IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`1
`
`Slotted ALOHA for High-Capacity
`Voice Cellular Communications
`
`Michele Zorzi, Student Member, IEEE, and Silvano Pupolin, Senior Member, IEEE
`
`Abstract—Slotted ALOHA is proposed as a multiple access scheme for
`
`high capacity voice cellular communications in mobile radio environment.
`The performance of such a system, in the presence of fading and shadowing,
`is evaluated for both Mobile-to-Base and Base-to-Mobile links, in terms of
`number of supported conversations per cell, under some constraints on
`maximum tolerable delay. The numerical results show that a system of this
`sort can compete with other multiaccess schemes currently considered, such
`as TDMA, FDMA, and even CDMA. A heuristic stability analysis is also
`presented, showing that the proposed system does not suffer from instability
`problems.
`
`I. INTRODUCTION
`
`In recent times, the demand for mobile radio services has been
`increasing at an astonishing rate. Service providers are therefore
`required to accommodate more users in the same bandwidth, and
`the need for the systems to have a higher capacity and to utilize
`the spectrum resources more efficiently is clear. Moreover, these
`systems must provide service to a large population of users,
`distributed at random on the ground, that require the use of a
`communications channel rather unfrequently. Traditionally, the
`multiple access (MA) schemes used are TDMA and FDMA [1],
`in which there is no contention for the channel, resulting in a
`demand-based fixed-assignment strategy. The overall capacity
`is increased by organizing the system as a cellular structure,
`reusing the same frequencies in different cells, sufficiently apart
`from each other in order to keep the interference at a tolerable
`level [2].
`Digital voice communication, however, has some features
`that make this philosophy not efficient. For example, voice
`activity is limited to about the 35-40% of the time [3]: in fixed
`assignment strategies the remaining time would be wasted. The
`implementation of high capacity voice communications systems
`calls for more sophisticated multiple access protocols.
`Recently, Gilhousen et al. [4] have proposed code-division
`multiple access (CDMA) as a way to utilize more efficiently the
`available bandwidth, showing that it can perform significantly
`better than TDMA and FDMA. This is due basically to the fact
`that spread spectrum modulation has the capability to reduce
`the interference, and therefore it is no longer necessary any
`separation between cochannel cells, i.e., the same frequency
`range can be used in all cells, while maintaining a satisfactory
`transmission quality. This makes CDMA remarkably better than
`fixed assignment techniques.
`
`This work has been partially supported by the “Consiglio Nazionale delle Ricerche - Progetto
`Finalizzato Trasporti 2”, and by MURST, Italy.
`Part of this work has been presented at IEEE ICC’93, Geneva, Switzerland, 23-26 May
`1993.
`Michele Zorzi is with the Dipartimento di Elettronica e Informazione, Politecnico di Milano
`- ITALY.
`Silvano Pupolin is with the Dipartimento di Elettronica e Informatica, Universit`a di Padova
`- ITALY.
`
`In this paper, we develop an average performance analysis of
`another
`popular MA
`technique,
`namely
`slotted
`ALOHA, applied to a cellular voice environment; the propa-
`gation model takes into account Rayleigh fading, log-normal
`shadowing, inverse power loss law, random distribution of the
`mobiles in the two-dimensional space. The motivation of this
`study is that, although slotted ALOHA (along with an entire
`class of protocols originated by it) is very popular as a multiac-
`cess protocol, its application to this particular environment, i.e.,
`packet voice transmission in cellular systems, in the presence of
`fading and shadowing, has never been considered, to the best
`of our knowledge. The recent success of CDMA vs. TDMA
`and FDMA shows that random access protocols can do better
`than fixed assignment schemes; therefore, we believe that it is
`meaningful to explore more deeply the possibility of employ-
`ing random access in cellular, studying techniques other than
`CDMA. Since ALOHA was the first (and the simplest) random
`access protocol, it is a reasonable choice to develop an analysis
`and to give some results referring to it. In fact, if this simple
`approach proves to be promising, we believe that the remarkable
`expertise about the ALOHA protocol and its variations will en-
`able us to build on the basic idea here presented, and to develop
`more efficient schemes and more rigorous analyses. However,
`we are aware that some sophisticated techniques, developed for
`different environments, may be unsuitable in this context. For
`instance, CSMA/CD, which in the Ethernet standard achieves a
`throughput close to one, has been shown to suffer from some dif-
`ficulty in radio communications, in the presence of the so-called
`hidden-terminal problem [5].
