IEOR 4703: 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.
(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
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 Spring '07
 sigman
 Normal Distribution, Variance, Probability theory, Random walk, Lundberg

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