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 $=0,1,...,10.$