Several tasks are submitted to a computer system with two processors, $1$ and $2,$ working in

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

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

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

$0,1,...,10$ 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 $1.$ It is assumed that if a task is sent to $$

$,=$

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

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

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

Let $()$ be the total number of customers having departed from the system at time $,$ and $1()$

and $2()$ be the queue lengths at processors $1$ and $2,$ 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 $=0,1,...,10$ showing arrivals and departures $($assume that

$1(0)=2(0)=(0)=0).$

b$)$ Construct a table with the values of $1(),2()$ and $()$ for all $=0,1,...,10.$

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

$2\mathrm{.}0$ time units. Consider an event$-$driven model with event set $=\{,1,2\},$ where $=$

arrival, $=$ departure from processor $$

$,=1,2.$ Construct a table with the values of

$1$

$(),2(),(),(),$ where $1(),2(),()$ are the queue lengths and cumulative

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

of the $$

$$ 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