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) = .003x2, 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:  tA;  ti = time of next service completion by server i, i = 1,2

System State: (n1,n2),  ni = # in i-th queueing system, i = 1,2

Counters:  NA = number of arrivals by time t

ND = number of departures by time t

Output:  A1(k) = arrival time of customer k

A2(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  g1(t) = 1 - ½t-1½ for 0 < t < 2  (use rejection method)

1. Second Server: service time has pdf   g2(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(U1) 0.29 2.59 1.13 0.53 0.41 0.56 1.91 0.27 0.03 1.07 0.04 U2 0.72 0.49 0.63 0.63 0.17 0.43 0.62 0.67 0.39 0.63 0.37

(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?