TIME VARIANT POWER CONTROL IN
`CELLULAR NETWORKS
`
`Michael Andersin ∗ and Zvi Rosberg †
`
`(August 1996)
`
`—————————————-
`Abstract
`
`We study the transmission power control in a cellular network where users mobility results
`in a time varying gain matrix. A framework for evaluating the channel quality is specified,
`and an asymptotic representation of the link gain evolution in time is obtained. Then, a
`variant of a standard Distributed Constrained Power Control (DCPC) which copes with
`user mobility is derived. These two power controls, as well as constant-received power and
`constant-transmitted power controls are compared with respect to their outage probabilities
`in a Manhattan-like microcellular system. The comparison reveals that the classical DCPC
`algorithm has an outage probability close to one, unless some counter-measures are taken.
`The time variant algorithm however, copes well with users mobility and provides a close to
`an optimal scale up factor for the Signal to Interference Ratio (SIR) target. Furthermore,
`the time variant algorithm provides a substantial reduction in outage probability compared
`to the other algorithms above.
`
`Keywords: PCS, Wireless, Power Control, Time Variant Gain Matrix, Mobility.
`
`∗Telia Mobile, SE-131 86 Nacka Strand, Stockholm, Sweden.
`†Haifa Research Lab., Science and Technology, MATAM, 31905 Haifa, Israel.
`
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`—————————————-
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`1
`
`Introduction
`
`Transmitter power control has proven to be an efficient method to control cochannel interfer-
`
`ence in cellular PCS, and to increase bandwidth utilization. Power control can also improve
`
`channel quality, lower the power consumption, and facilitate network management functions
`
`such as mobile disconnection, hand-offs, base-station selection and admission control.
`
`Power control algorithms can be sub-divided into two main classes. One is the constant-
`
`received-power control, where transmitters adapt their power to meet some received power
`
`target at the receiver. The other is the quality-based power control, where the transmitters
`
`adapt their power to meet some signal quality target at the receiver. Quality-based power
`
`control has been shown to outperform constant-received-power control [32], and it has been
`
`extensively studied for narrow-band and wide-band systems.
`
`Centralized and distributed algorithms with continuous power levels, non-random in-
`
`terferers, and Signal to Interference (SIR) quality measure, have been developed and their
`
`convergence properties have been investigated in [1, 2, 8, 12, 13, 14, 15, 18, 22, 23, 25, 32, 33].
`
`Distributed algorithms with continuous power levels, random interferers, and Signal to In-
`
`terference (SIR) objectives, have been studied in [24, 27]. Distributed power control with
`
`discrete power levels and SIR quality measure, has been studied in [4, 31], and with continu-
`
`ous power control and Bit Error Rate (BER) quality measure, in [20]. Resource management
`
`functions combined with power control have been also investigated. A combination with
`
`mobile admission has been studied in [5, 9]; a combination with base station selection in
`
`[19, 29]; and a combination with mobile disconnection and hand-off in [4]. Notably is the
`
`study in [30], where sufficient conditions have been derived for the convergence of power
`
`control algorithms, which unifies most of the known converging results.
`
`In all the studies above, it has been assumed that the power control converges much
`
`faster compared to the changes in the link gains due to mobility. This assumption
`
`has motivated a snapshot evaluation of the algorithms (where link gains are fixed in time),
`
`which implies an under estimation of the quality measure target. (see e.g., [6]). To compen-
`
`sate this under estimation, coarse over-allocation of bandwidth is being used for designing
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`a cellular network. In future PCS environments, bandwidth would be more carefully allo-
`
`cated and users mobility will have a greater impact on the system performance. Hence, the
`
`snapshot analysis will not provide the desired system design parameters, and users mobility
`
`should be taken into account.
`
`A preliminary study of time variant power control in [6], reveals that the quality measure
`
`target must be set significantly higher than the target which is determined under the snapshot
`
`assumption. The study however, does not provide any concrete rule to determine the actually
`
`required quality target. Determining this value is a primary engineering problem in power
`
`control and it is the main objective of the current paper. The authors are not aware of any
`
`previous results on this design problem.
