Hindawi Publishing Corporation
EURASIP Journal on Wireless Communications and Networking
Volume 2011, Article ID 738386, 5 pages
doi:10.1155/2011/738386
Research Article
An Efficient Method for Proportional Differentiated Admission
Control Implementation
Vladimir Shakhov
1
and Hyunseung Choo
2
1
Institute of Computational Mathematics and Mathematical Geophysics of SB RAS, Prospect Akademika Lavrentjeva, 6,
Novosibirsk 630090, Russia
2
School of Information and Communication Engineering, Sungkyunkwan University, Chunchun-Dong 300, Jangan-Gu,
Suwon 440-746, Republic of Korea
Correspondence should be addressed to Hyunseung Choo, [email protected]
Received 14 November 2010; Accepted 11 February 2011
Academic Editor: Boris Bellalta
Copyright © 2011 V. Shakhov and H. Choo. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
The admission control mechanism inspired in the framework of proportional differentiated services has been investigated.
The mechanism provides a predictable and controllable network service for real-time traffic in terms of blocking probability.
Implementation of proportional differentiated admission control is a complicated computational problem. Previously, asymptotic
assumptions have been used to simplify the problem, but it is unpractical for real-world applications. We improve previous
solutions of the problem and offer an efficient nonasymptotic method for implementation of proportional differentiated admission
control.
1. Introduction
Efficient implementation of admission control mechanisms

B formula. We then use it for a proportional differentiated
admission control implementation and consider some alter-
native problem statements for an admission control policy.
In Section 4, we present the results of numerous experiments
with the proposed method. Section 5 is a brief conclusion.
2. Problem Statement
Let us consider the concept of admission control inspired in
the framework of proportional differentiated services. In the
above paper [3], whose notation we follow, PDAC problem
is defined as
δ
1
B
1

ρ
1
, n
1

=
δ
2
B
2

ρ
2
, n
2

/b
i
, C
i
is an allotted partition of the link
capacity, b
i
is a bandwidth requirement of class i
connections, and
x is the largest integer not greater
than x;
(v) B(ρ
i
, n
i
): is the Erlang loss func tion, that is, under the
assumptions of exponential arrivals and general ses-
sion holding times [5], it is the blocking probability
for traffic of class i, i
= 1, , K.
It needs to find C
1
, C
2
, , C
K
taking into account known
δ
i
, ρ


≈
δ
2
B
2

ρ
2
, n
2

≈··· ≈
δ
K
B
K

ρ
K
, n
K

. (3)
But, even in this case, the above problem is difficult and
complex combinatorial problem. For its simplification, the
following asymptotic approximation has been used [3]. If the
capacity of link and the offered loads are increased together:
n
−→ ∞ , ρ −→ ∞ ,(4)

and ρ
i
>C
i
/b
i
, i = 1, , K. Under these conditions, the
asymptotic approximation of the Erlang B formula has been
used and (1) has been replaced by simplified equations as
follows:
δ
1

1 −
C
1
b
1
ρ
1

=
δ
2

1 −
C
2
b
2

1

,(9)
then
1
−
n
ρ
<B

ρ, n

< 1 −
n
ρ
+
. (10)
Proof. Here and below, we use the fol lowing designation:
β

ρ, n

=
1 −
n
ρ
. (11)
Assume that ρ>n. First, we rewrite the Erlang B formula
B


n

i=0

n
ρ

n−i
. (13)
Taking into account properties of geometrical progres-
sion, we have
1
B

ρ, n

≤
n

i=0

n
ρ

n−i
<
1
β

ρ, n

ρ/n

− ρ

1 −

ρ/n

. (16)
Transform this as follows:
UB
=
ρ

ρ − n +2

−
n

ρ − n +2

+ n
ρ

ρ − n +2

. (17)
It implies
UB
= 1 −

−
n
ρ
+
. (21)
From the inequality (20), we obtain the condition (9).
The proof is completed.
Note that the approximate formula (6) can provide the
required accuracy
 in the case of ρ<n+1/.Actually,if
 = 0.01, n = 200, then the required accuracy is reached
for ρ
= 270 < 300. Thus, the condition (9)issufficient
but not necessary. It guarantees the desired accuracy of the
approximation for any small
 and n.
3.2. PDAC Solution. Assume that the solution (C
1
, C
2
,
, C
K
) of the PDAC problem satisfies inequalities ρ
i
>
C
i
/b
i

