(Solution) IEOR E4404 Simulation, Spring 2015 March 31, 2015 Assignment 4 Due Date: April 14, 2015 Problem 1. Consider A Life Insurance Company That Works As... | Snapessays.com


(Solution) IEOR E4404 Simulation, Spring 2015 March 31, 2015 Assignment 4 Due date: April 14, 2015 Problem 1. Consider a life insurance company that works as...


Could you help with this simulation hw. This one is really intense.IEOR 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

 


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