3: Consider the two discrete time markov chains shown in the following figure. Assume you start at state 1. You are interested in the long-term (limiting) probability of finding the chain in state 3). (a) For which of the two chains is the above probability higher? Justify your answer. (b) Which of the two chains converges to its stationary distribution faster? Justify your answer. (c) You want to download a large file, and you are offered 4 options of servers to choose from. All servers have the same arrival rate for traffic, Poisson with rate 2 = 0.5, but differ in terms of the service times: Server 1: file download times are exponentially distributed with mean 1 (time unit). Server 2: all files take exactly 1-time unit to download. Server 3: file download times are uniformly distributed in (0,2). Server 4: file download times are either exactly 0.2 with probability 0.9 or 8.2 with probability 0.1. Which server would you prefer if you would want to minimize the (queueing) time you wait until your file starts downloading? Rank all 4 servers in terms of your preference. . . 002 0.04 0.2 1 2 3 3 0.02 0.02 0.1 0.1 0.98 0.94 0.98 0.9 0.7 0.9 (a) (b) 3: Consider the two discrete time markov chains shown in the following figure. Assume you start at state 1. You are interested in the long-term (limiting) probability of finding the chain in state 3). (a) For which of the two chains is the above probability higher? Justify your answer. (b) Which of the two chains converges to its stationary distribution faster? Justify your answer. (c) You want to download a large file, and you are offered 4 options of servers to choose from. All servers have the same arrival rate for traffic, Poisson with rate 2 = 0.5, but differ in terms of the service times: Server 1: file download times are exponentially distributed with mean 1 (time unit). Server 2: all files take exactly 1-time unit to download. Server 3: file download times are uniformly distributed in (0,2). Server 4: file download times are either exactly 0.2 with probability 0.9 or 8.2 with probability 0.1. Which server would you prefer if you would want to minimize the (queueing) time you wait until your file starts downloading? Rank all 4 servers in terms of your preference. . . 002 0.04 0.2 1 2 3 3 0.02 0.02 0.1 0.1 0.98 0.94 0.98 0.9 0.7 0.9 (a) (b)