−
C
1
b
1
ρ
1
= δ
i

1 −
C
i
b
i
ρ
i

, i = 2, , K. (22)
According to the transitivity property, any solution of the
PDAC problem under condition (8) is also a solution of the
PDAC problem under condition (22). Therefore,
C
i
= b
i
ρ
i

1+

1
K

j=2
b
j
ρ
j
δ
j
, S
2
=
K

j=2
b
j
ρ
j

1
δ
j
− 1

. (25)
Thus, the formulas (23)–(25) provide the implementa-
tion of proportional differentiated admission control.
It is clear that for some values C, b

2
= b
1
ρ
1
S
1
+
K

j=2
b
j
ρ
j
, (27)
we derive
C<CL
1
= b
1
ρ
1
⎛
⎝
1 −

K
j
=2

δ
K
= min
i
δ
i
> 1 −
C
1
b
1
ρ
1
. (30)
By substituting the expressions (24) for the C
1
into (30),
we get after some manipulations the following inequality:
C>CL
2
=
K−1

j=1
b
j
ρ
j

1 −

1
, CL
2
)
<C<
K

j=1
b
j
ρ
j
. (34)
It follows from the theorem that the approximation
(6)isapplicableevenforn
= 1andanysmall > 0
if ρ>1/
 − 1. In spite of this fact, the solution above
cannot be useful for small values of the ratio C
i
/b
i
. In this
case, the loss function B(ρ
i
, n
i
) is sensitive to fractional part
dropping under calculation n
i

= 1, , K. Each class i is
characterized by a worst-case loss guarantee α
i
[7, 8].
4 EURASIP Journal on Wireless Communications and Networking
Consider the following optimization problem:
min
K

i=1
n
i
,
B

ρ
i
, n
i

≤
α
i
, α
i
∈
(
0, 1
]
, i

K
)of
the problem (35) satisfies the mentioned condition
B

ρ
i
, n
∗
i

=
α
i
. (36)
If we designate δ
i
= 1/α
i
, then we get
δ
i
B

ρ
i
, n
∗
i


. (38)
Note that in practice the solution n
∗
i
is not usually
integer; thus, it has to be as follows:
arg min

n
i
∈ N | n
i
≥

ρ
i
(
1
− α
i
)

, i = 1, , K. (39)
We now consider the optimization of routing in a
network through the maximization of the revenue generated
by the network. The optimal routing problem is formulated
as
max
K


is a revenue rate of class i traffic. Obviously, the Erlang loss
function B(ρ, n) is an increasing function of ρ. Therefore, the
optimal solution (ρ
∗
1
, ρ
∗
2
, , ρ
∗
K
) of the problem (40), (41)
satisfies the following condition:
B

ρ
∗
i
, n
i

=
α
i
. (42)
Hence, the problem (40), (41) can be reduced to the
problem (1) as well. Under the approximation, the optimal
solution takes the form
ρ
∗

i
, n
i
)
1 130887 1022 0.0803 0.0803
2 129786 1013 0.0877 0.079
3 128409 1003 0.0961 0.0769
4 126639 989 0.108 0.0756
5 124279 970 0.1243 0.0746
4. Performance Evaluation
Let us illustrate the approximation quality. The difference
Δ(ρ, n)
= B(ρ, n) − β(ρ, n) is plotted as a function of offered
load in Figure 1. If the number of channel n is relatively
small then high accuracy of approximation is reached for
heavy offered load. Let us remark that heavy offered load
corresponds to high blocking probability. Generally, this
situation is abnormal for general communication systems,
but the blocking probability B(n, ρ) decreases if the number
of channels n increased relative accuracy
. Let us designate
ρ
∗
= n +1/. If the approximation (2) is admissible for ρ
∗
then it is also admissible for any ρ>ρ
∗
. In Figure 2, the
behavior of losses function B(n, ρ
∗

δ
i

1 −
C
i
b
i
ρ
i

=
0.0704, i = 1, ,5. (45)
It is easy to see that
max
i=1, ,5

B

ρ
i
, n
i

−

1 −
C
i
b



< 0.01.
(46)
If K
= 10, δ
i
= 1 − 0.05(i − 1), i = 1, , 10, and other
parameters are the same then
max
i, j



δ
i
B

ρ
i
, n
i

− δ
j
B

ρ
j
, n

n = 100
n = 200
n = 300
Figure 1: Approximation quality as a function of the offered load.
0.5
0.4
0.3
0.2
0.1
0.07
0.05
0.04
0.03
0.02
0.01
0.007
0.005
0.004
0.003
0.002
0.001
Loss function, B(n, ρ
∗
)
Number of channels, n
12345710
×10
4
 = 0.01
 = 0.001

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