- Email: [email protected]
Fluid analysis of CDMA cellular systems
Ramaswami and P.E. Wirth (Editors) 9 1997 Elsevier Science B.V. All rights reserved.
ITC 15 / V.
13
F l u i d A n a l y s i s of C D M A C e l l u l a r S y s t e m s Rajesh S. Pazhynnur a and Alexander Stolyar b and Philip J. Fleming c aMotorola, Inc., 1501 West Shure Drive, Arlington Hts, IL 60004; pazhynnr~cig.mot.com bAT&T Labs-Research, 600 Mountain Avenue, 2A-405, Murray Hill, NJ 0794; [email protected] CMotorola, Inc., 1501 West Shure Drive, Arlington Hts, IL 60004; fleming~cig.mot.com We formulate the power control problem for a fluid limit model of the reverse link (mobile-to-base station) of an IS-95 CDMA system [12] with arbitrary placement of base stations and an arbitrary mobile density and solve it through an efficient numerical procedure. The model yields estimates of base site characteristics such as median noise rise, other-cell interference, and the mean number of best-served mobiles as a function of system load. The results are compared with known numerical and simulation results. 1. I n t r o d u c t i o n Code Division Multiple Access (CDMA) cellular telephony systems are being deployed as an efficient method for using limited radio spectrum to meet the teletraffic capacity requirements of cellular mobile and personal communications systems (PCS) [5],[10]. Capacity analysis for CDMA systems presents unique problems in stochastic modeling not found in more traditional wireless communications systems. (See [2] for an overview.) In Frequency Division Multiple Access (FDMA) and Time Division Multiple Access (TDMA) systems service quality is typically given as a grade of service (GoS) based on the ratio of blocked calls to the offered load of calls requesting service. In these systems, the quality of the channel is assumed not to vary significantly over the hold-time of the call. For analog system this implies a sufficiently high signal-to-noise ratio and for digital systems it implies sufficiently low Bit Error Rate (BER), or similarly, Frame Error Rate (FER). Teletraffic engineering for such a system consists of specifying a GoS and then determining the offered traffic the system (or proposed system) can support at that GoS. In this sense, FDMA and TDMA are channel limited. In contrast, CDMA systems are interference limited. Although there are a limited number of codes and there are a fixed number of channel elements (consisting of demodulators, etc.) at the base site, these sorts of limitations are typically not exceeded in practice. The cause of capacity limitation comes from the fact that all conversations use the entire spectrum allocation simultaneously and any particular conversation can be reconstructed only if the correct code word is known. This makes the capacity of CDMA systems 'soft' in the sense that as more users are added, more interference is introduced which can cause voice reproduction quality to degrade gradually past the point of ac-
14
ceptability before any of the physical limits are reached. The quality of the voice link is maintained by sophisticated power control algorithms and soft handoff methods in which multiple transmitter/receivers are used to enhance the signal to interference and noise ratio (SINR). The principal metrics for determining reverse link interference are noise rise and the other-cell interference factor. Noise rise is a measure of received signal strength at a base station and is defined as the ratio of ambient noise plus user-generated interference to the ambient noise. This random quantity is typically quoted using the dB scale. The othercell interference factor at a base station is the ratio of the mean interference generated by mobiles best-served by other base stations to the mean interference generated by mobiles best-served by the given base station. It is typically used to extrapolate system capacity from the analysis of a single base station in isolation (e.g., [13]). Because telephone traffic is stochastic, the channel quality of the voice links, both the reverse link (mobile-to-base) and forward link (base-to-mobile), is inherently stochastic. In addition, IS-95 CDMA [12] systems use a variable rate speech encoder which decreases the mean level of interference but increases the variance. The aim of this paper is to present a 'fluid limit' approximation for estimating the SINR on the reverse link for a specified system configuration. The system configuration is specified as a 2-dimensional mobile density function, a set of base station locations, and a radio link propagation model or a table representing propagation loss values obtained via measurements. For this system we obtain estimates for median noise rise, other cell interference factors and mean number of mobiles controlled (A mobile is controlled or best served by the base station where its SINR equals its target SINR) by a base station. In Section 2 we describe in detail the system model. In Section 3 we introduce the fluid approximation, describe the power control algorithm, give analytical expressions for estimating the quality metrics described earlier and provide an efficient method for numerically evaluating the fluid model. Based on the analytical expressions obtained in Section 3 we numerically investigate two examples of CDMA systems in Section 4 and find that for most of the base stations the median noise rise is within 1 dB of simulation results. The model we present can be used to model real deployments of CDMA cellular and PCS systems. Since many if not most IS-95 CDMA system are reverse-link limited, this method has broad practical application for system planning. We extend the numerical results of Viterbi, Viterbi and Zehavi [14] concerning other-ceU interference in a system with base stations placed on an infinite regular hexagonal grid and uniformly distributed mobile locations. As mentioned before, our model allows for arbitrary placement of base stations and arbitrary mobile density. Furthermore, we also consider the power control problem which is studied in [14]. For the system studied in [14] our results are very close to the results in [14]. We also demonstrate usefulness of our approach by analyzing a city wide deployment of CDMA cellular system. 2. The System M o d e l We consider a model with N base stations arranged in a fixed but arbitrary manner on a bounded set S, S C ~2. The base stations are specified by their location (in S), antenna
15
bearing, antenna height, and the receiver antenna gain. In the case of a sectorized cell, each sector is represented by a unique base station. For example, a three sector base site is represented in our model by three base stations with the same location but different antenna bearings. The mobiles are distributed according to a mobile density p(s), s E S. In particular, the probability of finding a mobile in a small area dA(s) around the point s is given by p(s)dA(s). We consider a propagation loss model where the attenuation of a signal transmitted from a point s, s E S, to base station j, (1 _ j _< N), is given by
