stochastics

Problem 1 [10 points]

Random variables, {Tj : j ≥ 1} are independent with a common exponential density function,

g(t) = λ · exp(−λ · t) for t > 0,

with λ =

5

per hour. Introduce the sums,

Wk =
k∑

j=1

Tj and W0 = 0.

Consider a process, N = {N(t) : t ≥ 0} defined as follows:

[N(t) = n] ⇐⇒ [Wn ≤ t < Wn+1]

1. Derive E [W1 |W

3

≤ 1 < W4]

2

. Evaluate expectation E [W1 |W4 = 2]

Show answers in minutes, please!

Solution

2

Problem 2 [10 points]

Random variables, {Tj : j ≥ 1} are independent with a common exponential density function, g(t) =
λ · exp(−λ · t) for t > 0, with λ = 5 per hour. Introduce the sums,

Wk =
k∑
j=1
Tj and W0 = 0.

Consider a process, N = {N(t) : t ≥ 0} defined 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 = {N(t) : t ≥ 0} such that the
first arrival time, W1 = S1, has E [S1|N(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 first 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 first 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 λ = 3 per hour, and service times are independent exponentially
distributed with µ−1 = 5 minutes.

1. Derive the average queue length, E [X(t)], assuming that the process X = {X(t) : t ≥ 0} follows the
stationary distribution.

2. Evaluate expected busy time.

Solution
6

Problem 10 [10 points]

Consider a Poisson process, N = {N(t) : t ≥ 0} with rate λ = 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 < W

11

2. Find conditional expectation of the W4, given that the tenth arrival occurred exactly at W10 = 5.5.

Solution
11