Use the Generalized Birth Death theorem of Chapter 10 to prove that Eqs. (7.3.4)-(7.3.5) are the solution to the Markov Chain of Section 7.3.
Suppose that the data center uses a load balancer to equally distribute the load among the M machines. Assume that the overall arrival rate of requests to the data center is g requests/sec. If j machines are in operation, each sees an average arrival rate of g/j requests/sec. Assume that the average response time of a request once at an operational machine is given by S/(1 (g/j).S) where S is the average service time of a request at a machine. Assume also that the load balancer does not route requests to failed machines. Assume further that if a machine fails, any requests in execution at that machine are lost and can be ignored for the purpose of this exercise. Give and expression for the average response time of a request at the data center.
Use the Chap7-MarkovModel.XLS MS Excel workbook to draw graphs of the probability that exactly P_{j} (j = 0, ···, M) machines are operational as a function of the ratio l/m. Use M = 120 and draw curves for N = 2, 5, and 10.
Use the Chap7-MarkovModel.XLS MS Excel workbook to generate a table of MTTR for various values of the ratio l/m assuming M = 120 and N = 5.