`
`This paper studies the “slow” power control problem in a cellular network where link
`
`gains vary in time according to a slow fading process which is exponentially correlated in
`
`time, [17]. An asymptotic representation of the link gain evolution in time is derived, and
`
`a framework to evaluate the channel quality in a time varying system is specified. In spite
`
`of the dynamic problem complexity, we derive a simple distributed time-dependent power
`
`control algorithm which successfully copes with users mobility. The algorithm enhances a
`
`previously proposed Distributed Constrained Power Control (DCPC) algorithm, [15], and
`
`requires only three additional system parameters. One is the maximum velocity of a mobile,
`
`the second is the log normal variance of the shadow fading, and the third is the correlation
`
`distance of the shadow fading. These three parameters can be a priori estimated by the
`
`system operator, therefore resulting in an algorithm that can be applied in practice.
`
`Our numerical examples reveal that the DCPC algorithm has an outage probability close
`
`to one, unless some counter-measures are taken. One possible counter-measure is to bound
`
`the transmission power from below. Another, is to scale up the quality measure target. In
`
`the latter case, it is not clear however, with how much to scale up. The time dependent
`
`algorithm which we develop, copes with this situation and provides a close to an optimal scale
`
`up factor. The algorithm also provides a substantial improvement in the spectrum utilization
`
`compared to the DCPC algorithm enhanced with a lower bound on the transmission power,
`
`the constant-transmitted power algorithm, and the constant-received power algorithm.
`
`In Section 2 we present the time variant system model, and in Section 3 we derive the
`
`power control algorithm. Numerical results are evaluated in Section 4, and final conclusions
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`are given in Section 5.
`
`—————————————-
`
`2 System Model
`
`Consider a generic channel in a cellular network which is being accessed by N transmitters,
`
`where each one of them is communicating with exactly one receiver. For the uplink case,
`
`the transmitters are the mobiles and the receivers are their corresponding base stations. For
`
`the downlink case, their roles are reversed.
`When transmitter j (1 ≤ j ≤ N ) is transmitting at time t, it uses a power of pj(t) ≤ pj,
`where pj is the maximum transmission power for transmitter j. Given that at time t, the
`link gain between transmitter j and receiver i is gij(t) (1 ≤ i, j ≤ N ), the received signal
`(cid:80)
`power at receiver i is gii(t) pi(t). The interference power experienced by receiver i at time
`j:j(cid:54)=i gij(t) pj(t) , (1 ≤ i ≤ N ), where νi > 0 is a time independent background
`noise power.
`
`t, is νi +
`
`Define the Signal to Interference Ratio at receiver i at time t, SIRi(t), by
`
`SIRi(t) =
`
`νi +
`
`(cid:80)
`
`gii(t) pi(t)
`j:j(cid:54)=i gij(t) pj(t)
`
`(1 ≤ i ≤ N ) .
`
`,
`
`(1)
`
`The SIR is a standard measure for channel quality, which is highly correlated with its error
`
`rate. Let γi be the SIR target for the channel between transmitter i and its corresponding
`
`receiver. We say that channel i is supported at time t, if
`
`SIRi(t) ≥ γi .
`
`(2)
`
`To incorporate mobility of the transmitters or the receivers (uplink or downlink), which
`results in time variant link gains, we have to specify the link gain processes (gij(t) | t ≥ 0),
`(1 ≤ i, j ≤ N ).
`
`We focus on a relatively slow power control algorithms with 1-100 power updates per
`
`second. Such rates are too slow to track fast multipath fading (usually modeled by a fast
`
`time varying Rayleigh process). Hence, we assume that the multipath fading is resolved by
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`appropriate coding and interleaving techniques. Power control algorithms with update rates
`
`of 100-10000 updates per second (which includes multipath fading) has been studied in [26].
`
`For every time instant t, the link gain is modeled as a product of a distance dependent
`
`propagation loss, and a slow shadow fading component. That is,
`
`i The factor Lij(t) is modeled as,
`
`gij(t) = Lij(t) · Sij(t) .
`
`Lij(t) = D−α
`ij (t) ,
`
`(3)
`
`(4)
`
`where Dij(t) is the distance between transmitter j and receiver i at time t, and α is a
`
`propagation constant. The factor Sij(t) is assumed to be log-normally distributed with a
`
`log-mean of 0 dB, and a log-variance of σ2 dB. That is,
`
`Zij(t) def=
`
`10
`σ
`
`log10 Sij(t) ,
`
`is the standard normal random variable.
`
`We assume that the link gain processes are mutually independent, and the evolution of
`each process (gij(t) | t ≥ 0) is governed by the following correlated process.
