Could you help with this simulation hw? This one is intenseIEOR E4404 Simulation, Spring 2015
March 31, 2015
Assignment 4
Due date: April 14, 2015
Problem 1.
Consider a life insurance company that works as follows. Customers arrive according to
a Poisson process with rate
?
, and let the inter-arrival times be
X
1
,X
2
,...
. The
i
th individual stays
in the system for an amount of time
T
i
, which is uniformly distributed in the interval [0
,?
]. Thus, the
n
th individual leaves the system at time
X
1
+
···
+
X
n
+
T
n
.
T
i
are assumed to be i.i.d. Moreover, we
assume that
U
i
’s and
T
i
’s are independent. We wish to estimate the number of people who have left the
system in the time interval [0
,t
]. Let’s call it
M
(
t
).
(a) Argue that the expected number of people who have left the system by time
t
is equal to
Z
t
?
P
(
T
?
t
-
s
)
ds,
where
T
is uniformly distributed in the interval [0
,?
].
(b) Suppose that
t
=
?
. What is the distribution of the number of customers who are still in the
system at time
t
, assuming that the system is empty at time zero?
Hint
: Use the thinning theorem.
(c) Suppose that the company has zero customer at time zero, and that at time
?
the company has 5
customers in its portfolio. Conditional on this information, explain how to simulate the distribution
of the remaining times of these customers in the system.
Problem 2. (Two queues in series)
In this problem, we are interested in simulating a system with
two servers in series (sometimes called a
tandem
system).
The system consists of two servers with in?nite waiting room. Each customer must be ?rst served by
server 1, and upon completion of service at 1, goes over to server 2. An arriving customer is served by
server 1 immediately upon arrival if server 1 is free, otherwise she joins the queue at server 1; similarly
at server 2. Both servers serve customers in the order at which they arrive, i.e., both are ?rst-come-?rst-
serve (FCFS).
Let the system be empty at time 0. Let
A
n
be the arrival times of the
n
th customer,
S
(1)
n
his service
requirement at server 1, and
S
(2)
n
his service requirement at server 2. Let
W
(1)
n
and
W
(2)
n
be the waiting
times
in queue
of the
n
th customer at servers 1 and 2 respectively.
1
2
?
1
?
2
(a) Show that
W
(1)
n
+1
= [
W
(1)
n
+
S
(1)
n
-
(
A
n
+1
-
A
n
)]
+
for all
n
.
(b) Let
D
n
be the departure time of the
n
th customer from server 1. Show that
D
n
=
A
n
+
W
(1)
n
+
S
(1)
n
,
and
W
(2)
n
+1
= [
W
(2)
n
+
S
(2)
n
-
(
D
n
+1
-
D
n
)]
+
for all
n
.
4-1
This question was answered on: May 23, 2022
Solution~00021147719023.zip (25.37 KB)
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