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STAT2005: Introduction to Stochastic Processes

ASSIGNMENT 3: October 2022

Question 1. [6 marks].

(a)  Using the moment generating function, find the mean and variance of the exponential distribution with rate parameter λ > 0, written Exponential(λ).

(b)  Prove the memoryless property of the exponential distribution: that if X has an expo- nential distribution,then

P[X > s + t1X > s] = P[X > t]

for all s, t > 0.

(c)  In the lecture notes we have seen that if X1 , . . . , Xn  are independent exponential ran-

dom variable with positive rate parameters λ1 , . . . , λn , respectively, then

Xmin  = min(X1 , . . . , Xn ) ~ Exponential(λ+ )

and J, the random variable dened by J = j if Xj  < Xk for all k  j, satises

λj

P[J = j] =

where

λ+  =ì λk .

k=1

Here, you are asked to prove that:

Xmin and J dened above are independent random variables.

Question 2. [12 marks]. A factory has produced m televisions which are for sale. Televi- sion i works for a random time Xi , where the Xi are independent and identically distributed with common distribution Exponential(λ) where λ > 0 is the rate parameter. Let N(t) de-

note the number of televisions that have failed by time t.

(a)  Calculate the probability P[N(t) = r], where r = {0, 1, . . . , m}.

(b)  Let τ1 denote the (random) time to the first failure. What is the distribution of τ1 ?

(c)  Let τ2 denote the time to the second failure. What is the distribution of τ2 _ τ1 ?

(d)  As defined, N(t) is a pure birth process. Justify this claim. What is the relevant state space? For each state, what is the birth rate?

(e)  Find the expectation and variance of the random variable

T = inf{t : N(t) = m},

t>0

where inf means min but without the assumption that a minimal element exists. Hint: you may nd results in Question 1 useful.

Question 3. [12 marks].

(a)  For what values of A, B , C and D is the matrix below a Q-matrix?

 ì                                        (1)

(D    0     0    _1|

Assume that for the remainder of this question these choices for A, B , C and D have been made.

(b)  Determine the one-step transition matrix for the jump chain. What is the communica- tion class structure of the jump chain?

(c)  Solve the set of equations

πQ = 04

where 04   =  (0, 0, 0, 0) is a row vector of zeros, π  =  (π1 , π2 , π3 , π4 ) and Q is the Q-matrix in (1).

(d)  Assuming that {X(t)}t0 is a Markov process on the state space S = {1, 2, 3, 4}, with Q-matrix given in (1), compute

lim P[X(t) = 21X(0) = 1].

t↓Q

(e)  Determine the stationary distribution of the jump chain.

Question 4.  [10 marks].  Consider a job shop that has 3 identical machines and 3 tech- nicians. Suppose that the amount of time each machine operates before breaking down is exponentially distributed with parameter 0. 1, while the time that a technician takes to fix a broken machine is exponentially dsitributed with parameter 0.5. Suppose that all the times to breakdown and times to repair are independent random variables and let X(t) denote the number of operating machines at time t > 0.

(a)  Determine the Q-matrix of the Markov process.

(b)  Write down theforward differential equations in terms of

pj (t) = P0j(t) = P[X(t) = j1X(0) = 0],     j = 0, 1, 2, 3.

(c)  Obtain the equilibrium probabilities pj  = limtQ pj , j = 0, 1, 2, 3.

(d)  What is the average number of busy technicians in the long run?