Evaluation of Call Performance in Cellular Networks With Generalized ...

3002

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

Evaluation of Call Performance in Cellular Networks With Generalized Cell Dwell Time and Call-Holding Time Distributions in the Presence of Channel Fading Suwat Pattaramalai, Valentine A. Aalo, Senior Member, IEEE, and George P. Efthymoglou, Member, IEEE

Abstract—Call-completion probability, call-dropping probability, and handoff rate are important performance measures of wireless networks. In this paper, we study the joint effect of channel fading and handover failure on these performance measures. For the case of Rayleigh and lognormal fading channels and for generalized distributions of the cell dwell time and the call-holding time, we derive simple closed-form expressions that closely approximate these performance metrics. The results are given in terms of the moment-generating function (MGF) of the distribution for the call-holding time and may be useful in the cross-layer design and the optimization of wireless networks. Index Terms—Call-completion probability, call-holding time, cell dwell time, handoff rate.

N OMENCLATURE K M NR R1 SK Tc Ti Vi UM Pc PDH PDlink flink pf plink p0 vm τf τm τlink

Random number of cells that are traversed during one call. Random number of good-link periods in one call. Level crossing rate. Residual life of a call in the first cell. Random sum of cell dwell times in one call. Call-holding time. ith cell dwell time [independent identically distributed (i.i.d.) T ]. ith good-link period (i.i.d. V ). Random sum of good-link periods in one call. Call-completion probability. Call-dropping probability due to handoff failure. Call-dropping probability due to link breakdown. Frequency of link breakdown. Probability of handoff attempt failure. Probability of (radio) link breakdown. Probability that a new call is blocked. Mobile velocity. Fade duration of the received signal. Allowed minimum fade duration by the system. Fade duration that is longer than τm .

Manuscript received April 1, 2008; revised July 7, 2008. First published September 12, 2008; current version published May 29, 2009. The review of this paper was coordinated by Prof. Y. B. Lin. S. Pattaramalai is with the Faculty of Engineering, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand (e-mail: [email protected]). V. A. Aalo is with the Department of Electrical Engineering, Florida Atlantic University, Boca Raton, FL 33431 USA (e-mail: [email protected]). G. P. Efthymoglou is with the Department of Digital Systems, University of Piraeus, 18534 Piraeus, Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2008.2005637

I. I NTRODUCTION

T

HE EXPLOSIVE demand for wireless services in recent years has led to the development and the deployment of a new generation of wireless systems that offer a variety of high-speed wireless applications, including voice and data services, multimedia services, navigation services, text-video messaging, and Internet browsing [1]. To satisfy the quality-ofservice (QoS) requirements, these systems encounter problems that are related to the user’s mobility in the cell, i.e., outage due to fading over the wireless link and handoff failure due to the scarcity of spectral resources as the user moves through the coverage area of the cellular system. In a wireless network, the instantaneous received signal suffers from short-term fading, which is usually modeled by the Rayleigh, Rician, or Nakagami probability density functions (pdfs), and from the slow variation of its local mean power (known as long-term fading) that is usually modeled by the lognormal distribution. A parameter that is commonly used to quantify the effect of channel fading on the performance of a wireless network is the minimum time (duration) that the received signal level stays below a preset threshold, causing the call to be dropped by the base station. The minimum duration for an outage event is derived using a level crossing analysis, and it usually spans widely different timescales for the short- and long-term fading channels [2], [3]. In a cellular system, the coverage area is divided into cells, and a portion of the total available channels is assigned in each cell. Neighboring cells are assigned different groups of channels to minimize the interference between cells [1]. Two timedependent random parameters that are usually encountered in the study of cellular network performance are the cell dwell time and the call-holding time [4]–[16]. The cell dwell time of a mobile user is a random variable that characterizes the time that the mobile user spends in a cell, and it depends on many factors, including the cell size, the terrain, the speed, and the direction of the user. Although it has usually been assumed to follow an exponential distribution, more generalized distributions that provide better fit to measured data have recently been employed [4], [5]. Furthermore, the call-holding time mainly depends on the call type, i.e., voice, data, Web browsing, etc. Although, for voice calls, this random variable has been assumed in the past to follow an exponential distribution, more generalized distributions have been proposed for the new types of calls in the new generation of wireless networks [6]–[13]. The call performance analysis of homogeneous wireless networks under the generalized cell dwell time and call-holding

0018-9545/$25.00 © 2009 IEEE

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

time has been extensively studied by Fang et al. [8]–[12]. Their approach is based on the residue theorem and requires that the Laplace transform of the distribution of the call-holding time be a rational function. However, in some cases of practical interest, the Laplace transform of the call-holding time distribution may not be a rational function [6], [7]. For example, when the callholding time has a gamma distribution with a noninteger shape parameter, its Laplace transform does not have a rational form, and the residue theorem approach is not applicable. A new approach based on the concept of random sums has recently been proposed by Pattaramalai et al. [13]; they have derived simple closed-form expressions for some useful wireless network performance metrics when the call-holding time follows a generalized distribution. However, the effect of channel fading was not considered in [8]–[13], although it is well known that channel fading has a detrimental effect on any wireless network performance [14], [15]. The effect of Rayleigh fading in the physical link on the teletraffic performance of the wireless network was studied through the minimum outage duration in [14]. However, the formulas that are presented for the handoff call arrival rate, the call-completion probability, and the forced termination probability due to link breakdown and handoff failure are given in integral form and may not be easy to use in practice. In this paper, we study the call performance of wireless networks operating in a fading link with generalized cell dwell time and call-holding time distributions. Specifically, based on the results in [14], we extend the analysis in [13] that uses the compound random sum approach to evaluate the effect of physical link impairments on the call performance of a wireless network. For Rayleigh and lognormal shadow fading, we derive simple closed-form expressions that closely approximate the call-completion probability and the call-dropping probabilities due to link breakdown and handoff failure. The results, which are given in terms of the moment-generating function (MGF) of the call-holding time distribution, only require that the first two moments of the cell dwell time be finite and, therefore, are applicable to a wide range of call-holding time and cell dwell time distributions. The rest of this paper is organized as follows. In Section II, we describe the system model, whereas some mathematical preliminaries are presented in Section III. In Section IV, the probability of link breakdown in Rayleigh fading and lognormal shadowing is derived. The performance measures are derived in Section V, whereas some numerical results are presented in Section VI. Finally, concluding remarks are given in Section VII. II. S YSTEM M ODEL Fig. 1 illustrates the timing diagram for a call that is successfully connected to the wireless network and experiences successful handoffs, as well as survives bad-link periods until it ends at an arbitrary cell. From this figure, it is evident that the two processes that affect the call performance in a wireless network are the handoff event, with probability of failure pf due to unavailable spectral resources in the cell that the user moves into, and the link breakdown event, with probability plink due to channel fading. These two processes are assumed to be

3003

Fig. 1. Time diagram for a complete call.

