Performance Modelling of W-CDMA Networks Supporting Elastic and Adaptive Traffic Georgios A. Kallos, Vassilios G. Vassilakis, Ioannis D. Moscholios and Michael D. Logothetis* WCL, Dept. of Electrical & Computer Engineering, University of Patras, 265 04 Patras, Greece *Corresponding author:
[email protected]
Abstract-We propose a new model, named Wideband Threshold Model (WTM) for the analysis of WCDMA networks supporting elastic and adaptive traffic. Mobile users generate Poisson arriving calls that compete for the acceptance to a W-CDMA cell under the complete sharing policy. A newly arriving call can be accepted with one of several possible Quality-of-Service (QoS) requirements depending on the resource availability in the cell. We propose an approximate method, which is an extension of the Kaufman-Roberts algorithm, for the calculation of call blocking probabilities in the uplink direction. The accuracy of the proposed approximation is verified by simulation results. Keywords: Quality-of-Service; Call Blocking Probability; W-CDMA; Elastic Traffic; Adaptive Traffic.
1.
Introduction
The call-level performance modelling of 3rd generation (3G) wireless networks is important for the resource allocation among different services, the avoidance of too costly over-dimensioning of the network and the prevention, through traffic engineering mechanisms, of excessive throughput degradation. Despite of its importance, the call-level performance modelling and QoS assessment remains an open issue, due to the presence of elastic and adaptive traffic. The Universal Mobile Telecommunication System (UMTS) is the proposal for 3G wireless networks in Europe. The existing 2nd generation (2G) systems like GSM are designed primarily for voice services. UMTS networks, however, aim at supporting wide range of voice and data services. The air interface used in UMTS is the Wideband Code Division Multiple Access (W-CDMA). It uses the Direct Sequence CDMA (DSCDMA) technique and supports very high bit rates, up to 2Mbps [1]. Herein we distinguish three types of traffic: stream, elastic and adaptive. Stream traffic is generated by calls that have fixed resource and holding time requirements, which cannot be reduced at any time (e.g. realtime audio or video). Elastic and adaptive traffic is generated by calls that may have different possible resource requirements depending on the resource availability. The holding time of an elastic call is strongly related to the resources allocated to this call, while the holding time of an adaptive call is always constant and independent of the resources allocated to this call. The well-known Erlang Multi-rate Loss Model (EMLM) is used for the analysis of traditional networks supporting only stream traffic. A recurrent algorithm developed by Kaufman and Roberts (K-R algorithm) facilitates the calculation of call blocking probabilities in the EMLM [2], [3]. Since then, several modifications of this algorithm were proposed for wired and mobile networks [4]-[8]. In [4], a blocked call can retry many times, requesting for less resources each time. In [5], calls arrive to the link with several possible resource requirements and their request is made according to thresholds, which indicate the total number of occupied resources. The Connection-Dependent Threshold Model (CDTM) [6] generalizes the retry and threshold models (as well as the EMLM) by individualizing the thresholds among different services. The abovementioned models are proposed for wired connection-oriented networks with elastic traffic and they do not consider the resource allocation scheme of W-CDMA wireless networks. In [7], a stream traffic model for W-CDMA is presented where the calculation of call blocking probabilities in the uplink of a W-CDMA cell is based on an extension of the K-R algorithm. In [8], an extension of [7] is presented in order to model elastic traffic. In this model, elastic calls may change the occupied resources while in-service, however it is not possible to have different resource requirements upon arrival. In this paper, we develop an analytical model for W-CDMA networks supporting both elastic and adaptive traffic. Calls of different services arrive to a W-CDMA cell with several resource (QoS) requirements, depending on the total number of occupied resources at the time of arrival. Based on this model, we present an approximate recurrent algorithm for the calculation of call blocking probabilities in the uplink direction. The remainder of the paper is as follows: In section 2 we briefly review the EMLM and the K-R algorithm. In section 3 we review the CDTM. In section 4 we propose the Wideband Threshold Model; in section 4.1
1
the description of the model is given, while in sections 4.2 to 4.4 we calculate the local blocking probabilities, the state probabilities and the call blocking probabilities, respectively. In section 5 we present an application example and compare the analytical to simulation results. We conclude in section 6.
2.
Review of the Erlang Multi-rate Loss Model (EMLM) 2.1.
