JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 26, 267-278 (2010)
Partial Loss Probability of OVSF Single-code Assignment
with Space-limited Batch Arrival Queues: M[2X]/M/c/c
JUI-CHI CHEN
Department of Computer Science and Information Engineering
Asia University
Taichung, 413 Taiwan
E-mail: [email protected]
In a WCDMA Node-B, orthogonal variable spreading factor (OVSF) codes are both
valuable and limited. Many efforts are being made to create OVSF code-assignment
schemes that support a maximum number of users. Here the author (a) proposes a batch
arrival and partial loss queuing model to evaluate OVSF single-code assignment system
performance, and (b) derives expressions for calculating two important system performance measures (partial loss probability and bandwidth utilization) that respectively denote subscriber service quality and operator profit. Results from a simulation test suggest
that the proposed model has utility in the construction of WCDMA networks.
Keywords: queuing models, partial loss probability, bandwidth utilization, WCDMA
networks, OVSF code
1. INTRODUCTION
Orthogonal variable spreading factor (OVSF) code transmission supports various
wideband services with low and high data rates in Wideband-CDMA (WCDMA) networks [1]. Both forward and reverse WCDMA links can only be assigned single OVSF
codes to match user-requested data rates [3-7]. Since Node-B (3G base station) OVSF
codes are valuable and limited, 3G operators must utilize them as efficiently as possible.
Each Node-B is serving of a few cells (sometimes only one) and an OVSF code-assignment system. A lots of OVSF code-assignment schemes have been proposed and
tested to support as many users as possible [1, 7-13]. Call blocking probability (CBP)
and bandwidth utilization (BU), which respectively represent subscriber quality of service (QoS) and operator profit, are two significant performance measures for any OVSF
code-assignment system. In this paper I will describe a proposal for a batch arrival partial
loss model, M[2X]/M/c/c, to measure OVSF single-code assignment system performance
and to obtain expressions of partial-loss CBP (also referred to as partial loss probability,
or PLP) and BU. The goal is to establish a model that can be applied to the utilization
maxi-mization problem to gain maximum profit under a specific QoS constraint by determining the optimal number of OVSF codes in a Node-B, as well as for constructing
WCDMA networks.
Cromie et al. [16-18] have developed two related models, MX/M/c and MX/M/c/c,
while Tseng et al. [10] developed another OVSF single-code model. However, Cromie et
al. obtained the steady-state results, delay, queue length behavior and waiting time distributions of the loss system, but did not discuss the CBP and BU. Although Tseng’s
Received May 9, 2008; revised April 21, 2009; accepted June 20, 2009.
Communicated by Tei-Wei Kuo.
267
JUI-CHI CHEN
268
model was developed to calculate the CBP and BU, it has been adopted only when the
maximum spreading factor of the OVSF codes is less than 32, due to the large number of
states in its state diagram. However, the number of states rises only linearly with the increasing number of rate resources in the proposed model, which can therefore be applied
to solve the equilibrium equations.
The remainder of this paper is organized as follows: a description of the OVSF single code system is presented in section 2 and the proposed model is described in section
3. The focus of section 4 is on performance measures. Simulation results and theoretical
analysis verification are offered in section 5, and conclusions are given in section 6.
2. OVSF SINGLE CODE
2.1 OVSF Code Generation
Channelization is the WCDMA spreading operation through which data symbols
are transformed into numbers of chips; the number of chips per data symbol is called the
spreading factor. Channelization codes are OVSF codes that preserve orthogonality between channels at different rates. As illustrated in Fig. 1, a code tree recursively generates
codes based on a modified Walsh-Hadamard transform [2]. An OVSF code is designated
k
as Cch
, SF , where SF denotes the spreading factor, k the code number, subscript ch the
channelization, and 1 ≤ k ≤ SF = 2n.