`The basic contribution of this paper is to explore the possibility
`of using ALOHA random access in a cellular voice packet mobile
`radio network and to develop a simplified performance analysis
`for such a system. As mentioned, this is a new application,
`since in the previous literature the ALOHA systems are studied
`in different situations. In many papers, such as [6; 7], capture is
`not considered: this is clearly unacceptable in this environment
`where, because of the propagation characteristics (e.g., the near-
`far effect and the random fluctuations of the received power),
`the advantages of a capture mechanism can be fully exploited.
`In other papers, such as [8; 9; 10; 11], capture ALOHA is
`presented, with reference to a single receiver:
`this is also not
`applicable to the present context, where the presence of many
`cells is a basic feature of the system. Papers like [12], presenting
`PRMA for cellular voice, suffer from this fundamental limitation
`as well. The concept of cellular ALOHA requires to consider
`different spatial distributions for the intended user and for the
`interferers. In fact, not all colliding users are contending: those
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`PETITIONERS EX. 1016 PAGE 1
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`2
`
`(carrier frequency). Therefore, even though the transmission
`quality is enhanced, the required bandwidth is significantly in-
`creased (more specifically, it must be multiplied by the number
`of cells in a cluster). For a typical cluster configuration, the
`number of cells is 7. The throughput per cell in TDMA/FDMA
`is at most equal to the voice activity, i.e., 35-40%, as already
`
`of the total capacity, so that the overall throughput is at most
`5-6%. This means that a random access scheme, with complete
`frequency reuse, performs better as soon as its throughput per
`cell is greater than 5-6%, and this is often the case.
`Packets are assumed to have fixed length, and to fit into a slot.
`Voice is digitalized and packetized by using some standard tech-
`nique of redundancy reduction. In such a case, voice packets
`are generated periodically during voice activity, and the dura-
`
`observed, but the available capacity in a single cell is 1/7' 14%
`tion of a period is called frame. LetZ be the number of slots
`bidimensional Poisson point process with intensity(cid:21) (users per
`All users transmit the same amount of power,PT (i.e., no power
`the slot the Signal-to-Interference plus Noise Ratio,SNR, is
`greater than a fixed threshold, denoted byb. The value ofb
`
`transmitting from within the cell are contending, whereas those
`transmitting from outside are just interfering. In this view, the
`cellular approach with complete frequency reuse seems to be
`unique. Also, very often slotted ALOHA is considered for data
`transmission, where delay, which is the fundamental constraint
`in the present context, is not a major concern. In this view, the
`analysis presented in [10], although one of the few cases in which
`intercell interference is taken into account, is not applicable here.
`We remark that the analysis presented in this paper is an
`average analysis, i.e., does not take into account the dynamic
`behaviour of the network; at the end of the paper, we deal
`briefly with the stability issues, although in a simplified and not
`completely rigorous manner. There are several reasons for this
`choice. First, the main goal, as already stated, is to explore the
`feasibility of a scheme of this sort, and therefore a rough analysis
`appears a convenient choice. Second, a complete and rigorous
`analysis, in a real-world context, where some basic assumptions
`are not verified, is too complex. Moreover, even though the
`throughput analysis can probably be done rigorously, the study
`of the delay turned out to be not feasible, without recurring to
`simulation; our opinion is that in this first approach a rough
`analytical study is better than the presentation of the results of a
`sophisticated (and somewhat obscure) simulation, which would
`not be of theoretical interest anyway. The stability issues and
`a more rigorous approach remain our primary concerns, and a
`topic we are currently doing research on. Finally, as has been
`shown in several papers [13; 14; 15], there exist stabilization
`schemes, which enable the system to achieve the performance
`predicted by the static (i.e., average) analysis. Even though
`designed for different situations, our sense is that they might be
`successfully employed, possibly with some modifications, in the
`present context as well.
