Uplink blocking probability calculation for cellular systems with WCDMA radio interface and finite source population

Mariusz Głąbowski*, Maciej Stasiak*, Arkadiusz Wiśniewski and Piotr Zwierzykowski*

* Institute of Electronics and Telecommunications, Poznań University of Technology, ul. Piotrowo 3a, 60-965 Poznań, Poland,

e-mail: stasiak@et.put.poznan.pl PTK Centertel, ul Górecka 30, 60-201 Poznań, Poland

Abstract The paper presents and describes the uplink blocking probability calculation method in cellular systems with WCDMA radio interface and finite source population. The method is based on the model of the full availability group with multi-rate traffic streams. The proposed method can be easily applied to 3G network capacity calculations during planning stages of its design.

1. Introduction Universal Mobile Telecommunication System (UMTS) that uses WCDMA radio interface is one of

the standards proposed for third generation cellular technologies (3G). The standard has been adopted in Europe and some Asian countries. According to the ITU recommendations, the 3G system should include services with circuit switching and packet switching, transmit data with the speed of up to 2 Mbit/s, and ensure access to multimedia services [Weso 02].

In the GSM system, the maximal number of subscribers serviced by one cell was unequivocally defined and depended on the number of used frequency channels. Because the GSM system was assumed to perform voice services only, all calculations of the network capacity could be done on the basis of the known dependencies worked out by Erlang. Capacity calculations of WCDMA radio interface, due to the possibility of resource allocation for different classes of traffic, is much more complex. Moreover, all users serviced by a given cell make use of the same frequency channel and a differentiation of the transmitted signals is possible only and exclusively by an application of orthogonal codes [Faru 96]. However, due to multipath propagation occurring in a radio channel, not all transmitted signals are orthogonal with respect to one another, and, consequently, are received by users of the system as interference adversely and considerably affecting the capacity of the system. Additionally, the increase in interference also arises from the users serviced by other cells of the system that make use of the same frequency channel as well as by the users making use of the adjacent radio channels. To ensure appropriate level of service of the UMTS network it is thus necessary to limit interference by decreasing the number of active users or the allocated resources employed to service them. It is estimated that the maximal usage of resources of radio interface without decreasing the level of service will be at 50 – 80% [Holm 00].

This paper presents the uplink blocking probability calculation method for cellular systems with WCDMA radio interface and finite source population. To achieve this, the Kaufman-Roberts recursion used for the calculation of blocking probability for a full-availability group has been adopted and appropriately modified. It is expected that the presented method, due to taking into account decreasing number of sources, can yield more accurate results than traditional Erlang approach.

The paper has been divided into five sections. Section 2 discusses basic dependencies describing radio interface load for the uplink direction. Section 3 presents an analytical model applied to blocking probability calculations for the uplink direction and finite source population. The following section includes a comparison of the results obtained in the calculation with the simulation data for a system comprising seven cells. The final section sums up the discussion.

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2. Uplink load factor for WCDMA radio interface WCDMA radio interface offers enormous possibilities in obtaining large capacities, however,

imposes many limits as regards the acceptable level of interference in the frequency channel. In every cellular system with spreading spectrum the radio interface capacity is drastically limited due to the occurrence of a few types of interference [Laih 02], namely:

co-channel interference within a cell – coming from the concurrent users of a frequency channel within the area of a given cell, outer co-channel interference – from the concurrent users of the frequency channel working

within the area of adjacent cells, adjacent channels interference – from the adjacent frequency channels of the same operator or

other cellular telecommunication operators, all possible noise and interference coming from other systems and sources, both broadband and

narrowband . Accurate signal reception is possible only when the relation of energy per bit Eb to noise spectral

density N0 is appropriate [Holm 00]. A too low value of Eb / N0 will cause the receiver to be unable to decode the received signal, while a too high value of the energy per bit in relation to noise will be perceived as interference for other users of the same radio channel.

