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：電機資訊學院
開課系所︰資訊所、網媒所
考試日期（年月日）︰2006 Jan 16
考試時限（分鐘）：3小時
是否需發放獎勵金：是
試題 :
Final Exam.
1. We have 2 homogeneous multiprocessors. Each processor is subjected
to failure with rate 0.2. We have a single repair facility with
rate 0.6. Jobs arrive at rate Poisson with rate 0.3. Whenever there
is no processor available, all jobs are lost. When k, for k=1 or 2
processors are functional, the service rate of customers is given
by k*0.6.
a. Draw the state transition diagram.
b. Determine if the system is stable or not and explain why.
2. For M/M/c queueing system, given a customer is queued, please find out
his/her waiting time dist. is P(D|D>0) ~ exp(c \mu - \lambda)
3. Consider an M/G/1 system with bulk service. Whenever the server becomes
free, he accepts 2 customers from the queue into service simultaneously,
or, if only one is on queue, he accepts that one; in either case, the
service time for the group (of size 1 or 2) is taken from B(x). Let
qn be the number of customers remaining after the nth service instant.
Let vn be the number arrivals during the nth service.
Define B*(s), Q(z), and V(z) as transforms associated with the random
variables x, q, v as usual, Let \rho = \lambda x/2
a. Using the method of imbedded Markov chain, find E(q) in term of
\rho, P(q=0) = p_0
b. Find Q(z) in term of B*(.), p_0, p_1 = P(q=1)
c. Express p_1 in terms of p_0
4. We have 2 systems. The first system is an M/M/3 queue with arrival rate
3\lambda and service rate \mu while the second system is an M/M/1 queue
with arrival rate 3\lambda and service rate 3\mu. What system yields the
smallest expected customer response time?
5. Consider a non-preemptive system and two customer class A and B with
respective arrival and service rate \lambda_a = 0.3, \mu_a 0.6, and
\lambda_b = 0.1, \mu_b = 0.3; As \mu_a > \mu_b, show that the average
delay per customer
T = \frac{\lambda_a T_a+\lambda_b T_b}{\lambda_a+\lambda_b} is smaller
when the priority of class A > the priority of class B than the priority
of class B > the priority of class A
6. Consider an M/G/1 queueing system in which service is given as follows.
Upon entry into service, a coin is tossed, which has probability p
of giving Heads. If the result is Heads, the service time for that customer
is 0 seconds. If Tails, his service time is drawn from the following
uniform distribution: f(x) = 1/(b-a), if a<x<b; otherwise f(x)=0
a. Find the average service time x
b. Find the variance of service time
c. Find the expected waiting time
d. Find W*(s)
7. Consider an M/G/1 system in which a departing customer immediately joins
the queue again with probability 0.3, or departs forever with probability
q = 0.7.
Given the arrival rate is \lambda, service rate is \mu, service is FCFS,
and the service time for a returning customer is independent of his
previousu service time. Please determine the stable condition for this
system (in term of \lambda, \mu)
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