Aj(s) = rj(s)-Ze axj, (1) for some ~, rj(s), Xj is N(0,1) - a Gaussian random variable with zero mean and unit variance, a = qa(log 10)/10, where a is the standard deviation (in dB) of the lognormal shadowing component of the signal attenuation, and 0 < q < 1 is a constant. (A typical value of q is 1/vf2.) We assume that all components of the vector {Xj, j --- 1 , . . . , N } are independent, and vectors {X/} corresponding to different mobiles are independent. In the numerical section we focus on two special cases (power law and field measurement) of the above propagation loss model. For the power law model # in (1) represents the propagation exponent, and rj(s) is given by
rj(s) = c(j, s)b(s ),
(2)
where ~j(s) is the distance from mobile at s to base station j, and c(j, s) is some function which accounts for impact of sectorization and antenna gain on the propagation loss. For the field measurement model rj(s) represents a propagation value based on field measurements. R e m a r k . We assume that the mobile is always controlled/served by the station that can best serve it, i.e., the best station always belongs to the set of base stations a mobile is in soft handoff with. For convenience, we also assume that there are no constraints on the power transmitted by the mobile. A similar analysis applies for the more general case of constrained powers. In the remainder of this paper we consider a fluid limit of the above system and solve the power control problem for the limiting fluid limit system. 3. P o w e r C o n t r o l P r o b l e m for t h e F l u i d Limit o f t h e S y s t e m A fluid limit system is obtained from the original one by scaling a mobile's required SINR by a (small) factor c > 0 and simultaneous scaling of the mobile density by a (large) factor 1/c. As c $ 0 we obtain a fluid limit system in which the interference levels at the base stations are not random but deterministic. The deterministic interference levels in the limiting system can serve as an approximation to the medians of the interference levels in the original system. The power control problem for the fluid limit system is solved through an iterative procedure. Let H(n) - ( H i ( n ) , . . . , H N ( n ) ) T denote the total interference (powers) received at the N base stations after the nth iteration. Let U(n) = ( U l ( n ) , . . . , UN(n)) T denote the interference (excluding the background noise) at the N base stations after the nth iteration. Let B = (b,..., b)T, where b is the background noise at each base station. Then, for all n _>_0 H ( n + 1) = B + U ( n + 1) =: S + P ( H ( n ) ) =: P ( H ( n ) ) .
(3)
16
Note that U(n + 1) is calculated from H(n) (via the operator P) by assuming that at the end of the nth iteration the total interference at all base stations is given by H(n). A fixed point H* of the operator t5 is called an (optimal) solution to the power control problem. It can be shown (see [16]) that if a solution H* exists then it is unique and for any "initial condition", H(0), the above algorithm converges to H*, i.e., H* = lim,_.oo H(n). In the next subsection we discuss the choice of the initial vector H (0). We now obtain expressions for the operator P, i.e., given H(n) we give expressions for obtaining U(n + 1) and thus H(n + 1). For the remainder of this subsection for notational simplicity we denote H(n) and U(n) as H and U respectively. Recall that the best base station for a mobile located at point s is one where it achieves the maximum SINR. Let B(s) denote the best base station. Then, B(s) is given by
argmax
A./
h/rJ.
(4)
( The notation argmax means "The argument from a specified set where the expression (Aj(s)/Hj) attains its maximum".) Henceforth we assume that the best base station for the mobile located at s will always be one of the K, K _ N, closest base stations. This assumption is not necessary for deriving the following analytical expressions for P. However, restricting the choice of candidates for the best base station to K closest base stations reduces the numerical computations considerably. In Example 4.1 we investigate the affect of K on the othercell interference factor. Let Af(s) contain exactly the closest K base stations. Then, (4) is replaced by
B(s)
=
ar gmax j E ./V'(s) Aj(s)/Hj.
We assume that in the original system, the SINR for a mobile at s at its best base station B(s) is given by a random variable F(s) with mean 7- Therefore, the power received by base station j from the mobile located at s is given by
Hs(s)r(s)As(s)lAs(s)(s). Let Uj,k be the power (interference) received by base station j from all mobiles controlled by base station k in the fluid limit system. Note that Uj~/is the in-cell interference and Uj,k, k # j is the other-cell interference. Then, Uj - ~k Uj,k and
(5)
= gk7 fs E [Aj(s)/Ak(s)I;,(s)] p(s)dA(s)
(0)
where I~ (s) is given by
[I T_ i e H(s),i # k Remark. Note in (5) we scale the mobile density and the SINR by a parameter c. As c --+ 0, the original system 'converges' to a limiting fluid system where U and H are
17 not random but are deterministic. In this paper we do not prove the convergence to the fluid limit but focus on the power control problem for the fluid limit. The power control problem results ([3],[16]) apply directly to the fluid model and do not depend on the fluid model being established as a limit of the original system. After some simplification (6) can be rewritten as the following. For j = 1 , . . . , N, k ~ j,
us,s
gj~ f
f
oo
1-1:
1
Q ( - # / a log(ri,j(s)) + x)] dxp(s)dA(s),
(7) u~,~ = g~v~ ~'-/~ rj,~(s)-"/{j~.(,)}z{k~.(,)} ~[ ~
n Q ( - u I ~ log(ri,k(s)) + x + a) i E N'(s),i ii=k, i # j
Q(-u/~log(rj,k(s)) + z + 2<~))]dxp(s)dA(s) +HkTea2 fs ri'k(s)-"Iucz(~)}I{ke~fO)}
f f V< ~ [-=:I~rI # k Q(-ul~l~ ~ i ~ Ar(s),i
+ x +~) ] dxp(s)dA(s),
(8) where rj, k(s) - rj(s)/rk(s) and Q(-) is the tail distribution of a normal random variable with zero mean and unit variance. Note, (7) and (8) specify the operator P. Based on H*, the other-cell interference factor (also known as the f factor) for base station j is given by
S~ = u; - v;,j v;b
'
(9)
and the mean number of mobiles controlled by base station j (i.e., those mobiles which have the best SINR at base station j) is given by
L~
=
u;b H;')"
(10)
Similar expressions were derived by Viterbi, et. al.[14] for the case of hexagonal grid and uniform mobile density p(s) = p. 3.1. Choosing the initial interferences H(0) To speed up the convergence of H(n) to the solution H*, it is important to choose the vector of initial interferences ("initial state") H(0) as 'close' to H* as possible. We choose H(0) as a non-negative solution of the following linear vector equation: H(O) = 7ZH(O) + B
(11)
where the non-negative matrix Z = {zj}} is defined as follows:
(12)
18
where
ar gmax B(s) = j e Af(s) Aj(s).