The relation Eb / N0 for the j-th user can be calculated as follows [Holm 00]:

jtotal

j

jjj

b

PIP

RN −⋅

⋅ ν0

WE=

, (1)

in which the following notation has been adopted: Pj – received signal power from the j user, W – chip rate of spreading signal, vj – activity factor of the j user, Rj – bit rate of the j user, Itotal – total received wideband power including thermal noise power. The mean power of the j-user can be determined with the help of the equation (2):

totalj

jjj

b

totalj IL

RNE

⋅=

⋅⋅

+

ν0

1 WIP = , (2)

where Lj is the load factor for the user of j class connection.

jj

j

b

R

NE

ν⋅⋅

+

0

1

1j WL = . (3)

Table 1 shows sample values Eb/N0 for different traffic classes and the matching values of the load-factor L.

Once the load factor of a single user has been established, it is possible to determine the total load for the uplink connection:

∑=ULη , (4) =

N

jjL

1

where, N is the number of users.

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Table 1. Examples of Eb/N0 and vk for different traffic classes and the matching load factor Lk

Type of service (k) Speech Video call Data Data

W [Mchip/s] 3.84

Rk [kb/s] 12.2 64 144 384

Vk 0.67 1 1 1

(Eb/N0) k [db] 4 2 1.5 1

Lk 0.0053 0.0257 0.0503 0.1118

The above relation is true only when we deal with a system that consists of a single cell. In fact,

however, there are many cells available, in which the generated traffic influences the capacity of radio interface of other cells. Thus, the relation (4) should be complemented with an element that would take into consideration interference coming from other cells [Holm 00]. To achieve this, a variable i is introduced, which is defined as other cell interference over own cell interference. The total load for the uplink can thus be rewritten as:

( )=η . (5) ∑=

+N

jjUL Li

11

The bigger the load of a radio link, the higher level of the noise generated in the system. The increase in the noise ∆nr is defined as the relation of the total noise received in the system to the thermal noise and is described by the following equation:

ULN

totalnr P η−

=1

1I=∆ , (6)

where PN is the thermal noise. When the load of the uplink approaches unity, the matching increase in noise tends towards infinity.

Therefore, it is assumed that the actual maximal usage of the resources of a radio interface without lowering the level of the quality of service will amount to about 50 – 80% [Laih 02].

3. Model of the system Before admitting a new connection in the systems with WCDMA radio interface, admission control

needs to check whether the admittance will not sacrifice the quality of the existing connections. The admission control functionality is located in RNC (Radio Network Controller) where the load information from the surrounding cells can be obtained. The admission control algorithm estimates the load increase that the establishment of a new call would cause in the radio network [Holm 00]. This is done not only in the access cell, but also in the adjacent cells to take the inter-cell interference effect into account. The new call is rejected if the predicted load exceeds particular thresholds set by the radio network planning either in uplink or downlink [Laih 02].

3.1. Basic assumptions Our model consists of seven cells with omni directional antennas, as it is shown in Figure 1. We

apply load factor Lj for the user of j class connection to estimate whether a new call can be admitted or blocked. We assume, for example, that the new call, which generates Lj load in the access cell, will generate part of that value in the surrounding cells (Lj≤Lj’). The problem of estimating Lj’ is not the subject of this paper. The measurements or theoretical calculations that might be the subject of further research can estimate

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these values. In our model we assume that each call which generates load Lj in access cell will cause load Lj’= Lj/2 in the neighbouring cells.

1

23

45

6

7'jL

'jL

'jL

'jL

'jL

new call

jL

'jL

Figure 1. A seven-cell set model of wireless cellular network

We assume that the model of full availability group servicing a mixture of the multi-rate traffic streams will approximate a radio interface which services the load generated in that cell and the load from the neighbouring cells (Figure 2) [Stas 04].

cell 21 V

cell 11 V

cell 31 V

cell 41 V

cell 51 V

cell 61 V

cell 71 V

12

34

567

ia ,11

ia ,14

ia ,1

ia ,77

ia ,66

ia ,55

ia ,44

ia ,33

access cell

Figure 2. A set of full availability groups as the model of a wireless network

For Figure 2 the following notation is used: V - is the cell capacity, azz,k - is the mean traffic of class k offered to the system by the users in the cell z and azr,k - is the mean traffic of class k offered in the cell r by the users of the cell z.