Fig. 2. Good-link period definition.

independent of each other. In Fig. 1, the upper time line shows the cell dwell times and the call-holding time that characterize the handoff process, whereas the lower time line shows the physical link parameters. The call-holding time is the duration from the time that a call is initiated to the instant that the call is ended by the mobile user, without the call being abruptly terminated by the wireless network. Let random variable Tc denote the call-holding time with the pdf given by fTc (t). Furthermore, the cell dwell time is defined as the time duration that a mobile user resides in the ith cell and is denoted by Ti (i = 1, 2, . . . , K − 1, K). The call is assumed to end in the (random) Kth cell. We assume that the cell dwell times are independent identically distributed (i.i.d.) nonnegative random variables, with common pdf fT (t) and finite first and second moments E(T ) and E(T 2 ), respectively. Moreover, a new call does not always begin at the beginning of the first cell. Indeed, we assume that a new call is initiated and successfully connected at an arbitrary time in the first cell, so that the actual duration of the call in the first cell, i.e., R1 , is the residual life of the cell dwell time in the first cell [17]. Finally, let the random variable Vi (i = 1, 2, . . . , M − 1, M, . . .) denote a good-link period, which is defined as the time duration that the received signal power at the mobile unit is above the required threshold for satisfactory QoS. We assume that the call starts in the first good-link period and ends in the (random) M th good-link period. In addition, as shown in Fig. 2, the good-link periods also include the duration that the received signal power stays below the threshold, provided that these time intervals are less than the allowed minimum outage duration τm that results in link breakdown in the system. Throughout this paper, we assume that all good-link periods are i.i.d. nonnegative random variables with finite mean E(V ). Finally, p0 denotes the probability that a new call is blocked when trying to connect to the base station due to the lack of available channels in the cell. In Section III, we present some

3004

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

preliminary results that will be used in the derivation of call performance metrics in subsequent sections. III. M ATHEMATICAL P RELIMINARIES Let Xi (i = 1, 2, 3, . . .) be a sequence of independent but not necessarily identical nonnegative random variables, and let  X N = 1, 2, 3, . . . be an integer. We define YN = N i=1 i as the partial sum of random variables Xi . A. N Is a Constant Let N = n be a known constant and {Xi }ni=1 be a sequence of i.i.d. nonnegative random variables with a common cumulative distribution function (cdf) FX (x). Then, the probability distribution of the partial sum is given by n∗ (x) Pr(X1 + X2 + · · · + Xn ≤ x) = FX

(1)

n∗ (x) is the n-fold convolution of FX (x). In renewal where FX theory [18], the random variable Xi may define the interrenewal time of a random process. In this case, the average number of renewals in time interval (0, x) is known as the renewal function and denoted by H(x), and the process is called an ordinary renewal process with a renewal function that is given by [18]

H(x) =

∞ 

(n−1)∗

FX1 ∗ FX

(x)

(3)

n=1

where FX1 is the cdf of X1 and ∗ denotes the convolution operation. In particular, if the distribution of X1 is the survivor function of FX (x), i.e., FX1 (x) = [1 − FX (x)]/E(X), then the process is called an equilibrium renewal process, and the renewal function becomes [18] H(x) =

 ∞   1 − FX (x) n=1

Pr(YN ≤ x) =

∞ 

Pr(N = n) Pr(X1 + X2 + · · · + Xn ≤ x).

n=1

(5)

The compound random sum in (5) has been extensively studied in many areas of applied probability, including reliability theory, risk theory, and queuing theory in computer networks [19]–[25]. For the case that N follows the geometric distribution, i.e., it models the number of independent Bernoulli trials (each with probability q of success) that are needed before the first success, then its pmf is given by [26] Pr(N = n) = q(1 − q)n−1 ,

n = 1, 2, 3, . . .

E(X)

(n−1)∗

∗ FX

(x)

(4)

where E(X) is the mean of Xi , i = 1. In cellular networks, when Xi denotes the cell dwell time, then the renewal function is the average number of handoffs in time interval (0, x). In [16], the equilibrium renewal process approach is used to derive the probability mass function (pmf) of the number of handoffs per call in a cellular network. The results show that when there is no new or handoff call in the wireless network, the average number of handoffs is equal to the call-to-mobility factor, which is given by the ratio of the mean of the call-holding time to the mean of the cell dwell time [13], [16].

(6)

where q is the probability that the event succeeds, and (1 − q) is the probability that the event fails. In this case, the compound random sum in (5) is called a geometric compound sum and is given by Pr(YN ≤ x) =

∞ 

q(1−q)n−1 Pr(X1 + X2 + · · · + Xn ≤ x). (7)

(2)

When the random variable X1 has a different distribution from Xi (i = 1), the process is known as a delayed renewal process, and the renewal function becomes [18] ∞ 

When N  is an integer-valued random variable, the partial sum YN = N n=1 Xn is a compound random sum, and its distribution is given by [19]–[21]

n=1 n∗ FX (x).

n=1

H(x) =

B. N Is a Random Variable

In wireless networks, the geometric compound sum may be used to study performance metrics such as the call-completion probability and the handoff rate, where q denotes the probability of handoff failure [8]–[15]. A closed-form expression for the geometric compound random sum is not available, except for some special cases. For example, it is well known that the geometric random sum of i.i.d. exponential random variables is also distributed exponentially [21]. For a fixed value of x, as shown by the following theorem, the exponential distribution also results for asymptotically small values of q. Theorem 1: Let {X1 , X2 , . . . , XN } be a sequence of independent random variables, each with finite mean. Let m1 = E(X1 ) and {Xi }N i=2 be a sequence of i.i.d. random variables, each with first moment μ1 = E(X) < ∞. Finally, let N be a geometric random variable with parameter q, independent of small values of q, Xi , i = 1, 2, . . . , N . Then, for sufficiently  the geometric random sum YN = N X has an exponential i=1 i distribution with mean (qm1 + (1 − q)μ1 )/q. That is  Pr(YN ≤ x)  1 − exp

−qx qm1 + (1 − q)μ1

 as q → 0. (8)

Proof: The proof is given in Appendix A.  The limit distribution in (8) has been used in [13] to compute the call-completion probability and the handoff rate in a cellular system, and has been shown to provide very accurate results. Note that, in the special case when sequence {Xi }N i=1 is i.i.d., i.e., μ1 = m1 , (8) reduces to the so-called Renyi limit theorem [19].