(~3)
Equation (12) can be rewritten as follows. For j = 1 , . . . , N, k # j, zJ'i =
fs f_,o
e -=2/2
lI
o~ I{/ez(,)} v / ~ i E Af(s), i r j
O(-p/alog(ri j(s)) + xldxp(s)dA(s), (14)
z~,~ = e ~2 f~ r ~ , k ( s ) - " ~ ~ ( s ) ~ ~ ( ~ ) ~
/~e -=~/2
+ + oo ~ / ~ i e Af(s),i ~ k, i r jQ(-p/alog(ri,k(s)) x OL) Q(-p/alog(rj,k(s)) + x + 2a))dxp(s)dA(s)
f_~ e -x2/2
o~ V ~
II
ieN(s),i#k
Q ( - # / a log(ri,k(s)) + x + a)dxp(s)dA(s).
Compare equations (12), (13), (14), and (15), to equations (5), (5), (7), (8), respectively. It is known from the Perron-Frobenius theory (see, for example, [11], Section 2.1) that a non-negative solution to (11) exists and is unique if and only if A0 < 1/7, where A0 is the largest real eigenvalue of the matrix Z. Equation (11) is common in the power control literature. (See [17], [4], [6], [9] and references therein.) Usually, it determines a solution to the power control problem in cases when there is a fixed number of mobiles in the system, the propagation losses from each mobile to each base station are fixed, and the best base station for each mobile is fixed. Our model is more general, but the nature of equation (11) is essentially the same. Namely, a non-negative solution of (11) is the exact solution to the power control problem for our model in the case where the best base station for a mobile is always the base station to which the mobile has the lowest propagation loss. It can be shown using the power control theory results (see [16] and [3]) that the nonnegative solution H(0) of (11) (if it exists!) majorizes the solution to the power control problem H*. Moreover, the sequence of the vectors H ( n ) , n -- 0, 1,2,... obtained by iterations is monotonically decreasing and converges to H*, i.e., H(n) $ H*,
n -+ oo
(16)
This property is attractive, since even an approximate solution obtained after a finite number of iterations majorizes the actual solution H*. Numerical results suggest that the iterations converge very fast. 3-4 iterations are typically sufficient and even H(0) is a reasonably close approximation of H*. 3.2. N u m e r i c a l I n t e g r a t i o n Note from (7) and (8) that the computation of the in-ceU and other-ceU interference involves numerical integration. In particular, there is an outer integral (over ,5) and
19 an inner integral. In this subsection we focus on computational aspects of the inner integral. The outer integral is computed through a standard method for a two dimensional integral. The inner integral is obtained as follows. Consider the R.H.S of (7) and (8). The integrand of the inner integral is of the form e-X2F(x,s) for some F. Such integrals can be numerically computed by Gauss-Hermite integration methods (see [7] for details). The method is summarized below. The integral is computed by the following approximation. n
f ~ ~-~:E(~)~ ~ ~,F(~,) cr
(17)
1=0
where, xt are the roots of the Gn+l(X) = (-1)n+Xe x"
Hermite polynomial of order
n
dn+l d~-+~ (~-~")'
(~s)
and the corresponding weights At are obtained by
~' = f_~~
(~-
e-='a,+~(x) dx. ~,)(a',+~(~,))
(19)
For a given n, the roots x~ and the weights Az can be precomputed (see table in [1]). Furthermore, (17) is exact F(x) is a polynomial of degree 2n + 1 or less. (See [7] for a discussion on the corresponding numerical error associated with Gauss-Hermite integration methods.) Thus, (7), (8) can be rewritten (approximately) by the following. For
j , k = l , . . . , N , k:/:j, Ui,i =
Hk~' fs/{/egO)} ~ A'/~J(x',
s)p(s)dA(s),
(20)
l--0
Ui'k = HkTea2fs I{ie~f(')}I{ke~f(')}ri'k(s)-" n
Y~ AtFk(xt, s)Q(#/c~log(rkZ(s)) + xk + 2a)p(s)dA(s) l--0 n
+Hk'ye"2 fs ri'k(s)-UI{ir162
y~ AzFk(xt, s)p(s)dA(s),
(21)
l--0
where
P~(~, ~) = n,~c,),,~ Q(uI,~ log(~,,(~) + ~ + ~)/vff~, Fk(x, s) = IIiesc(,),ickQ(#/alog(ri,~(s)) + x + a ) / v ~ . 4. N u m e r i c a l W o r k EXAMPLe. 4.1. In this example we consider the simple case where base stations are arranged in a regular hexagonal grid and the mobiles are distributed (spatially) uniformly. For a fixed value of a, and K, we compute the other-cell interference factor fJ and the noise rise Hj/b at each base station j. We repeat this for different values of a and K (see Table 1 ). Viterbi, et. al. [14] studied an identical model and present a table similar to
20
Table 1 but for a = 0, 2 , . . . , 12 and K - 1 , . . . , 4. Our results match those reported in [14],[15]. Observe that while g - 4 is sufficient (for computing the f-factor) for small values of a larger values of K yield additional accuracy especially when a > 8. Values of a greater than 8 may be applicable to dense urban areas.
