Stochastics

Problem 1 [10 points]
Random variables, fTj : j  1g are independent with a common exponential density function,
g(t) = 
exp(
t) for t > 0;
with  = 5 per hour. Introduce the sums,
k
X
Wk =
Tj and W0 = 0:
j=1
Consider a process, N = fN(t) : t  0g dened as follows:
[N(t) = n] () [Wn  t < Wn+1]
1. Derive E [W1 jW3  1 < W4]
2. Evaluate expectation E [W1 jW4 = 2]
Show answers in minutes, please!
Solution
2
Problem 2 [10 points]
Random variables, fTj : j  1g are independent with a common exponential density function, g(t) =

exp(
t) for t > 0; with  = 5 per hour. Introduce the sums,
k
X
Wk =
Tj and W0 = 0:
j=1
Consider a process, N = fN(t) : t  0g dened as follows: [N(t) = n] () [Wn  t < Wn+1]
1. Derive expectation of W5; given that W2 = 1 (in hours).
2. Evaluate expectation of the ratio, (W5=W2)
Show answers in minutes when appropriate, please!
Solution
3
Problem 4 [10 points]
Consider a small service with arrivals described as a Poisson process, N = fN(t) : t  0g such that the
rst arrival time, W1 = S1; has E [S1jN(0) = 0] = 6 minutes, or (0.1) of an hour.
1. Find conditional expected value for a number of customers arrived by the end of rst hour, given
that by t = 3 hours there were ten customers.
2. Evaluate expected number of customer by t = 3 hours, given that by the end of rst hour there were
four customers.
Solution
5
Problem 5 [10 points]
Consider a queuing system, M=M=1 with one server and parameters such that customer arrivals are
described by a Poisson process with  = 3 per hour, and service times are independent exponentially
distributed with 
1
= 5 minutes.
1. Derive the average queue length, E [X(t)]; assuming that the process X = fX(t) : t  0g follows the
stationary distribution.
2. Evaluate expected busy time.
Solution
6
Problem 10 [10 points]
Consider a Poisson process, N = fN(t) : t  0g with rate  = 2 arrivals per hour. Introduce arrival times,
W0 = 0 and Wk = min [t  0 : N(t) = k] for k  1:
Assume that inspection occurs at t = 5:5 hours.
1. Evaluate conditional expectation of the forth arrival, given that the W10  5:5 < W11
2. Find conditional expectation of the W4, given that the tenth arrival occurred exactly at W10 = 5:5:
Solution
11