`
`Let v be the average mobile velocity, and t0 be an arbitrary time reference. For every
`t > 0, we assume that (Zij(t) | t ≥ 0) is a stationary Gaussian process with an exponential
`correlation function given by,
`
`E[Zij(t0 + t)Zij(t0)] = e− vt
`X ,
`
`(5)
`
`where X is the effective correlation distance of the shadow fading. The parameter X is
`
`environment dependent and describes how rapid the fading correlation is decreasing as a
`
`function of distance.
`From (5), we can represent the evolution of (Zij(t) | t ≥ 0) by
`(cid:179)
`
`X + Nij(t) ·
`Zij(t0 + t) = Zij(t0) · e− vt
`
`1 − e− 2vt
`
`X
`
`(cid:180) 1
`
`2
`
`,
`
`(6)
`
`where {Nij(t)} are independent standard normal random variables. variables,
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`Observe that for every pair (i, j), process. Sij(t) , t ≥ 0, are identically distributed
`random variables. The time variant shadow fading process with the exponential correlation
`
`function in (5), has been proposed in [17] based on field experimental data.
`
`For notational convenience, we introduce the normalized velocity,
`
`u =
`
`2v
`X
`
`.
`
`The evolution in (6) then becomes,
`
`2 + Nij(t) ·
`Zij(t0 + t) = Zij(t0) · e− ut
`
`(cid:179)
`
`(cid:180) 1
`
`2
`
`.
`
`1 − e−ut
`
`(7)
`
`(8)
`
`Assuming that the mobile moves with a constant velocity v, we can use the power expansion
`of the functions x−α, e−x and 10x, to obtain
`
`and
`
`Lij(t0 + t) = Lij(t0) + o(ut) ,
`
`(cid:179)
`
`(cid:180)
`1 + c · (ut)1/2 · ·Nij(t)
`
`· 10o((vt)1/2) ,
`
`Sij(t0 + t) = Sij(t0)
`
`(9)
`
`(10)
`
`where c = σ
`10 ln(10), and o(x) is a function of x with the property limx→∞ o(x)/x = 0.
`
`To facilitate the derivation of a time variant power control, we use the following asymp-
`
`totic representation with respect to (ut)1/2 (the standard deviation scale). For notational
`clarity, we adopt the convention of a ≈ b to denote an equality a = b+o(x1/2) or a = b·co(x1/2).
`
`From (3)-(10), it follows that
`
`gij(t0 + t) ≈ gij(t0)(1 + c · (ut)1/2 · Nij(t)) .
`
`(11)
`
`Remark 2.1 Note that c · (ut)1/2 · Nij(t) is normally distributed with mean 0 and standard
`deviation c · (ut)1/2. Thus, the link gain means within a short time interval are practically
`the same.
`
`—————————————-
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`3 Time Variant Power Control
`
`In this section, we propose a time variant version of the Distributed Constrained Power
`
`Control (DCPC) from [15]. We start by showing the limitation of the DCPC in a time
`
`variant system.
`
`When the link gains vary in time, DCPC updates the power according
`
`(cid:110)
`
`¯pi,
`
`γi
`gii(t)
`
`(cid:179)
`
` min
`
`pi(t + dt) =
`
`pi(t) ,
`
`(cid:80)
`
`νi +
`
`j:j(cid:54)=i gij(t) pj(t)
`
`(cid:180)(cid:111)
`
`if i ∈ U (t),
`
`,
`
`otherwise ,
`
`(12)
`
`where U (t) is an arbitrary set of transmitters. Observe that any asynchronous power update
`
`is allowed (subject to some week conditions which exclude infinitely long intervals where a
`If U (t) = {1, . . . , N} for every update instance t, then we
`get the synchronous DCPC algorithm. Otherwise, we get an arbitrary asynchronous version
`
`power is not being updated).
`
`(ADCPC).
`
`Also note, that the right element in the right-hand-side of the power iteration is the SIR
`
`target times the ratio between the interference power (including the background noise) at
`
`receiver i, and the link gain gii(t). Since the interference power can be measured, and gii(t)
`
`can be detected by the transmitter from the base station pilot signal (assuming a reciprocal
`
`system), this algorithm can be implemented in a distributed manner.