Table 1 Relative Other-Cell Interference Factor f. (# - 4, q - l/v/2) a
K-1
K-2
K=3
K-4
K-5
K-6
0 2 4 6 8 10 12
0.466 0.518 0.713 1.210 2.545 6.610 21.200
0.466 0.439 0.479 0.577 0.806 1.355 2.794
0.466 0.433 0.452 0.491 0.571 0.748 1.174
0.466 0.433 0.452 0.488 0.549 0.663 0.915
0.466 0.433 0.452 0.488 0.545 0.639 0.826
0.466 0.453 0.452 0.488 0.542 0.625 0.774
EXAMPLE 4.2. We consider a cellular system deployed in a major city and compute the noise rise, f-factor, and mean number of mobiles best-served for each base station. We validate these results with those obtained from a CDMA simulator which includes detailed modeling of both the forward and reverse links. The system is comprised of a mixture of omni and sectorized cells totaling 167 base sites (for this example, we will call both a sector of a sectorized cell and an omni cell a sector). The mobile density is based on field measurement and provided through an external file. Propagation loss information is provided through field measurement data. Figure 1 shows the comparison of noise rise between the simulator and the fluid limit model (FLM). The sectors have been renumbered so that the simulator results are monotonic for easier visual comparison. Although there are some sectors where the FLM differs considerably from the simulation result, the FLM tracks both the noise rise and f-factor quite well for most stations. 5. C o n c l u s i o n In this paper we present a fluid limit model of the reverse link in a CDMA cellular system that allows arbitrary base station placement, an arbitrary mobile density and a general radio propagation model. We show that the model is accurate and since numerical evaluation is computationally efficient the model can be used for analysis of city-wide CDMA deployments. The capacity (and coverage) of many CMDA systems is determined by the reverse link and so a reverse link model is a useful system engineering tool. However, much work remains in the area of CDMA cellular teletraffic engineering. For example, we have not captured the effect of mobile motion on the interference metrics. (See [8] for a simulation study of this aspect.) Also, we have not dealt with the forward link which is more difficult to treat mathematically than the reverse link as a result of the complexities of soft handoff and differences in the forward link power control algorithm.
21
I
o "
--H --H
2'o
4'o
6'0
8'0
Base
~o
station
~#o
14o
~;o
(Simulation) (Analysis)
Figure 1. Noise Rise for a CDMA System with 167 sectors. The sectors are renumbered so that their noise rises are monotonically increasing.
REFERENCES 1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover Publications, 1964. 2. D.E. Everitt. Traffic engineering the radio interface for cellular mobile networks. Proceedings of the IEEE, September 1994. 3. P.J. Fleming and A.L. Stolyar. Convergence properties of the CDMA reverse link power control algorithm. Submitted, 1995. 4. G.J. Foscini and Z. Miljanic. A simple distributed autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology, 1993. 5. K.S. Gilhousen, I.M. Jacobs, R. Padovani, A.J. Viterbi, L.A. Weaver, and C.E. Wheatley. On the capacity of a cellular CDMA system. IEEE Transactions on Vehicular Technology, May 1991. 6. S.V. Hanly. Information capacity of radio networks. Ph.D. Thesis. University of Cambridge, August 1993. 7. F.B. Hildebrand. Introduction to Numerical Analysis. Dover Publications, 1974. 8. G. Labedz, R. Love, K Stewart, and B. Menich. Predicting real-world performance for key parameters in a CDMA cellular system. VTC 1996, 1996. 9. D. Mitra. An asynchronous distributed algorithm for power control in cellular radio systems. WINLAB-93, 1993. 10. D.L. Schilling. Broadband spread spectrum multiple access for personal and cellular communications. Proc. IEEE Vehicular Technology Con/erence, 1993.
22
11. E. Seneta. Non-negative matrices. George Allen and Unwin Ltd., 1973. 12. TIA/EIA/IS-95. Mobile Station-Base Station Compatibility Standard .for Dual-Mode Wideband Spread Spectrum Cellular System. Telecommunications Industry Association, 1993. 13. A. Viterbi and A. Viterbi. Erlang capacity of a power controlled CDMA system. IEEE Journal on Selected Areas in Communications, August 1993. 14. A. Viterbi, A. Viterbi, and E. Zehavi. Other-cell interference in cellular power controlled CDMA. IEEE Transactions on Communications, March 1994. 15. A.J. Viterbi. CDMA: Principles of Spread Spectrum Communicatio~ AddisonWesley, 1995. 16. R.D. Yates. A framework for uplink power control in cellular radio systems. IEEE Journal on Selected Areas in Communications, September 1995. 17. J. Zander. Performance of optimum transmitter power control algorithm in cellular radio systems. IEEE Transactions on Vehicular Technology, 41, 1992.