C 1ch , 4 = {1,1,1,1}
C 1ch , 2 = {1,1}
C ch , 4 = {1,1,−1,−1}
2
C ch ,1 = {1}
1
C ch , 4 = {1,−1,1,−1}
3
C ch , 2 = {1,−1}
2
C ch , 4 = {1,−1,−1,1}
4
SF=1
C 1ch , 8
C ch2 ,8
C ch3 ,8
C ch4 , 8
C ch5 , 8
C ch6 ,8
C ch7 , 8
C ch8 ,8
SF=8
SF=4
SF=2
Fig. 1. Code tree for generating OVSF codes.
Variable spreading factors are used for low and medium-high data rates. Data transmission spreading factors in a reverse link range from 4 to 256 and in a forward link from
4 to 512 (with restrictions on the use of factor 512). The maximum spreading factor SFmax
normally equals system capacity. Without loss of generality, the data rates described in
later sections are multiplied by the basic data rate Rb, where Rb denotes an OVSF code
with SFmax. All codes with the same SF spreading factor are located at the same level
PARTIAL LOSS PROBABILITY OF OVSF SINGLE-CODE ASSIGNMENT
269
log2(SF) in the code tree. Restated, any code at level log2(SF) is associated with the data
SF
rate SFmax Rb .
2.2 Code-limited Capacity Test
A request rate Ri for call i can be expressed as the polynomial Ri = ∑ j = 0 rj 2 j , where
n
rj ∈ {0, 1}; n = log2(SFmax); 1 ≤ Ri ≤ SFmax; and Ri is the value of a multiplication of Rb.
Before assigning a code, the Node-B must perform an interference-limited test or a codelimited test to measure its available system capacity. In the interference-limited test, the
Node-B checks its reverse link capacity via the non-blocking condition used in [14]:
k
other − cells k
∑ vi Ebi Ri + ∑ ∑ vi ( j ) Ebi ( j ) Ri + N 0 × W ≤ I0 × W ,
i =1
j
(1)
i =1
where k denotes the number of users per cell; vi voice activity; W system bandwidth; Ri
data rate; Eb bit energy; N0 thermal noise density, and I0 total acceptable interference
density.
In the code-limited test, system capacity equals SFmax × Rb in a single Node-B. Here
the non-blocking condition can be defined as
k
∑ R j ≤ SFmax × Rb .
(2)
j =1
The limited number of OVSF codes means that Node-B may run out of codes. If the
above inequality is not met, call i is partially blocked. The partial loss probability (PLP)
denotes the average partial-loss CBP of incoming call requests during a sufficiently long
time period.
2.3 Single Code Assignment
OVSF code assignment schemes are designed to support as many users as possible.
In a single-code assignment scheme, user equipment (UE) transmits a signal with a variable data rate on only one channel, and therefore needs only one RAKE receiver. A UE
equipped with one RAKE receiver can only convey a single-code rate Ri with ∑ n r j ≤ 1.
j =0
Therefore, call i with ∑ n r j > 1 is assigned a slightly higher single-code rate, expressed
j =0
as
ξ ( Ri ) = 2 ⎣⎢log 2 (2*Ri −1)⎦⎥.
(3)
3. PARTIAL LOSS M[2X]/M/c/c FOR THE OVSF SINGLE-CODE
ASSIGNMENT SYSTEM
The number of OVSF codes with the maximum spreading factor c = SFmax usually
JUI-CHI CHEN
270
equals Node-B system capacity − in other words, Node-B has a total of cRb rate resources.
A single-code call (i.e., single customer) can request multiple channels ranging from 1 to
2K to meet its requirements. Accordingly, a scenario in which a new call assigned ξ(Ri) =
2kRb (equals 2k OVSF codes with SFmax) can also be viewed as 2k new calls, each of
which is assigned an OVSF code with Rb simultaneously; this allows the case to be
treated as a group arrival with size 2k. Furthermore, any call served with 2kRb can be
viewed as 2k basic rate calls released simultaneously − all with the same service (call
holding) time. In this scenario the case must be treated as a bulk service system problem.