`The paper is organized as follows: in Section 2, the system
`model is described, and in Section 3 the throughput and delay
`analyses are developed. Section 4 takes into account the effect
`of noise (neglected up to that point), whereas Section 5 presents
`some numerical results. Some considerations about stability
`(Section 6) and the Conclusions complete the paper.
`
`II. SYSTEM MODEL
`
`The protocol used by our system is slotted ALOHA [6], in
`the sense that, every time a user has a packet, this is transmitted
`in the next slot, regardless the behavior of the other users.
`It
`is also cellular, because there are many base stations and each
`terminal transmits packets to the closest one, which will relay
`them towards their destinations. Note that, in reality, slotted
`ALOHA as a multiple access protocol is used only on the mobile-
`to-base channel; in the base-to-mobile direction, in fact, the base
`station takes care of all the transmissions in its cell, and the
`resulting scheme is very much like TDMA.
`The frequency reuse is complete, i.e. the same frequency band
`is used in every cell (all cells are cochannel). This fact enables us
`to overcome a major problem exhibited by the fixed assignment
`schemes, i.e., spectral efficiency. In fact, in TDMA and FDMA,
`adequate spacing between cochannel cells is unavoidable, in or-
`der to keep the interference at tolerable levels. On the other
`hand, this choice requires that the cells be grouped into clusters,
`and that within a cluster all cells transmit on a different channel
`
`contained in a frame. A wideband channel is used, and each
`packet must be successfully transmitted within a given number
`of frames to meet the delay constraints, otherwise it is discarded.
`Users are assumed to be distributed on the ground according to a
`
`unit area per slot), up to an infinite distance from the receiver.
`This assumption, although not strictly verified, turns out to be
`reasonable in the presence of a large population of users [16].
`Also, with many users, the Poisson assumption is a good model
`of a finite population (binomial distribution) as well.
`The propagation model is the same as in [17]. Due to multi-
`ple reflections, each signal at the receiver is modeled as the su-
`perposition of two orthogonal Gaussian components (Rayleigh
`fading), so that its envelope turns out to be a Rayleigh random
`variable (r.v.); as a result, the signal power is exponentially dis-
`tributed. Another random effect in the propagation is shadow-
`ing, which is a slow random fluctuation of the average received
`power, due to weather conditions, terrain roughness and the pres-
`ence of obstacles: in a popular model [10], based on the available
`measurements, this effect is described by a log-normal r.v., i.e.,
`a r.v. whose representation in logarithmic units (e.g., in dB) is
`Gaussian. Finally, the fact that the average received power is a
`decreasing function of the transmitter-receiver distance must be
`taken into account, by means of a deterministic path loss law.
`In this paper, we will assume an inverse loss law, defined later.
`
`control is used).
`Acknowledgements are neglected, in the sense that a transmit-
`ter is assumed to know instantaneously whether its transmission
`has been successful or not. The base stations are able to cap-
`ture “strong” packets, and the capture effect, due to the random
`locations of users, is enhanced by the random effects in prop-
`agation (Rayleigh fading and log-normal shadowing) [18]. We
`assume that a packet is correctly received if and only if during
`
`depends on the modulation format and on the coding scheme
`used, if any. Being this kind of system typically interference
`limited, noise is neglected in most of the following analysis: in
`Section 4, formulas in the presence of noise are given, and in
`Section 5.2 some numerical results are presented, showing that
`this assumption is in fact reasonable and that a more complete
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`PETITIONERS EX. 1016 PAGE 2
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`

`e(cid:24)0Ar(cid:0)(cid:17)
`(1)
`PT;
`ulation, is given byP0=(cid:11)2
`where(cid:11)2
`Sincee(cid:24)0 represents the random fluctuation around the area
`mean, due to the log-normal shadowing, the r.v.(cid:24)0 is Gaussian
`with zero mean and variance(cid:27)2. In this context,(cid:27) is expressed
`spread; often, in the literature, the dB spread,(cid:27)dB, is used in-
`stead, when decimal logarithms are employed. Of course,(cid:27) =
`0.1(cid:27)dB log10. The values that(cid:27) can take are usually between
`1.3 and 3 (6 to 13 dB [17]). The factorAr(cid:0)(cid:17)
`deterministic dependence of the power on the distance,r0;(cid:17) can
`take a value between 2 and 4, whereas the constantA depends
`on the heigths of the antennas and on the carrier frequency.PT
`In the same way, the power from thei-th interferer can be
`Ii=(cid:11)2ie(cid:24)iAr(cid:0)(cid:17)iPT;
`(2)
`the constantA is assumed to be the same for all users.