`
`In this paper, we consider a SIR based power control algorithm. An alternative approach
`
`is to use a Bit Error Rate (BER) based algorithm. Although BER is more directly connected
`
`to the user perceived quality than SIR is, it has the following deficiency. In practical systems
`
`bit errors are rare events. This makes BER estimators highly inaccurate, especially within
`
`the short time intervals that are imposed by fast power control updates.
`
`In practice, the interference and the link gain of the allocated channel i, are evaluated
`
`by sampling and averaging. Thus, the implemented DCPC is actually
`
` min
`
`pi(t + dt) =
`
`(cid:110)
`
`¯pi,
`
`γi
`gii(t)
`
`(cid:179)
`
`νi +
`
`(cid:80)
`
`(cid:180)
`
`j:j(cid:54)=i gij(t) pj(t)
`
`(t)
`
`(cid:111)
`
`if i ∈ U (t),
`
`,
`
`(13)
`
`pi(t) ,
`where {gij(t)} are averages of the link gains in a small time interval around t.
`
`otherwise ,
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`Under the assumption that link gains are fixed in time (i.e., gij(t) = gij), it has been
`
`shown in [15], that the iterated powers in (12) converge from every initial set of powers, to
`
`the following unique and positive fixed-point solution (p1, p2, ..., pN ) of
`
`¯pi,
`
`νi +
`
`(cid:88)
`
`j:j(cid:54)=i
`
`γi
`gii
`
` , } ,
`
`gij pj
`
`pi = min
`
`(1 ≤ i ≤ N ) .
`
`(14)
`
`When all the channels can be supported, this power vector is the minimum power vector
`
`(component-wise) which supports them at every time instant t. That is, SIRi(t) = γi
`
`for
`
`every i and t.
`
`However, in a cellular network mobiles change their positions, resulting in random and
`
`non-stationary link gains. To demonstrate the DCPC limitation in such an environment,
`
`consider the SIR values under the converging power vector in a slightly more favorite case
`
`where the link gain means are stationary in time. Assume that the sample averages have
`a highly statistically significant level so that the {gij(t)} averages in (13) are practically
`the same as the theoretical stationary means (to be denoted by {gij}). In such a case, the
`iterated powers converge to the unique and positive fixed-point solution of
`(cid:88)
`
`¯pi,
`
`νi +
`
`γi
`gii
`
`pi = min
`
` ,
`
`gij pj
`
`j:j(cid:54)=i
`
`(1 ≤ i ≤ N ) .
`
`(15)
`
`Observe however, that even in this case the equality in (15) involves only the mean link
`
`gains. Since the actual link gains are distributed around the means, the probability that
`
`each SIRi(t) is below the SIR target could be too high. Therefore, it is very likely that none
`
`of the mobiles are supported. This indeed turned out to hold true in our numerical results.
`
`To address this limitation we consider a more general case where the link gain means are
`
`not necessarily stationary, but the approximation in (11) holds true for every time reference t0
`
`and small t. Fix an arbitrary time reference t0 (where the power vector is (p1(t0), . . . , pN (t0))),
`
`and examine the iterated powers in (13) for t > t0, given the link gain matrix realization
`
`at time t0. (In probability theory terminology, we examine the iterated powers given the
`
`sub-σ-field at time t0.)
`
`From (11), gij(t0 + t) is random with respect to any realization instance gij(t0), and
`its variance increases linearly with t. Thus, up to some threshold t∗, the samples of the
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`interference and the link gain of the allocated channel may produce highly statistically
`
`significant estimates of the means. Therefore, given the link gain matrix at time t0, we may
`
`from Remark 2.1, practically use the following equalities:
`gij(t0 + t) ≈ gij(t0) ,
`
`(0 ≤ t ≤ t∗) .
`
`(16)
`
`Assume an ideal condition where the iterated power vector always converges within a time
`interval of length t∗, under some convergence stopping rule. (The time horizon t∗, will serve
`as a tuning parameter in our time-variant power control algorithm.) Under these conditions,
`it follows from (13), (15) and (16) that (p1(t0 + t∗), p2(t0 + t∗), . . . , pN (t0 + t∗)) is a fixed-point
`solution of
`
`pi(t0 + t∗) ≈ min
`
`¯pi,
`
`νi +
`
`(cid:88)
`
`j:j(cid:54)=i
`
`γi
`gii(t0)
`
` ,
`
`gij(t0) pj(t0 + t∗)t∗)
`
`(1 ≤ i ≤ N ) ,
`
`(17)
`
`for every realization of a gain matrix and a power vector at time t0.