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

IV. L INK B REAKDOWN P ROBABILITY IN A F ADING E NVIRONMENT It is well known that a wireless link is inherently timevarying and susceptible to performance degradation due to signal fading, interference, and noise in the physical link [27]. When the link is degraded severely, a call in progress will be terminated and disconnected from the base station. In [14], an analytical model to study the interaction between the Rayleigh fading in the physical channel and the wireless network performance was introduced through the probability of link breakdown, which transfers the effects of physical layer characteristics (e.g., carrier frequency, Doppler frequency, and fade margin) to higher layer performance metrics of the wireless network. Moreover, the probability of link breakdown in a fading channel is usually studied in terms of the minimum link outage duration [28]. Outage probability is an important measure of the quality of a wireless link. It represents the probability that the instantaneous received signal-to-noise-and-interference ratio (SNIR) falls below a preset threshold. However, in practice, it is not the instantaneous drop of the SNIR below the threshold that is really important, but the duration that it stays below the threshold [2], [15]. Consequently, a link breakdown or outage event may be defined as the event that the received SNIR stays below the system threshold for a time period that is longer than minimum duration τm . As shown in Fig. 2, fade duration τf is the time that the received signal stays below the required threshold. Therefore, the allowed minimum fade duration τm is the minimum value of τf that the system can tolerate without losing its connection to the network. Furthermore, link breakdown duration τlink is defined as fade duration τf given that τf is greater than or equal to τm . The pdf of link breakdown duration is given by [2]  fτf (τlink ) fτlink (τlink ) = Pr(τf ≥τm ) , τlink ≥ τm (9) 0, otherwise where fτf (τ ) = −d Pr(τf ≥ τ )/dτ and 1 − Pr(τf ≥ τ ) are the pdf and the cdf of τf , respectively. Then, the average duration of link breakdown can be evaluated as

normal shadowing). However, the minimum link outage duration in a lognormal shadowing environment has a much larger timescale (in the order of seconds) than the one due to Rayleigh fading, which is in units of milliseconds [2], [3]. Therefore, in what follows, we separately analyze the link breakdown probability for Rayleigh fading and lognormal shadowing.

A. Probability of Link Breakdown Due to Rayleigh Fading In this section, we assume a pure multipath fading environment, where fast fluctuations due to multipath fading dominate in the received signals, and the effect of lognormal shadowing can be ignored. Using level crossing analysis, the link breakdown may be studied as a problem of a single Rayleigh signal envelope, which fades below the required signal target [3], [14]. It is well known that average fade duration τ¯f  E(τf ) for a Rayleigh-faded signal is given by [27] τ¯f =

τlink fτlink (τlink )dτlink .

exp(ρ2 ) − 1 √ 2πfd ρ

(13)

whereas the level crossing rate is given by [27] NR =

√

2πfd ρ exp(−ρ2 )

(14)

where fd is the maximum Doppler frequency, and ρ = Rreq /RRMS is the ratio of the required amplitude level to the local RMS of the instantaneous received signal amplitude. Also, the cdf of fade duration τf is given by [29] Pr(τf ≥ τ ) = 2

τ¯  f

τ



2 2 τ¯f 2 τ¯f 2 exp − I1 (15) π τ π τ

where I1 (·) is the modified Bessel function of the first kind [30]. Substituting (9) and (15) in (10), the average duration of link breakdown is given by E(τlink ) =

∞ E(τlink ) =

3005

(10)

τm

τ¯f Pr(τf ≥ τm ) 



 ∞ −2 2 2 exp (−u)d · I1 u πu2 πu2

(16)

τm /¯ τf

The frequency of link breakdown is given by [2] flink = {Level crossing rate} × {Probability that crossing leads to link breakdown} = NR Pr(τf ≥ τm )

τf . The integral in (16) can be evaluated in where u = τlink /¯ closed form as (see Appendix B for details) (11)

where NR is the level crossing rate [3]. Finally, the probability of link breakdown is given by [2] plink = flink E(τlink ).

(12)

The received signal in a mobile radio environment suffers from short-term (Rayleigh) fading and long-term fading (log-

E(τlink ) =

⎧ ⎪ ⎪ ⎨

1 τm τm   + 2 2 2 2 ⎪ τ ¯ τ¯f f ⎪ ⎩ exp −2 I1 π2 τm π τm  ⎫ 2 τ¯f ⎪ ⎪ I0 π2 τm ⎬ − 2 . (17) ⎪ τ¯f ⎪ ⎭ I1 π2 τm

3006

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

bility of link breakdown plink versus τm in a Rayleigh fading channel for a carrier frequency of 1900 MHz with different values of mobile speed vm and fade margin F . The figure shows that the value of plink decreases as the mobile moves faster or the fade margin increases. Therefore, for a mobile user moving at 30 mi/h (1 mi/h = 0.0447 m/s) and for the minimum duration of link breakdown of 5 ms, plink is 2% when F = 8 dB and improves to 0.3% when F = 12 dB. Furthermore, Fig. 3(b) shows the effect of increasing the carrier frequency from 850 to 1900 MHz on the value of plink . We observe that, in a Rayleigh fading environment, lower carrier frequencies result in higher values for plink . B. Probability of Link Breakdown Due to Lognormal Shadowing The link breakdown condition in lognormal shadowing is usually formulated on a decibel scale as the level crossings of a Gaussian process with variance σ 2 about a required threshold [2], [28]. The level crossing rate of the received signal in a lognormal fading channel is given by [2] NR =



vm γ2 exp − 2 2πdc 2σ

(19)

where dc is the correlation distance [2], and γ is the fade margin in decibels. The pdf of fade duration below the level τf is given by [2]   fτf (τ ) = (2λ)τ exp −λτ 2

Fig. 3. Probability of link breakdown in a Rayleigh fading channel (a) against the minimum duration of link breakdown for fc = 1900 MHz with different values of speed vm and fade margin F , and (b) against the fade margin for vm = 30 mi/h with different values of carrier frequency fc and minimum fade duration τm .

