UNIVERSITY OF SOUTHERN CALIFORNIA, SPRING 2008
1
Lecture Notes 5
EE 549 — Queueing Theory
Instructor: Michael Neely
I. BACKLOG AND DELAY BOUNDS FOR LEAKY BUCKET
(
r, σ
)
INPUTS
Recall that
X
(
t
)
∼
(
r, σ
)
if
X
[
t
1
, t
2
]
≤
r
(
t
2

t
1
) +
σ
for all intervals
[
t
1
, t
2
]
. Consider a single server work
conserving queue, and assume:
•
X
(
t
)
∼
(
r, σ
)
input stream
•
single server queue
•
constant server rate
μ
•
initially empty system
(
U
(0) = 0)
Fact 1:
(Queue Bound) If
r
≤
μ
, then
U
(
t
)
≤
σ
for all time
t
.
Proof:
Fix a particular time
t
. We want to show that
U
(
t
)
≤
σ
. If
U
(
t
) = 0
, then we are done. Else,
U
(
t
)
>
0
¡
and so we are in a busy period that started at some time
t
b
, where
t
b
≤
t
. Thus:
U
(
t
)
=
X
[
t
b
, t
]

μ
·
(
t

t
b
)
≤
r
(
t

t
b
) +
σ

μ
·
(
t

t
b
)
≤
σ
Fig. 1.
An illustration of the unﬁnished work bound for leaky bucket inputs.
Fig. 1 illustrates the bound on unﬁnished work.
Corollary 1:
(Delay Bound) If service is FIFO then delay
≤
σ
μ
.
Proof:
Take any packet
i
. Let
t
i
be the time of the packet arrival, and let
U
(
t
i
)
be the unﬁnished work at this
time (which includes all bits that were in the queue just before the packet arrived, plus the total number of bits in
packet
i
). Then under FIFO, the delay of packet
i
is exactly
U
(
t
i
)
/μ
, which is less than or equal to
σ/μ
.
Fact 2:
(Summing Leaky Bucket Inputs) If
X
1
(
t
)
∼
(
r
1
, σ
1
)
and
X
2
(
t
)
∼
(
r
2
, σ
2
)
, then
X
1
(
t
) +
X
2
(
t
)
∼
(
r
1
+
r
2
, σ
1
+
σ
2
)
.
Proof:
X
[
t, t
+
T
]
=
X
1
[
t, t
+
T
] +
X
2
[
t, t
+
T
]
≤
(
r
1
T
+
σ
1
) + (
r
2
T
+
σ
2
)
=
(
r
1
+
r
2
)
T
+ (
σ
1
+
σ
2
)