課程名稱︰系統效能評估 課程性質︰選修 課程教師︰周承復 開課學院：電資學院 開課系所︰資工所、網媒所 考試日期（年月日）︰2015.05.06 考試時限（分鐘）：120 試題 : Mid-term exam. Perf. 2015 (120 min) 1. (25%) Consider a system with two computers. Two jobs: a and b are submitted to that system simultaneously. Job a and b are assigned directly to each computer. What is the probability that job a is still in a system after job b finished? i. (5%) the service time for each job is exactly 10 mins ii. (5%) the service time for each job are i mins with prob. 1/3, i = 1, 2, 3 iii. (5%) the service time for job a is exponential with mean 15 mins, the service time for job b is exponential with mean 10 mins? iv. (10%) the service time for job a is uniformly distributed [0, 20 mins], and the service time for job b is uniformly distributed [0, 10 mins] 2. (20%) Consider the failure of a link in a communication network. Failures occur according to a Poisson process with rate 2.4 per day. Find a) P[time between failures ≦ 10 days] b) P[5 failures in 7 days] c) Expected time between 2 consecutive failures d) P[0 failures in next day] 3. (20%) Consider the following probability transition matrix for states 1, 2, and 3. P = 0.6 0.2 0.2 0.1 0.8 0.1 0.6 0.0 0.4 Prove that the chain is irreducible and determine the steady-state probabilities. 4. (15%) f''(t) - 5f'(t) + 6f(t) = t, f(0) = 0, f'(0) = 0; find out f(t) 5. (10%) The number of transactions arriving into a database system forms a Poisson process with rate λ. The database consists of 2 distinct files. An arriving transaction requests 1st file with probability p, and 2nd file with probability 1-p. With the usual independence assumption, prove that the number of requests directed to 1st file forms a Poisson process of rate λp. 6. (15%) Is it true that (if not, please explain the reason) : i. N(t) ＜ n <=> S_n ＞ t ii. N(t) ≦ n <=> S_n ≧ t iii. N(t) ＞ n <=> S_n ＜ t 7. (10%) We wish to determine the maximum call rate can be supported by one telephone booth. Assume that the mean duration of a telephone conversation is 3 min, and that no more than a 3-min (average) wait for the phone may be tolerated; what is the largest amount of incoming traffic can be supported? --

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