By using (14) and (15), the frequency of link breakdown in (11) is given by  2  √ τ ¯ τ¯f 2 f flink = 2 2πfd ρ exp(−ρ2 ) exp − τm π τm   2 2 τ¯f · I1 . (18) π τm The probability of link breakdown in a Rayleigh fading channel can then be computed using (17) and (18) in (12). Note that the result in (17) agrees with [31], which was derived using an alternative approach, and is considerably simpler than the infinite series result given in [3] and [14]. For the Rayleigh fading channel, Fig. 3 plots link breakdown probability plink for specific values of several wireless link parameters, including fade margin F = −20 log10 ρ, mobile velocity vm , carrier frequency fc , maximum Doppler frequency fd = vm · fc /(3 × 108 ) Hz, and minimum fade duration τm that will result in an outage event. Fig. 3(a) shows the proba-

(20)

where λ = (1/2)(γvm /2σdc )2 . Using (9) and (20) in (10), the average duration of link breakdown can be shown to be given by [2], [28]    2 √ π exp λτm Q( 2λτm ) E(τlink ) = τm + λ

(21)

√ ∞ 2 where Q(x) = (1/ 2π) x e−t /2 dt is the tail of the Gaussian integral [30]. Similarly, the frequency of link breakdown for a lognormal shadowing channel is given by [2] flink =



vm 4d2 2 exp −λ τm + 2c . 2πdc vm

(22)

Fig. 4(a) plots plink = flink E[τlink ] versus τm in the presence of lognormal shadowing. Notice that the unit of τm is in seconds (a much larger timescale than the case of the Rayleigh fading channel, which is in milliseconds for the same value of plink ). For example, when plink is 2%, τm is about 2.5 s for mobile velocity of 30 mi/h, correlation distance dc = 30 m, and fade margin γ = 12 dB (where, for the lognormal fading channel, we assumed that σ = 6 dB). The value of plink can be reduced by increasing γ. Fig. 4(b) shows plink versus γ for several values of vm and τm . We observe that mobile user velocity vm has a larger impact on plink than τm .

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

3007

probability that a call is not initially blocked and connected to the wireless network, while the call-holding time Tc is jointly less than SK and UM . Therefore, we have Pc = (1 − p0 ) Pr(Tc < SK , Tc < UM ).

(23)

Since SK and UM are independent processes, we have Pc = (1 − p0 ) Pr(SK > Tc ) Pr(UM > Tc ) ∞ = (1 − p0 ) {[1 − Pr (SK ≤ x|Tc = x)] 0

· [1 − Pr (UM ≤ x|Tc = x.)]} fTc (x) dx. (24)

For a given value of the call-holding time, i.e., Tc = x, the conditional distribution of SK is given by Pr (SK ≤ x|Tc = x) ∞  = Pr(K = n) Pr (R1 +T2 +· · ·+Tn ≤ x|Tc = x)

(25)

n=1

and the conditional distribution of UM is Pr (UM ≤ x|Tc = x) ∞  = Pr(M = n) Pr (V1 +V2 +· · ·+Vn ≤ x|Tc = x) .

(26)

n=1

In a cellular network, the probability that the random variable K is equal to n is the probability that the call has completed (n − 1) successful handoffs, and the nth handoff attempt fails. Therefore, the random variable K follows a geometric distribution with parameter pf . Its pmf is given by [32] Fig. 4. Probability of link breakdown in a lognormal shadowing channel with dc = 30 m (a) against the minimum duration of link breakdown with different values of speed vm and fade margin γ and (b) against the fade margin with different values of minimum fade duration τm and speed vm .

V. E VALUATION OF W IRELESS N ETWORK P ERFORMANCE In this section, we analyze the performance of a wireless network using the concept of the geometric compound sum. In particular, we evaluate important performance measures, including the call-completion probability, the call-dropping probability (due to link breakdown or handoff failure), and the average number of handoffs. A. Call-Completion Probability Call-completion probability Pc is the probability that a new call is successfully connected to the cellular network and is not dropped until the mobile user ends the call during a good-link period in any cell. Let SK = R1 + T2 + · · · + TK and UM = V1 + V2 + · · · + VM be the sums of K i.i.d. cell dwell times and M i.i.d. good-link periods, respectively. Since K and M are independent random variables, SK and UM are random sums of independent random variables. As shown in Fig. 1, a call is assumed to end in the Kth cell during the M th good-link period. The call-completion probability is the

Pr(K = n) = pf (1 − pf )n−1 ,

n = 1, 2, 3, . . .

(27)

and its mean is E(K) = 1/pf . Similarly, the probability that the random variable M is equal to n is the probability that the call has passed through (n − 1) good-link periods, and the nth bad-link period results in link breakdown. Therefore, the random variable M also follows a geometric distribution, but with parameter plink and corresponding pmf given by Pr(M = n) = plink (1−plink )n−1 ,

n = 1, 2, 3, . . . .

(28)

Using (27) and (28), it follows that the conditional distributions of SK and UM are, respectively, given by Pr(SK ≤ x|Tc = x) =

∞ 

pf (1 − pf )n−1 Pr (R1 + T2 + · · · + Tn ≤ x|Tc = x)

n=1

(29) Pr (UM ≤ x|Tc = x) =

∞ 

plink (1 − plink )n−1

n=1

× Pr (V1 + V2 + · · · + Vn ≤ x|Tc = x) .

(30)

3008

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

We observe that (29) and (30) can also be written in an alternate form as follows (see Appendix C for details):

∞

1−Pr(SK ≤ x|Tc = x) =

∞ 

Pc = (1 − p0 )

(1−pf )

0

· Pr(R1 +T2 +· · ·+Tn < x < R1 +T2 +· · ·+Tn+1 |Tc = x) (31) 1−Pr (UM ≤ x|Tc = x) ∞ 

{exp(−ax) exp(−bx)} fTc (x)dx

n

n=0

=

where b = plink /E(V ). Substituting (33) and (34) in (24), the call-completion probability is given by

where MTc (t) = dom variable Tc .

