Design and Analysis of Computer Networks
Homework Assignment 2
1. Suppose that a rare disease has an average incidence of 1 in every 1000 persons. Assume that
members of the population are affected independently and the number of affected follows Poisson
distribution. Find the probability of k cases in a population of 10,000 for k=0,1,2 respectively.
2. If new cases of West Nile in New England are occurring at a rate of 2 per month, then
(a) What’s the probability that exactly 4 cases will occur in the next 3 months?
(b) What’s the probability that exactly 6 cases will occur in the next 3 months?
3. Customers arrive at a restaurant according to a Poisson process with rate 10 customers/hour.
The restaurant opens daily at 9:00 am. Calculate the following:
(a) When the restaurant opens at 9:00 am, the workers need 30 min to arrange the tables and chairs.
What isthe probability that they will finish the arrangement before the arrival of the first customer?
(b) Given that a new customer arrived at 9:13 am, what is the expected arrival time of the next
(c) If a customer arrive at restaurant at 2:00 pm, what is the probability that the next customer will
arrive before 2:10 pm.
4. Consider a small bank with one teller. Customers arrive to the bank according to a Poisson
process with rate 8 customers per hour. The teller provides all kinds services for the customers.
Each customer takes on average 5 minutes to service. Assume that the service time is exponentially
distributed. In steady-state, calculate the following:
(a) What is the probability that the teller is idle?
(b) What is the average number of customers waiting for service?
(c) On average, how long will a customer spend in the bank to complete his service?
(d) What is the probability that there are more than 5 customers in the bank?
5. Consider a continuous-time Markovian system with discouraged job arrivals. Jobs arrive to a
server according to a Poisson process, with an arrival rate of one job per 7 seconds. The jobs
observe the queue. They do NOT join the queue with probability lk if they observe that there are k
jobs in the queue (This only refers to the number of jobs in the queue. The job being serviced, if
any, is not included in this number.). lk = k/4 if k < 4, or 1, otherwise. The service time is
exponentially distributed with mean time of 6 seconds.
(a) Please draw the state transition diagram for this queueing system;
(b) Write the Balance Equation for each state. If this is a birth-death process, please only write the
Detailed Balance Equations;
(c) Determine the stationary distribution of the number of jobs in the system, and also calculate
the mean number of jobs in the system;
(d) When the system becomes stationary, in an interval of 100 seconds, on average how many jobs
enter the system (hint: when the system is stationary, the average number of jobs entering the
system equals to the average number of jobs finishing their service then leaving the system)?
6. Consider a gas station located on a highway with five pumps. Cars arrive at the gas station
according to a Poisson process at rate 50 cars/hour. Any car able to enter the gas station stops by
one of the available pumps. If all pumps are occupied, the car will not enter the gas station and
will just leave. Each car takes an exponential amount of time to refill, and the average refill time
is 5 minutes.
(a) Draw the state transition diagram for the gas station.
(b) Determine the stationary distribution of the number of cars in the system.
(c) What is the probability that an arriving car will NOT be able to enter the gas station to refill?
In 24 hours, on average how many cars cannot enter the gas station and thus have to leave (If you
are the owner of the gas station, this is the business you will lose)?
(d) Now consider that you have bought a small parking lot right beside the gas station, so that a
car can stop there and wait for any pump becomes available. Suppose that there are in total 2 spaces
in the parking lot. An arriving car will not enter the gas station and instead leave immediately if
all parking spaces are occupied. How many business will be lost in 24 hours in this case?
7. Consider a communication link with a constant rate of 4.8kbit/s. Over the link we transmit two
types of messages, both of exponentially distributed size. Messages arrive in a Poisson fashion
with rate 10 messages/second. With probability 0.5 (independent from previous arrivals) the
arriving message is of type 1 and has a mean length of 300 bits. Otherwise a message of type 2
arrives with a mean length of 150 bits. The buffer at the link can at most hold one message of type
1 or two messages of type 2. A message being transmitted still takes a place in the buffer.
(a) Draw the state transition diagram for the system. Note that in this case the state cannot simply
be defined as the number of messages in the system, as messages are of different types.
(b) Determine the average times in the system for accepted messages of type 1 and 2, respectively.
(c) Determine the message loss probabilities for messages of type 1 and 2.