`
`Consider a channel i, where the approximated equality in (17) is obtained by
`
`pi(t0 + t∗) ≈ γi
`gii(t0)
`
`νi +
`
`(cid:88)
`
`j:j(cid:54)=i
`
` .
`
`gij(t0) pj(t0 + t∗)
`
`Had the link gains been constant, the channel would have been supported from time (t0 + t∗)
`and on. However, in a time varying case, the powers at time (t0 + t∗) are appropriate only for
`the gain matrix at time t0. To cope with this out-dated condition, we propose the following
`
`modification to the DCPC algorithm.
`
`Accounting for the random link gains, the channel quality requirement has to be prob-
`
`abilistic. We require that for every time reference t0, the conditional probability given the
`
`link gains and powers at time t0, will satisfy
`Pt0 (SIRi(t0 + t∗) ≥ γi) ≥ 1 − β .
`(18)
`Here, Pt0 (Y ∈ A) = P (Y ∈ A | {gij(t0)}, {pi(t0)}), and β is a given positive parameter.
`Relaxing the standard notion in (2), we say that channel i is supported at time t if and only
`
`if (18) is satisfied. The selected parameter β is the outage probability.
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`Note that under the ideal convergence conditions above, if we could modify the DCPC
`
`algorithm in such a way that (18) would hold for every t0, then the channel quality would
`have been satisfied for every time instant t > t∗. This objective is carried out in the remaining
`of this section, in which we specifically derive a scale up factor to the SIR target which is
`
`used in the DCPC.
`
`First, we use the approximation in (11) to project appropriate percentiles for the link
`gains at time t0 + t∗. In practice, the cochannel interference is dominated by a small number
`of interferers, to be denoted by N0 (usually 2 or 3). In this situation, even when the dominant
`
`interferers are unknown, we may still regard the interference as if it is being generated by
`
`N0 transmitters. Let 0 < β1 < 1 and 0 < β2 < 1, be two parameters such that
`β2 (1 − β1) = 1 − β ,
`where (1 − β) is the SIR quality parameter in (18).
`We may now define the Time Variant Power Control (TVPC) with parameters (β1, β2, t∗).
`0 percentile of the normal random variable N (0, c2 · u· t∗), and
`Let ξ1(t∗) be the 1− (1− β1)
`(ξ2(t∗) − 1) be its β2 percentile. That is,
`(cid:179)
`(cid:180)
`
`N (0, c2 u t∗) ≥ ξ1(t∗)
`(cid:179)
`
`= 1 − (1 − β1)
`(cid:180)
`
`= β2 .
`
`N (0, c2 u t∗) ≥ ξ2(t∗) − 1
`
`(19)
`
`(20)
`
`(21)
`
`1N
`
`0 ,
`
`1N
`
`P
`
`and
`
`TVPC Algorithm
`
`P
`
`For any given parameters (β1, β2, t∗), every transmitter i updates its transmission
`(cid:179)
`(cid:180)(cid:111)
`(cid:110)
`(cid:80)
`power according to
`
`¯pi,
`
`γi
`ξ2(t∗) gii(t)
`
`νi + (1 + ξ1(t∗))
`
`j:j(cid:54)=i gij(t) pj(t)
`
`if i ∈ U (t),
`
`,
`
` min
`
`pi(t+dt) =
`
`pi(t) ,
`
`otherwise .
`
`(22)
`
`Remark 3.1 In our numerical examples below we have used N0 = 4. The difference how-
`
`ever, in the scale up factor compared to the case with N0 = 2, is at most 3%.
`
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`Observe that under TVPC, powers are being updated as under DCPC but with a SIR target
`of γi (1+ξ1(t∗))/ξ2(t∗) rather than γi, and a background noise of νi/(1+ξ1(t∗)) rather than νi.
`Also observe that TVPC is a power control algorithm that aims at maintaining the outage
`
`probability below a certain level β, whereas the DCPC algorithm aims at balancing the SIR
`
`values. Hence, TVPC is derived from a more practical objective function than DCPC is.