`k, this power is
`PI=kXi=1
`Ii:
`(3)
`TheSNR at the receiver is given by
`P0PI+W;
`SNR=
`(4)
`whereW is the thermal noise power. The packet success prob-
`Ps=P[SNR>b]=P(cid:20)P0PI+W>b(cid:21);
`(5)
`whereb is the thresholdSNR.
`LetG=(cid:21)(cid:25)R2 the average number of attempted transmis-
`sions to a base station during a slot:R is the radius of the cells
`value ofG and for a given mobile-base distance,r0, is computed
`Ps((cid:16))=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:27)2p
`2(cid:25)(cid:27)e(cid:0)J((cid:24);(cid:16));
`(6)
`J((cid:24);(cid:16))=U(cid:16)e(cid:0)2(cid:24)=(cid:17)
`(7)
`(cid:16)=(cid:21)(cid:25)r2
`=G(cid:26)2
`(8)
`U= 2(cid:25)(cid:17) cosec
`2(cid:25)(cid:17)b2=(cid:17)e2((cid:27)=(cid:17))2;
`(9)
`and(cid:26)=r0=R is the normalized mobile-base distance. Note that
`Ps does not depend on the location of the intended user,(cid:26), and
`
`ability is defined as
`
`A. Mobile-to-base link (MB)
`
`It is shown
`(modeled as circular for analytical convenience).
`in Appendix A that the packet success probability, for a given
`
`as:
`
`where
`
`0
`
`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`3
`
`0
`
`0
`
`0 is an exponentially distributed r.v. with unit mean.
`
`in logarithmic natural units, and will referred to as the natural
`
`accounts for the
`
`0
`
`is the transmitted power.
`
`expressed as
`
`where the symbols have the same meaning as in (1); in particular,
`
`The total interference power at the receiver is the (incoherent)
`sum of the contributions of all the interferers; given their number,
`
`analysis leads to substantially the same results.
`
`III. PERFORMANCE ANALYSIS
`
`When studying MA systems, two are the relevant performance
`indices: throughput and delay. These parameters typically form
`a trade-off to be solved in the system design. In fact, when the
`network is loaded with a traffic close to the maximum sustain-
`able, the probability of a successful transmission is small (many
`collisions), and therefore a large number of retransmissions are
`needed. On the other hand, when the network is lightly loaded
`most of the transmissions are successful, but many slots go by
`without being used, and the throughput is decreased. In the fol-
`lowing analysis, which is an average analysis, we will assume
`to be in stationary conditions. In studying the performance, we
`will treat separately the two links: mobile-to-base and base-to-
`mobile.
`The (normalized) throughput per cell is defined as the average
`number of successfully received transmissions per slot. As will
`be defined more precisely in the following, it can be expressed
`as the offered load per cell (average number of attempted trans-
`mission per slot in a cell) multiplied by the average probability
`of success of a transmission (which is a decreasing function of
`the offered load).
`In the two directions, these quantities are
`computed differently, even though the concept of throughput is
`substantially the same.
`The delay is strongly related to the probability of packet suc-
`
`packet must be transmitted until it is successfully received can
`
`In our analysis of voice
`and depends on the user’s position.
`communications, we are interested in throughput as a perfor-
`mance measure to be maximized, whereas delay is a constraint
`to be met, rather than a performance measure itself.
