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）︰2011.11.22
考試時限（分鐘）：120
試題 :
Performance Evaluation Midterm 11/22/2011 (120min)
1. (25%) Consider the M/G/1 system with the difference that each busy period
is followed by a single vacation interval. Once this vacation is over, an
arriving customer to an empty system starts service immediately. Assume that
vacation intervals are independent, identically distributed, and independent
of the customer inter-arrival and service times. Prove that the average
＿＿ ＿＿
λx^2 V^2
waiting time in queue is W = ────+ ── , where I is the average length
2(1-ρ) 2I
of an idle period, and show how to calculate I.
2. (20%) Consider a computer system that has a total of n disk drives and each
disk fails at an exponential rate of λ. There are 3 repairmen and each of
them independently fixes failed drives at an exponential rate of μ. (i.e.,
one failed disk is fixed by only one repairman).
(a) draw the state transition diagram.
(b) what is the average number of failed disk drives?
3. (20%) A telephone company establishes a direct connection between two cities
expecting Poisson traffic with rate 30 calls/min. The duration of calls are
independent and exponentially distributed with mean 3 min. Inter-arrival
times are independent of call durations. How many circuits should the
company provide to ensure that an attempted call is blocked (because all
circuits are busy) with probability less than 0.01? It is assumed that
blocked calls are lost (i.e., a blocked call is not attempted again).
4. (15%) 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 when the service time is exponential with mean 1/μ for job a
and the service time is exponential with mean 1/λ for job b?
5. (20%) Consider a system that is identical to M/M/1 except that when the
system empties out, service does not begin again until 2 customers are
present in the system (λ=1 is arrival rate and μ=4 is service rate). Once
service begins it proceeds normally until the system becomes empty again.
Find the steady-state probabilities of the number in the system, the average
number in the system, and the average delay per customer.
6. (20%) Consider the failures of a link in a communication network. Failures
occur according to a Poisson process with rate 4.8 per day. Find
(i) P[time between failures ≦ 10 days]
(ii) P[10 failures in 20 days]
(iii) Expected time between two consecutive failures.
(iv) P[0 failures in next day]
(v) Suppose 6 hours have elapsed since the last failure. Then, find the
expected time to next failure.