∞ 0

n=0

· Pr(V1 +V2 +· · ·+Vn < x < V1 +V2 +· · ·+Vn+1 |Tc = x). (32) Note that (31) and (32) evaluate the respective conditional probabilities based on the analysis in [14]. Therefore, we have established that the analysis based on the random sum concept is equivalent to that in [14]. The analysis in [14] and other previous performance analyses of wireless networks [8]–[12] rely on the residue theorem, which requires that the Laplace transform of the call-holding time distribution be a rational function. In this paper, we compute the call-completion probability by using the exponential approximation of the geometric compound random sum given by Theorem 1. This approach has been shown to result in very simple and accurate expressions for evaluating the call-completion probability and other performance measures of wireless networks [13]. Based on renewal theory, the random variable R1 is the residual life of the call in the first cell [17], and its mean can be shown to be given by E(R1 ) = E(T 2 )/2E(T ). Therefore, using the limit theorem in (8), the conditional probability in (29) can be closely approximated as lim Pr(SK ≤ x|Tc = x)  ∞  = lim pf (1 − pf )n−1

PDH = (1 − p0 ) Pr(Tc > SK , UM > SK ).

x≥0

Noting that the processes Tc , SK , and UM are independent, we have

· ESK [Pr(Tc > s|SK = s) Pr(UM > s|SK = s)] ∞

(33)

where a = pf [pf E(R1 ) + (1 − pf )E(T )]−1 . Also, the compound random sum in (30) can be accurately approximated as lim Pr(UM ≤ x|Tc = x)  ∞  plink (1 − plink )n−1 = lim

plink →0



n=1

· Pr(V1 + V2 + V3 + · · · + Vn ≤ x|Tc = x)  1 − exp(−bx),

x≥0

(36)

= (1 − p0 ) 

n=1

Call-dropping probability is the probability that a call is successfully connected to the cellular network, but it is terminated by the system while the mobile user has not completed the call. The call may get disconnected from the base station as a result of handoff failure due to limited resources in the new cell or as a result of link breakdown due to channel fading. 1) Call-Dropping Probability Due to Handoff Failure: The call-dropping probability due to handoff failure PDH is the probability that a call, which is initially connected to a wireless network, is unsuccessfully handed over to another cell during a good-link state. We assume that the call fails to handoff from the Kth to the (K + 1)th cell while the call is in the M th good-link period. Then, PDH is the probability that the call is not blocked initially, and the sum of the first K cell dwell times is less than the call-holding time and, at the same time, less than the sum of the first M good-link periods. Therefore, we have

· Pr(R1 + T2 + T3 + · · · + Tn ≤ x|Tc = x)

plink →0

exp(tx)fTc (x)dx is the MGF of the ran-

PDH = (1 − p0 )ESK [Pr(Tc > s, UM > s|SK = s)]

pf →0

 1 − exp(−ax),

(35)

B. Call-Dropping Probability

(1−plink )n

pf →0

= (1 − p0 )MTc (−[a + b])

(34)

= (1 − p0 )

(1 − Pr(Tc ≤ s|SK = s)) 0

· (1 − Pr(UM ≤ s|SK = s)) fSK (s)ds.

(37)

In (37), ESK [·] denotes the expectation over the distribution of the random sum SK . Using the fact that, for sufficiently small values of plink , the random sum UM has an exponential distribution, we may rewrite (37) as ∞ PDH = (1 − p0 )

(1 − FTc (s|SK = s)) 0

· exp(−bs)fSK (s)ds.

(38)

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

Also, for sufficiently small values of pf , since SK is an exponentially distributed random variable with mean 1/a, we obtain ∞ PDH = (1−p0 ) (1−FTc (s|SK = s))·exp(−bs)

3009

Using a similar argument as before, we obtain ∞ PDlink = (1 − p0 )

(1 − FTc (u|UM = u)) · exp(−au) 0

· b exp(−bu)du

0

· a exp(−as)ds ⎧ ⎫ ∞ ⎨ 1 ⎬ = (1−p0 )a − FTc (s|SK = s)exp(−[a+b]s)ds . ⎩a+b ⎭



1 − = (1 − p0 )b a+b

FTc (u|UM = u) 0



0

· exp (−[a + b]u) du

(39) Using the following Laplace transform property for a nonnegative random variable T [30]:

∞

= (1 − p0 )

b a+b

{1 − MTc (−[a + b])} .

(44)

∞ FT (t) exp(−st)dt = MT (−s)/s

(40)

0

we obtain the final result as

a PDH = (1 − p0 ) {1 − MTc (−[a + b])} . a+b

(41)

2) Call-Dropping Probability Due to Link Breakdown: The call-dropping probability due to link breakdown PDlink is the probability that a call that is initially connected to the wireless network is dropped due to poor link quality when there are enough system resources for successful handoffs. In this case, we assume that the call has gone through M consecutive goodlink periods, and that the (M + 1)th period is a bad-link period that causes link breakdown, resulting in the call, which is now in the Kth cell, to be dropped. Then, PDlink is the probability that the call is not initially blocked and has a call-holding time that is greater than the sum of the first M good-link periods, which is also less than the sum of the first K cell dwell times, i.e., PDlink = (1 − p0 ) Pr(Tc > UM , SK > UM ).

(42)

C. Average Number of Handoffs The average number of handoffs per user call is an important performance parameter for wireless network dimensioning and management. If H denotes a random variable that represents the number of handoffs that are performed during one call in the network, then the average number of handoffs per call is given by E[H] =

∞ 

n Pr(H = n)

(45)

n=1

where Pr(H = n) is the probability that the number of handoffs H is equal to n. This is the probability that, while the link remains in a good state, a call that is not initially blocked does not end in the (K = n)th cell, but requires a handoff to the (n + 1)th cell. This implies that the sum of the first K cell dwell times is less than the call-holding time and the sum of the first M good-link periods. Therefore, we have Pr(H = n) = (1 − p0 ) Pr(SK < Tc , SK < UM ) Pr(K = n). (46) Substituting (46) in (45) and using (36), we have

By assuming, again, that Tc , SK , and UM are independent, we have PDlink = (1 − p0 )EUM [Pr(Tc > u, SK > u|UM = u)]

E[H] =

∞ 

n(1 − p0 ) Pr(SK < Tc , SK < UM )

n=1

· Pr(K = n)

= (1 − p0 ) · EUM [Pr(Tc > u|UM = u) Pr(SK > u|UM = u)] ∞ = (1 − p0 ) [(1 − Pr(Tc ≤ u|UM = u))

= (1 − p0 ) Pr(SK < Tc ) Pr(SK < UM ) ·

∞ 

n Pr(K = n)

n=1

0

· (1 − Pr(SK ≤ u|UM = u))] fUM (u)du. (43)

= PDH E[K] =

PDH . pf

Note that this relationship agrees with the result in [12].

