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]