IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 12, DECEMBER 1999 329
Non-Reinitialized Fully Distributed
Power Control Algorithm
Jui Teng Wang and Tsern-Huei Lee
Abstract—A fully distributed power control (FDPC) algorithm
has been recently proposed for cellular mobile systems. In the algorithm, the connection which has the smallest initial carrier-to-interference (CIR) ratio is removed if CIR requirements are not satisfied afterL iterations of power control. The transmitter power levels of surviving connections are then reset to the maximal allowed values and the algorithm is executed again. We prove in this paper that, if the transmitter power levels are not reset after a connection is removed, then a feasible power set can be found faster and the power levels employed are smaller.
Index Terms—Distributed algorithms, power control.
I. INTRODUCTION
T
RANSMITTER power control is a common technique which can be used to reduce interference and allow as many receivers as possible to obtain satisfactory reception. Many power control algorithms have recently been proposed and analyzed [1]–[9]. In general, one can categorize power control algorithms into centralized and distributed. Centralized power control can achieve optimum outage probability [1], [5], [7] but requires link gains between all mobile users and the base station. Thus centralized power control is not feasible for a large network or an environment where link gains change rapidly. Some distributed power control algorithms which use only local carrier-to-interference ratio (CIR) information were studied [2], [3], [5], [6]. Among these algorithms, the fully distributed power control (FDPC) algorithm was reported in [6] to outperform others in finding a feasible power set, i.e., a power set which can meet the CIR requirements. In the FDPC algorithm, all users start with the maximal allowed transmitting power levels. If no feasible power set is found after iterations, the connection with the minimal initial CIR is removed. After the connection is removed, the algorithm is reinitialized, i.e., all surviving connections reset their transmitting power levels to the maximal allowed values and the algorithm is executed again. In this paper, we formally prove that, if the transmitter power levels are not reset after a connection is removed, then a feasible power set can be found faster and the power levels employed are smaller.Manuscript received November 5, 1998. The associate editor coordinating the review of this letter and approving it for publication was Prof. M. D. Zoltowski. This work was supported by National Science Council, Taiwan, R.O.C., under Grant NSC 88-2218-E-009-045.
The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 1089-7798(99)09924-X.
II. SYSTEM MODEL
We assume that there are connections in a cellular mobile network and consider the reverse link. (The results can also be used in the forward link.) Let represent the transmitting power of the th mobile user and denote its thermal noise. Assume that is the base station it is assigned to. As a result, the received CIR for the th user is given by
where represents the link gain between the th mobile user and the base station Let denote the CIR requirement for the th user. For all the users to meet their CIR
require-ments, we must find a power set such
that and for all As in [4], such
a power set is called a feasible power set. Given a configuration
specified by if there exists a
feasible power set , then this configuration is said to be feasible.
III. POWER CONTROL PROCEDURE
Let denote the initial transmitter power set.
Also, let and denote the
transmitter power set and the set of received CIR in the th iteration, respectively. The power control procedure of the FDPC algorithm is described below. In the procedure, represents the maximal allowed transmitter power for the th mobile user.
The power control procedure of FDPC Algorithm is
and
for all where
Proofs of the following properties of the FDPC power control procedure can be found in [6].
Property 1: for all and
Property 2: If then for all IV. REMOVAL CRITERIA
It is possible that, after iterations of power control, no feasible power set is found. In this case, the connection with
330 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 12, DECEMBER 1999
the smallest received initial CIR is removed. Let
represent the set of initial CIR. For convenience, every iterations are counted as a round and the round number is denoted by A new round is begun each time a reset occurs. The FDPC algorithm including removal procedure can be described as follows.
Step 1: Let and
for all .
Step 2: Execute at most iterations of the FDPC power control procedure.
Step 3: Stop if a feasible power set is found. Else, remove
connection which has the smallest initial CIR (i.e.,
for all .
Step 4: Let and for all
connection and go to Step 2.
Notice that the transmitting power levels are reset to the maximal allowed values for all surviving connections in Step 4. For convenience, we call such an algorithm the reinitialized FDPC (R-FDPC) algorithm. An alternative choice, which will be referred to the nonreinitialized FDPC (NR-FDPC)
algorithm, is to let and go to Step 2. It has
to be pointed out that the connection removed by both R-FDPC and NR-FDPC algorithms in Step 4, if necessary, are the same in every round. We prove in the following that the NR-FDPC algorithm performs better than the R-FDPC algorithm.
