We propose a distributed algorithm for power control in cellular, wideband networks for bursty tra c sources ... geometric rate. The condition is not ..... It appears that the algorithm is generic, i.e., it quite possibly applies in frameworks, composed.
A DISTRIBUTED POWER CONTROL ALGORITHM FOR BURSTY TRANSMISSIONS ON CELLULAR, SPREAD SPECTRUM WIRELESS NETWORKS Debasis Mitra and John A. Morrison AT&T Bell Laboratories Murray Hill, New Jersey 07974
ABSTRACT We propose a distributed algorithm for power control in cellular, wideband networks for bursty trac sources, such as data and speech with silence detection. The algorithm adapts power at mobiles on the basis of only local measurements of the mean and variance of the interference. The rst part of the paper introduces a probabilistic quality of service speci cation and gives an asymptotic framework for estimating orders of magnitudes of the quantities which a ect it. The second part gives the algorithm and the condition for its convergence at a geometric rate. The condition is not burdensome.
1. INTRODUCTION A major challenge facing designers of services on cellular wireless networks is the ecient integration of data trac, which belongs to the general category of trac sources which transmit in relatively short bursts and are silent at other times. Speech with silence detection belongs to this category, although its parameters are di erent from data. We propose a distributed algorithm for power control in cellular, wideband networks for such stochastic trac sources, in which the adaptation of power is driven by local measurements of the mean and variance of the interference. The paper is in two parts. The rst part introduces a probabilistic quality of service speci cation and gives an asymptotic framework for estimating the orders of magnitudes of various quantities which a ect it. The second part gives the algorithm and an analysis of its dynamic behavior. The condition for geometric convergence is obtained and, surprisingly, it is not burdensome. The condition is only slightly more demanding than what is obtained by substituting mean activity values in the condition derived in prior work with sources which are on all the time. The algorithm and analysis are new. Indeed, we are not aware of any prior work which considers the design of power control algorithms explicitly for bursty trac sources.
We consider a cellular network with many sources or mobiles in each cell, each transmitting for only a, possibly small, fraction of time. The transmission framework is spread spectrum, which implies that the interference to any user's transmission at any time is the superposition of the e ects of all other current transmissions in the network. The statistical multiplexing that is inherent to such a wideband system would appear to make it a natural choice for the bursty trac conditions which are considered here. A considerable amount of prior work has been done on power control in distributed [ZAN92a, ZAN92b, BPO92, HAN92, FMI93, HAN93, GZY94], distributed-asynchronous [MIT93], and centralized [GVG93] frameworks. Hanly [HAN92, HAN93] considers a cellular model with interference uctuations, rather than burstiness, i.e., he assumes that the power control algorithm can adapt to the interference uctuations. Also, of particular interest is work on power control in spread spectrum systems [NAL83]. Recent work in which power control is combined with network management is noteworthy [CBP94, HAN95, YAT95]. On-o sources are considered in [VVI93, MMO94].
2. MODEL An active mobile m in cell j transmits at power pjm , and the gain to its cell site is gjm , so that the received power at the cell site is gjm pjm . We assume that the received power from all mobiles in any cell is common and given by Pj , for cell j , i.e.,
Pj , gjm pjm 8 m 2 cell j :
(1)
We prefer to work with the quantities fPj g since they are fewer than fpjm g and this leads to a simpler analytic treatment. However, it should be straightforward to recover information on pjm from Pj via (1). Let Xjm be the activity indicator for mobile m in cell j , i.e., Xjm 2 f0; 1g and Xjm = 1 if and only if the mobile is active at a point in time. Let the received power at cell site j due to an active mobile m in cell k (k 6= j ) be gjm pkm . Let Ijk denote the received power at cell j due to all mobiles in cell k (k 6= j ), i.e.,
Ijk = We de ne
X
m2 cell k