However, in the interest of taking a long-term average approach and wanting to reduce
model complexity, I simulated the 2k basic rate calls as a high rate single-code call, with
the added assumption that customers arrive in groups according to a Poisson process with
mean group-arrival rate λ and probability sequence {x2k} governing group size. λ2k thus
represents the batch arrival rate with a group size of Poisson customer stream 2k, where
λ2k = x2kλ; ∑ K x2k = 1; λ = ∑ K λ2k ; 0 ≤ k ≤ K, and k, K ∈ {Positive Integers}. The 2K
k =0
k =0
rate is the maximum request rate. Average group size is denoted as g = ∑ K 2k x k ,
2
k =0
where the probability density function of x2k may be any countable distribution. Accordingly, a new call assigned ξ(Ri) = 2kRb can be considered a batch arrival with size 2k. The
service time for all customers is assumed to have an independent exponential distribution
with parameter 1/μ ; service discipline is first-come, first-serve in terms of group order.
Fig. 2 presents a flow-in/flow-out diagram for state m in the single-code assignment system, where 2K ≤ m ≤ c − 2K. Incoming flows include (m + 1)μ from state m + 1 and all
possible λ2k batch arrivals with size 2k, where 0 ≤ k ≤ K. From a flow-out perspective,
state m has rate mμ that flows to state m − 1; λ1 flows to state m + 1, λ2 to state m + 2, λ2
to state m + 4, λ2k to state m + 2k, and λ2K to state m + 2K.
λ2
λ1
…
λ2
m
λ1
λ4
K
…
λ2
K
…
λ2
m+1 m+2 m+3
m+4 … m+2K …
mμ
(m+1)μ
Fig. 2. Flow-in flow-out diagram for state m in OVSF single-code assignment system.
Additionally, the system has no buffer (i.e., queue length = 0), making it a loss system. If a new call finds that the available capacity in the corresponding Node-B cannot
satisfy its rate requirement, it is assigned the available rate − in other words, it is partially
blocked. Hence, a partial-loss single-code system can be modeled on the partial loss
batch arrival model M[2X]/M/c/c, whose state-transition diagram is shown in Fig. 3 (where
1 ≤ m ≤ c − 1). If c ≥ 2K, then state 0 has the outflow rate λ = ∑ K λ k . When m < 2K,
k =0 2
the maximum batch-arrival size from a previous state is given by 2⎣⎢log 2 m ⎦⎥. Therefore, the
maximum rate of incoming arrivals into state m should be λ2k, with k = min(⎣log2m⎦, K). If
m ≤ c − 2K, then the outgoing arrival rates from state m are given by λ2k, where 0 ≤ k ≤ K
(Fig. 2). However, the arrivals will only have partial success in being served with ratio (c
− m)/2k when c − 2K < m ≤ c – 1. The outgoing arrival rates from state m are therefore
λ2
0
λ1
…
λ A ,k = min ( ⎣log m ⎦,K)
2
λB,k=K
λA,k=1
λA,k=0
…
λE,h=3
λE,h=2
…
λ4
λE,h=2K
K
…
λ2
271
m
…
…
PARTIAL LOSS PROBABILITY OF OVSF SINGLE-CODE ASSIGNMENT
λB,k=1
λB,k=0
λE,h=1
…
…
c
cμ
mμ (m+1)μ
c−m
h
λ A = λ2k λB = min( k ,1)λ2 k λE = min( k ,1)λ2 , K ≥ k ≥ ⎡log 2 h ⎤
2
2
Fig. 3. State-transition diagram for M[2X]/M/c/c partial loss OVSF single code assignment system.
μ
k
given by min((c − m)/2k, 1)λ2k, 0 ≤ k ≤ K. Finally, each dotted line moving toward state c
in Fig. 3 denotes all possible partial loss batch arrivals from some previous state (c − h),
where 1 ≤ h ≤ 2K. Each state from c − 2K to c – 1 always possesses at least one arrival to
state c. For example, the c – 1 state has a full arrival rate of λ1 and K partial-loss arrivals
to state c at a rate of λ2k/2k, where 1 ≤ k ≤ K.
