课程名称︰系统效能评估
课程性质︰选修
课程教师︰周承复
开课学院：电资学院
开课系所︰资工所、网媒所
考试日期（年月日）︰
考试时限（分钟）：
试题 :
Midterm of Performance Modeling 2010
1. (20%) Consider a system with 2 components. We observe the state of the
system every hour. A given component operating at time n has probability
0.3 of failing before the next observation at time n+1. A component that
was in a failed condition at time n has a probability 0.6 of being repaired
by time n+1, independent of how long the component has been in a failed
state. The component failures and repairs are mutually independent events.
Let Xn be the number of components in operation at time n. {Xn| n=0,1,2,…}
is a discrete time homogeneous Markov chain with state space I={0,1,2}.
Determine its transition probability matrix P, and draw the state diagram.
Obtain the stead-state probability vector, if it exists.
2. Consider a server with Poisson job-arrival stream at an average rate of 60
per hour. Determine the probability that the time interval between
successive job arrivals is
(a) Longer than 4 min
(b) Shorter than 8 min
(c) Between 2 and 6 min
3. We are given two independent Poisson arrival streams {Xt| 0≦t＜∞} and
{Yt| 0≦t＜∞} with respective arrival rate λx and λy. Show that the
number of arrivals of the Yt process occurring between 2 successive
arrivals of Xt process has a modified geometric distribution with parameter
λx / (λx + λy). (A random variable z is said to have a modified geometric
pmf, specified by p_z(i) = p(1-p)^i for i=0,1,2,…
4. 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 0.6 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) = 1/10, if 0 < x < 10; 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)
5. 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 r = λx/2.
(a) Using the method of imbedded Markov chain, find E(q) in term of r,
P(q=0) = p0.
(b) Find Q(z) in term of B*(.), p0, p1 = P(q=1).
6. Customers arrive at a fast-food restaurant at a rate of five per minutes and
wait to receive their order for an average 5 minutes. Customers eat in the
restaurant with prob. 0.5 and carry out their order without eating with
prob. 0.5. A meal requires an average of 20 minutes. What is the average
number of customers in the restaurant?
7. Empty taxis pass by a street corner at a Poisson rate of 2 per minutes and
pick up a passenger if one is waiting there. Passengers arrive at the street
corner at a Poisson rate of 1 per minute and wait for a taxi only if there
are fewer than 4 persons waiting; otherwise, they leave and never return.
Find the average waiting time of a passenger who joins the queue.