`
`Thus, under the ideal conditions above, it follows from (15), (16) and (22), that
`
`¯pi,
`
`pi(t0+t∗) = min
`
`γi
`ξ2(t∗) gii(t0)
`
`νi + (1 + ξ1(t∗))
`
`(cid:88)
`
`j:j(cid:54)=i
`
` ,
`
`gij(t0) pj(t0 + t∗)
`
`(1 ≤ i ≤ N ) ,
`
`(23)
`
`for every realization of a gain matrix and power vector at time t0.
`
`Observe that the updated power for channel i is a function of the gains at time t0. This is
`
`a result of averaging over many samples taken around time t, and our asymptotic properties
`
`in (16).
`
`Let Ei be the event that for channel i the equality in (23) is attained by,
`
`pi(t0 + t∗) =
`
`γi
`ξ2(t∗) gii(t0)
`
`νi + (1 + ξ1(t∗))
`
`(cid:88)
`
`j:j(cid:54)=i
`
` .
`
`gij(t0) pj(t0 + t∗)
`
`Ignoring elements whose magnitude is the order of o(ut)1/2, it follows from (11), (16), (19),
`
`(20) and (21) that follows )
`Pt0 (SIRi(t0 + t∗) ≥ γi | Ei) ≥ Pt0 (gii(t0 + t∗) ≥ ξ2(t∗) gii(t0)) ·
`·Pt0 (gij(t0 + t∗) ≤ (1 + ξ1(t∗)) gij(t0) | ∀ j ∈ N0)
`= P ( N (0, c2 u t∗) ≥ ξ2(t∗) − 1 ) · [P ( N (0, c2 u t∗) ≤ ξ1(t∗) )]N0
`= β2 · (1 − β1) = 1 − β .
`
`This is the condition we were aiming at. Thus, under the TVPC algorithm every transmitter
`
`whose power converges to a value below the maximum transmission power, is supported. The
`
`reader should not confuse between the property given in (24) and the outage probability of
`
`channel i. The latter is upper bounded by P (Ei), and it depends on the cell plan, reuse
`
`factor and SIR target.
`
`(24)
`
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`Note that the TVPC algorithm as defined in (22), requires the knowledge of both the
`
`interference power and the noise power, which may be difficult to measure in practice. A
`
`more practical version of the TVPC algorithm is the following one, which requires only the
`
`sum of the powers above. It is similarly formulated except for a noise scale up in (22), by a
`factor of (1 + ξ1(t∗)). Let
`
`then every transmitter i updates its transmission power according to
`
`ξ(t∗) =
`(cid:179)
`
` min
`
`pi(t + dt) =
`
`pi(t) ,
`
`(cid:110)
`¯pi, γiξ(t∗)
`
`gii(t)
`
`1 + ξ1(t∗)
`ξ2(t∗)
`(cid:80)
`
`,
`
`(25)
`
`(26)
`
`(cid:180)(cid:111)
`
`if i ∈ U (t),
`
`,
`
`otherwise .
`
`νi +
`
`j:j(cid:54)=i gij(t) pj(t)
`
`As above, for every channel i where
`
`pi(t0 + t∗) =
`
`γiξ(t∗)
`gii(t0)
`
`νi +
`
`(cid:88)
`
`j:j(cid:54)=i
`
` ,
`
`gij(t0) pj(t0 + t∗)
`
`it is straightforward to show that
`Pt0 (SIRi(t0 + t∗) ≥ γi | Ei) ≥ gij(t0) | β2 · (1 − β1) = 1 − β .
`
`(27)
`
`Remark 3.2 In an interference limited system, the noise power is much smaller than the
`
`cochannel interference power. Thus, one may expect only marginal differences in the trans-
`
`mission powers and the outage probabilities between the two versions of the TVPC algorithm.
`
`This is indeed supported by our numerical examples.
`
`As mentioned above, the conditional probabilities in (24) and (27) are not the outage
`
`probabilities. It is well recognized that analytical derivation of the system outage probability
`
`is intractable, and therefore we derive it by a simulation described in the next section.
`Note that TVPC differs from DCPC by the scale up factor ξ(t∗) which has the follow-
`ing two degrees of freedom for the design. A fraction β that reflects the error correction
`capability, and an expected time horizon t∗ for power stabilization. From Equation (24) one
`may observe that the scale up factor is determined by the following system parameters: the
`
`number of interferers N0, the normalized velocity u, and the log variance of the shadow
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`fading σ2. The actual number of interferers N0, is not crucial in practice as pointed out in
`
`Remark 3.1. Thus, TVPC (compared with DCPC) requires only the mobile speed v, the
`
`effective correlation distance X, and the log variance of the shadow fading σ2.
