1. The number of hits on a popular Web page follows a Poisson process. The average time
between arrivals is 6 seconds (or .1 minutes). One begins observation at exactly noon
tomorrow (WOI standard time).
(a.) What is the distribution of the number of hits in the first 2 minutes? State a random
variable, and give the name of the distribution and the value(s) of parameter(s). What
is the probability of 8 or less hits in the first minute?
(b.) What is the distribution of the time until the first hit? State a random variable and
give the name of the distribution and the value(s) of parameter(s). Compute the
probability that the time until the first hit exceeds 10 seconds.
(c.) Find the mean and the variance of the time until the third hit.
(d.) Evaluate the probability that the time till the third hit exceeds 24 seconds.
2. The drive-through at Fancy Fast Food has one window and enough space for an infinitely
long waiting line. Customers arrive at a rate of 6 per hour. The average service time is
2 minutes. Inter-arrival times and service times are independent exponentially distributed
random variables. Using what you know about M/M/1 queuing systems, answer the following questions.
(a.) What is the (large t) probability that the drive-through is empty? (In other words,
what is the steady-state probability that there are no customers in the drive-through?)
(b.) What is the (steady-state) expected value of the time a customer spends in the system
(time waiting plus time in service)?
(c.) The attendant arrives at 12:00pm. What is the distribution of the number of customers
that arrive in the first 3 hours? give the name of the distribution and the value(s) of
parameters. How many customers does the attendant expect will arrive in the first 3
hours?
3. A small communication system has two processors and a buffer that holds at most one
message. (The total capacity of the communication system is 3 messages). Messages arrive
to the system at a rate of 2 per minute. Each processor decodes messages at a rate of 1
per minute.
(a.) Draw a state diagram to represent the communication system. Show all possible states
along with the corresponding birth/death rates.
(b.) Let X be the steady-state (or large t) number of messages in the system. Write down
the pmf and cdf of X.
(c.) What is the (large t) expected number of messages in the buffer ? (The buffer is the
queue or “waiting line.”)
(d.) What is the expected time that a message spends in the buffer? If you were unable to
answer part (c.), you may assume that the expected number of messages in the buffer
is .1 messages. (Hint: Use Little’s Law).
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4. Messages arrive to a communication system according to a Poisson process. The average
time betwen arrivals is 5 minutes. The system begins to operate at 8:00am.
(a.) What is the distribution of the time until the first message arrives? State a random
variable, and give the name of the distribution and the value(s) of parameter(s). What
is the probability that the first message arrives before 8:15am? (7 pts.)
(b.) What is the distribution of the number of messages that arrive in the first ten minutes?
State a random variable, and give the name of the distribution and the value(s) of
parameter(s). What is the probability that no messages arrive in the first ten minutes?
(c.) What is the distribution of the time until the 6th message arrives? State a random
variable, and give the name of the distribution and the value(s) of the parameter(s).
How long do you expect to have to wait for the arrival of the 6th message?
Now suppose that the system has two processors for decoding messages and holds at most
four messages. (If a new message arrives when there are already four messages in the
system, the new message is lost). Assume that the decode times of the two processors are
independent, exponentially distributed random variables. Processor A decodes messages
at a rate of 1 per minute, and Processor F decodes messages at a rate of 2 per minute.
Because Processor F is faster than Processor A, if there is only one message in the system,
then Processor F decodes that message.
(d.) Draw a state diagram to represent the system. Indicate all possible states along with
the appropriate birth and death rates.
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