Complexity Analysis of Optimal Stationary Call
Admission Policy and Fixed Set Partitioning Policy
for OVSF-CDMA Cellular Systems
Daniel Lee
Muhammad Naeem
Chingyu Hsu
Presentation Outline
„
Background
„
System Model
„
Call Admission Control Schemes
„
Complexity Of Proposed Schemes
„
Numerical Results
„
Summary
Background
¾
¾
In Wideband CDMA orthogonal variable spreading factor
(OVSF) are codes for providing different channels for
different users with different data rates
Well known OVSF code generation
C
C
1
4
[
]
= C ,C
= [+ , + , + , +
C
1
2
[
= C ,C
= [+ , + ]
C 11
= [+
1
1
1
1
]
]
C
1
1


1
2
[
1
n −1
2
,C
1
2
2 k −1
n
2
C 2k n − 1
,C
k
n−1
2
n−1
]
]
2
4
3
4
[
]
= C 22 , C 22
= [+ , − , + , −
C
n
= C
1
=  C 21 , C 2 


= [+ , + , − , − ]
C
2
2
=  C 11 , C

= [+ , − ]
C
1
2
1
2
C
=
• • •
]
4
4
C
[
]
2k
n
2
= C

k
n −
2 1
,C
k
2
n −1
n
n −1
2
,C
n
2
n − 2


2
=  C 22 , C 2 


= [+ , − , − , + ]
C
2n
n
2
= C

•Nodes with the depth = l from the root of the tree have code
words of code length = 2l.


Example
4R
2R
R
Resource constraint:
∑
M −1
j =0
2 kj ≤ C
j
k1
k0
Example of capacity region,, Cmax=32
System Model
¾
Random arrivals of call requests in each class can be
modeled by a stochastic process – (Poisson process).
¾
For class j
• Average arrival rate is λj
• Average call duration μj.
¾
In response to each arrival of call request, the network
operator must make a call admission control (CAC)
decision (whether to accept or reject the call request), and
the decision should be made based on the resource
constraint.
¾The objective can be to maximize average throughput, or
minimize blocking probability, etc.
System Model
¾
We allow code re-assignment
Main point
• The greedy Call Admission Policy is not
necessarily optimal.
Optimal policy
• The system can be modeled by a Markov decision
process, so an optimal policy can be obtained by
optimizing a Markov Decision Process.
– Linear Programming
– Policy iteration
– etc.
{( k0 , k1,", kM −1 ) ∑ i=0
M −1
2 i ki ≤ C , ki ∈ Z +
}
Previous Results: optimal policy
• An optimal policy can have significantly better
performance than the greedy policy
– According to small-scale cases
• Large-scale: large C, large number of classes, etc.
– Computational complexity of linear programming
(e.g., number of variables)
{( k0 , k1,", kM −1 ) ∑ i=0
M −1
2i ki ≤ C , ki ∈ Z +
}
Previous results: suboptimal policy
• Even a fixed set policy can have
significantly better performance than the
greedy policy.
Optimal Fixed Set Partitioning
¾OVSF codes are partitioned into mutually exclusive groups of
codes and uniquely assigns a group to each service class.
¾Gj the number of codes in the subset assigned with class j,
where j = 0, 1, 2,…, M−1.
¾Once vector G ≡ (G0,G1,…., GM-1) is determined, the actual
partition of the set is trivial.
¾The numbers of codes in the subsets G0, G1,…, GM-1 satisfy
∑
M −1
j =0
2 j Gj = C
¾Average Through put of this scheme
TF ( G ) = ∑ k =0 ( 2k R ) (1 − Pk ) λk µk
M −1
where
( λk µk ) n!
n
Pk =
∑ (λ
Gk
n =0
k
µk ) n !
n
Optimal Fixed Set Partitioning
Greedy within fixed sets
Complexity Analysis
Size of the feasible set: the number of vectors ( G0 , G1 ," , GM −1 )
that satisfy constraint
∑
M −1
j =0
2 j Gj = C
Combinatorial Analysis and
Recursion
¾GM−1 can have any value from 0, 1,…, C/2M−1. if GM−1 is xM. Then,
GM−2 can be 0, 1, 2, 3, …, 2(C/2M-1- xM) and so on.
¾The relation between adjacent classes has a recursion
Let Γ m ( v ) be the number of non-negative integral vectors
( G0 , G1 ,", GM −1 )
that can occupy the space taken by v codes
of largest codes i.e., satisfying constraint ∑ i =0 2i Gi = v 2m.
m
Then,
Γ m ( v ) = ∑ x =0 Γ m−1 ( 2 ( v − x ) ) = ∑ i =0 Γ m−1 ( 2i )
v
Γ0 ( v ) = 1, v = 0,1, 2,3,....
v
Functions Γ 0 ( v ) , Γ1 ( v ) ,..., Γ M −1 ( v ) can be evaluated from the recursion.
The number of vectors ( G0 , G1 ," , GM −1 ) that satisfy constraint
∑
M −1
j =0
2 j G j = C is Γ M −1 ( C 2 M −1 ) .
Numerical Results
Numerical Results
Optimization of Markove Decision
process through LP
In the case of LP, its complexity is closely related to the size of
state space,
{( k0 , k1,", kM −1 ) ∑ i=0
M −1
2 i ki ≤ C , ki ∈ Z +
}
¾Let χm(v) be the number of non-negative integer vectors (k0,
m
k1,…, km) that satisfy the constraint
2i k ≤ v 2 m
∑
i =0
i
¾The number of vectors (k0, k1,…,km-1) that satisfy this constraint
is χm-1(2(v-xm)). Therefore, we have the following recursion:
∑
m −1
i =0
2i ki ≤ ( v − xm ) 2m = 2 ( v − xm ) 2m−1
χ m ( v ) = ∑ i =0 χ m−1 ( 2i )
v
χm(v) and Γm(v) have an identical
recursion
The size of the state space
{( k , k ,", k
0
1
) ∑ i =0
M −1
M −1
2 ki ≤ C , ki ∈ Z
i
= χ M −1 ( C 2 M −1 ) = Γ M ( C 2 M −1 )
+
}
Numerical Results
Complexity
¾Fixed Set Partitioning (FSP)
The complexity of FSP operation is characterized as O(1)
lookups
¾DCA-CAC
The complexity of DCA-CAC is also O(1)
In summary, in both FSP and DCA,
the computational complexity of
online operations is O(1) per call
arrival.
Thank You.