The first step is to solve the probability that the number of customers in system N(t)
(i.e., the number of assigned basic-rate resources) equals m at some time t. This can be
expressed as
Pm(t) = P[N(t) = m],
(4)
which can be viewed as the transition probability in state m at time t. Each OVSF singlecode call (i.e., each group in the proposed model) is considered independent, and can
request a variable number of basic-rate resources (e.g., 2k) to meet its data-rate requirement. It is reasonable to assume that the interarrival times between any two continuous
calls have a memoryless negative exponential distribution independent of time t. The
time-independent discrete distribution for the requested data rates − for instance, the
geometric distribution − is also considered. Keep in mind that any Node-B system capacity (number of states) is limited, therefore the proposed model is assumed to be an ergodic continuous time Markov Chain with a finite number of states. Accordingly, the
resulting differential-difference equations are written as:
K
dPo (t )
= − λ Po (t ) + μ P1 (t ), λ = ∑ λ2k .
dt
k =0
(5)
K
K
⎛
⎞
dPm (t )
= − ⎜ mμ + ∑ λ2k ⎟ Pm (t ) + ∑ λ2k Pm − 2k (t ) + (m + 1) μ Pm +1 (t ),
dt
k =0
k =0
⎝
⎠
2 K ≤ m ≤ c − 2 k.
(6)
⎢⎣log 2 m ⎥⎦
K
⎛
⎞
dPm (t )
= − ⎜ mμ + ∑ λ2k ⎟ Pm (t ) + ∑ λ2k Pm − 2k (t ) + (m + 1) μ Pm +1 (t ),
dt
k =0
k =0
⎝
⎠
1 ≤ m < 2 k ≤ c − 2 k.
(7)
JUI-CHI CHEN
272
K
K
⎛
⎞
dPm (t )
c−m
= − ⎜ mμ + ∑ k λ2k ⎟ Pm (t ) + ∑ λ2k Pm − 2k (t ) + (m + 1) μ Pm +1 (t ),
dt
k =0 2
k =0
⎝
⎠
2K ≤ c − 2K < m ≤ c − 1. (8)
K
2
K
dPc (t )
h
= − cμ Pc (t ) + ∑ ∑ min( k , 1)λ2k Pc − h (t ) .
dt
2
h =1 k = ⎢⎡log 2 h ⎥⎤
(9)
Combining Eqs. (6), (7) and (8) produces
min( ⎣⎢log 2 m ⎦⎥ , K )
K
⎛
⎞
dPm (t )
c−m
= − ⎜ mμ + ∑ min( k ,1)λ2k ⎟ Pm (t ) +
∑ λ2k Pm−2k (t ) + (m + 1)μ Pm+1 (t )
dt
2
k =0
k =0
⎝
⎠
1 ≤ m ≤ c − 1.
(10)
Assume that pm = lim Pm (t ).
(11)
t →∞
Accepting the existence of the limit in Eq. (11) makes it possible to set dPm(t)/dt as t
→ ∞ equal to zero, and therefore we immediately derive
0 = − λPo + μP1, λ =
K
∑ λ2 .
(12)
k
k =0
⎛
0 = − ⎜ mμ +
⎝
K
c−m
k =0
2k
∑ min(
2K
0 = − c μ pc + ∑
min( ⎣⎢ log 2 m ⎦⎥ , K )
⎞
, 1)λ2k ⎟ pm +
∑ λ2k pm − 2k + (m + 1)μ pm +1 ,
k =0
⎠
1 ≤ m ≤ c − 1.
(13)
K
∑
min(
h =1 k = ⎢⎡log 2 h ⎥⎤
h
2k
, 1)λ2k pc − h .
(14)
The following steady-state equations can be used to obtain the equilibrium probabilities pm for state m of the model.
K
λp0 = μp1, λ = ∑ λ2k .
(15)
k =0
min( ⎣⎢ log 2 m ⎦⎥ , K )
K
⎛
⎞
c−m
⎜ mμ + ∑ min( k ,1)λ2k ⎟ pm =
∑ λ2k pm−2k + (m + 1)μ pm+1 ,
2
k =0
k =0
⎝
⎠
1 ≤ m ≤ c − 1.
2K
c μ pc = ∑
K
∑
h =1 k = ⎢⎡log 2 h ⎥⎤
min(
h
2k
,1)λ2k pc − h .
