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This question relates the queueing network with feedback of Exercise 15.2.3 to the two node tandem queueing network without feedback shown in Figure 15.18. In this tandem queueing network, the first node represents the CPU of the previous example and the second represents the I/O device. The arrival process is Poisson with rate γ. Each arriving process requires the CPU only once, and holds it for an average of S1 seconds. On exiting the CPU, each process must go to the I/O device and spends an average of S2 seconds there after which it departs never to return.

The following values for S1 and S2 are given:

(a) Show that the utilization factor in this tandem model is the same as the utilization factor in the feedback model of Exercise 15.2.3.

(b) Show that the response time (average time a process spends in the network) is the same in both models.

(c) Show that the probability distribution of customers is the same in both networks.

(d) Generate a counter-example to show that the waiting time distribution is not the same in both networks. Hint: Choose μ1 μ2.

(e) How should S1 and S2 be interpreted in terms of CPU and I/O requirements?

Exercise 15.2.3

Consider the queueing model shown in Figure 15.17. This may be used to model a computer CPU connected to an I/O device as indicated on the figure. Processes enter the system according to a Poisson process with rate γ, and use the CPU for an exponentially distributed amount of time with mean 1/μ1. Upon exiting the CPU, with probability r, a process uses the I/O device for a time that is exponentially distributed with mean 1/μ2, or with probability 1 − r, it exits the system. Upon exit from the I/O device, a process again joins the CPU queue. Assume that all service times, including successive service times of the same process at the CPU or the I/O device are independent. Find the mean number of customers at the CPU and at the I/O device. What is the average time a process spends in the system? What is the utilization at the CPU and at the I/O device?