Lassettre and Scherr [40] used the machine repair queueing system to model the performance of the OS/360 time-sharing option (TSO), a system that allowed programmers to develop and run programs from a terminal in a time-sharing mode. It was, of course, the progenitor of the current TSO, that runs under the MVS operating system. (Allan Scherr received the Association of Computing Machinery's Grace Murray Hopper award in 1975 for some earlier work (Scherr [52]), in which he used this same queueing system to model the MIT Project MAC time-sharing system called CTSS for Compatible Time-Shared System.) For their model, Lassettre and Scherr identified the parameters $E[O], W, W_s$, and $K$ of the machine repair system with the parameters of the TSO model as follows: $E[O]$ is the average user time, that they defined as the mean elapsed time from the request for input made by the processing program to the completion of the requested input by the user. $W$ is the average response time that they defined as the mean length of the time interval that begins with the completion of input and ends when the program processing this input finishes and requests more input. Thus, taken together, the user time and the response time make up a complete cycle or interaction. $W_S$ is the mean system service time, that is, the mean of the time required by the system to execute the program that processes the input entered by the user. $K$ is the average number of users actively interacting with the system.
(a) Use Little's law to show that the mean number of interactions per second is given by
$$
\lambda=\frac{K}{E[O]+W} .
$$
(b) Show that the mean response time is given by
$$
W=\frac{K W_s}{1-p_0}-E[O],
$$
where $p_0$ is the probability that there are no requests for system service pending.
(c) Show that the average number of users waiting for a response is
$$
L=\frac{K W}{W+E[O]} .
$$
(d) Show that the mean response time when there is a large number of users ( $K$ is large) is approximately $K W_s-E[O]$. Assume that the CPU is the computer system bottleneck.
(e) The curve $W / W_s$ versus $K$ has the asymptote $W / W_s=K-$ $E[O] / W_S$ for large $K$. We see that for small values of $K$ there is an asymptote to the curve, that is the horizontal line $W / W_S=1$. The two asymptotes intersect at the point $\left(K^*, 1\right)$ where
$$
K^*=\frac{W_s+E[O]}{W_s} .
$$
Kleinrock [32] calls the value of $K^*$ the saturation number, but Lassettre and Scherr call the point $\left(K^{* *}, 0\right)$ where the line $K W_s-E[O]$ intersects the $x$-axis the saturation point and claim it is a fair approximation of the capacity of a time-sharing system. Thus,
$$
K^{* *}=\frac{E[O]}{W_s} .
$$
Lassattre and Scherr use an $E[O]$ value of 35 seconds based on measurements for users of terminals such as Teletypes and IBM $2741 \mathrm{~s}$. Using this value of $E[O]$, they reported that for one TSO system with 60 active terminals the mean response time, $W$, was 5 seconds, with $W_s$ equal to 0.8 seconds. ${ }^{18}$ If CRT type terminals are acquired for this system that provide an $E[O]$ of 10 seconds and the throughput remains the same as before, what is the mean response time $W$ ? What is $K^{* *}$ for this system?