Combining Eqs. (15), (16) and (17) yields
(16)
(17)
PARTIAL LOSS PROBABILITY OF OVSF SINGLE-CODE ASSIGNMENT
273
p1 = p0 λ/μ, and
(18)
K
⎡⎛
c−m
pm +1 = ⎢⎜ mμ + ∑ min( k ,1)λ2k
2
⎢⎣⎝
k =0
min( ⎣⎢log 2 m ⎦⎥ , K )
⎤
⎞
−
p
⎟ m
∑ λ2k pm−2k ⎥⎥ (m + 1)μ ,
k =0
⎠
⎦
1 ≤ m ≤ c −1.
K
⎡ 2K
⎤
h
pc = ⎢ ∑ ∑ min( k ,1)λ2k pc − h ⎥ c μ ,
2
⎣⎢ h =1 k = ⎢⎡log 2 h ⎥⎤
⎦⎥
(19)
(20)
which can be used for verification.
However, the formal closed-form solution for pm is difficult to express. Recursive
programs are not always capable of solving equations due to an overabundance of recursive levels for a large c value. An iterative computer procedure can be used to derive the
solution as follows:
Let p0* = 1 ; then p1* = p0* (λ μ ) = λ μ .
(21)
min( ⎣⎢log 2 m ⎦⎥ , K )
K
⎡⎛
⎤
⎞
c−m
λ2k pm* − 2k ⎥ (m + 1) μ ,
pm* +1 = ⎢⎜ mμ + ∑ min( k ,1)λ2k ⎟ pm* −
∑
2
⎢⎣⎝
⎥⎦
k =0
k =0
⎠
1 ≤ m ≤ c − 1.
According to the conservation relation
all states can be written as
c
∑ pi* ,
pm = pm*
∑
c
i =0
(22)
pi = 1 , the equilibrium probabilities of
0 ≤ m ≤ c.
(23)
i =0
Furthermore, Eq. (20) was verified as always being true when the equilibrium probabilities of all states are solved (Table 2).
4. PERFORMANCE MEASURES
PLP and BU can be derived forwardly once the equilibrium state probabilities are
known. At this point, if a new single-code call with data rate 2k determines that the available capacity in the corresponding Node-B is insufficient for its rate requirements, it is
partially blocked. Accordingly, the model’s PLP can be expressed as
α=
12
λ
K
−1
∑
K
∑
i = 0 k = ⎡⎢log 2 (i +1) ⎤⎥
2k − i
2k
λ2k pc −i ,
where c = SFmax ≥ 2K; K ≥ 0, and λ =
(24)
K
∑ λ2 .
k
k =0
For instance, if K = 2, then. α = pc +
3
2 λ2 + 4 λ4
λ1 + λ2 + λ4
1
2
λ4
pc −1 + λ + λ4
1
2 + λ4
1
λ4
pc − 2 + λ + λ4
1
2 + λ4
pc −3 .
(25)
JUI-CHI CHEN
274
Subsequently, the average number of customers (also known as the average system
length, or ASL) in the model is
c
L = lim N (t ) = ∑ ipi .
t →∞
(26)
i =0
Since the queue size equals zero, the ASL equals the mean number of busy servers in the
model. When the system is observed for a long time period, the average BU can be written as
β = L/c.
(27)
Furthermore, the average system waiting time (system delay) is expressed as
W = L/[ g λ (1 − α )] ,
(28)
which conforms to Little’s rule [15].
5. THEORETICAL AND SIMULATION RESULTS
The 3G WCDMA channelization environment was simulated to test the usability of
the proposed model. The simulated environment consisted of a Node-B with an OVSF
code table and OVSF single-code allocator, several UEs with various multimedia transmission requirements, a traffic generator, and a statistics recorder. Only one RAKE receiver was assumed to be embedded in each UE. The Node-B had a limited cRb bps capacity and code table size c (maximum spreading factor). The system capacity test was
code-limited. The optimal dynamic greedy method presented by Minn and Siu [11] was
adopted for the single-code allocation scheme. Call requests were generated with various
data rates and call durations according to the following parameters: call arrival process,
Poisson with a mean arrival rate of λ calls per unit time (i.e., interarrival time followed a
negative exponential distribution with a mean of 1/λ time units); call duration (call holding time), a negative exponential distribution with a mean of 1/μ time units; and transmission rate (group size) for each call request, a discrete distribution with a mean of g Rb
bps and a variance between 1Rb and 2KRb bps. Each simulation consisted of over 1,000,000
calls, which allowed for the collection and averaging of long-term PLP, BU, and other
metric data.