ITC 15 / V.
13
F l u i d A n a l y s i s of C D M A C e l l u l a r S y s t e m s Rajesh S. Pazhynnur a and Alexander Stolyar b and Philip J. Fleming c aMotorola, Inc., 1501 West Shure Drive, Arlington Hts, IL 60004; pazhynnr~cig.mot.com bAT&T Labs-Research, 600 Mountain Avenue, 2A-405, Murray Hill, NJ 0794; [email protected] CMotorola, Inc., 1501 West Shure Drive, Arlington Hts, IL 60004; fleming~cig.mot.com We formulate the power control problem for a fluid limit model of the reverse link (mobile-to-base station) of an IS-95 CDMA system [12] with arbitrary placement of base stations and an arbitrary mobile density and solve it through an efficient numerical procedure. The model yields estimates of base site characteristics such as median noise rise, other-cell interference, and the mean number of best-served mobiles as a function of system load. The results are compared with known numerical and simulation results. 1. I n t r o d u c t i o n Code Division Multiple Access (CDMA) cellular telephony systems are being deployed as an efficient method for using limited radio spectrum to meet the teletraffic capacity requirements of cellular mobile and personal communications systems (PCS) [5],[10]. Capacity analysis for CDMA systems presents unique problems in stochastic modeling not found in more traditional wireless communications systems. (See [2] for an overview.) In Frequency Division Multiple Access (FDMA) and Time Division Multiple Access (TDMA) systems service quality is typically given as a grade of service (GoS) based on the ratio of blocked calls to the offered load of calls requesting service. In these systems, the quality of the channel is assumed not to vary significantly over the hold-time of the call. For analog system this implies a sufficiently high signal-to-noise ratio and for digital systems it implies sufficiently low Bit Error Rate (BER), or similarly, Frame Error Rate (FER). Teletraffic engineering for such a system consists of specifying a GoS and then determining the offered traffic the system (or proposed system) can support at that GoS. In this sense, FDMA and TDMA are channel limited. In contrast, CDMA systems are interference limited. Although there are a limited number of codes and there are a fixed number of channel elements (consisting of demodulators, etc.) at the base site, these sorts of limitations are typically not exceeded in practice. The cause of capacity limitation comes from the fact that all conversations use the entire spectrum allocation simultaneously and any particular conversation can be reconstructed only if the correct code word is known. This makes the capacity of CDMA systems 'soft' in the sense that as more users are added, more interference is introduced which can cause voice reproduction quality to degrade gradually past the point of ac-
14
ceptability before any of the physical limits are reached. The quality of the voice link is maintained by sophisticated power control algorithms and soft handoff methods in which multiple transmitter/receivers are used to enhance the signal to interference and noise ratio (SINR). The principal metrics for determining reverse link interference are noise rise and the other-cell interference factor. Noise rise is a measure of received signal strength at a base station and is defined as the ratio of ambient noise plus user-generated interference to the ambient noise. This random quantity is typically quoted using the dB scale. The othercell interference factor at a base station is the ratio of the mean interference generated by mobiles best-served by other base stations to the mean interference generated by mobiles best-served by the given base station. It is typically used to extrapolate system capacity from the analysis of a single base station in isolation (e.g., [13]). Because telephone traffic is stochastic, the channel quality of the voice links, both the reverse link (mobile-to-base) and forward link (base-to-mobile), is inherently stochastic. In addition, IS-95 CDMA [12] systems use a variable rate speech encoder which decreases the mean level of interference but increases the variance. The aim of this paper is to present a 'fluid limit' approximation for estimating the SINR on the reverse link for a specified system configuration. The system configuration is specified as a 2-dimensional mobile density function, a set of base station locations, and a radio link propagation model or a table representing propagation loss values obtained via measurements. For this system we obtain estimates for median noise rise, other cell interference factors and mean number of mobiles controlled (A mobile is controlled or best served by the base station where its SINR equals its target SINR) by a base station. In Section 2 we describe in detail the system model. In Section 3 we introduce the fluid approximation, describe the power control algorithm, give analytical expressions for estimating the quality metrics described earlier and provide an efficient method for numerically evaluating the fluid model. Based on the analytical expressions obtained in Section 3 we numerically investigate two examples of CDMA systems in Section 4 and find that for most of the base stations the median noise rise is within 1 dB of simulation results. The model we present can be used to model real deployments of CDMA cellular and PCS systems. Since many if not most IS-95 CDMA system are reverse-link limited, this method has broad practical application for system planning. We extend the numerical results of Viterbi, Viterbi and Zehavi [14] concerning other-ceU interference in a system with base stations placed on an infinite regular hexagonal grid and uniformly distributed mobile locations. As mentioned before, our model allows for arbitrary placement of base stations and arbitrary mobile density. Furthermore, we also consider the power control problem which is studied in [14]. For the system studied in [14] our results are very close to the results in [14]. We also demonstrate usefulness of our approach by analyzing a city wide deployment of CDMA cellular system. 2. The System M o d e l We consider a model with N base stations arranged in a fixed but arbitrary manner on a bounded set S, S C ~2. The base stations are specified by their location (in S), antenna
15
bearing, antenna height, and the receiver antenna gain. In the case of a sectorized cell, each sector is represented by a unique base station. For example, a three sector base site is represented in our model by three base stations with the same location but different antenna bearings. The mobiles are distributed according to a mobile density p(s), s E S. In particular, the probability of finding a mobile in a small area dA(s) around the point s is given by p(s)dA(s). We consider a propagation loss model where the attenuation of a signal transmitted from a point s, s E S, to base station j, (1 _ j _< N), is given by
