2. Preliminary
We present preliminary analysis of the time-based admission control without
buffering requests in an idealized transmission environment. The purpose is to
illustrate the spirit of this study and some limitations of underlying system parameters.
In Section 2.1, we outline the subject of the chapter. The data rate function adopted in
analysis is presented in Section 2.2. In Section 2.3, we define a measure of
performance improvement and then present analysis for the measure. Numerical
results from the analysis are discussed in Section 2.4.
2.1 Preliminary
We consider an ideal wireless environment where data over forward link can be
transmitted at any rate according to a continuous rate function. The rate function is
assumed to be deterministic depending on distance-related path-loss exponent and the
distance from BST to a target MT. Under this environment and the condition of system
stability, our primary focus is the factor of improvement in serviced request rates; that
is, the ratio of the rate of serviced requests when the forward-link data service system
is under time-based admission control to that when the system is operated without
admission control. Although the forward link transmission system is considerably
simplified, it is enough for us to look at the impact of underlying system parameters.
2.2 Data Rate Function
A standard model for peak rate function is
N SINR E
C W
o b /
= , (2.1)
where Eb is the energy per information bit, No the total interference and noise
power spectrum density, W the forward link bandwidth, SINR the signal to interference
and noise power ratio at a target MT, and C the data rate for the MT [15]. When the
spectral efficiency in bits/sec/Hz is less than one, Eb/No can be regarded as a constant
for all data rate, according to Shannon’s capacity limit in AWGN channels. Thus, the
data rate for the MT is proportional to its received SINR, which is essentially a random
variable depending on distance-related path-loss, fading, shadowing, and interference
power. For a high data rate system [15], it is more appropriate to assume the rate
function C = f(SINR), where f is typically expressed as
1 0
), 1
( log )
(SINR =W 2 +ηSINR ≤η<
f , (2.2)
limited by the Shannon capacity function and modem performance η. However, we
adopt a deterministic peak rate function [9],
⎪⎩( )
⎪⎨
⎧ ≤
×
= otherwise.
if , ) 1
( 0
0
0 α
r r
r C r
r
C (2.3)
where r is the distance from the BST to a target MT, C0 the maximum data rate that
can be achieved, α the path-loss exponent, and r0 the maximum distance at which the
MT can receive C0 data rate. The path-loss component typically has a value between 2
and 5. Note that intercell interference, fading, and shadowing effects are not
considered in rate function C(r).
2.3 Performance Analysis
Let Γ be a time threshold used as a criterion for admission of forward link data
traffics. That is, if a service request for air time resources is larger (resp. smaller) than
Γ, it is blocked (resp. accepted). Traffic arrives in batch. Let V be a random variable
representing the volume (size), in bits, of a request. Let σ=E[V], the expected batch
size. Assume that the distribution of batch size is independent of MT’s spatial location.
Then, the air time resource required to service a request for a target MT at a distance r
from the BST is V/C(r). The probability of blocking the service request, denoted by
b(r,Γ), is
⎭⎬
⎫
⎩⎨
⎧ >Γ
=
Γ C(r)
r V
b( ,σ, ) Pr . (2.4)
Assume that arrivals of service requests are uniformly distributed in two dimensional
space with rate λ per unit area. We use Λ(λ,σ,Γ) to indicate the total rate of admitted
service requests. Then, we have
∫ − Γ
= Γ
Λ R rdr b r
0
)) , , ( 1 ( 2 )
, ,
(λ σ λ π σ , (2.5)
where R is the radius of a cell.
On the other hand, the total admitted traffic load in terms of required service
time, represented by ρ(λ,σ,Γ), can be computed by
)) , , ( 1 )(
( 2
) , , (
0∫ 0∫ ( )|{ ( ) } − Γ
=
Γ λ π ∞ <Γ σ
σ λ
ρ R rdr zdF z b r
r CV r
CV , (2.6)
where |{ }
) ( )
(r CVr <Γ
CV
F (z) is the conditional cumulative distribution function of
requested air time V/C(r).
We can assume that the forward-link service system is a work-conservative
server with service rates depending on the MT in service. Our objective is to
maximize Λ(λ,σ,Γ) subject to the constraint of system stability, which requires
admitted traffic load ρ(λ,σ,Γ) < 1. In order to see the effect of time-based admission
control, we define ψ(σ,Γ) as the factor of improvement in the rate of serviced request
under the stability constraint; that is,
) ,
, (
) , , ) (
,
( * *
∞
→ Γ Λ
Γ
= Λ
Γ λ σ
σ σ λ
φ , (2.7)
where Λ* indicates the maximum rate of (2.5) subject to the stability constraint. Note
that Λ*(λ,σ,Γ→∞) is the maximum rate of serviced requests, maximum departure rate,
when the system is operated without admission control.
To compute Λ*(λ,σ,Γ→∞), we first find the largest possible value of λ by letting
ρ(λ,σ,Γ→∞) < 1 in (2.6) and then use the maximum λ in (2.5). We thus have
1
0 2
*
) ( ) 2
, , (
−
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ ∫
=
∞
→ Γ
Λ R
r C R σ rdr σ
λ . (2.8)
In fact, the term ( )
0 ( )
2
∫ 2
R r C R
σ rdr in (2.8) is the mean air time required for the system
without admission control to service a request.