(47)

3010

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

Fig. 5. Call-completion probability versus call-to-mobility factor for pf = p0 = 2% with different values of plink and ζ.

Fig. 6. Call-dropping probability due to handoff failure versus call-tomobility factor for pf = p0 = 2% with different values of plink and ζ.

VI. N UMERICAL R ESULTS The performance measures that are considered in this paper are plotted against the call-to-mobility factor, which is defined as the ratio of the mean call-holding time and the mean cell dwell time, i.e., E(Tc )/E(T ). For illustration purposes, call-holding time Tc is assumed to follow an arbitrary twoparameter gamma distribution with parameters η and ξ and an MGF given by MTc (t) = (1 − ξt)−η [32]. Cell dwell times Ti (i = 1, 2, . . .) are i.i.d. random variables and also follow a two-parameter gamma distribution with arbitrary parameters α and β. Therefore, the mean cell dwell time is given by E(T ) = αβ, and the corresponding second moment is E(T 2 ) = β 2 α(α + 1) [32]. It follows that the mean of the actual time that the call resides in the first cell is E(R1 ) = β(α + 1)/2. Finally, the mean of good-link period E(V ) is specified in terms of the (quality) link-to-mobility factor, which is defined by the ratio ζ  E(V )/E(T ). In this section, we will validate the analytical model, present the effects of wireless link breakdown and handoff failure, and illustrate the effect of call-to-mobility and link-to-mobility factors. We present an example assuming that Tc ∼ Gamma(η, ξ = 1.5) and Ti (i = 1, 2, . . .) ∼ Gamma(α = 0.5, β = 1.5). For each value of the call-tomobility factor, the value of shape parameter η is calculated using the equation E(Tc )/E(T ) = ηξ/αβ. In addition, a simulation model implementing Fig. 1 for the time diagram of a typical call in the presence of the handoff procedure as well as wireless link breakdown was devised in MATLAB. Similar to the analytical model, the two processes affecting the call performance, i.e., the potential handoff failure with probability pf and the possible link breakdown with probability plink , are assumed to be independent since they are triggered by independent factors [14]. Figs. 5–8 show the analytical and simulation results for the probabilities of call completion, call dropping due to handoff failure, call dropping due to link breakdown, and the average number of handoffs per call, respectively, versus the call-to-mobility factor for different values of plink (0%, 1%,

Fig. 7. Call-dropping probability due to link breakdown versus call-tomobility factor for pf = p0 = 2% with different values of plink and ζ.

and 2%) and for two values of the link-to-mobility factor ζ (ζ = 0.5 and 1.5). We observe that the call-completion probability, the call-dropping probability due to handoff failure, and the average number of handoffs decrease with higher values of plink and shorter duration of the mean good-link period (smaller ζ). However, the reverse observation is made in Fig. 7 with respect to the call-dropping probability due to link breakdown, that is, when the wireless physical link is degraded seriously (higher plink ) or when the signal experiences short duration of average good-link periods (smaller ζ), the average number of calls dropped because of link breakdown is higher, as expected. Notice also from Figs. 6 and 7 that, while the calldropping probability due to handoff failure is reduced when there are more bad-link periods (smaller ζ), this reduction is slower than the increase in the call-dropping probability due to link breakdown. This observation agrees with the result shown in Fig. 5; that is, the call-completion probability decreases with smaller values of ζ. Finally, the simulation results (the points) are shown to be in close agreement with the analytical results (the lines), which validates the proposed

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

3011

Xi , i = 2, 3, . . . , N . Then, the MGF of qYN = (q given by

N i=1

Xi ) is

MqYN (−t) = E [exp(−t qYN )]    N  = E exp −tq Xi =

∞ 

i=1

 k−1 Pr(N = k)E[e−tqX1 ] E[e−tqX ]

k=0

(A-1) where we have used the fact that the nonnegative random variables X2 , . . . , XN are i.i.d. When N is a geometric random variable, we then have MqYN (−t) =

Fig. 8. Average number of handoffs versus call-to-mobility factor for pf = p0 = 2% with different values of plink and ζ.

= qMX1 (−qt)

(A-2)

Next, we substitute the series expansion for the MGF as follows: m2 (qt)2 + · · · MX1 (−qt) = 1 + m1 qt + 2   where mi = E X1i μ2 (qt)2 + · · · MX (−qt) = 1 + μ1 qt + 2 where μi = E(X i ). When the parameter q is sufficiently small, we have (A-3), shown at the bottom of the page. Ignoring terms of order q 2 and higher, (A-3) may be written as lim {MqYN (−t)}   1 + qm1 t + · · · = lim μ q→0 1 − (1 − q)μ1 t − 2 q(1 − q)t2 + · · · 2 ⎧ ⎫ ⎨ ⎬ 1 = lim μ q (μ1 m1 + 22 )t2 +··· ⎭ q→0 ⎩ 1 − [(1 − q)μ1 + qm1 ] t + 1+qm1 t+··· 1  (A-4) 1 − [(1 − q)μ1 + qm1 ] t

q→0

which is the MGF of the exponential distribution with mean [(1 − q)μ1 + qm1 ]. Note that, in (A-4), we have considered only terms that are linear in t when q is sufficiently small. 





