Call Blocking Probabilities in a W-CDMA Cell with Fixed Number of ...

Call Blocking Probabilities in a W-CDMA Cell with Fixed Number of Channels and Finite Number of Traffic Sources Georgios A. Kallos, Vassilios G. Vassilakis, and Michael D. Logothetis WCL, Dept. of Electrical & Computer Engineering, University of Patras, Patras, Greece e-mails: {kallos, vasilak, m-logo}@wcl.ee.upatras.gr

Abstract—This paper focuses on the Call Blocking Probabilities calculation in a W-CDMA cell with fixed number of channels and finite number of traffic sources. To this end, the use of the Engset Multirate Loss Model (EnMLM) is proposed in the uplink direction. To apply the EnMLM we need to extend it by incorporating the so called local blockings. Arriving calls compete for their successful admission to a W-CDMA cell. The call admission depends on the availability of the required channels. To analyze the system, we formulate an aggregate onedimensional Markov chain, and based on it, we determine the system state probabilities by an efficient recurrent formula. Consequently, we determine the call blocking probabilities in the uplink direction. Although the proposed model is approximate, its accuracy is found to be quite satisfactory. The evaluation is done through simulation. Moreover, the model performs better than the existing model which assumes infinite number of traffic sources in the cell.

I.

INTRODUCTION

The Wideband Code Division Multiple Access (WCDMA) is the underlying air interface used in the Universal Mobile Telecommunications System (UMTS). Unlike the pre-existing 2nd generation mobile networks (e.g. GSM), the UMTS was primarily designed to provide multimedia services with data rates higher than 144 Kbps. Improved speed data rates and higher throughput are achieved through effective interference control mechanisms, fast power control and soft handover. The performance modeling of 3rd generation mobile networks has to include the heterogeneous environment and the multiple services they support in order to prevent the over-dimensioning of the network or a degradation of the end-user QoS [1]. The Call Blocking Probabilities (CBP) calculation in WCDMA networks has been done by using several modifications of the well-known Kaufman-Roberts (K-R) recursion [2]-[3]. Special peculiarities of these networks such as soft blocking and activity factors have been incorporated into the K-R recursion to finally form a recurrent formula that achieves approximate but efficient calculation of the system state probabilities and consequently CBP. The proposed recursions correspond to the Erlang Multi-rate Loss Model (EMLM) used for the analysis of wired networks with random (Poisson) arriving calls, i.e. when infinite number of traffic sources that generate incoming calls is assumed. In [4] an efficient algorithm based on the K-R recursion is presented to estimate the CBP of multiple service-classes in the uplink

direction of a W-CDMA cell, for Poisson arriving calls. In [5], another model is presented for CDMA networks, for Poisson arriving calls and elastic service-classes (the inservice calls may alter their assigned bandwidth). As far the number of users roaming in the cell is concerned, it is certainly more realistic to consider finite number of traffic sources (than infinite), due to the limited coverage area of the cell. This is especially true in medium and small sized cells. In [6] an extension of [4] is presented in order to include finite traffic source population (quasi-random call arrival process). In [7] another model with finite number of traffic sources is proposed. It calculates CBP in the uplink direction based on the reduced load approximation method. The model presented in [8] computes CBP both in the uplink and downlink direction by exploiting the Delbrouck’s algorithm [9]; thanks to the latter, a more general than a Poisson or quasi-random call arrival process can be considered. The above mentioned models ([4]-[8]) do not have a product form solution since the system is described by an irreversible Markov Chain. In this paper, we propose the use of the EnMLM [10] for the efficient calculation of the CBP in the uplink direction of a W-CDMA cell with fixed number of channels and finite number of traffic sources. The EnMLM has been used in [6], where not a certain number of channels are accommodated in the W-CDMA cell. Herein, the total chip rate of 3.84 Mcps is divided into a fixed number of channels. A user must seize one or more channels to convey the call. The same consideration is used in [8]. To analyze the system in the uplink, we formulate an aggregate one-dimensional Markov chain, in which a state represents the total number of occupied channels in the cell. Based on the Markov chain we determine the state probabilities and consequently the CBP. We show that the resultant CBP of the proposed analytical model are significantly lower than that of the corresponding model with infinite traffic source population. This proves the necessity of the proposed model. The accuracy of the proposed model is verified through simulation. The analysis presented for the uplink direction can be applied for the downlink as well. This paper is organized as follows: In section II we present the theoretical analysis of our model. In section III we propose the algorithm for calculating the state probabilities and CBP; to clarify the derivation of the state probabilities, we assume both infinite (subsection III.A) and finite number

of traffic sources (subsection III.B) in the cell; consequently we determine the CBP, in subsection III.C. Section IV is the evaluation section, where numerical results are presented. We conclude in section V. II. THEORETICAL MODEL

B. Channel Capacity and Bandwidth requirement Having calculated the maximum number of calls for each service-class, N k , we are interested in interpreting this number into channels for each service-class and determine the system Capacity in channels. First, we define the spread data rate of service-class k, R s,k , as the proportion of W which is utilised by one call of service-class k [8]: R

