Reference no: EM133044983

Jobs arrive at a processing center according to a Poisson distribution with rate of 1 job per day. However, the center has waiting space for only 2 jobs and so an arriving job finding 2 other jobs waiting goes away. At most 1 job per day can be processed, and the processing of this job must start at the beginning of the day. Thus, if there are any jobs waiting for processing at the beginning of the day, then one of them is processed that day, and if no jobs are waiting at the beginning of the day then no jobs are processed that day.

a) Let Xn denote the number of jobs at the center at the beginning of day n. Model {Xn, n ≥ 1}, i.e. the evolution of the number of jobs at the beginning of the day, as a discrete-time Markov chain by providing its corresponding graphical representation (i.e., states and transition probabilities). In addition, provide the corresponding transition probability matrix.

b) Simulate the evolution of the number of jobs at the center at the beginning of the day to estimate the expected fraction of days on which not a single job can be accepted and the expected number of jobs waiting for processing at the beginning of the day. Consider a minimum of 10 30-day replications and provide the corresponding confidence intervals. Begin the simulation with only one job at the center at the beginning of the day.

c) Determine the stationary (i.e., steady state) probabilities associated with this processing center model. Using these probabilities, compute the long-term fraction of days on which not a single job can be accepted and the long-term average number of jobs waiting for processing at the beginning of the day. Compare the results with those obtained in part b) and explain any differences.

d) What other quantities of interest would you consider to evaluate the performance of the system described above?

e) What changes would need to be made to your simulation model from part b) to evaluate the impact of an increase in the waiting space on the performance measures mentioned above. Note: You don't need to modify your model or create a new one; just explain how you would do it (i.e., what would need to change).

f) Describe at least two advantages and two disadvantages of using simulation vs. steady state analysis to studying the performance of this type of systems.