Several tasks are submitted to a computer system with two processors, $P_1$ and $P_2,$ working in

parallel. The process of submitting tasks can be described in discrete time by $a(t),t=0,1,2,$dots

where $a(t)=1$ if a task is submitted at time $t$ and $a(t)=0$ otherwise $($at most one task can

be submitted in each time step$).$ Suppose such a process is specified for the time interval $t=$

$0,1,$dots,$10$ as follows: $\{1,1,1,0,1,0,1,1,0,0,1\}.$ When a task is seen by the computer system,

the following rule for deciding which of the two processors to use is applied: Alternate between

the two processors, with the first task going to $P_1.$ It is assumed that if a task is sent to $P_i,i=$

$1,2,$ and that processor is busy, the task joins a queue of infinite capacity. The processing time

of a task at $P_1$ alternates between $4$ and $1$ time units $($starting with $4),$ whereas the processing

time at $P_2$ is always $2$ time units.

Let $y(t)$ be the total number of customers having departed from the system at time $t,$ and $x_1(t)$

and $x_2(t)$ be the queue lengths at processors $P_1$ and $P_2,$ respectively $($including a task in

process$).$ If one or more events occur at time $t,$ the values of these variables are taken to be just

after the event occurrence$($s$).$

a$)$ Draw a timing diagram with $t=0,1,$dots,$10$ showing arrivals and departures $($assume that

$x_1(0)=x_2(0)=y(0)=0).$

b$)$ Construct a table with the values of $x_1(t),x_2(t)$ and $y(t)$ for all $t=0,1,$dots,$10.$

c$)$ Suppose we now work in continuous time. Arrivals occur at times $0\mathrm{.}1,0\mathrm{.}7,2\mathrm{.}2,5\mathrm{.}2$ and $9\mathrm{.}9.$

The processing time at $P_1$ now alternates between $4\mathrm{.}2$ and $1\mathrm{.}1,$ whereas that of $P_2$ is fixed at

$2\mathrm{.}0$ time units. Consider an event$-$driven model with event set $E=\{a,d_1,d_2\},$ where $a=$

arrival, $d_i=$ departure from processor $P_i,i=1,2.$ Construct a table with the values of

$x_1(k),x_2(k),y(k),t(k),$ where $x_1(k),x_2(k),y(k)$ are the queue lengths and cumulative

number of departures after the kth event occurs, $k=1,2,$dots, and $t(k)$ is the time instant

of the $k^{th}$ event occurrence. If two events occur at the same time, assume that a departure

always comes before an arrival. Compare the number of updates required in this model to a

time$-$driven model with a time step of magnitude $0\mathrm{.}1$ time units.