Several tasks are submitted to a computer system with two processors, P1 and P2, working in parallel. The process of submitting tasks can be described in discrete time

by a(t), t = 0,1,2,... where a(t) = 1 if a task is submitted at time t and a(t) = 0 oth- erwise (at most one task can be submitted in each time step). Suppose such a process is specified for the time interval t = 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 P1. It is assumed that if a task is sent to Pi, i = 1,2, and that processor is busy, the task joins a queue of infinite capacity. The processing time of a task at P1 alternates between 4 and 1 time units (starting with 4), whereas the processing time at P2 is always 2 time units.

Let y(t) be the total number of customers having departed from the system at time t, and x1(t) and x2(t) be the queue lengths at processors P1 and P2, respectively (in- cluding 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,...,10 showing arrivals and departures (assume that x1(0) = x2(0) = y(0) = 0).

(b) Construct a table with the values of x1(t), x2(t) and y(t) for all t = 0, 1, . . . , 10.

(c) Suppose we now work in continuous time. Arrivals occur at times 0.1, 0.7, 2.2, 5.2 and 9.9. The processing time at P1 now alternates between 4.2 and 1.1, whereas that of P2 is fixed at 2.0 time units. Consider an event-driven model with event set E = {a,d1,d2}, where a = arrival, di = departure from processor Pi, i = 1, 2. Construct a table with the values of x1(k), x2(k), y(k), t(k), where x1(k),x2(k),y(k) are the queue lengths and cumulative number of departures after the kth event occurs, k = 1,2,..., and t(k) is the time instant of the kth 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 0.1 time units.