Problems: Problem 1: Consider a renewal system, with cycle durations having cumulative distributed function F and density f. The spread at time t is the duration of the cycle containing t. Determine the equilibrium density of spread.
Problem 2: Consider a computer system that needs to implement some form of control for how jobs from various users are allowed to access system resources. Jobs that are granted access to the system are immediately dispatched for processing. User i generates new jobs into the system according to a Poisson process of rate Ai. Access control is enforced by way of permits, with each new job requiring a permit in order to enter the system. User i starts with a total number of permits equal to with permits being refreshed, i.e., reset to every T time units. Note that this means that user i never has more than Ni permits, and wastes any permit that was not used before a refresh takes place.
Assuming that when permits have been exhausted, incoming jobs are dropped, derive first an expression for the rate at which the jobs are dropped expected, and next identify a method to determine how Ni should be set to ensure that the fraction of dropped jobs is at most €, 0 < < 1.
Problem 3: A subway station has both local and express service, on opposite sides of the same plat¬form. Local trains arrive every 5 minutes (constant), and express trains arrive every 15 minutes (constant), scheduled so that every third local train arrives simultaneously with an express train. Both trains stop at your destination, with transit times of 17 minutes for a local train and 11 minutes for an express train.
You arrive at random on the station platform. Your objective is to minimize your expected travel time, E(T) say, from your arrival epoch at the station until you reach your destina¬tion.
1. What is the waiting time distribution until the next local train arrives? The next ex¬press train?
2. What is the probability that the next local train arrives alone, i.e., without an express? What is the probability that the next two locals arrive alone? Briefly explain. (Hint: Interpret probabilities as fractions of time.)
3. If the next local train arrives alone, should you board that train or wait for an express?
4. Given your decision in part 3, find E(T).
5. If local and express trains are boarded from different platforms, on which platform should you wait? (Running between platforms is not allowed.)
Problem 4: An absent-minded professor schedules two student appointments for the same time. The appointment durations are independent and exponentially distributed with mean thirty minutes. The first student arrives on time, but the second student arrives five minutes late. What is the expected time between the arrival of the first student and the departure of the second student?
Problem 5: Suppose X1, X2, X3, . . . are independent random variables each with finite first and second moments. Also, suppose X2, X3, X4, ... are identically distributed with mean EX. X1 does not have a bounded support, i.e. there does not exist a constant c such that 1 X 11.
Ein = i xz Ex.