To determine the blocking probability of a new call appearing in the z-cell, it is necessary to take into consideration the load generated by the call in the neighbouring cells. Therefore we propose a new analytical model, which can be used for the calculation of uplink blocking probability in UMTS network. In the model, we have used the idea of the model of the switching node servicing a mixture of multi-rate unicast and multicast traffic streams presented in [Stas 98].

3.2. Analytical model 3.2.1. Full-availability group with infinite population of traffic sources

Let us consider the full-availability group with different multi-rate traffic streams. The full-availability group is a discrete link model that uses complete sharing policy [Cost 96]. This system is an example of a state-independent system in which the passage between two adjacent states of the process associated with a given class stream does not depend on the number of busy bandwidth units in the system. Let us assume that the system services call demands having an integer number of the so-called BBU’s (Basic

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Bandwidth Units) [Cost 96]. The total capacity of the group is equal to V BBU’s. The assumed BBU in our model is, for example, 0.0001 of load factor (The idea of demanding resources by a call as the load factor was presented [Stae 03]). The group is offered m independent classes of Poisson traffic streams having the intensities: λ1, λ2, ..., λm. The class k call requires Lk BBU to set up a connection (we assume that Lk

corresponds to the load Lj generated by the user of j-class connection). The holding time for calls of particular classes has an exponential distribution with the parameters: µ1, µ2, ..., µm. Thus the mean traffic offered to the system by the class k traffic stream is equal to:

k

kk µ

λa = . (7)

A multi-dimensional service process occurring in a system with different multi-rate traffic streams can be approximated by the one-dimensional Markov chain characterised by a product form solution. In the full-availability group model, Markov chain can be described by the Fortet-Grandjean recursion [Fort 64] which is generally known as the Kaufman-Roberts recursion [Kauf 81], [Robe 81]:

Pn = ∑ , (8) ( kk

m

kk LnPLan −

=1)( )

where P(n) is the state probability in the multi-rate system.

n-1

L2y2(n+1) L2y2(n+2)

a2L2 a2L2

L1y1(n) L1y1(n+1) L1y1(n+2)

a1L1 a1L1 a1L1 n n+2 n+1

Figure 3. Fragment of a diagram of the one-dimensional Markov chain in a multi-rate system (m=2, L1=1, L2=2)

The blocking probability bk for the class k traffic stream can be written as follows:

∑=b . (9) +−=

V

LVnk

k

nP1

)(

The diagram presented in Figure 3 can be characterised by the Kaufman-Roberts recursion for the system with two call streams (m=2, L1=1, L2=2). The yk(n) symbol denotes reverse transition rates of a class k service stream outgoing from state n. These transition rates for a class k stream are equal to the average number of class k calls serviced in state n.

From the equation (8) it results that the knowledge of the parameter yk(n) is not required for the determination of the occupancy distribution in the full-availability group with multi-rate traffic generated by infinite population of traffic sources. However, the value of this parameter in the given state of the group is the basis of the proposed method of the occupancy distribution calculation in the group with finite population of traffic sources. Accordingly, let us consider a part of the one-dimensional Markov chain diagram constructed for a system with multi-rate traffic (Figure 3). As stated earlier, the yk(n) symbol denotes reverse transition rates of a class k service stream outgoing from state n. This parameter determines – for class k traffic stream – the average number of class k calls serviced in state n and can be determined on the basis of the following reasoning [Stas 00].