To find the factor ψ in (2.7), we repeat the above steps, by first obtaining the
largest possible value of λ from (2.6) and substituting the result for λ in (2.5), and then
divide (2.5) by (2.8). We thus obtain
)) , , ( 1 )(
( 2
) ( )) 2
, , ( 1 ( 2 )
, (
0 0 |{ }
0 2
0
) ( )
( − Γ
∫ ∫
∫
∫ − Γ
=
Γ ∞
Γ
< σ
π
σ σ
π σ
φ
r b z
zdF rdr
r C R r rdr
b rdr
R
R R
r CV r CV
. (2.9)
Given cell radius R, the average blocking probability, denoted by b(σ,Γ), is
0 2
) , , ( 2
) ,
( R
r b rdr b
R
π σ π
σ ∫
Γ
=
Γ . (2.10)
Assume that the request size V is exponentially distributed with mean σ. Then,
the expressions for (2.9) and (2.10), respectively, become
∫ ⎥⎦⎤
⎢⎣⎡ − + Γ
∫
∫ −
=
Γ − Γ
Γ
−
R C r
R R
r C r C
r e rdrC
r C R e rdr
rdr
0
) (
0 2
0
) 1
( ) 1
2 (
) ( ) 2
1 ( 2 )
,
( ( )
) (
σ σ
σ
σ
σ σ
φ (2.11)
and
2 0
) (
2 ) ,
( R
e rdr b
R C r
= ∫ Γ
Γ
− σ
σ . (2.12)
2.4 The Effect of Underlying System Parameters
To investigate the effects of varying admission time-threshold on the
improvement factor of serviced request rate ψ(σ,Γ) and on average blocking
probability b(σ,Γ), we set normalized close-in radius r0=1, path-loss exponent σ=2,
maximum peak service rate C0=2457.6Kbps, and consider two mean request sizes,
σ=81920 bits and σ=40960 bits, and three possible cell coverage radii, R=2, 3, 4.
Numerical data for (2.11) and (2.12) are illustrated in Figs. 2-1 (a) and (b), respectively.
It can be seen that the improvement factor and average blocking probability both
decrease with admission time threshold. In order to obtain a high improvement factor,
the admission time threshold must be small and service requests arrive at high rate.
This also gives rise to a very high average blocking probability, which implies that the
admission control is very selective and thus not feasible. Considering the random
fluctuation of wireless forward link data loads, it is however possible to significantly
buffering long air-time service requests, instead of dropping them, when traffic arrival
rates are high temporally or spatially. This is what we will study latter in the thesis.
Comparing results for σ=81920 bits and σ = 40960 bits in Figs 2-1 (a) and (b),
we see that arrival traffics with larger mean data sizes suffer more blocking
probabilities but have more potential for obtaining higher improvement factors. In fact,
they provide more chances for admission controller to discriminate against services for
long air-time requests. Comparing results for R=2,3, and 4 in Figs 2-1 (a) and (b), we
also see that the effect of large cell coverage is similar to that of larger request size
distribution, because of lower service rate and hence longer service air-time
requirement for MTs at far field. Therefore, large cell coverage or large request size
distribution gives rise to more dynamic service air-time requirements, which more or
less imposes the requirement of queueing service requests on using time-based
admission control.
As to the effects of varying path-loss exponents, we set r0=1, R=3,
C0=2457.6Kbps, and consider two mean request sizes, σ=81920 bits and σ =40960 bits,
and three possible path-loss exponents, α=2, 3, 4. Numerical results for (2.11) and
(2.12) are illustrated in Figs. 2-2 (a) and (b), respectively. It can be seen that both the
improvement factor and average blocking probability increase with path-loss exponent
α. In fact, the effect of larger α is lower peak service data rate and longer service air
time requirement, similar to the effect of increasing cell coverage discussed
previously.
Figure 2. 1 (a) The factor of improvement in serviced request rate ψ(σ,Γ) versus admission time threshold Γ (second); (b) average blocking probability b(σ,Γ) versus admission time threshold Γ (second), for normalized close-in radius r0=1, cell coverage radii R=2,3,4, path-loss exponent α=2, and maximum peak service rate C0=2457.6Kbps.
(a) (b)
1e-05 0.0001 0.001 0.01 0.1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Average blocking probability
Admission time_threshold(sec) R=2;Sigma=81920
R=2;Sigma=40960 R=3;Sigma=81920 R=3;Sigma=40960 R=4;Sigma=81920 R=4;Sigma=40960 0
1 2 3 4 5 6 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The factor of improvement
Admission time_threshold(sec) R=2;Sigma=81920 R=2;Sigma=40960 R=3;Sigma=81920 R=3;Sigma=40960 R=4;Sigma=81920 R=4;Sigma=40960
Figure 2. 2 (a) The factor of improvement in serviced request rate ψ(σ,Γ) versus admission time threshold Γ (second); (b) average blocking probability b(σ,Γ) versus admission time threshold Γ (second), for normalized close-in radius r0=1, cell coverage radius R=3, path-loss exponent α=2, 3, 4, and maximum peak service rate C0=2457.6Kbps
(a) (b)
0.0001 0.001 0.01 0.1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Average blocking probability
Admission time_threshold(sec) Alpha=2;Sigma=81920
Alpha=2;Sigma=40960 Alpha=3;Sigma=81920 Alpha=3;Sigma=40960 Alpha=4;Sigma=81920 Alpha=4;Sigma=40960 0
2 4 6 8 10 12 14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
The factor of improvement
Admission time_threshold(sec) Alpha=2;Sigma=81920 Alpha=2;Sigma=40960 Alpha=3;Sigma=81920 Alpha=3;Sigma=40960 Alpha=4;Sigma=81920 Alpha=4;Sigma=40960