gjm pkm Xkm = Mjk ,
X gjm X gjm X ( g p ) = P km km km k g g Xkm :
m2 cell k km
m2 cell k km
X gjm g Xkm (1 j J; 1 k J ) :
m2 cell k km
2
(2) (3)
Now, for any active mobile m in cell j , Xjm = 1 and the interference to its transmission is, using (1), X X Ij = Pj Ijk + j (4) Xjm + 0
m 2 cell j ?fmg 0
k:k6=j
where j (1 j J ) is the local receiver noise power at cell j . From (2), (3) and (4) we obtain
Ij =
J X k=1
Pk Mjk ? Pj + j
(1 j J ) :
(5)
The quality of service requirement in terms of the carrier-to-inference ratio is
Pj =Ij (1 j J )
(6)
where is a prespeci ed, typically small number. It will be slightly more convenient to write (6) as Pj ~ I~j (1 j J ) (7) where ~ = =(1 + ), and
I~j ,
J X k=1
Pk Mjk + j
(1 j J ) :
(8)
Typically is small and ~ . Now consider the stochastic trac model of a bursty mobile to be on-o , i.e., in cell j it is on or active with probability wj :
Pr[Xjm = 1] = wj ; Pr[Xjm = 0] = 1 ? wj ; 8 m 2 cell j :
(9)
On account of the random source behavior, the quality of service requirement has to be probabilistic, with probability of compliance at best less than 1. Let the requirement be
j 1 ? Lj where
(1 j J )
j , Pr[Pj ~I~j ]
(1 j J )
(10) (11)
and fLj g are given parameters. From (9) we have
E (Xkm) = wk ; var(Xkm ) = wk (1 ? wk ); 8m 2 cell k : 3
(12)
Hence, from (3),
E (Mjk ) =
X gjm g wk = Kk Gjk wk ;
(13)
m2 cell k km
where Kk is the number of mobiles in cell k, and
X gjm g :
Gjk = K1
(14)
k m2 cell k km
Note, in particular, that Gjj = 1. From (3) and (12), we obtain var(Mjk ) = where
X gjm 2 w (1 ? w ) = K H 2 w (1 ? w ) ; k
gkm
m2 cell k
Hjk = K1 2
k
k jk k
X gjm 2 : g
k m2 cell k
km
k
(15) (16)
We assume that gjm =gkm are uniformly bounded for m 2 cell k. Since Xkm 2 f0; 1g, it follows that the random variables gjm Xkm =gkm are uniformly bounded for m 2 cell k. Hence, from the Lindeberg theorem [FEL68], the central limit theorem holds for Mjk , as Kk ! 1. Hence q (17) Mjk = Kk Gjk wk + Hjk Kk wk (1 ? wk )Zjk ; where Zjk is asymptotically normally distributed, with zero mean and unit variance, as Kk ! 1. We note that Zjk (1 k J ) are independent random variables.
3. ASYMPTOTICS, ORDERS OF MAGNITUDES We introduce a natural scaling, which allows us to make order of magnitude estimates and to develop meaningful and ecient approximations by dropping negligibly small terms. Inherent to wideband systems in which the bandwidth is shared by a large number of users is the following scaling: (18) = Ka ; Kj = j K (1 j J )
where a = O(1) and j = O(1) as K ! 1. A natural choice for the large parameter K is the PJ sum of all mobiles in the network, i.e., K = Kj . Let
j =1
Pj = K1 Pbj + p1 Qj K
(1 j J )
(19)
where Pbj and Qj are O(1). The orders of magnitudes of the rst order (dominant) and second order terms in the expansion of Pj are dictated by consistency, as will be seen. 4
Now let us investigate the implications of (18) and (19) on the terms Pj and ~I~j appearing in the quality of service speci cations, (7), (10) and (11). By substituting (17), (18) and (19) in the expression for I~j in (8), it may be veri ed that
"X # q X a a b b ~ Gjk k wk Qk + Hjk k wk (1 ? wk )Zjk Pk +O(1=K 2) : ~Ij = K Gjk k wk Pk + j + K 3=2 k k (20)
To achieve Pj & ~ I~j (1 j J ), we compare coecients of, rst, 1=K and, second, 1=K 3=2 and obtain the systems of inequalities given below in (21) and (22). These constitute sucient conditions to give Pj ~ I~j , to within O(1=K 2).
Pbj ? a and where
X
Qj ? a Yj , a
X k
k
Gjk k wk Pbk aj (1 j J )
X k
Gjk k wk Qk Yj (1 j J )
q
Hjk k wk (1 ? wk )Pbk Zjk (1 j J ) :
(21) (22) (23)
Equation (21) is a system of relations free of random variables, which is in contrast to (22). We rst treat (21) before returning to (22). In matrix form (21) is [I ? F]Pb = a ; (24) where
Fjk , aGjk k wk = Gjk Kk wk ;
(25)
which will be recognized to be mean values of dominant terms in the interference. Also, (24) is similar in algebraic structure to relations for power levels necessary to combat co-channel interference in narrowband systems, see for instance, [FMI93] and [MIT93]. We assume that F is an irreducible matrix and since it is also nonnegative, it has an eigenvalue of maximum modulus, called the Perron-Frobenius eigenvalue, which is real, positive and simple. Denote this eigenvalue by rF . We recall the following result quoted in [MIT93]. The following statements are equivalent: (i) rF < 1 :
(ii) A solution to (24) with Pb = 0 exists.
