ELEC 3703   Queuing Theory
			    Practice Exercise 1


1.  Determine the mean and variance of a binomial random variable with
    parameters B(n,p), where n is a positive integer and 0 <= p <= 1.


2.  Show that an exponential random variable is memoryless.


3.  Let X and Y be the exponential random variables with parameters a and b
    respectively.  Find the distribution of the random variable Z = min(X,Y).


4.  (a)  Compute the z-transform of:
	 (i)  a binomial random variable with parameters B(n,p), and;
	 (ii) a Poisson random variable with mean c.

    (b)  By using the results of (a), compute the mean and variance of the
	 binomial and Poisson random variables.


5.  Let X_1, X_2, ..., X_n be Poisson random variables with mean c_i,
    i = 1, 2, ..., n.  By using z-transform, show that
    Y = X_1 + X_2 + ... + X_n is also Poisson.


6.  (a)  A network node N receives packets from two different nodes, A and B.
	 Packets arriving from A and B are described by a Poisson distribution
	 with parameters c_1 and C_2, respectively.  What is the probability
	 density function (p.d.f.) of the arrival process at node N?
	 Hint: use the result from Question 5.

    (b)  Given that a packet arrived at t_1, what is the probability that no
	 packet arrives in the interval (t_1,t_2]?

    (c)  Given that a packet arrived at time t_0, and no packet arrived at
	 time (t_0,t_1), what is the probability that no packet arrives in the
	 interval (t_1,t_2]?