Let and denote
respectively the transmitter power set and the set of received CIR in the th iteration of round for the R-FDPC algorithm.
Similarly, let and
represent those sets for the NR-FDPC algorithm.
Lemma 1: Assume that a connection has to be removed at
the end of round If then
for all and
Proof: Since for any surviving connec-tion in the NR-FDPC algorithm, we have
where represents the connection removed at the end of round Therefore, Lemma 1 is true.
Lemma 2: Assume that, at the beginning of round n, the
following two conditions hold:
(i) for all users ;
(ii) if for any user .
We have, for all iterations of round :
(iii) for all users ;
(iv) if for any user .
Proof: We prove Lemma 2 by mathematical induction.
By assumption, (iii) and (iv) are true for . Assume
that the lemma is true for Consider the case
If then, according to the FDPC
algorithm, we have
Besides, since implies
we get
By hypothesis, we have for all users and
thus for all users On the other hand,
if then we have
Therefore, (iii) is true for The remaining work is to show that (iv) is true for
Assume that and According
to Property 2, we have and
Therefore, all we have to prove is that
together with imply
Assume that and
Since for all users we get
Consequently, (iv) is also true for This completes the proof of Lemma 2.
The meaning of Lemma 2 is that if, at the beginning of a round, the power levels employed in the NR-FDPC algorithm are smaller than or equal to those employed in the R-FDPC algorithm and, moreover, connection satisfies its CIR requirement in the NR-FDPC algorithm if it is so in the R-FDPC algorithm, then the same conditions hold after every iteration of the round. Based on Lemmas 1 and 2, we obtain the following theorem.
Theorem 1: It holds for all that:
(i) for all and ;
(ii) if then for all and .
The proof for Theorem 1 is similar to that for Lemma 2 and thus is omitted. It is noted that, with the results of Lemma 2, one needs only prove Theorem 1 for .
A consequence of Theorem 1 is that the NR-FDPC al-gorithm employs smaller power levels and finds a feasible power set faster than the R-FDPC algorithm. Numerical results presented in the following section show that the NR-FDPC algorithm may result in a much smaller outage probability than the R-FDPC algorithm.
WANG AND LEE: NON-REINITIALIZED FULLY DISTRIBUTED POWER CONTROL ALGORITHM 331
Fig. 1. A 19-cell CDMA cellular network.
Fig. 2. Outage probability against number of users.
V. NUMERICAL RESULTS
In this section, we study a CDMA cellular network which is composed of 19 cells, as shown in Fig. 1. The locations of users are uniformly distributed in this network and the reverse link is considered. The chip rate is chosen to be 1.2288 Mb/s, the same as that of current IS-95 cellular CDMA system. The data rate is 9.6 kb/s, and hence, the processing gain is 128. To obtain a bit error probability of 10 it was reported [10] that the required is 7 dB. Since
the CIR requirement for all users is set to 14 dB here.
The link gain is modeled as where
is the attenuation factor, is the distance between the th mobile user and the base station and is a constant that models the large scale propagation loss. The attenuation factor models power variation due to shadowing. is assumed to be independent, log-normal random variables with 0 dB expectation and dB log-variance.
The parameter value of in the range of 4–10 dB and the propagation constant in the range of 3–5 usually provide good models for urban propagation [11]. In our simulations, we
choose and as in [10].
The number of iterations is chosen to be eight. The outage probability is defined as the ratio of the number of removed connections to the number of total connections. Numerical results were obtained by means of computer simulation for 20 000 independent configurations. In Fig. 2, we plot the outage probability against the number of users. It can be seen that the NR-FDPC algorithm results in a much smaller outage probability than the R-FDPC algorithm. In this figure, the curve for NR-FDPC represents the outage probability for the nonreinitialized FDPC algorithm in which the connection removed in round is the one which has the smallest CIR after one iteration of the round. It can be seen that outage probabilities for NR-FDPC and NR-FDPC algorithms are close to each other.
VI. CONCLUSION
We prove in this paper that the nonreinitialized FDPC algorithm employs smaller power levels and finds a feasi-ble power set faster than the reinitialized FDPC algorithm. Simulation results reveal that the NR-FDPC algorithm may result in a much smaller outage probability than the R-FDPC algorithm. One possible further research topic which is currently under investigation is to study the performance of removal algorithms based on other criteria such as the maximum received interference.
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