 1 + m1 qt + m22 (qt)2 + · · ·   ∴ lim {MqYN (−t)} = lim μ2 2 q→0 q→0 2 (qt) + · · · + q 1 + μ1 qt +   1 + qm1 t + m22 (qt)2 + · · · = lim μ q→0 1 − (1 − q)μ1 t − 2 q(1 − q)t2 − · · · 2 q  1 − 1 + μ1 qt +

[(1 − q)MX (−qt)]k−1

k=1

A PPENDIX A P ROOF OF T HEOREM 1



∞ 

qMX1 (−qt) . = 1 − (1 − q)MX (−qt)

VII. C ONCLUSION

In this section, we prove the exponential limit theorem given in (8). To this end, let MX1 (−t) denote the MGF of the random variable X1 and MX (−t) be the MGF of the random variable

q(1 − q)k−1 MX1 (−qt) [MX (−qt)]k−1

k=1

analysis for the performance of a call operating over a wireless network.

In this paper, we have proposed an analytical technique that accounts for the effect of channel fading in the evaluation of performance metrics, such as call-completion probability and call-dropping probability in wireless networks. The effects of short-term fading (Rayleigh fading) and long-term fading (lognormal shadowing) have been separately considered and determine the type of applications and link recovery mechanisms that should be adopted by the wireless network. Performance metrics have been defined using the concept of random sums. The derived expressions have been given in terms of the MGF of the call-holding time distribution and only require that the first two moments of the cell dwell time distribution and the mean of the good-link period be finite. Call-completion probability and call-dropping (by handoff failure or link breakdown) probability have been derived and computed. The derived expressions show the relationship between the network call performance metrics and the physical link characteristics, and will be useful for the cross-layer design and optimization of wireless networks.

∞ 

μ2 2 2 (qt)

+ ···



(A-3)

3012

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009

A PPENDIX B D ERIVATION OF (17)

Simplifying the term inside the bracket, we have

By performing the integration by parts in (16), we have τ¯link

τ¯f = Pr(τf ≥ τm )



∞ (−u)d τm τ ¯f

2 exp u



−2 πu2

I1



2 πu2





 ∞  −2 2 τ¯f 2  = exp I1 (−u)   τm Pr(τf ≥ τm ) u πu2 πu2 ∞ + τm τ ¯f

τ¯f = τm + Pr (τf ≥ τm )

∞

τm τ ¯f

2 exp u



2 exp u

−2 πu2



−2 πu2

I1

2 πu2

I1

τ ¯f



2 πu2

du



×Pr(R1 +T2 +· · ·+Tn ≤ x|Tc = x) ∞ 1  pf (1−pf )n−1 Pr(R1 +T2 +· · ·+Tn ≤ x|Tc= x) pf n=1   ∞  n (1−pf ) Pr(R1 +T2 +· · ·+Tn ≤ x|Tc = x) = 1+

−

n=1

du

−

∞ 

(1−pf)n−1 Pr(R1 +T2 +· · ·+Tn ≤ x|Tc = x). (C-3)

n=1

(B-1) where we have used (15) to obtain the first term. Denoting the integral in the second term above by I and making the substitution x = (2/π)u−2 , we have  τ¯f 2 2 π τm I= x−1 exp(−x)I1 (x)dx 0

      2 2 2 2 τ¯f 2 τ¯f −2 τ¯f I1 + I0 . = 1 − exp π τm π τm π τm (B-2) Substituting (B-2) into (B-1) gives the desired result in (17). A PPENDIX C P ROOF OF (31) AND (32)

Merging the terms in the bracket and substituting k = n − 1 in the other term, we have 1 − Pr(SK ≤ x|Tc = x) =

∞ 

(1 − pf )k Pr(R1 + T2 + · · · + Tk ≤ x|Tc = x)

k=0

−

∞ 

(1 − pf )k Pr(R1 + T2 + · · · + Tk+1 ≤ x|Tc = x).

k=0

(C-4) Note that k = 0 results in the empty set in pmf, which is equal to 1. Finally, combining the two terms in (C-4) gives the desired result of (31), i.e., 1 − Pr(SK ≤ x|Tc = x)

Here, we show that (29) and (31) are equivalent. From (29), we have

=

∞ 

(1 − pf )k · Pr(R1 + T2 + · · · + Tk < x < R1

k=0

+ T2 + · · · + Tk+1 |Tc = x). (C-5)

1 − Pr(SK ≤ x|Tc = x) ∞  =1− pf (1 − pf )n−1



n=1

× Pr(R1 + T2 + · · · + Tn ≤ x|Tc = x).

1−Pr(SK ≤ x|Tc = x)  ∞ (1−pf )  · pf (1−pf )n−1 = 1+ pf n=1

(C-1)

Without loss of generality, we may add and subtract the same term to the right-hand side of (C-1) to obtain 1−Pr(SK ≤ x|Tc = x)  ∞  pf (1−pf )n−1 Pr(R1 +T2 · · · Tn ≤ x|Tc = x) = 1−  ∞ 1  n−1 + pf (1−pf ) Pr(R1 +T2 · · · Tn ≤ x|Tc = x) pf n=1 n=1

∞ 1  − pf (1−pf )n−1 Pr(R1 +T2 · · · Tn ≤ x|Tc = x). pf n=1

(C-2)

R EFERENCES [1] M. Schwartz, Mobile Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2005. [2] N. B. Mandayam, P. C. Chen, and J. M. Holtzman, “Minimum duration outage for CDMA cellular systems: A level crossing analysis,” Wireless Pers. Commun., vol. 7, no. 2/3, pp. 135–146, Aug. 1998. [3] J. Lai and N. B. Mandayam, “Minimum duration outages in Rayleigh fading channels,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1755–1761, Oct. 2001. [4] M. M. Zonoozi and P. Dassanayake, “User mobility modeling and characterization of mobility patterns,” IEEE J. Sel. Areas Commun., vol. 15, no. 7, pp. 1239–1252, Sep. 1997. [5] F. Khan and D. Zeghlache, “Effect of cell residence time distribution on the performance of cellular mobile networks,” in Proc. IEEE Veh. Technol. Conf., May 1997, vol. 2, pp. 949–953. [6] D. E. Duffy, A. A. McIntosh, M. Rosenstein, and W. Willinger, “Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks,” IEEE J. Sel. Areas Commun., vol. 12, no. 3, pp. 544–551, Apr. 1994.

