IEOR 4703: Solutions to Homework 6
Refer to the Lecture Notes 8 (Importance Sampling) (and Class Lecture 7) for the basics
needed for this assignment. (Only Problems 3(c)(d) requires programming/simulating.)
1. Consider the random walk
R
k
= Δ
1
+
· · ·
+Δ
k
, R
0
= 0, in which the iid Δ
i
are of the
form Δ
i
=
S
i

T
i
, where
{
S
i
}
are iid exponential at rate
μ
and independently the
{
T
i
}
are iid exponential at rate
λ
. This yields the famous M/M/1 queue (service
times
S
, interarrival times
T
) with arrival rate
λ
and service time rate
μ
.
We
assume (stability) that
ρ
=
λ/μ <
1 which is equivalent to the random walk having
negative drift,
E
(
S

T
)
<
0.
Our objective is to use importance sampling to
compute
P
(
D > b
), equivalently
P
(
M > b
). (Here
D
denotes stationary customer
delay, and
M
the maximum of the random walk.)
(a) Show that the Lundberg constant
γ >
0,
E
(
e
γ
Δ
) = 1
,
always exists, and find its value.
SOLUTION:
E
(
e
γ
Δ
) =
μ
μ

γ
×
λ
λ
+
γ
= 1
,
has solution
γ
=
μ

λ
and
γ >
0 as
λ < μ
(b) Letting
f
(
x
) denote the density function of Δ =
S

T
, and letting
g
(
x
) =
e
γx
f
(
x
), the alternative density for use in importance sampling, show that
in fact
g
(
x
) is the density of

Δ =
T

S
. This means that the alternative
density has the effect of simply swapping the two rates
λ
and
μ
! (Thus the new
value of
ρ
is now greater than 1.) That is, under
g
, the service times become
iid exponential at rate
λ
and interarrival times become iid exponential at rate
μ
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 sigman
 Normal Distribution, Variance, Probability theory, Random walk, Lundberg

Click to edit the document details