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）︰ 考試時限（分鐘）： 試題 : Performance Evaluation 2003 Fall Final Exam 1. We have 2 systems. The first system is an M/M/2 queue with arrival rate 2λ and service rate μ while the second system is an M/M/1 queue with arrival rate 2λ and service rate 2μ. What system yields the smallest expected customer response time? 2. In our system, there are k machines and a single repairman. Each machine breaks down after a time that is exponentially distributed with parameter α. When a breakdown occurs, a request is send to the repairman for fixing it. Requests are buffered. It takes an exponentially distributed amount of time with parameter μ for the repairman to repair a machine. What is the probability p(i) that i machines are up? What is the overall failure rate? 3. There are 2 independent Poisson processes. One has mean arrival rate λ_1 and the other has mean arrival rate λ_2. Prove that the superposition of 2 independent Poisson processes with rate λ_1 + λ_2. 4. Consider a non-preemptive system and 2 customer classes A and B with respective arrival and service rate λ_a, μ_a and λ_b, μ_b; if μ_a > μ_b, show that the average delay per customer λ_a T_a + λ_b T_b T = ────────── is smaller when the priority of class A > the λ_a + λ_b priority of class B than the priority of class B > the priority of class A. 5. M/G/1 queue with random-size batch arrivals. Consider the M/G/1 system with the difference that customers are arriving in batches according to a Poisson process with rate λ. Each batch has n customers, where n has a given distribution and is independent of customer service time. Show that the waiting time in the queue is ＿＿＿ ＿ ＿＿ ＿ λn X^2 X (n^2 - n) W = ─────+ ─────── 2(1-p) 2 n(1-p) ↑這個n上面有一橫 6. Let X_L and X_R be independent Poisson distribution random variables, each with mean G. Find a. P[X_L ＝ 0 | X_L + X_R ≧ 2] b. P[X_L ＝ 1 | X_L + X_R ≧ 2] c. P[X_L ≧ 0 | X_L + X_R ≧ 2] d. P[X_L ＝ 1 | X_L ＝ 1, X_L + X_R ≧ 2] e. P[X_L ＝ i | X_L ＝ 0, X_L + X_R ≧ 2], i ≧ 2 f. P[X_L ＝ i | X_L ≧ 2, X_L + X_R ≧ 2] --

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