課程名稱︰隨機程序及應用
課程性質︰選修
課程教師︰葉丙成
開課學院：電資學院
開課系所︰電信所
考試日期（年月日）︰2010.1.13
考試時限（分鐘）：180
是否需發放獎勵金：是，謝謝
（如未明確表示，則不予發放）
試題 :
1. (2%) Write down your name on the top of every answer sheet.
2. (8%) Consider an experiment involving two independent Poisson processes
with rate λ1 and λ2. Let X1(k) and X2(k) be the times of the kth arrival
in the 1st and the 2nd processes, respectively. Show that
n+m-1 n+m-1 λ1 k λ2 n+m-1-k
P{X1(n) < X2(m)} = Σ ( ) (-------) (-------)
k=n k λ1+λ2 λ1+λ2
3. Suppose that a branch office of a bank is open between 10 A.M. and 2P.M.,
and that the pattern of the customer arrivals follows a nonhomogeneous
Poisson process. Assume that, between 10 and 11 A.M., customers arrive at
a constant Poisson rate of 10 per hour. Between 11 A.M. and 12 noon, the
rate increases steadily to 20 per hour and remains constant until 1:00 P.M.
From 1:00 to 2:00 P.M., the rate decreases steadily to 0.
(a) (6%) Plot λ(t), the arrival rate of the nonhomogeneous Poisson process
from 10 A.M. to 2 P.M.
(b) (6%) What is the probability distribution of the number of customers in
a day?
(c) (6%) What is the probability that no customers will arrive in the
period between 1:45 and 2:00 ?
4. Consider a single-server queue system in which the service times are expo-
nential at rate 1 per minute. Suppose that the potential customers arrive
at a Poisson rate λ per minute. However, if a potential customer finds
upon arriving that there are already n customers in the system, the proba-
bility is 1/(n+1) that he joins the queue and n/(n+1) that he leaves
immediately.
(a) (6%) Give the rate transition diagram.
(b) (8%) Calculate the steady state probabilities for this system.
(c) (6%) What is the average number of customers in the system, under
steady state condition?
5. Empty taxis pass by a street corner at a Poisson rate of two per minute and
pick up a passenger if one is waiting there. Passengers arrive at the street
corner at a Poisson rate of one per minute and wait for a taxi only if there
are less than four persons waiting; otherwise they leave and never return.
(a) (6%) Give the rate transition diagram.
(b) (6%) Calculate the steady state probabilities of the system.
(c) (6%) Find the expected number of people waiting for a taxi at the street
corner.
(d) (8%) Alice arrives at the street corner at a given time. Find her
expected waiting time, given that she joins the queue. Assume that the
process is in steady state.
(TA補充: 計程車一次只載一人，且先到的會先被載走)
6. Consider an M/M/r queue with arrival rate λ and service rate μ that
contains both patient and impatient customers at its input. If all servers
are busy, patient customers join the queue and wait for service, while
impatient customers leave the system instantly. Let p represent the proba-
bility of an arriving customer to be patient.
(a) (6%) Give the rate transition diagram for the queue. Define state n as
the state where total number of customers in the queuing system is n.
(b) (8%) Show that when pλ/rμ < 1, the steady state distribution in the
system is given by
n
(λ/μ)
┌ ----------p0 , n ＜ r
│ n!
Pn = ┤ ,
│ r
│ (λ/μ) n-r
└ ----------(pλ/rμ) p0 , n ≧ r
r!
where
1
p0 = ------------------------------------ .
r-1 (λ/μ)^n (λ/μ)^r
Σ ----------- + -------------
n=0 n! r!(1-pλ/rμ)
7. Consider a machine that works for an exponential amount of time having mean
1/λ before a breaking down; and suppose that it takes an exponential amount
of time having mean 1/μ to repair the machine.
(a) (6%) Give the rate transition diagram that describes the condition of
the machine.
(b) (6%) Give the Kolmogorov's backward and forward equations for the
process described in (a).
(c) (8%) If the machine is in working condition at time 0, then what is the
probability that it will be working at time t ?
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