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)

























(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?