Model description
In the EMLM, a system with C units of a resource accommodates Poisson arriving calls of K different categories (services). Calls compete for the available resource units (r.u.) under the complete sharing policy [2]. Each service k (k=1,…,K) call requests upon arrival rk r.u. If they are available, the call is accepted in the system and the rk r.u. are occupied by the call for a time, exponentially distributed with mean µk-1. Otherwise the call is blocked and lost. The EMLM is widely used for the analysis of traditional telecommunication networks supporting stream traffic. The system can be a transmission link of certain capacity. The link capacity corresponds to the shared resource. For example if we define an r.u. equal to the transmission rate of 64 Kbps, then a 1.28 Mbps link consists of C=20 r.u. A voice call of R1=64 Kbps will request upon arrival r1= 1 r.u., while a video call of R2=256 Kbps will request r2=4 r.u. 2.2.
Local balance equation and resource share
The system state j ( j=0,…,C) is defined as the total number of r.u. occupied by the calls. The probability that the system is in state j is denoted by q( j). In the EMLM the following local balance equation exists between adjacent system states [2]:
αk q ( j - rk ) = Yk ( j )q ( j )
(1)
where ak = λk µk-1 is the offered traffic load of service k and Yk( j) is the average number of service k calls in state j. We can calculate the resource share (proportion of the resource occupied by calls of a specific service) of service k calls in state j ( j>0), Pk( j) as follows: (1) ⇔
αk rk q ( j - rk ) Yk ( j ) rk = jq ( j ) j
⇔
αk bk q ( j - rk ) = Pk ( j ) jq ( j )
(2)
In Fig. 1. we show the state transition diagram for the EMLM. In this example, a link of capacity C=4 is considered and the resource requirement of service k calls is rk=1.
Figure 1. State transition diagram for the EMLM
2.3.
State probabilities
To calculate the un-normalized state probabilities, qˆ ( j ) , the well-known K-R algorithm is used [2], [3]:
qˆ ( j ) =
1 K ∑ αk rk qˆ ( j - rk ), for j = 1,...,C j k=1
(3)
where qˆ (0) =1 and qˆ ( j ) =0 for j…> µk2-1 > µk1-1. The k
k
holding time of service k calls is assumed to be exponentially distributed. The pair (rk1, µk1-1) is used from service k calls when the number of occupied r.u. at the call arrival is j ≤ Jk1, where Jk1 is the lowest threshold of the service k. The pair (rkl, µkl-1), (for l>1), is used from service k calls when Jkl< j ≤ Jkl+1, where Jkl and Jkl+1 are two successive thresholds of the service k. The pair (rkS , µkS -1) is used from service k calls when JkS k
k
k
-1< j ≤ C- rkS . Finally, a service k call is blocked when C- rkS < j ≤ C. The offered traffic load of service k
k
k
calls with resource requirement rkl is defined as: akl = λk µkl-1. The total offered traffic load is equal for every pair (rkl, µkl-1) and is defined as: “the product of the offered traffic load by the required r.u. per call”, aklrkl [9]. In Fig. 2 we show the basic principle of the CDTM. A service k call with one threshold, Jk1=1024 Kbps (or Jk1= 16 r.u.) and two contingency transmission rate requirements, Rk1= 384 Kbps (rk1= 6 r.u.) and Rk2= 128 Kbps (rk2= 2 r.u.) is accommodated to a transmission link of capacity C= 2048 Kbps (or C=32 r.u.).
Figure 2. Principles of the CDTM
3.2.
Assumptions
In order to derive an approximate recurrent formula for the CBP determination in the CDTM, the following assumptions are necessary [6]: • Local balance: we assume that the local balance equation exists between adjacent system states. • Upward migration approximation: Calls accepted in the system with their maximum resource requirement are negligible within a space, called upward migration space. More precisely, the mean number of calls, Yk1(j), with requirement rk1 in state j is negligible when J1 + rk1 < j ≤ C ; the latter region is related to the variable δk1(j), defined below in (5). • Migration approximation: Calls accepted in the system with other than the maximum resource requirement are negligible within a space, called migration space. More precisely, the mean number of calls, Ykl(j) (l>1) with requirement rkl in state j is negligible when 0 1) ⎪ δk1 ( j ) = ⎨1, when j ≤ J k1 + rk1 and rk2 > 0 ⎪ ⎩0, otherwise
(5)
⎧⎪1, when ( J kl + rkl < j ≤ J kl+1 + rkl ) and (rkl > 0) δkl ( j ) = ⎨ , for l >1 ⎪⎩0, otherwise
(6)
3.3.
