrm{~s}\), and a constant transfer time equal to \(0.01 \mathrm{~s}\). After a service completion at the disk, the job always requires a CPU burst. The average arrival rate of jobs is \(0.8 \mathrm{job} / \mathrm{s}\) and the system does not have enough main memory to support multiprogramming. Solve for the average response time using the \(M / G / 1\) model. In order to compute the mean and the variance of the service time distribution, you may need the results of the section on random sums in Chapter 5.