**Math 286/366 Extra Credit**

**Spring, 2005**

** **

**1. Compound
Poisson Surplus Process.**

Claims come into an insurance company according to a Poisson
process with mean 5 claims per year. The claims are independent and identically
distributed with pdf f(x) = .003x^{2},
0 < x < 10.

What is the expected value of a single claim? The expected total amount of claim payments in one year?

The insurance company collects an annual premium c = (4/3)E[S], where S is the random variable that is the sum of all claim payments in a year. The company begins the year with a surplus of 5. Assuming that the premium is collected continuously, the “surplus process” U(t) has the form

U(t) = 5 + ct - S(t), 0 < t < 1,

where S(t) is the total of claim payments at time t (t in years).

We say “ruin occurs” if the value of U(t) falls below zero at any time during the year. Design an algorithm or run a simulation to determine the probability that ruin occurs.

You may use an Excel spreadsheet.

**2. Two Servers in
Series**.

Example: McDonald’s drive-up windows. (A bank is NOT an example)

Event list: t_{A}; t_{i} = time of next service
completion by server i, i = 1,2

System State: (n_{1},n_{2}), n_{i} = # in i-th queueing system, i
= 1,2

Counters: N_{A}
= number of arrivals by time t

N_{D} = number of departures by
time t

Output: A_{1}(k)
= arrival time of customer k

A_{2}(k) = arrival time of customer
k at server 2

D(k) = departure time of customer k

Simulate two servers in series for customers arriving in the first two minutes (all arrivals get complete service). Put your results in a matrix as we did in class. Arrivals are Poisson(2).

1.
First Server: service time has pdf g_{1}(t) = 1 - ½t-1½ for
0 < t < 2 (use rejection method)

- Second
Server: service time has pdf g
_{2}(t) = .7u(t) + .3h(t), where u(t) is the pdf for the Uniform(0,1) distribution and h(t) is the pdf for the Exponential(1/2) distribution. (use composition method)

Use the following table as needed (left to right)

-ln(U |
.29 |
2.59 |
1.13 |
.53 |
.41 |
.56 |
1.91 |
.27 |
.03 |
1.07 |
.04 |

U |
.72 |
.49 |
.63 |
.63 |
.17 |
.43 |
.62 |
.67 |
.39 |
.63 |
.37 |

(a) What is the average time spent in the system?

(b) Calculate the average time spent waiting for service (not being served). What additional output variables would be helpful?