Aj(s) = rj(s)-Ze axj, (1) for some ~, rj(s), Xj is N(0,1) - a Gaussian random variable with zero mean and unit variance, a = qa(log 10)/10, where a is the standard deviation (in dB) of the lognormal shadowing component of the signal attenuation, and 0 < q < 1 is a constant. (A typical value of q is 1/vf2.) We assume that all components of the vector {Xj, j --- 1 , . . . , N } are independent, and vectors {X/} corresponding to different mobiles are independent. In the numerical section we focus on two special cases (power law and field measurement) of the above propagation loss model. For the power law model # in (1) represents the propagation exponent, and rj(s) is given by
rj(s) = c(j, s)b(s ),
(2)
where ~j(s) is the distance from mobile at s to base station j, and c(j, s) is some function which accounts for impact of sectorization and antenna gain on the propagation loss. For the field measurement model rj(s) represents a propagation value based on field measurements. R e m a r k . We assume that the mobile is always controlled/served by the station that can best serve it, i.e., the best station always belongs to the set of base stations a mobile is in soft handoff with. For convenience, we also assume that there are no constraints on the power transmitted by the mobile. A similar analysis applies for the more general case of constrained powers. In the remainder of this paper we consider a fluid limit of the above system and solve the power control problem for the limiting fluid limit system. 3. P o w e r C o n t r o l P r o b l e m for t h e F l u i d Limit o f t h e S y s t e m A fluid limit system is obtained from the original one by scaling a mobile's required SINR by a (small) factor c > 0 and simultaneous scaling of the mobile density by a (large) factor 1/c. As c $ 0 we obtain a fluid limit system in which the interference levels at the base stations are not random but deterministic. The deterministic interference levels in the limiting system can serve as an approximation to the medians of the interference levels in the original system. The power control problem for the fluid limit system is solved through an iterative procedure. Let H(n) - ( H i ( n ) , . . . , H N ( n ) ) T denote the total interference (powers) received at the N base stations after the nth iteration. Let U(n) = ( U l ( n ) , . . . , UN(n)) T denote the interference (excluding the background noise) at the N base stations after the nth iteration. Let B = (b,..., b)T, where b is the background noise at each base station. Then, for all n _>_0 H ( n + 1) = B + U ( n + 1) =: S + P ( H ( n ) ) =: P ( H ( n ) ) .
(3)
16
Note that U(n + 1) is calculated from H(n) (via the operator P) by assuming that at the end of the nth iteration the total interference at all base stations is given by H(n). A fixed point H* of the operator t5 is called an (optimal) solution to the power control problem. It can be shown (see [16]) that if a solution H* exists then it is unique and for any "initial condition", H(0), the above algorithm converges to H*, i.e., H* = lim,_.oo H(n). In the next subsection we discuss the choice of the initial vector H (0). We now obtain expressions for the operator P, i.e., given H(n) we give expressions for obtaining U(n + 1) and thus H(n + 1). For the remainder of this subsection for notational simplicity we denote H(n) and U(n) as H and U respectively. Recall that the best base station for a mobile located at point s is one where it achieves the maximum SINR. Let B(s) denote the best base station. Then, B(s) is given by
argmax
A./
h/rJ.
(4)
( The notation argmax means "The argument from a specified set where the expression (Aj(s)/Hj) attains its maximum".) Henceforth we assume that the best base station for the mobile located at s will always be one of the K, K _ N, closest base stations. This assumption is not necessary for deriving the following analytical expressions for P. However, restricting the choice of candidates for the best base station to K closest base stations reduces the numerical computations considerably. In Example 4.1 we investigate the affect of K on the othercell interference factor. Let Af(s) contain exactly the closest K base stations. Then, (4) is replaced by
B(s)
=
ar gmax j E ./V'(s) Aj(s)/Hj.
We assume that in the original system, the SINR for a mobile at s at its best base station B(s) is given by a random variable F(s) with mean 7- Therefore, the power received by base station j from the mobile located at s is given by
Hs(s)r(s)As(s)lAs(s)(s). Let Uj,k be the power (interference) received by base station j from all mobiles controlled by base station k in the fluid limit system. Note that Uj~/is the in-cell interference and Uj,k, k # j is the other-cell interference. Then, Uj - ~k Uj,k and
(5)
= gk7 fs E [Aj(s)/Ak(s)I;,(s)] p(s)dA(s)
(0)
where I~ (s) is given by
[I T_ i e H(s),i # k Remark. Note in (5) we scale the mobile density and the SINR by a parameter c. As c --+ 0, the original system 'converges' to a limiting fluid system where U and H are
17 not random but are deterministic. In this paper we do not prove the convergence to the fluid limit but focus on the power control problem for the fluid limit. The power control problem results ([3],[16]) apply directly to the fluid model and do not depend on the fluid model being established as a limit of the original system. After some simplification (6) can be rewritten as the following. For j = 1 , . . . , N, k ~ j,
us,s
gj~ f
f
oo
1-1:
1
Q ( - # / a log(ri,j(s)) + x)] dxp(s)dA(s),
(7) u~,~ = g~v~ ~'-/~ rj,~(s)-"/{j~.(,)}z{k~.(,)} ~[ ~
n Q ( - u I ~ log(ri,k(s)) + x + a) i E N'(s),i ii=k, i # j
Q(-u/~log(rj,k(s)) + z + 2<~))]dxp(s)dA(s) +HkTea2 fs ri'k(s)-"Iucz(~)}I{ke~fO)}
f f V< ~ [-=:I~rI # k Q(-ul~l~ ~ i ~ Ar(s),i
+ x +~) ] dxp(s)dA(s),
(8) where rj, k(s) - rj(s)/rk(s) and Q(-) is the tail distribution of a normal random variable with zero mean and unit variance. Note, (7) and (8) specify the operator P. Based on H*, the other-cell interference factor (also known as the f factor) for base station j is given by
S~ = u; - v;,j v;b
'
(9)
and the mean number of mobiles controlled by base station j (i.e., those mobiles which have the best SINR at base station j) is given by
L~
=
u;b H;')"
(10)
Similar expressions were derived by Viterbi, et. al.[14] for the case of hexagonal grid and uniform mobile density p(s) = p. 3.1. Choosing the initial interferences H(0) To speed up the convergence of H(n) to the solution H*, it is important to choose the vector of initial interferences ("initial state") H(0) as 'close' to H* as possible. We choose H(0) as a non-negative solution of the following linear vector equation: H(O) = 7ZH(O) + B
(11)
where the non-negative matrix Z = {zj}} is defined as follows:
(12)
18
where
ar gmax B(s) = j e Af(s) Aj(s).