Local balance equation and resource share
Due to the assumptions of Section 3.2 the following local balance equation exists between adjacent states: αkl δ kl ( j ) q ( j - bkl ) = Ykl ( j )δ kl ( j ) q ( j )
(7)
where q( j) is the probability that the system is in state j. The resource share of service k calls with requirement rkl in state j (j>0), Pkl( j) is determined by: αkl rkl δ kl ( j )q ( j - rkl )
(7 ) ⇔
jq ( j )
3.4.
=
Ykl ( j )rkl δ kl ( j ) j
αkl bkl δ kl ( j )q ( j - rkl )
⇔
jq ( j )
= Pkl ( j )
(8)
State probabilities
The un-normalized state probabilities are calculated by the following recurrent formula: S
qˆ ( j ) =
1 K k ∑ ∑ ak rk δ k ( j )qˆ ( j − rkl ) j k =1 l =1 l l l
for j = 1,...,C
(9)
where qˆ (0) =1 and qˆ ( j ) =0 for j1) and the calls’ mean holding time strongly depends on the QoS level. • Adaptive type: services that have more than one QoS levels (Sk>1) and the calls’ mean holding time is the same for every QoS level. The arrival rate of service k calls is Poisson with mean λk. The service kl calls’ holding time is exponentially distributed with mean µkl -1. For elastic services it holds: µkS -1 >…> µkl-1>…> µk2-1 > µk1-1, while k
for adaptive services: µkS -1 =…= µkl-1=…= µk2-1 = µk1-1. The offered traffic load of a service kl is defined as: k
akl = λk µkl-1. For the purposes of our analysis, we express later in the paper the different service’s QoS requirements as different resource requirements. 4.1.1 Interference and call admission control We assume perfect power control – i.e. at the BS, the received power from each service kl call is the same and equal to Pkl [1]. Since in W-CDMA systems all users transmit within the same frequency band, a single user “sees” the signals generated by all other users as interference. We distinguish the intra-cell interference, Iintra, caused by users of the reference cell and the inter-cell interference, Iinter, caused by users of the neighbouring cells. We also consider the existence of the thermal noise, PN, which corresponds to the interference of an empty system. The call admission control (CAC) in W-CDMA systems is performed by measuring the noise rise, NR which is defined as the ratio of the total received power at the BS, Itotal to the thermal noise power, PN : NR =
I total I intra + I inter + PN = PN PN
(11)
When a new call arrives, the admission control estimates the noise rise and if it exceeds a maximum value, NRmax , the new call is blocked and lost. 4.1.2 User activity A user, during his call’s duration, alternates between transmitting and silent periods. This behavior is characterized by the activity factor vk, which represents the fraction of the call’s holding time during which the user is occupying system resources. Obviously we have 0 nmax )
(19)
In order to calculate the LBP of (19) we can use (15)-(18). We notice that the only unknown parameter is the inter-cell interference, Iinter. Similarly to [11], we model Iinter as a lognormal random variable (with parameters µI and σI), that is independent of the intra-cell interference. (An alternative approach is to model the inter-cell interference as a ratio of the intra-cell interference [12], [13]). Hence, the mean, E[Iinter] and the variance, Var[Iinter] of Iinter are calculated by (20) and (21): E[ Iinter ] = e
µI +
σI2
(20)
2
2
Var[ Iinter ] = (eσ I − 1)e2 µ I +σ I
2
(21)
Consequently, because of (16), the inter-cell load, ninter will also be a lognormal random variable. Its mean, E[ninter] and the variance, Var[ninter] are calculated by:
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E[ninter ] = e
µn +
σ n2 2
=
1 − nmax E[ Iinter ] N0
(22)
2 2 1 − nmax 2 Var[ ninter ] = (eσ n − 1)e 2 µn +σ n = ( ) Var[ I inter ] N0
(23)
where µn and σn are the parameters of ninter, which can be found by solving (22) and (23):
µ n = ln( E[ I inter ]) −