`
`for TVPC. correlation distance of the shadow fading. How can these parameters be
`
`estimated in practice? The mobile speed can be taken as the maximum speed, a case that
`
`reflects a worst case scenario (see the results in Figure 8). One may also take the the actual
`
`instantaneous speed during the power control process, by applying good real-time speed
`
`estimators (see e.g. [7, 28]). The effective correlation distance and the log normal variance of
`
`the shadow fading can be taken from field measurements in the area where the cellular system
`
`is installed. Note that in general, urban environments have higher normalized velocity than
`
`rural environments, in spite of their higher vehicular speed.
`Note that there are many combinations of ξ1(t∗) and ξ2(t∗) satisfying (19), and yielding
`the same value of ξ(t∗). Furthermore, the feasible ξ(t∗) may range from a minimum value
`denoted by ξmin(t∗), to infinity. A question then rises, which one is best. Since a too
`high quality target may result in too high transmission powers, and consequently, too high
`
`interference powers and larger outage, one may argue for selecting the minimum required
`SIR target ξmin(t∗), which covers the individual channel variability. As we will see in the
`next section, this strategy is close to the optimal one.
`
`To summarize, we may state that the essence of the TVPC algorithm is in its computation
`
`of the scale up factor required when mobility is taking into account. It can be referred to as
`
`the transmission power cost of mobility.
`
`4 Numerical Results
`
`In this section we evaluate the performance of the TVPC algorithm in a microcellular sys-
`
`tem. Although the vehicular speed is typically lower in this environment compared with
`
`a macrocellular (rural) environment, the normalized velocity in Equation (7) is higher due
`
`to a much smaller effective correlation distance. Since the scale up factor is monotonically
`
`increasing with the normalized velocity, we obtain higher scale up factors for microcellular
`
`environments. Hence, our numerical results present a worst case scenario for the proposed
`
`algorithm.
`
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`We compare its performance with that of DCPC, fixed transmission power (no power
`
`control), and constant received power control. The evaluation is made for several practical
`
`SIR target values.
`
`Manhattan-like microcellular environment
`
`This is a typical metropolitan environment consisting of building blocks of a square shape.
`
`Streets are running between the building blocks in two directions, horizontal and vertical.
`
`In our simulation we assume that each block is of length 100 m. We further assume that
`
`radio-waves can propagate only along the streets.
`
`We study the power control algorithms for two different cell plans. The first one is an
`
`Asymmetric Half Square (AHS) cell plan, depicted in Figure 1. The cluster size Nc = 3, and
`
`the line-of-sight (LOS) reuse distance is DLOS = 3. This cell plan is denoted by AHS(1,1,3),
`
`in agreement with the notation in [16]. (The notation from there is extended to include also
`
`the cluster size.)
`
`The second cell plan (Figure 2) which we consider is an Asymmetric Half Square cell
`
`plan with cluster size Nc = 4. The corresponding LOS reuse distance is DLOS = 4, and
`
`the cell plan is denoted by AHS(1,1,4). From our numerical results it appears that the
`
`outage probability curve as a function of the SIR target in AHS(1,1,4), is a shift of the curve
`
`obtained for AHS(1,1,3). This is explained by the fact that the distance between two LOS
`
`interferers is also a shift of each other. Therefore, most of our results are presented only for
`
`the AHS(1,1,3) case.
`
`In both cell plans, one base station is placed in every street corner at lamp-post level.
`
`Base stations use omnidirectional antennas and the cell size is assumed to be half a block
`
`in all four directions. In the simulation, we take 48 cochannel cells for AHS(1,1,3), and 64
`
`cochannel cells for AHS(1,1,4). For each cell plan we use a fixed channel assignment scheme
`
`which divides the cells into Nc different channel groups.
