Stat 650 Spring 2007 HW7 Assignment, Due Friday, May 11, 2007.
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The reading for this HW set consists of these specific 
sections of Chapter 8: 5 and 6.1, plus material on 
Queueing from the Durrett book (Chapter 4, Secs. 4-6).

Six homework problems to solve and hand in. 
BUT NOTE THAT AS OF  5/10/07, PROBLEMS #2 AND THE PART 
OF DURRETT #8.25 (A) ASKING YOU TO JUSTIFY THE UNIQUENESS 
OF SOLUTIONS OF FORWARD AND BACKWARD EQUATIONS WILL BE 
TREATED AS OPTIONAL (ie will be graded together as a signle
extra-credit problem). SO YOU ONLY HAVE 5 PROBLEMS TO 
HAND IN.


p.367: #8.3.2. (Note that this HMC does NOT 
  have uniformly bounded rates q_i.)

#2. For the Birth-Death process given in 8.3.2, 
  show that for all parameter values the 
  forward and backward equations for  P(t)  both 
  hold and have the same unique solutions  P(t). 

  (The probabilistic intuition behind this is that
  when the rates do not increase with  n  any 
  faster than proportionally to  n , the expected
  time to hit  N  is at least of order  log(N), 
  i.e., it increases to infinity, preventing 
  explosions.)

HINT: there is now a handout on the main web-page 
indicating a general differential-equation method for
establishing existence and uniqueness of the forward and 
bacrkward equation solutions under the condition 
sup_i q_i < infty
but that condition does not hold in this problem. What 
you are meant to do here is to establish by the form of 
the equations themselves that the P(t) solutions exist 
and are unique.

#3. For the 3-state (states 1,2,3) continuous time 
  HMC with transition intensity matrix

                   (   -2     2     0   )
          Q   =    (    2    -3     1   )
                   (    2     2    -4   )

(a) Find the limiting probability that at a large fixed 
  time  t , the chain is in state 2 .

(b) Find the long-run proportion of time that the chain
  is in state 3 and state 2 will be the next state 
  visited.

(c) Find the average time taken between successive times 
  of entering state 2 in cycles which visit state 1
  exactly once.


#4. Suppose that  N(t)  is a Poisson(10) process starting 
at  0 , with successive jump times T_1, T_2, ...  
Let A_1, A_2 ,... be {0,1}-valued random 
variables with conditional probabilities

P(A_k = 1 | A_1, ..., A_{k-1}))
            =  .8  if  sum_{j=1}^{k-1} A_j  is even, and  
            =  .3  if  sum_{j=1}^{k-1} A_j  is odd.

Let  X(t) = sum_{j=1}^N(t) A_j mod 2 be the indicator that 
the cumulative sum of the first k=N(t)variables  A_j  is 
odd.  Show that X(t) is a continuous time  HMC. Find its 
transition intensity matrix Q and its transition 
probability function  P(t) .


Two more Qeueing related problems are assigned 
from Durrett's Chapter 4 : 

(Durrett #8.25, p.204) [Queue with impatient customers.]
Customers arrive at a single server according to a
Poisson process with rate lambda, and require an exponential 
amount of service with rate  mu . Customerrs waiting in line 
are impatient and if they are not in service they each 
independently leave at rate  delta   without regard to their 
position in the queue.  (a) Show that for any delta > 0 the 
system has a stationary distribution. Also justify as well as 
you can that the backward and forward equations for this HMC 
hold and have a unique solution. (b) Find the stationary 
distribution in the very special case where  delta=mu.

(Durrett #8.33, p.205) Two barbers share a common waiting 
room which has  N  chairs.  Barber  i  gives service at rate  
mu_i  and has customers that  arrive at rate  lambda_i < mu_i.  
Assume that customers always wait for the barber they came to 
see even when the other is free, but they go away if their 
preferred barber is occupied and the waiting room is full. Let 
N_{t,i} be the number of customers for barber  i  that are 
waiting or being served. Find the stationary distribution 
for (N_{t,1}, N_{t,2}).

HINT: try solutions pi with probability mass at allowed 
(n_1,n_2) states proportional to  a^(n_1) * b^(n_2) .