ln(1 + CV [ I inter ]2 ) + ln(1 − nmax ) − ln( PN ) 2
σn = ln(1 + CV [ Iinter ]2 )
(24) (25)
The coefficient of variation, CV [Iinter] is defined as: CV [ I inter ] =
Var[ I inter ]
(26)
E[ I inter ]
Note that (19) can be rewritten as:
1− β kl (nintra ) = P(ninter ≤ nmax − nintra − Lkl )
(27)
The Right Hand Side (RHS) of (27), is the cumulative distribution function (CDF) of ninter. It is denoted by P(ninter ≤ x)=Fn(x) and can be calculated from: ln x − µ n 1 Fn ( x) = [1 + erf ( )] 2 σn 2
(28)
where erf(•) is the well-known error function. Hence, if we substitute x= nmax - nintra- Lkl into (28), from (27) we have: ⎧1- Fn ( x), x ≥ 0 xj 4.3.3 Incorporation of local blockings. In W-CDMA systems, due to the inter-cell interference, blocking of a service kl call may occur at any state j with a probability LBkl( j). This is called local blocking factor (LBF) and can be calculated from (33): LBkl ( j ) =
j
∑ β kl ( c )Λ(c | j )
(33)
c =0
Note that for j=0 we have LBkl(0)=βkl(0). In Fig. 4 we show the state transition diagram for the WTM. We see that, due to the local blockings, the transition rates from lower states to higher, are reduced by the factor 1- LBk( j) in comparison to the EMLM (see Fig. 1). The highest reachable state in the diagram is denoted by jmax.
Figure 4. State transition diagram for the WTM
4.3.4 Determination of the resource share. The service kl resource share in state j, Pkl( j), is derived from (2) by incorporating the LBFs and the parameters delta of (5) and (6): Pkl ( j ) =
αk (1 - LBkl ( j - rkl ))rkl δ kl ( j )q ( j - rkl )
(34)
jq ( j )
4.3.5 Calculation of state probabilities. The un-normalized state probabilities are given by extending (3) due to the presence of local blockings: S
qˆ ( j ) =
1 K k ∑ ∑ αk (1 - Lkl ( j - rkl ))rkl δkl ( j )qˆ ( j - rkl ), for j = 1,..., jmax j k =1 l=1 l
(35)
where qˆ (0) =1 and qˆ ( j ) =0 for j1
(37)
5.
Evaluation
In this, section an application example is presented. We compare the analytical versus simulation CBP results for the WTM in order to show its accuracy. The simulation results have been obtained as mean values from 6 runs with confidence interval of 95%. However, the resultant reliability ranges of our measurements are small enough and therefore we present only the mean CBP results. 5.1 Application example
We consider a W-CDMA system with three services: voice, data and video. The traffic parameters used for each service are as follows (see also Table 1): • Voice service requires a transmission rate of R11=12.2 Kbps which can be reduced to R12=8.4 Kbps depending on the threshold J11= 0.7. The activity factor is chosen to be v1=0.5 and the •
required BER parameter is (Eb/N0)1=5dB. This service is adaptive since the reduction of the transmission rate does not affect the holding time. Data service requires a transmission rate of R21=64 Kbps which can be reduced to R22=32 Kbps depending on the threshold J21= 0.6. The activity factor is chosen to be v2=1.0 and the required
•
BER parameter is (Eb/N0)2=4dB. This service is elastic since the reduction of the transmission rate corresponds to the same increase of the holding time. Video service requires a transmission rate of R31=144 Kbps which can be reduced to R32=128 Kbps and to R33=112 Kbps depending on the thresholds J31= 0.4 and J32= 0.6. The activity factor
is chosen to be v2=0.3 and the required BER parameter is (Eb/N0)2=3dB. This service is adaptive since the reduction of the transmission rate does not affect the holding time. We take measurements for eight different traffic load points (x-axis of Fig.5). Each traffic load point corresponds to some values of the offered traffic load of three considered services as it shown in Table 2. In this example the mean value for the inter-cell interference is: E[Iinter] = 3*E-18 mW and CV [Iinter]=1.