Each state of the one-dimensional Markov chain in the multi-rate system (Figure 3) satisfies the following state equation:

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. (10) ∑ ∑∑ ∑= == =

++−=

+

m

k

m

kkkkkkk

m

k

m

kkkkk LnyLLnPLanyLLanP

1 11 1)()()()(

From the equation (8) it results that the sum of service streams outgoing from state n towards the lower states (states with lower number of busy BBU's) expressed in BBU's, is equal to n:

∑=n . (11) =

m

kkk nyL

1)(

According to the formulae (8) and (11), the equation (10) can be expressed as follows:

∑ ∑ . (12) = =

++=m

k

m

kkkkkkk LnPLnyLnPLa

1 1)()()(

Expression (12) is the equation of statistical equilibrium between the total stream outgoing from state n towards higher states and the total service stream entering state n from higher states. This equation holds only when local balance equations for call streams of individual traffic classes are satisfied:

(kk yLPa )())( kkkk LnPLnnL ++= . (13)

Based upon the equation (13), the reverse transition rate for class k calls in (n+Lk) state is equal to:

=P

Ly )( . (14) >+

≤+++

VLnforVLnforLnnPa

nk

kkkkk 0

)(/)(

Formula (14) determines the average number of class k calls serviced in the state n+Lk.

3.2.2. Full-availability group with finite population of traffic sources

Let us consider now the group servicing multi-rate traffic generated by finite population of sources [Glab 04]. Let us denote as Nk the number of sources of class k whose calls require Lk for service.

The input traffic stream of class k is build by the superposition of Nk two-state traffic sources which can alternate between the active state ON (the source requires Lk) and the inactive state OFF (the source is idle). The class k traffic offered by idle source is equal to:

k kk µα /Λ= , (15)

where Λk is the mean arrival rate generated by an idle source of class k and 1/µk is the mean holding (service) time of class k calls. In the considered model we have assumed that the holding time for the calls of particular classes has an exponential distribution. Thus, the mean traffic offered to the system by the idle class k traffic sources is equal to:

a kkkk nN α)( −= , (16)

where nk is the number of active (in-service) sources of class k. Interrelation between the offered traffic load and the number of in-service sources of a given class

(eq. 16) makes the direct application of the Kaufman-Roberts recursion (8) for determination of occupancy distribution in the considered system impossible.

The paper [Glab 04] proposes a new approximate method which enables us to make the mean value of traffic offered by class k dependent on the occupancy state (the number of occupied BBU's) of the group, and thereby determination of the system with finite population of sources by the Kaufman-Roberts recursion.

Let us notice that reverse transition rate yk(n) of class k determines the average number of class k calls serviced in the state of n busy BBU's. In the proposed method, it is assumed that the number of in-service nk sources of class k in the state of n BBU's being busy is approximated by the parameter yk(n), i.e.

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(n )() nyn kk ≈ . (17)

Such an approach assumes that the average number of the given class calls being serviced in a given state of the group with infinite population of traffic sources is approximate to the average number of calls being serviced in the same state in the case of finite source population. Thereby, the parameter yk(n) can be determined by the equation (14), where probabilities P(n) are calculated by the Kaufman-Roberts recursion, under initial assumption that the offered traffic load is not dependent on the number of in-service sources and is equal to:

Na kkk α= . (18)

The determined values yk(n) enables us to depend the mean value of the offered traffic on the occupancy state of the group in the following manner:

( ) kkkk nyNn α)()( −a ' = , (19)

Eventually, the approximated recurrent equation which determines the occupancy distribution in the full-availability group with multi-rate traffic and a finite population of sources can be calculated by the formula:

∑ −= anP ' . (20) =

−m

kkkkk LnPLLnn

1

'' )()()(

I the proposed model, it is assumed that a new call is rejected when the increase in the load, both in the access cell and the neighbouring cells, will exceed acceptable thresholds. This means that servicing process for a new call occurring in the access cell and the adjacent cells, are mutually dependent. Therefore, the blocking probability Bzz,k of the class k call occurring in the access cell z also depends on the blocking probability Bzh,k of those calls in the adjacent cells (h). The blocking probability of class k call coming from the cell z in the cell h ( h≠z ) can be expressed in the following function:

. (21) ( )),(),...,,(,),,(),...,,( ',,

'1,1,,,1,1,, mzhmzhzhzhmhhmhhhhhhkzh LaLaLaLafB K=

Thus, the blocking probability of the class k traffic streams coming from the z cell, in the h cell depends on “internal” traffic streams – generated in the cell h, i.e. the traffic streams ahh,k and on the contributing traffic streams from the cell z. For better clarity, it is assumed the radio interface load of each cell h ( h≠z ) is constant (i.e. 25 percent). This assumption makes it possible to express the probability Bzh,k as only dependable on the traffic streams which influence the call h from the access cell z. Therefore we obtain:

{k fB = , (22) }

)

),(),...,,( ',,

'1,1,, mzhmzhzhzhzh LaLa

The probabilities Bzh,k can be determined on the basis of the full-availability group servicing a mixture of multi-rate traffic streams (eq. 8 and 9). To determine the value azh,k influencing the load of cell h by the load generated in access cell z (for class k calls), we used the Fixed Point Methodology [Kell 89], [Stas 98]. According to the method, a given cell can be offered such a traffic which is not blocked in the neighbouring cells. This phenomenon leads to a decrease in the value of real traffic offered to a given cell and is called the thinning effect [Kell 89].

The class k traffic stream decreased by thinning effect which is offered to the cell h by the calls occurring in the access cell z has been named effective traffic. The traffic can be determined on the basis of the following equation:

∏= kaa , (23) (≠=

−u

rhr

kzrzkzh B1

,,, 1

where az,k is the real traffic class k offered to the system by the users form the cell z and u is the number of cells in the considered system.

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Let us noticed that to determine the value of the effective traffic azh,k of class k it is indispensable to know the blocking probability Bzh,k of the traffic of the same class in the neighbouring cells. Therefore, the iterative method is used to determine the values azh,k.

The known values of the blocking probability Bzh,k of class k in the cell h coming from cell z make it possible to determine the total blocking probability Bz,k of calls in the access cell z:

∏=kB . (24) (=

−−u

hkzhz B

1,, 11 )

4. Numerical results and simulations In order to confirm the proposed calculation method of uplink blocking probability in cellular system

with WCDMA radio interface and with finite number of traffic sources, the results of analytical calculations have been compared with the results of simulation experiments.

The study has been carried out for subscribers demanding a set of services. Figure 4 shows the mean blocking probability of a new call as a function of the mean load for speech and video calls (values in Table 1) for infinite number of sources with load of neighbouring cells of 25 percent. The assumed maximum usage of radio interface resources is equal to 80%. Figures 6-9 present blocking probabilities for a system with finite number of sources. The ratio of the number of traffic sources of all classes and the cell capacity expressed in BBU’s: {1, 2, 5, 10, 100}. =∑ =

VNmk k /1

All the presented results (Figures 4-9) show the robustness of the proposed method of blocking probability calculation. In each case, regardless of the offered traffic load and the ratio of the number of traffic sources, the results of the calculation are characterised by fair accuracy.

5. Conclusions The admission control in wireless network with WCDMA radio interface admits or blocks new calls

depending on the current load situation both in access cell and in neighbouring cells. A new call is rejected if the predicted load exceeds particular thresholds set by the radio network planning. In this paper, we present uplink blocking probability calculation method with finite number of sources, which employs analytical model of the node in multi-service network. In our model, we use load factor Lj to estimate whether a new call can be allowed or blocked. The proposed method allows us to model cellular system more accurately due to considering limited number of customers (traffic sources). The obtained results show that taking into account decreasing number of active sources in the system yields lower blocking probability than in models assuming infinite number of sources. This can be essential for services, which demand higher capacity.

The calculations are validated by a simulation. The proposed method can be easily applied to 3G network capacity calculations during planning stages of its design.

References [Cost 96] Broadband Network Teletraffic, Final Report of Action COST 242, (edited by

J.W. Roberts, V. Mocci, I. Virtamo), Published by the Commission of the European Communities, Springer Verlag, Berlin, 1996.

[Faru 96] S. Faruque: Cellular Mobile Systems Engineering, Artech House, Inc, 1996 [Fort 64] R. Fortet and C. Grandjean, Congestion in a loss system when some calls want several

devices simultaneously, Electrical Communications, vol. 39, no. 4, pp. 513–526, 1964, ITC-4.