Consider a W-CDMA system with a reference cell surrounded by other neighbouring cells. The mobile users roaming within the reference cell generate calls of K different service-classes. A. Maximum Number of Calls The power control equation for service-class k (i.e. the required Energy per bit, E b , over Noise spectral density, N 0 , to ensure a predefined QoS for every call-connection), is written as follows [11]: ⎛ Eb ⎞ pk ⎜ ⎟ =G − + N P p ( own k ) Pother + Pnoise ⎝ 0 ⎠k

(1)

Where G= W/v k R k is the processing Gain of service-class k with activity factor v k and data rate R k accommodated in the uplink of a W-CDMA cell of chip rate W= 3840 Kcps. p k is the received power from a user of service-class k, P o w n is the total received interference power from the users of the reference cell, P other is the total received interference power from the users of the other-neighbouring cells, a nd P noise is the background noise power. Solving (1) for p k we obtain: R

⎛ E ⎞ ( Pown + Pother + Pnoise ) pk = ⎜ b ⎟ ⎛E ⎞ ⎝ N0 ⎠k G+⎜ b ⎟ ⎝ N0 ⎠k

(2)

Assuming that only service-class k contributes to P o w n , it is implied that P o w n =p k N k where N k is the maximum number of calls of service-class k in the cell. Then, we deduce for (2):

Pown

⎛E ⎞ N k ⎜ b ⎟ ( Pother + Pnoise ) ⎝ N0 ⎠k = ⎛E ⎞ ⎛E ⎞ G +⎜ b ⎟ −Nk ⎜ b ⎟ N ⎝ 0 ⎠k ⎝ N0 ⎠k

Ptotal Pown + Pother + Pnoise 1 = = Pnoise Pnoise 1 − ηUL

(6)

⎡ Rs , j ⎤ ⎡W ⎤ = 384 channels and bk = ⎢ C=⎢ ⎥ channels ⎥ ⎢ bbu ⎥ ⎢ bbu ⎥

(7)

Note that this is a simple and approximate way to convert the chip rate W and the spread date rate R s,k into channels. It follows from (7) that the smaller the bbu is, the smaller is the error introduced by this approximation. Based on the previous analysis, in the next section we will derive a recurrent formula for the CBP calculation.

III. ALGORITHM FOR CALL BLOCKING PROBABILITIES CALCULATION

For the theoretical model described in Section II.B, in the case of Poisson arrival processes the K-R recursion computes the state probabilities, q(j) ([2]-[3]). In (8), j is the system state (i.e. the total number of occupied channels):

q( j) =

1 K ∑ ak bk q ( j − bk ), j = 1,...,C and q ( 0) = 1 j k =1

C

∑

(8) q(j) = 1

j =0

(3)

(4)

Substituting (3) in (4), we solve for N k : ⎛E ⎞ G+⎜ b ⎟ ( δ − 1) ⎝ N 0 ⎠k Nk = δ (1 − ηUL ) + δ ⎛ Eb ⎞ ⎜ ⎟ ⎝ N 0 ⎠k

W Nk

To quantify the relation between the uplink Capacity C, and the bandwidth requirement per call of each service-class k, b k , it is necessary to set a basic bandwidth unit, bbu. For instance, we can use a bbu equal to 10 Kcps. Then,

where

Now, consider the Noise Rise which is related to the uplink cell load, η UL , as follows [11]: NR =

Rs ,k =

(5)

Where δ= P o t h e r /P n o i s e . Using (5), we can calculate the maximum number of calls of service-class k given η UL and δ.

Where α κ =λ κ /μ κ is the offered traffic load of service-class k, and λ κ , μ κ are the mean arrival and mean service rate of service-class k respectively. However, (8) neglects the probability of a new call to be blocked in any system state j due to the effect of soft blocking. In the following, we refer to this probability as local blocking probability. In the following we modify (8) to incorporate the local blocking probabilities. A. State probabilities with local blocking probabilities for infinite number of sources We approximate the average number of channels, j’, which are occupied due to the effect of local blocking with an independent, lognormal distributed random variable with parameters given by [4],[8]:

μ=

Pother + Pnoise i +i / δ ⋅U ⋅ C = ⋅U ⋅ C , σ = μ Pown + Pother + Pnoise i + 1+ i / δ

(9)

Figure 1. State transition diagram for service-class k

where U is the average fraction of C utilized by the in-service calls and i= P o t h e r /P o w n . The local blocking probability of a newly arriving call while the system is in state j, is the state dependent blocking probability of service-class k:

q ( j) =

1 K ∑ ( N k − nk ( j ) + 1)α'k bk (1 - I j −bk ,bk ) q ( j - bk ) j k =1

for j =1,...,C where

I j ,bk = P ( j ' > C − ( j + bk −1) ) = 1 − CDF ( C − ( j + bk −1) )

1 K ∑ ak bk 1 − I j −bk ,bk q ( j − bk ) j k =1

(

)

(10)

(11)