Figure 5. CBP versus offered traffic load for the application example
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Table 1. Traffic parameters for the application example Voice Data Type Adaptive Elastic Transmission rates (Kbps) R11=12.2 and R12=8.4 R21=64 and R22=32
Video Adaptive R31=144, R32=128 and R33=112
Thresholds
J11= 0.7
J21= 0.6
J31= 0.4 and J32= 0.6
Activity factor BER parameter
v1=0.5 (Eb/N0)1=5dB
v2=1.0 (Eb/N0)2=4dB
v3=0.3 (Eb/N0)3=3dB
Table 2. Offered traffic load for the application example 1 2 3 Traffic load point Traffic load (erl)
Voice (α1) Data (α2) Video (α3)
2.0 1.0 0.75
6.0 2.0 1.0
10.0 3.0 1.25
4
5
6
7
8
14.0 4.0 1.5
18.0 5.0 1.75
22.0 6.0 2.0
26.0 7.0 2.25
30.0 8.0 2.5
In Fig. 5 we show the analytical and simulation results for three services versus the offered traffic load. The results show that the model’s accuracy is absolutely satisfactory, especially for low offered traffic load.
6.
Conclusions
We propose a new model for the analysis of a W-CDMA system supporting elastic and adaptive traffic. We provide a recurrent formula for the calculation of the call blocking probabilities. Simulations are used to verify the accuracy of the proposed calculations. We show by numerical examples that the accuracy of the new model is absolutely satisfactory. In this model, we assumed for each service infinite number of users that generate calls. This assumption restricts the model’s applicability to cells of very high density. In our future work we are going to investigate the modelling of finite number of users in the WTM – i.e. the number of users of each service will be limited. Acknowledgment Work supported by the research program PENED-2003 of the General Secretariat of Research and Technology of the Greek Ministry of Development. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13]
H. Holma and A. Toskala, eds., WCDMA for UMTS. John Wiley & Sons Ltd., 2002. J. Kaufman, “Blocking in a Shared Resource Environment”, IEEE Trans. Commun. COM-29 (10) (1981) 1474–1481. J. W. Roberts, “A Service System with Heterogeneous User Requirements”, in: G. Pujolle (Ed.), Performance of Data Communications systems and their applications, North Holland, Amsterdam, pp.423-431, 1981. J.S. Kaufman, “Blocking in a Completely Shared Resource Environment with State Dependent Resource and Residency Requirements”, Proc. IEEE INFOCOM’92, pp. 2224-2232, 1992. J.S. Kaufman, “Blocking with Retrials in a Completely Shared Resource Environment”, Performance Evaluation, 15, pp. 99-113, 1992. I. Moscholios, M. Logothetis and G. Kokkinakis “Connection Dependent Threshold Model: A Generalization of the Erlang Multiple Rate Loss Model”, Performance Evaluation, Vol. 48, issue 1-4, pp. 177-200, May 2002. D. Staehle and A. Mäder, “An Analytic Approximation of the Uplink Capacity in a UMTS Network with Heterogeneous traffic,” in 18th International Teletraffic Congress (ITC18), (Berlin), Sep 2003. G. Fodor and M. Telek, ”A Recursive Formula to Calculate the Steady State of CDMA Networks”, Proc. of International Teletraffic Congress 2005, Beijing, China, September 2005. H. Akimaru, K. Kawashima, “Teletraffic – Theory and Applications”, Springer-Verlag, 1993. K.W. Ross, “Multiservice Loss Models for Broadband Telecommunication Networks”, Springer, Berlin, 1995. A. Viterbi and A. Viterbi, “Erlang Capacity of a Power Controlled CDMA System,” IEEE Journal on Selected Areas in Communication, vol. 11, August 1993. D. Staehle, K. Leibnitz, K. Heck, B. Schröder, A. Weller, and P. Tran-Gia, “Approximating the Othercell Interference Distribution in Inhomogeneous UMTS Networks,” in Proc. IEEE VTC Spring, (Birmingham, AL), May 2002. Glabowski M, Stasiak M, Wisniewski A, Zwierzykowski P. “Uplink Blocking Probability Calculation for Cellular Systems with WCDMA Radio Interference and Finite Source Population”, Proc. of 2nd International Working Conference on Performance Modelling and Evaluation of Heterogeneous Networks (HET-NETs’04), Ilkley, West Yorkshire, U.K., 26–28 July 2004; 80/1–80/10. Iversen VB, Benetis V, Ha NT, Stepanov S. “Evaluation of Multi-service CDMA Networks with Soft Blocking”, Proc. of ITC Specialist Seminar, Antwerp, Belgium, August/September 2004; 223 227.
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