[Glab 04] M. Glabowski, M. Stasiak, An Approximate Model of the Full-availability Group with Multi-rate Traffic and Finite Source Population, accepted for publishing on 3rd Polish-German Teletraffic Symposium.

[Holm 00] H. Holma, A. Toskala, WCDMA for UMTS. Radio Access For Third Generation Mobile Communications, John Wiley & Sons, Ltd., 2000.

[Kauf 81] J.S. Kaufman: Blocking in a shared resource environment, IEEE Transactions on Communications, vol. COM-29, No. 10, 1981, s. 1474-1481.

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0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,45 0,65 0,85 1,05 1,25 1,45

b(i)

video call (calculations) video call (simulations)speech (calculations) speech (simulations)

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,40 0,55 0,70 0,85 1,00 1,15 1,30 1,45

b(i)

video call (calculations) video call (simulations)speech (calulations.) speech (simulations)

Figure 5. Blocking probability in the access cell for finite number of sources. Number of sources for both classes equals 800 (400/400). Load of

neighbouring cells equals 25%.

Figure 4. Blocking probability in the access cell for infinite number of sources. Load of

neighbouring cells equals 25%.

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,40 0,60 0,80 1,00 1,20 1,40

b(i)

video call (calculations) video call (simulations)speech (calculations) speech (simulations)

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,40 0,60 0,80 1,00 1,20 1,40

b(i)

video call (calculations) video call (simulations)speech (calculations) speech (simulations)

Figure 7. Blocking probability in the access cell for finite number of sources. Number of sources

for both classes equals 4000 (2000/2000). Load of neighbouring cells equals 25%.

Figure 6. Blocking probability in the access cell for finite number of sources. Number of sources for both classes equals 1600 (800/800). Load of

neighbouring cells equals 25%.

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0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,40 0,60 0,80 1,00 1,20 1,40

b(i)

video call (calculations) video call (simulations)speech (calculations) speech (simulations)

0,000001

0,000010

0,000100

0,001000

0,010000

0,100000

1,000000

0,40 0,60 0,80 1,00 1,20 1,40

b(i)

video call (calculations) video call (simulations)speech (calculations) speech (simulations)

Figure 8. Blocking probability in the access cell for finite number of sources. Number of sources

for both classes equals 8000 (4000/4000). Load of neighbouring cells equals 25%.

Figure 9. Blocking probability in the access cell for finite number of sources. Number of sources for both classes equals 40000 (20000/20000).

Load of neighbouring cells equals 25%.

[Kell 89] F.P. Kelly: Fixed Point Models of loss networks, Journal of Australian Mathematics Society, vol. B31, 1989, s. 319-378.

[Laih 02] J. Laiho, A. Wacker, T. Novosad: Radio Network Planning and Optimization for UMTS, John Wiley & Sons, Ltd., 2002.

[Robe 81] J.W. Roberts: A service system with heterogeneous user requirements, [in] Performance of Data Communications Systems and their Applications, (edited by G. Pujolle), North Holland Pub. Co., Amsterdam, 1981.

[Stae 03] D. Staehle, A. Mader: An Analytic Approximation of the Uplink Capacity in UMTS Network with Heterogeneous Traffic, University of Wurzburg, Report No. 310 May 2003.

[Stas 00] M. Stasiak, M. Głąbowski, A simple approximation of the link model with reservation by a one-dimesional Markov chain, Journal of Performance Evaluation, vol. 41, no. 2-3, pp. 195-208, July 2000.

[Stas 98] M. Stasiak, P. Zwierzykowski: Analytical model of ATM node with multicast switching, Proceedings of 9th IEEE Mediterranean Electrotechnical Conference, Tel-Aviv, Israel, May 18-20, 1998, vol. 2, s. 683-687.

[Stas 04] M. Stasiak, A. Wisniewski, P. Zwierzykowski: Blocking probability calculation in the uplink direction for cellular systems with WCDMA radio interface, accepted for publishing on 3rd Polish-German Teletraffic Symposium.

[Weso 02] K. Wesołowski: Mobile Communication Systems, John Wiley and Sons, 2002.

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