(~3)
Equation (12) can be rewritten as follows. For j = 1 , . . . , N, k # j, zJ'i =
fs f_,o
e -=2/2
lI
o~ I{/ez(,)} v / ~ i E Af(s), i r j
O(-p/alog(ri j(s)) + xldxp(s)dA(s), (14)
z~,~ = e ~2 f~ r ~ , k ( s ) - " ~ ~ ( s ) ~ ~ ( ~ ) ~
/~e -=~/2
+ + oo ~ / ~ i e Af(s),i ~ k, i r jQ(-p/alog(ri,k(s)) x OL) Q(-p/alog(rj,k(s)) + x + 2a))dxp(s)dA(s)
f_~ e -x2/2
o~ V ~
II
ieN(s),i#k
Q ( - # / a log(ri,k(s)) + x + a)dxp(s)dA(s).
Compare equations (12), (13), (14), and (15), to equations (5), (5), (7), (8), respectively. It is known from the Perron-Frobenius theory (see, for example, [11], Section 2.1) that a non-negative solution to (11) exists and is unique if and only if A0 < 1/7, where A0 is the largest real eigenvalue of the matrix Z. Equation (11) is common in the power control literature. (See [17], [4], [6], [9] and references therein.) Usually, it determines a solution to the power control problem in cases when there is a fixed number of mobiles in the system, the propagation losses from each mobile to each base station are fixed, and the best base station for each mobile is fixed. Our model is more general, but the nature of equation (11) is essentially the same. Namely, a non-negative solution of (11) is the exact solution to the power control problem for our model in the case where the best base station for a mobile is always the base station to which the mobile has the lowest propagation loss. It can be shown using the power control theory results (see [16] and [3]) that the nonnegative solution H(0) of (11) (if it exists!) majorizes the solution to the power control problem H*. Moreover, the sequence of the vectors H ( n ) , n -- 0, 1,2,... obtained by iterations is monotonically decreasing and converges to H*, i.e., H(n) $ H*,
n -+ oo
(16)
This property is attractive, since even an approximate solution obtained after a finite number of iterations majorizes the actual solution H*. Numerical results suggest that the iterations converge very fast. 3-4 iterations are typically sufficient and even H(0) is a reasonably close approximation of H*. 3.2. N u m e r i c a l I n t e g r a t i o n Note from (7) and (8) that the computation of the in-ceU and other-ceU interference involves numerical integration. In particular, there is an outer integral (over ,5) and
19 an inner integral. In this subsection we focus on computational aspects of the inner integral. The outer integral is computed through a standard method for a two dimensional integral. The inner integral is obtained as follows. Consider the R.H.S of (7) and (8). The integrand of the inner integral is of the form e-X2F(x,s) for some F. Such integrals can be numerically computed by Gauss-Hermite integration methods (see [7] for details). The method is summarized below. The integral is computed by the following approximation. n
f ~ ~-~:E(~)~ ~ ~,F(~,) cr
(17)
1=0
where, xt are the roots of the Gn+l(X) = (-1)n+Xe x"
Hermite polynomial of order
n
dn+l d~-+~ (~-~")'
(~s)
and the corresponding weights At are obtained by
~' = f_~~
(~-
e-='a,+~(x) dx. ~,)(a',+~(~,))
(19)
For a given n, the roots x~ and the weights Az can be precomputed (see table in [1]). Furthermore, (17) is exact F(x) is a polynomial of degree 2n + 1 or less. (See [7] for a discussion on the corresponding numerical error associated with Gauss-Hermite integration methods.) Thus, (7), (8) can be rewritten (approximately) by the following. For
j , k = l , . . . , N , k:/:j, Ui,i =
Hk~' fs/{/egO)} ~ A'/~J(x',
s)p(s)dA(s),
(20)
l--0
Ui'k = HkTea2fs I{ie~f(')}I{ke~f(')}ri'k(s)-" n
Y~ AtFk(xt, s)Q(#/c~log(rkZ(s)) + xk + 2a)p(s)dA(s) l--0 n
+Hk'ye"2 fs ri'k(s)-UI{ir162
y~ AzFk(xt, s)p(s)dA(s),
(21)
l--0
where
P~(~, ~) = n,~c,),,~ Q(uI,~ log(~,,(~) + ~ + ~)/vff~, Fk(x, s) = IIiesc(,),ickQ(#/alog(ri,~(s)) + x + a ) / v ~ . 4. N u m e r i c a l W o r k EXAMPLe. 4.1. In this example we consider the simple case where base stations are arranged in a regular hexagonal grid and the mobiles are distributed (spatially) uniformly. For a fixed value of a, and K, we compute the other-cell interference factor fJ and the noise rise Hj/b at each base station j. We repeat this for different values of a and K (see Table 1 ). Viterbi, et. al. [14] studied an identical model and present a table similar to
20
Table 1 but for a = 0, 2 , . . . , 12 and K - 1 , . . . , 4. Our results match those reported in [14],[15]. Observe that while g - 4 is sufficient (for computing the f-factor) for small values of a larger values of K yield additional accuracy especially when a > 8. Values of a greater than 8 may be applicable to dense urban areas.