`Indeed,
`there are strict limitations on the overall delay experienced by a
`voice packet (50-100 ms). In our analysis, we assume that, upon
`the arrival of the next packet at the transmitter, the old packet
`is discarded unless it has already been successfully transmit-
`ted. This strategy obviously forces the delay to be bounded (at
`
`packet gets lost. Subjective tests, reported in [19], indicate that
`
`good quality, even though there exist techniques to somewhat
`
`study, however, we will take 0.01 as the maximum acceptable
`
`The first quantity to be computed is the probability of a packet
`
`cess,Ps: given the offered traffic, the number of times that a
`be modeled as a geometric random variable of parameter 1(cid:0)Ps,
`most one frame), but introduces a positive probability,PL, that a
`PL should be kept below values of the order of 0.01 to provide
`“interpolate” packets, which admitPL as large as 5-10%. In our
`value ofPL.
`success,Ps; in general, it depends on the statistics of the in-
`terference, on theSNR at the receiver, and on the relationship
`between theSNR and the correct packet reception (in this con-
`text,SNR is the short-term Signal-to-Interference plus Noise
`alent to the event that theSNR at the receiver is greater than
`a threshold value, called outageSNR [17]. Therefore, in the
`
`Ratio). When powerful error control codes are used, however,
`with good approximation the correct packet reception is equiv-
`
`following, we will take the complement to one of the probability
`of outage [17] as the probability of correct reception.
`With the above model, the useful signal power, after demod-
`
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`4
`
`on the traffic,G, individually, but only on the product(cid:16)=G(cid:26)2.
`The average throughput,Smb, is obtained by averaging (6) over
`(cid:26), and multiplying it byG, to obtain
`Smb(G)=GZ
`2(cid:26)d(cid:26)Ps(G(cid:26)2)
`=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:25)(cid:27)Z
`2(cid:27)2p
`2(cid:26)d(cid:26)Ge(cid:0)J((cid:24);G(cid:26)2)
`=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:25)(cid:27) 1(cid:0)e(cid:0)GUe(cid:0)2(cid:24)=(cid:17)
`2(cid:27)2p
`:
`Ue(cid:0)2(cid:24)=(cid:17)
`Note that the limit ofSbm asG! is finite, and is equal to
`Smb( )=e2((cid:27)=(cid:17))2U
`=
`(11)
`2(cid:25)(cid:17) cosec 2(cid:25)(cid:17)b2=(cid:17):
`Note also that, taking the derivative ofSbm with respect toG,
`we obtainSmb(G)=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:27)2p
`2(cid:25)(cid:27)e(cid:0)GUe(cid:0)2(cid:24)=(cid:17);
`(12)
`which, of course, is always positive. Therefore,Sbm(G) is an
`increasing function of its argument,G.
`For every value of the offered load,G, the corresponding
`average throughput,Smb(G), can be computed from (10). Given
`the offered traffic,G, and the location of the intended user,(cid:26),
`the number of retransmissions (assumed independent),Nrt, is a
`geometrically distributed r.v. with parameter 1(cid:0)Ps(G(cid:26)2), and
`D= 1+NrtXi=1
`ki;
`(13)
`where theki’s are the random backoffs, which are necessary
`slot. We note that, due to the finite backoffski’s, the Poisson
`ki’s are independent and identically distributed uniform random
`variables between 1 andK, the mean,mD((cid:16)), and the variance,
`(cid:27)2D((cid:16)) of the delay (in slots) suffered by a user for a given value
`of(cid:16), are computed in Appendix B, to obtain (we neglect the half
`mD((cid:16))=E[D]= 1+K+ 1
`1(cid:0)Ps((cid:16))
`(14)
`Ps((cid:16))
`(cid:27)2D((cid:16))=Var[D]
`1(cid:0)Ps((cid:16))
`=K2(cid:0) 1
`Ps((cid:16))+
`+(K+ 1)2
`1(cid:0)Ps((cid:16))
`:
`P 2s((cid:16))
`
`1
`
`0
`
`1
`
`0
`
`1
`
`(10)
`
`the packet delay, in slots, can be expressed as:
`
`in order to avoid the certainty of another collision in the next
`
`It has been
`assumption about arrivals is not strictly verified.
