Homework Assignment 7

This assigment will not be collected and
graded. However, the material will be
covered on the final examination.

1. I go to the store and buy five light
bulbs. Each has an expected lifetime of
one year. What is the probability that 
all will be working after 6 months?
After one year? What is the probability
any will be working after 10 years?

2. Customers arrive at a fast-food 
restaurant at the rate of five per
minute, and wait to receive their
orders for an averagee of five
minutes. Customers eat in the restaurant
with a probability of 0.5 and carry out
their orders with a proability of 0.5.
A meal requires an average of 20 minutes
to eat. What is the average number of
customers in the restaurant?

3. A person enters a bank and finds all
four clerks busy serving customers.
There are no other customers in the bank,
so the person will start service as soon
as one of the customers in service leaves.
Customers have an independent, identical,
sxponential distribution of service times.

(a) What is the probability that the person
will be the last one to leave the bank,
assuming no other customers arrive?

(b) If the the average service time is one
minute, what is the average time the customer
will spend in the bank?

(c) Will the answer to (a) change if there
are some additional customers waiting in
a common queue, and the customers begin
service in the order of arrival?

4. An absent-minded professor schedules two
student appointments for the same time. The
appointment durations are independent
and exponentially distributed with mean
thirty minutes. The first student arrives
on time but the second arrives five minutes
late. What is the expected time between the
arrival of the first student and the departure
of the second?

5. Persons arrive at a taxi stand with room for
five taxis according to a Poisson process with
rate one per minute. A person boards a taxi upon
arrival if one is available and otherwise waits
in line. Taxis arrive at the stand according to
a Poisson process with rate two per minute.
An arriving taxi that finds the stand full
departs immediately; otherwise, it picks up
a customer if at least one is waiting, or
else joins the queue of waiting taxis. What 
is the steady state distribution of the taxi
queue size? What is the probabilty an arriving
customer will find a taxi waiting? What is the
average wait for a customer? Hint: you need
only a single queue.

6. A communication line capable of transmitting
at a rate of 50Kbps will be used to accomodate
10 sessions each generating Poisson traffic at
a rate of 150 packets/min. Packet lengths
are exponentially distributed with a mean length
of 1000 bits.

(a) For each session, find the average number of
packets in its queue, the average number in the
system and the average delay per packet when the
line is allocated to the sessions by using:

  (1) 10 equal capacity time-division multiplexed 
  channels.

  (2) statistical multiplexing

(b) Repeat (a) for the case where 5 of the sessions
transmit at a rate of 250 packets/min while the
other 5 at a rate of 50 packets/min.