Results from a theoretical analysis of the proposed model and the simulation of a
partial loss OVSF single code assignment system were compared for two scenarios: one
in which the arriving group size had a discrete uniform distribution (DUNI) and another
in which the size had a geometric distribution (GEOM). All countable distributions −
constant, discrete uniform, or geometric − could be applied to map arriving group size,
which is corresponding to user’s behavior. In addition, the call i requesting rate Ri was
assigned the OVSF single-code rate ξ(Ri). In the simulation, system parameters included
load g λ / μ and traffic intensity ρ = g λ / c μ .
PARTIAL LOSS PROBABILITY OF OVSF SINGLE-CODE ASSIGNMENT
275
1.2
1
PLP/BU
0.8
0.6
0.4
PLP-Theoretical
PLP-Simulation
BU-Theoretical
BU-Simulation
0.2
0
0.73
0.97
1.21
1.45
1.70
1.94
2.18
2.42
2.66
2.91
3.15
Traffic Intensity ρ = g λ /c μ
Fig. 4. Comparison of theoretical and simulation PLP and BU results (c = 256; K = 4; g = 6.2).
5.1 Batch Arrival Size in Discrete Uniform Distribution (DUNI)
PLP and BU were simulated and calculated with ρ values ranging from 0.73 to 3.15,
where c = 256; μ = 0.00125; and 0.0375 ≤ λ ≤ 0.175. Such small λ and μ values were
adopted to obtain accurate simulation results. High values of λ and μ with the same ratio
still result in the same outcome for theoretical analysis. Here the arriving group size was
distributed with DUNI; maximum group size was 16 (K = 4) and average arriving group
size g = 6.2 − that is, ξ(Ri) = 1, 2, 4, 8, or 16Rb and λ1 = λ2 = λ4 = λ8 = λ16 = λ/5. Small
λ and μ values were purposefully used to obtain accurate simulation results. It was assumed that high λ and μ values at the same ratio would produce the same outcome. Fig. 4
presents a comparison of the theoretical and simulated PLP and BU results; the horizontal axis denotes the traffic intensity given by ρ = g λ / c μ and the vertical axis denotes
the PLP and BU. The theoretical PLP and BU values are respectively shown as solid
lines with squares and circles, and the simulation PLP and BU values are respectively
depicted as dotted lines with crosses and plus signs. As shown, the PLP and BU values
in both cases increased with rising ρ, suggesting that the proposed model can be used in
practical applications. Numerical data from this comparison are shown in Table 1.
Table 1. Comparison of theoretical and simulation PLP and BU results by DUNI groupsize distribution (c = 256; K = 4; μ = 0.00125, and g = 6.2).
ρ
0.73
1.21
1.70
2.18
2.66
3.15
PLP
BU
ASL
Theoretical (α) Simulation Theoretical (β) Simulation Theoretical (L) Simulation
0.01013
0.01889
0.71743
0.68502
183.7
175.3
0.18372
0.18115
0.92499
0.89589
236.8
229.2
0.33342
0.33339
0.96112
0.94874
246.0
242.7
0.42645
0.43858
0.97314
0.96774
249.1
247.6
0.49031
0.51005
0.97917
0.97713
250.7
250.0
0.53802
0.56242
0.98286
0.98223
251.6
251.3
JUI-CHI CHEN
276
Table 2. Example of Eq. (20) verification and equilibrium probabilities for all states in
the proposed model with GEOM group-size distribution (K = 4).