Table 1 Relative Other-Cell Interference Factor f. (# - 4, q - l/v/2) a
K-1
K-2
K=3
K-4
K-5
K-6
0 2 4 6 8 10 12
0.466 0.518 0.713 1.210 2.545 6.610 21.200
0.466 0.439 0.479 0.577 0.806 1.355 2.794
0.466 0.433 0.452 0.491 0.571 0.748 1.174
0.466 0.433 0.452 0.488 0.549 0.663 0.915
0.466 0.433 0.452 0.488 0.545 0.639 0.826
0.466 0.453 0.452 0.488 0.542 0.625 0.774
EXAMPLE 4.2. We consider a cellular system deployed in a major city and compute the noise rise, f-factor, and mean number of mobiles best-served for each base station. We validate these results with those obtained from a CDMA simulator which includes detailed modeling of both the forward and reverse links. The system is comprised of a mixture of omni and sectorized cells totaling 167 base sites (for this example, we will call both a sector of a sectorized cell and an omni cell a sector). The mobile density is based on field measurement and provided through an external file. Propagation loss information is provided through field measurement data. Figure 1 shows the comparison of noise rise between the simulator and the fluid limit model (FLM). The sectors have been renumbered so that the simulator results are monotonic for easier visual comparison. Although there are some sectors where the FLM differs considerably from the simulation result, the FLM tracks both the noise rise and f-factor quite well for most stations. 5. C o n c l u s i o n In this paper we present a fluid limit model of the reverse link in a CDMA cellular system that allows arbitrary base station placement, an arbitrary mobile density and a general radio propagation model. We show that the model is accurate and since numerical evaluation is computationally efficient the model can be used for analysis of city-wide CDMA deployments. The capacity (and coverage) of many CMDA systems is determined by the reverse link and so a reverse link model is a useful system engineering tool. However, much work remains in the area of CDMA cellular teletraffic engineering. For example, we have not captured the effect of mobile motion on the interference metrics. (See [8] for a simulation study of this aspect.) Also, we have not dealt with the forward link which is more difficult to treat mathematically than the reverse link as a result of the complexities of soft handoff and differences in the forward link power control algorithm.
21
I
o "
--H --H
2'o
4'o
6'0
8'0
Base
~o
station
~#o
14o
~;o
(Simulation) (Analysis)
Figure 1. Noise Rise for a CDMA System with 167 sectors. The sectors are renumbered so that their noise rises are monotonically increasing.
REFERENCES 1. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover Publications, 1964. 2. D.E. Everitt. Traffic engineering the radio interface for cellular mobile networks. Proceedings of the IEEE, September 1994. 3. P.J. Fleming and A.L. Stolyar. Convergence properties of the CDMA reverse link power control algorithm. Submitted, 1995. 4. G.J. Foscini and Z. Miljanic. A simple distributed autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology, 1993. 5. K.S. Gilhousen, I.M. Jacobs, R. Padovani, A.J. Viterbi, L.A. Weaver, and C.E. Wheatley. On the capacity of a cellular CDMA system. IEEE Transactions on Vehicular Technology, May 1991. 6. S.V. Hanly. Information capacity of radio networks. Ph.D. Thesis. University of Cambridge, August 1993. 7. F.B. Hildebrand. Introduction to Numerical Analysis. Dover Publications, 1974. 8. G. Labedz, R. Love, K Stewart, and B. Menich. Predicting real-world performance for key parameters in a CDMA cellular system. VTC 1996, 1996. 9. D. Mitra. An asynchronous distributed algorithm for power control in cellular radio systems. WINLAB-93, 1993. 10. D.L. Schilling. Broadband spread spectrum multiple access for personal and cellular communications. Proc. IEEE Vehicular Technology Con/erence, 1993.
22
11. E. Seneta. Non-negative matrices. George Allen and Unwin Ltd., 1973. 12. TIA/EIA/IS-95. Mobile Station-Base Station Compatibility Standard .for Dual-Mode Wideband Spread Spectrum Cellular System. Telecommunications Industry Association, 1993. 13. A. Viterbi and A. Viterbi. Erlang capacity of a power controlled CDMA system. IEEE Journal on Selected Areas in Communications, August 1993. 14. A. Viterbi, A. Viterbi, and E. Zehavi. Other-cell interference in cellular power controlled CDMA. IEEE Transactions on Communications, March 1994. 15. A.J. Viterbi. CDMA: Principles of Spread Spectrum Communicatio~ AddisonWesley, 1995. 16. R.D. Yates. A framework for uplink power control in cellular radio systems. IEEE Journal on Selected Areas in Communications, September 1995. 17. J. Zander. Performance of optimum transmitter power control algorithm in cellular radio systems. IEEE Transactions on Vehicular Technology, 41, 1992.