EE602 HW6 -- Due November 27 

(1) A person enters a bank and finds all of the 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 independent, identical and exponential distribution of service time.
(a) What is the probability that the person will be the last to leave
the bank assuming that no other customers arrive?
(b) If the average service time is 1 minute, what is the average time the
person will spend in the bank?

(2) Empty taxis pass by a street corner at a Poisson rate of 2 per minute
and pick up a passenger if one is waiting there. Passengers arrive at
the street corner at a Poisson rate of 1 per minute and wait for a taxi
only if there are fewer than four persons waiting. Otherwise, they leave
and never return. Find the average waiting time of a passenger who
joins the queue. 

(3) A communication node A receives Poisson traffic from two other nodes,
1 and 2, at rates lambda_{1} and lambda_{2}, resepctively, and transmits
it, on a first-come first-serve basis, using a link with capacity C bits/sec.
The two input streams are assumed independent and their packet lengths
are identically and exponentially distributed with mean L bits. A packet
from node 1 is always accepted by A. A packet from node 2 is accepted only
if the number of packets in A (in queue or under transmission) is less
than a given number K > 0; otherwise, it is assumed lost.
(a) What is the range of values of lambda_{1} and lambda_{2} for which
the expected number of packets in A will stay bounded as time increases?
(b) Draw a state transition diagram of the queue at A. 
(c) For lambda_{1} and lambda_{2} in the range of part (a), find the
steady-state probability of having n packets in A.