Variable
c
g
ρ
μ
λ
λ1
λ2
λ4
λ8
λ16
p0
p1
p2
p3
p4
p5
p6
p7
p8
p9
p10
p11
Value
32
1.889129
0.59035281
0.0025
0.025
0.01591023
0.00585305
0.00215322
0.00079212
0.00029138
0.000002591384391330
0.000025913843913300
0.000134280239777639
0.000480125947152270
0.001331598943041586
0.003054041789427059
0.006031635253228073
0.010548106704975488
0.016671029123825232
0.024186702756380827
0.032611735780050846
0.041276389257241169
Variable
p12
p13
p14
p15
p16
p17
p18
p19
p20
p21
p22
p23
p24
p25
p26
p27
p28
p29
p30
p31
p32
Eq. (20)
Value
0.049449721157001934
0.056467342577214905
0.061829275495929793
0.065251159998193753
0.066668521585155266
0.066205194239147197
0.064094714118563892
0.060668163642458833
0.056290547421308901
0.051317833509918705
0.046065391875284196
0.040791373853337504
0.035690317792398847
0.030895693815301269
0.026441020419562436
0.022366273388008871
0.018690324715714638
0.015415641210329997
0.012420510979950216
0.009703653928406523
0.006923173253407345
0.006923173253407318
Table 3. Comparison of theoretical and simulation PLP and BU results with GEOM
group-size distribution (c = 32; K = 4; μ = 0.0025, and g = 1.889129).
ρ
0.59
1.18
1.77
2.36
2.95
3.54
PLP
BU
ASL
Theoretical (α) Simulation Theoretical (β) Simulation Theoretical (L) Simulation
0.0232
0.0392
0.5586
0.5371
17.9
17.2
0.1725
0.1977
0.8404
0.8185
26.9
26.2
0.3435
0.3563
0.9209
0.9105
29.5
29.1
0.4675
0.4700
0.9498
0.9452
30.4
30.2
0.5548
0.5528
0.9637
0.9623
30.8
30.8
0.6183
0.6113
0.9716
0.9712
31.1
31.0
5.2 Batch Arrival Size in Geometric Distribution (GEOM)
The arriving group size was assumed to have a GEOM distribution (λ1: λ2: λ4: λ8:
λ16 = 0.636409: 0.234122: 0.086129: 0.031685: 0.011655), where c = 32, K = 4, g =
1.889129, and μ = 0.0025, with λ ranging from 0.025 to 0.15 and ρ from 0.59 to 3.54.
To test the correctness of the equilibrium probabilities from Eqs. (18) and (19), Eq. (20)
was verified for all theoretical results using different parameter sets. The example in Table 2 shows the verification and equilibrium probabilities of all states in the proposed
model, where λ = 0.025 and K = 4. The data indicate close agreement between the equilibrium probability of state c and the verification of Eq. (20). Table 3 shows a compari-
PARTIAL LOSS PROBABILITY OF OVSF SINGLE-CODE ASSIGNMENT
277
son of the PLP and BU values from theoretical analysis and simulation.
According to the Poisson Arrival See Time Average property, 2k basic rate calls
with identical exponential service times can also be viewed as a single call with a 2k basic rate. The results indicate that both the theoretical analysis and simulation had good
approximate values, suggesting that the proposed model can be applied to partial loss
OVSF single code systems for purposes of performance analysis. However, addition
formal mathematical proofs are required.
6. CONCLUSIONS
This study demonstrated the effectiveness of the M[2X]/M/c/c batch arrival partial
loss model for analyzing partial loss OVSF single code assignment systems. Results
from a comparison of simulation and proposed model prediction data suggest that operators can apply the model to the utilization maximization problem of achieving the
optimum number of OVSF codes in a Node-B so as to gain maximum profit within a
specific QoS constraint. Results from a test simulation indicate that the model can assist
researchers in WCDMA network construction.
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Jui-Chi Chen (陳瑞奇) received his B.S. (1993) and M.S.
(1995) in Computer Science and Information Engineering from
National Chao Tung University and Ph.D. (2006) in Computer
Science from National Chung Hsing University. He is currently
an Assistant Professor in the Department of Computer Science
and Information Engineering of Asia University, Taiwan. His
research interests include wireless communications and computer networks.