PATTARAMALAI et al.: EVALUATION OF CALL PERFORMANCE IN CELLULAR NETWORKS

[7] E. Chlebus, “Empirical validation of call holding time distribution in cellular communications systems,” in Proc. 15th ITC, Washington, DC, Jun. 1977, pp. 1179–1188. [8] Y. Fang, I. Chlamtac, and Y. B. Lin, “Modeling PCS networks under general call holding time and cell residence time distributions,” IEEE/ACM Trans. Netw., vol. 5, no. 6, pp. 893–906, Dec. 1997. [9] Y. Fang, I. Chlamtac, and Y. B. Lin, “Call performance for a PCS network,” IEEE J. Sel. Areas Commun., vol. 15, no. 8, pp. 1568–1581, Oct. 1997. [10] Y. Fang, I. Chlamtac, and Y. B. Lin, “Channel occupancy times and handoff rate for mobile computing and PCS networks,” IEEE Trans. Comput., vol. 47, no. 6, pp. 679–692, Jun. 1998. [11] Y. Fang and I. Chlamtac, “Analytical generalized results for handoff probability in wireless networks,” IEEE Trans. Commun., vol. 50, no. 3, pp. 396–399, Mar. 2002. [12] Y. Fang, “Modeling and performance analysis for wireless mobile networks: A new analytical approach,” IEEE/ACM Trans. Netw., vol. 13, no. 5, pp. 989–1002, Oct. 2005. [13] S. Pattaramalai, V. A. Aalo, and G. P. Efthymoglou, “Call completion probability with Weibull distributed call holding time and cell dwell time,” in Proc. IEEE GLOBECOM, Washington, DC, Nov. 2007, pp. 2634–2638. [14] Y. Zhang and M. Fujise, “Performance analysis of wireless networks over Rayleigh fading channel,” IEEE Trans. Veh. Technol., vol. 55, no. 5, pp. 1621–1632, Sep. 2006. [15] Y. Zhang, M. Ma, and M. Fujise, “Call completion in wireless networks over lossy link,” IEEE Trans. Veh. Technol., vol. 56, no. 2, pp. 929–941, Mar. 2007. [16] R. M. Rodriquez-Dagnino and H. Takagi, “Counting handovers in a cellular mobile communication network: Equilibrium renewal process approach,” Perform. Eval., vol. 52, no. 2/3, pp. 153–174, Apr. 2003. [17] H. N. Hung, P. C. Lee, and Y. B. Lin, “Random number generation for excess life of mobile user residence time,” IEEE Trans. Veh. Technol., vol. 55, no. 23, pp. 1045–1050, May 2006. [18] D. R. Cox, Renewal Theory, 1967. Science Paperback. [19] B. V. Gnedenko and V. Y. Korolev, Random Summation: Limit Theorems and Applications. Boca Raton, FL: CRC, 1996. [20] T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels, Stochastic Processes for Insurance and Finance. Hoboken, NJ: Wiley, 1998. [21] V. Kalashnikov, Geometric Sums: Bounds for Rare Events With Applications. Norwell, MA: Kluwer, 1997. [22] E. S. Shiu, “The probability of eventual ruin in the compound binomial model,” ASTIN Bull., vol. 19, no. 2, pp. 179–190, Dec. 1989. [23] M. C. Carter, “A model for thunderstorm activity: Use of the compound negative binomial-positive binomial distribution,” Appl. Stat., vol. 21, no. 2, pp. 196–201, 1972. [24] I. B. Gertsbakh, “Asymptotic methods in reliability theory: A review,” Adv. Appl. Probab., vol. 16, no. 1, pp. 147–175, Mar. 1984. [25] N. Eshita, “An estimation of claims distribution,” ASTIN Bull., vol. 9, no. 1/2, pp. 111–118, Jun. 1977. [26] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [27] W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974. [28] J. Lai and N. B. Mandayam, “Fade margins for minimum duration outages in lognormal shadow fading and Rayleigh fading,” in Proc. 31st Asilomar Conf. Signals, Syst. Comput., Nov. 1997, vol. 1, pp. 609–613. [29] S. O. Rice, “Distribution of the duration of fades in radio transmissions: Gaussian noise model,” Bell Syst. Tech. J., vol. 37, pp. 581–635, May 1958.

3013

[30] I. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products, 6th ed. New York: Academic, 2000. [31] S. Nadarajah and S. Kotz “Comments on ‘Minimum duration outages in Rayleigh fading channels,’” IEEE Trans. Commun., vol. 55, no. 6, p. 1110, Jun. 2007. [32] M. Evans, N. Hastings, and B. Peacock, Statistical Distributions, 2nd ed. Hoboken, NJ: Wiley, 1993.

Suwat Pattaramalai was born in Bangkok, Thailand, in 1969. He received the B.E.E. degree in electrical engineering from the Chulalongkorn University, Bangkok, in 1991 and the M.Eng. and Ph.D. degrees in electrical engineering from Florida Atlantic University, Boca Raton, in 1996 and 2007, respectively. Since 1993, he has been with the faculty of the Department of Electrical and Electronics Engineering, King Mongkut’s University of Technology Thonburi, Bangkok. His research interests include digital communications, fading channels, performance analysis of wireless networks, and modeling of cellular systems.

Valentine A. Aalo (S’89–M’92–SM’05) was born in Nigeria on March 23, 1959. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Southern Illinois University, Carbondale, in 1984, 1986, and 1991, respectively. Since 1991, he has been with the faculty of Florida Atlantic University, Boca Raton, where he is currently a Professor with the Department of Electrical Engineering. He spent the summers of 1994 and 1995 with The Satellite Communications and Networking Group at Rome Laboratory, Griffiss Air Force Base, Rome, NY, as a Faculty Research Associate. His research interests include the areas of wireless communications, diversity techniques, adaptive array processing, satellite communications, and radar signal processing. Dr. Aalo is a member of Tau Beta Pi and several IEEE societies. He serves as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS.

George P. Efthymoglou (S’94–M’98) was born in Athens, Greece, on April 22, 1968. He received the B.S. degree in physics from the University of Athens in 1991 and the M.S. and Ph.D. degrees in electrical engineering from the Florida Atlantic University, Boca Raton, in 1993 and 1997, respectively. In 1997, he was with Cadence Design Systems, where he was engaged in modeling, simulation, and performance evaluation of third-generation (3G) wireless systems. Since 2002, he has been an Assistant Professor with the Department of Digital Systems, University of Piraeus, Piraeus, Greece. His research interests include the area of digital communication systems, with emphasis on the performance evaluation of 3G and fourth-generation cellular systems in fading channels.