(iii) [I ? F]?1 exists and is element-wise positive. 5
(26)
If (26) holds then a particular solution to (24) is
Pb = a[I ? F]?1 :
(27)
This solution is Pareto optimal since any other solution to (24) will have components which are at least as large and at least one component which is larger. Henceforth we shall assume that (26), which is to be interpreted as a mean-value capacity constraint, holds. Now we turn to (22) and (23). Note that Zjk (1 k J ) are independent and asymptotically normally distributed, with zero mean and unit variance. Hence Yj is asymptotically normally distributed, with zero mean and variance a2 j2 where
j2 ,
X k
Hjk2 k wk (1 ? wk )Pbk2 (1 j J ) :
(28)
To satisfy the quality of service requirement in (10) and (11), we pick Pb as in (27) and Q such that # " X (29) Pr Qj ? a Gjk k wk Qk Yj 1 ? Lj (1 j J ) : k
The latter condition is equivalent to the deterministic condition,
Qj ? a
X k
Gjk k wk Qk aj j
(1 j J )
(30)
where fj g is de ned as follows:
Z j 2 1 e?y =2dy (1 j J ) : (31) 1 ? Lj = p 2 ?1 That is, j is the multiple of the standard deviation of the asymptotic random variable Yj , which indicates how large the left-hand quantity in (30) has to be for the probabilistic requirement in (29) to be satis ed. In matrix form the system of inequalities in (30) is [I ? F]Q = au
(32)
where uj , j j and u = fuj g. The Pareto optimal solution is
Q = a[I ? F]?1u :
(33)
Note that Q > 0, assuming that j > 0 (1 j J ). In the nal part of this section we lift the veil of asymptotics and give relations without reference to the scaling in (18) and (19). Let A , 1 Pb and B = 1 Q ; (34)
K
K 3=2
6
so that, from (19),
P = A + B :
(35)
By multiplying (27) and (33) by 1=K and 1=K 3=2, respectively, we obtain relations for A and B: and where
[I ? F]A = ;
(36)
[I ? F]B = U(A) ;
(37)
Uj (A) , j Vj1=2(A) and Vj (A) =
X k
Hjk2 Kk wk (1 ? wk )A2k :
(38)
We will need to keep in mind that in the decomposition (35), B 0 : (48) Speci cally,
kxk = 1max jx j=w : j J j j
(49)
kU(P(n)) ? U(P)k ckP(n) ? Pk ;
(50)
It is shown in the Appendix that
where c is a constant, given by (62). It follows, from (46), (47) and (50), that
kP(n + 1) ? Pk (rF + c)kP(n) ? Pk :
(51)
We assume that (rF + c) < 1. Then, by induction,
kP(n) ? Pk [ (rF + c)]nkP(0) ? Pk ;
(52)
so that the convergence of P(n) to P is geometric. We note that in our asymptotic analysis, p with the scalings in (18) and the de nitions in (38), c = O(1= K ), and hence the condition for geometric convergence is only slightly more stringent than the condition rF < 1, corresponding to the substitution of mean activity values. 8
5. CONCLUSIONS We have given an algorithm in (44) for distributed power control in a cellular spread spectrum system. The application of the algorithm may be extended in a natural way to multiple classes of bursty trac, and even to combinations of constant and variable rate sources. It appears that the algorithm is generic, i.e., it quite possibly applies in frameworks, composed of assumptions, scalings, etc., other than the one considered here. Of course, its behavior and conditions for convergence may di er, thus necessitating separate analyses. An area for future study is the extension of the algorithm to asynchronous operations.
ACKNOWLEDGEMENT The authors gratefully acknowledge the bene t of some discussions with their colleague Stephen Hanly.
APPENDIX We here determine the constant c in (50). If xk and yk (1 k J ) are real, the CauchySchwarz inequality implies that
!1=2 X J J !1=2 X J X 2 xk yk2 xk yk :
(53)
3 J 2 J !1=2 J !1=2 2 X X X 4 x2k ? yk2 5 (xk ? yk )2 :
(54)
k=1
It follows that
k=1
Hence, from (41), we obtain j =1
[Uj (P) ? Uj (P)]2
J J X X 2 j =1
b2 where In other words,
k=1
k=1
k=1
J X
k=1
b2 = 1max kJ
J X j =1
j
J X
k=1
k=1
Hjk2 Kk wk (1 ? wk )(Pk ? Pk )2
(Pk ? Pk )2 ;
j2Hjk2 Kk wk (1 ? wk ); b 0 :
kU(P) ? U(P)k2 bkP ? Pk2 ; 9
(55) (56) (57)
where
0 J 11=2 X kxk2 = @ x2j A : j =1
But,
p
(58)
max jxj j kxk2 J 1max jx j : 1j J j J j
(59)
min wk kxk 1max jx j kxk2 : j J j
(60)
p p J 1max w max j x j =w = w kxk : kxk2 J 1max k j j kJ 1j J kJ k
(61)
Hence, from (49), we have 1kJ
Also, from (59), we obtain
The inequality (50) follows from (57), (60) and (61), where
p c = b J 1max w = min w ; kJ k 1j J j
and b is given by (56).
10
(62)
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