In (11), the factor (1 − I j −b ,b ) actually reduces the arrival rate k

∑ q( j) = 1 and q ( 0 ) = 1

(13)

j =0

where CDF(x) is the cumulative distribution function of the lognormal distribution. Thus a new connection of serviceclass k with bandwidth requirement b k is accepted only if the last required channel is obtained successfully. We assume that the rest of the channels are occupied simultaneously. The state probabilities computed in (8) can be modified to incorporate the state dependent blocking probabilities [8]: q( j) =

C

In (12) and (13), the number of calls of each service-class k, n k (j), in different system states j is approximated by the average number of service-class k calls, in state j, assuming infinite number of sources for each service-class k [12]:

nk ( j ) ≈ ak q( j − bk )(1 − I j −bk ,bk ) / q( j )

(14)

where a k and q(j) are the parameters of the corresponding infinite algorithm in (11). C. Call Blocking Probabilities Having calculated the state probabilities, the CBP of each service-class k is computed by adding all the state probabilities multiplied with the state dependent blocking probabilities:

k

λ k from state j-b k to state j.

C

Bk = ∑ q ( j ) I j ,bk

(15)

j =0

B. State probabilities with local blocking probabilities for finite number of sources We now limit the number of sources for each service-class to a finite number. To extend the calculation provided by (10) for finite number of sources within the cell, we assume that service-class’ k calls derive from N k sources with a Poisson arrival process as in (8). In this case, the following recurrent formula which is modification of (8) can be used to calculate the state probabilities [10]:

q ( j) =

1 K ∑ ( N k − nk ( j ) + 1)α'k bk q ( j - bk ) for j k =1

j =1,...,C where

C

∑ q( j) = 1 and q ( 0 ) = 1

(12)

j =0

where α′ κ = γ k μ k - 1 is the offered traffic-load per idle source of service-class k and n k (j) is the number of calls of service-class k in state j. Figure 1 illustrates the state transition diagram of a system with a single service-class k and C channels of Capacity. Similar to (11), a modification of (12) can be used to compute the state probabilities [6]:

IV. EVALUATION AND NUMERICAL RESULTS In this section, we compare the analytical versus the simulation results for the CBP. We evaluate the accuracy of both the infinite and finite algorithms (presented in subsections III.A and III.B respectively). For this reason, we present two case studies. Consider an uplink with η UL = 0.65, i=0.5 and δ =2. A consideration with 3 service-classes is made with parameters: R 1 =12.2Kbps, v 1 =0 .67 and ( E b /N 0 ) 1 =3d B, R 2 =144Kbps, v 2 =1.0 and (E b /N 0 ) 2 =2 .5d B and R 3 =384Kbps, v 3 =1 .0 and (E b /N 0 ) 3 =2dB. A comparison between analytical versus simulation CBP results which are the mean values from 6 runs with confidence interval of 95% is made. The resultant reliability ranges of the measurements are small enough and therefore we present only the mean CBP results. Figure 2 shows the CBP for different number of sources of each service-class that generate traffic. The ratio of the number of sources at the x-axis is kept constant and equal to 2:1:1 while the INF-point represents the case with infinite number of sources (corresponding to the case in Section III.A). The offered traffic load is constant for each serviceclass and equal to α k =N k α′ k . We consider a case where

α 1 =15 er l, α 2 =8 er l and α 3 =4 er l. As we see, the model’s accuracy is satisfactory for every case of the number of sources. We also observe that decreasing the number of sources, the CBP also drop thus showing the overestimation that is made by considering the infinite model. In the 2nd case study we consider the same uplink parameters and service-classes but the number of sources for each service-class is kept constant: N 1 = 200, N 2 =100 and N 3 =50. At the x-axis of Fig. 3 each point corresponds to a set of values for the offered traffic load for each service-class as shown in Table I. The results in Fig. 3 show that the model’s accuracy is satisfactory again.

V. CONCLUSION We present a new model for the call-level analysis of a WCDMA cell with fixed number of channels and finite number of sources. First, based on the chip rate of the W-CDMA carrier, we calculate the channel capacity of the system. The required number of channels for each service-class is determined according to the required bit rate of each serviceclass call. Then, we show how we modify the K-R recursion in order to include the local blockings, and afterwards we introduce the local blockings to the EnMLM. In the latter case, the calculation of system state probabilities is done through the use of the corresponding EMLM. The provided recurrent formula for the calculation of the CBP is verified through simulation results. The application examples prove the necessity of the new model, because it provides significantly lower CBP that the existing model for infinite traffic source population in the cell. Simulation results showed that the model’s accuracy is satisfactory.

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[4]

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[9]

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Table I.

Traffic Load Points for the 2nd case study

Traffic Load Point a 1 (erl) a 2 (erl) a 3 (erl)

1 13 7 4

2 14 7.5 4.25

3 15 8 4.5

4 16 8.5 4.75

5 17 9.0 5.0

6 18 9.5 5.25

Figure 2. CBP vs Number of sources for the 1st case study

Figure 3. CBP vs Traffic Load for the 2nd case study