`
`To model the large scale propagation loss, set (xi, yi) and (xj, yj) to be mobile i and base
`station j coordinates, respectively. Denote by x = |xi − xj| and y = |yi − yj|, the horizontal
`and the vertical distances, respectively, between the mobile and the base station. From [10],
`
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`the large scale propagation loss between mobile i and base station j can be modeled by
`
`16
`
`(cid:195)
`
`π2f 2
`c2
`
`xye
`
`(cid:179)
`
`−
`
`20WxWy
`xy
`
`(cid:180)
`
`Lij =
`
`+ x + y + 10
`
`(cid:33)2
`
`1 +
`
`(cid:118)(cid:117)(cid:117)(cid:116)(cid:181)
`
`x + y
`Ln
`
`(cid:182)(2n−4)
`
`(cid:195)
`
`+
`
`x2 + y2
`2
`Lm
`
`(cid:33)(m−2)
`
`
`
`−1
`
`,
`
`where c is the speed of light, f is the transmission frequency, and Wx and Wy are the
`
`street widths in the horizontal and vertical direction, respectively. The parameters n, Ln,
`
`m and Lm are all propagation constants, [10].
`
`In our simulation we use f = 900 M Hz,
`
`Wx = Wy = 25 m, n = 4, m = 25, Ln = 200 m and Lm = 700 m. From the measurement
`
`data in [17], we take the standard deviation of the shadow fading to be σ = 4 dB and the
`
`correlation distance X = 8.3 m.
`
`The median Signal to Noise Ratio (SNR) at a cell border under the maximum transmis-
`
`sion power is calibrated to 82 dB. That is, we take a strongly interference limited system.
`
`The starting position of the new mobiles are independently sampled from a uniform
`
`distribution over each cell area, and their travel directions (right, left, up or down) are
`
`chosen with equal probabilities. Moreover, mobiles move along the streets with a constant
`
`speed of 30 km/h. At street corners, they turn left or right, or continue straight ahead with
`
`equal probabilities.
`
`We further assume that call durations are independent and geometrically distributed
`
`with a mean of 120 seconds.
`
`Method of comparison
`
`The prime criterion by which we compare the algorithms is the outage probability evaluated
`
`by the following simulation. We maintain a fixed number of mobiles in the system by
`
`replacing every mobile which exits a cell, with a new one in a random location. A mobile
`
`disconnection and a mobile transition to another cell, are both regarded as mobile exits. The
`
`system is initialized with all mobiles transmitting according to the fixed-point power vector
`
`solution in (14), with respect to the instantaneous gain matrix. For every power control
`
`algorithm and SIR target, we simulate 10, 000 calls (i.e., mobile exits).
`For each mobile, we accumulate the proportion of time where SIRi(t) ≥ γi.
`If this
`proportion is greater than 1− β, then the mobile is supported, otherwise it is not supported.
`
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`In the simulations, we take β = 0.05. The outage probability under a given power control
`
`algorithm is estimated by the proportion of unsupported mobiles.
`
`Note that although TVPC is designed to achieve an outage probability below
`
`β, it may not be attainable for any load. What TVPC actually does achieve
`
`is the following. For those mobile whose short term average SIR(t) equals the
`
`scaled up SIR target, the probability of dropping below the non scaled up SIR
`
`target is less than β. At high loads, there will be mobiles for which their short
`
`term average SIR(t) cannot be equal to the scaled up SIR target. Those mobiles
`
`also contribute to the outage probability, which may therefore exceed β.
`
`We confine our examples to the uplink case and synchronous power updates. The time
`
`between two power updates is denoted by ∆t. We further assume a constant SIR target,
`γi = γt, for all mobiles. We compare the TVPC outage probability with that of a fixed
`
`transmission power (all transmitters use the maximum transmitter power), constant-received
`
`power control, and DCPC. Under the constant-received power control, the transmitters
`
`update their power according to,
`
`pi(t + ∆t) =
`
`C
`gii(t)
`
`,
`
`where the target power C is determined by a desired SNR of 63 dB, when the transmission
`
`power is less than the maximum value.
`
`Most notable is the fact that the classical DCPC algorithm has an extremely high outage
`
`probability. To shade some light on this result we also evaluate the ratio
`
`(cid:80)
`
`j:j(cid:54)=i gij(t + dt) pj(t + dt)
`gii(t + dt)
`
`(cid:33)
`
`(cid:195)
`
`νi +
`
`/
`
`(cid:195)
`
`νi +
`
`∆I =
`
`(cid:80)
`
`j:j(cid:54)=i gij(t) pj(t)
`gii(t)
`
`(cid:33)
`
`.
`
`This is the ratio between the desired updated power, and the actually updated power. Its
`
`symmetric distribution depicted below, explains the high outage probability under DCPC.
`
`We investigate the performance of the TVPC algorithm with different scale up factors.
`
`As will be seen in t