`observed, however, that when the arrival process is obtained
`by merging many streams, it can be very well approximated
`by means of a Poissonian stream [16]; therefore, we go on
`with the analysis under this assumption. If we assume that the
`
`slot due to the fact that packets may be generated at any point in
`time):
`
`2
`
`4
`
`12
`
`(15)
`
`D is a multinomial r.v. which, for the case of interest, i.e., when
`(cid:18)Z(cid:0)mD((cid:16))
`2(cid:27)D((cid:16))(cid:19);
`p
`PL((cid:16))=P[D>Z]' 1
`(16)
`whereZ is the number of slots in a frame.PL((cid:16)) is an increasing
`function of(cid:16), and therefore, for a givenG, is maximum at the
`boundary of the cell, i.e., for(cid:26)= 1 and(cid:16)=G. In order to guar-
`antee thatPL((cid:16))(cid:20) 0:01 everywhere, it is sufficient to require
`G(cid:20)G0, whereG0 is such thatPL(G0)= 0:01. Therefore, the
`given byS0=Smb(G0).
`p is the probability that a base station is transmitting a packet
`Ps(p;})=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:25)(cid:27) 18Yi=1
`1(cid:0)pJi((cid:24);})(cid:1);
`(cid:0)
`2(cid:27)2p
`(17)
`Ji((cid:24);})=Z (cid:0) dxe(cid:0)x2
`2(cid:27)2p
`1+e(cid:24)(cid:0)xbd(cid:17)i(}):
`(18)
`2(cid:25)(cid:27)
`Ps(p;}) depends explicitly onp, which has the meaning of
`offered traffic per cell (the same asG in the MB link), and on
`the location of the receiving mobile,}.di(}) is the distance of
`the point} from thei-th interfering base station, divided by the
`distance from the intended base station. Averaging (17) over},
`and multiplying it byp, we obtain the average throughput per
`Sbm(p)=pPs(p);
`(19)
`wherePs(p) is the average ofPs(p;}) over the (uniform) dis-
`packet is assigned at random to a slot. For a givenSbm, the
`probability that exactlyk slots in the frame are unsuccessful is:
`P[k unsuccessful]=(cid:18)Zk(cid:19)SZ(cid:0)k
`bm(1(cid:0)Sbm)k:
`(20)
`
`B. Base-to-mobile link (BM)
`The BM channel is different from MB in that the transmitters
`(the base stations) lie in fixed positions, and the receiver is
`located at random. Moreover, in the BM channel, all packets of
`a cell are managed by the same base station. Therefore, there
`are no collisions from within the cell, but only interference from
`outside. We consider in our analysis only the useful base station
`and the nearest 18 interfering base stations; this corresponds
`to assuming that all sites beyond the second ring around the
`intended cell contribute negligible interference. In this case, if
`
`several retransmissions are required, can be approximated by a
`Gaussian r.v., so that we can write, for the packet loss probability,
`
`erfc
`
`2
`
`maximum achievable throughput under the delay constraints is
`
`in a given slot (assumed the same for all stations), the success
`probability can be shown to be (see Appendix C for the details):
`
`where
`
`1
`
`cell on the BM link, given by
`
`tribution of the intended receiver over the cell.
`In order to evaluate the packet loss probability due to the
`delay constraint, let us consider the following. We assume that
`the transmissions in different slots are independent, and that a
`
`PETITIONERS EX. 1016 PAGE 4
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`

`IfN packets are to be transmitted in a frame, the number of
`lost packets, given thatk slots out ofZ are unsuccessful, is
`N(cid:0)(Z(cid:0)k), ifk>Z(cid:0)N , and is 0 ifk(cid:20)Z(cid:0)N . The
`probability that one ofN packets is lost whenn of them are lost
`isn=N . Therefore, we obtain:
`P[a given packet is lostjN]=
`(cid:18)Zk(cid:19)SZ(cid:0)k
`=ZXk=Z(cid:0)NN(cid:0)(Z(cid:0)k)
`bm(1(cid:0)Sbm)k
`N
`N(cid:0)jN(cid:18)Zj(cid:19)Sjbm(1(cid:0)Sbm)Z(cid:0)j:
`=NXj=0
`In the presence ofNconv conversations, each with voice activity
`(cid:12), the probability of havingN active users in a frame is given
`by:P[N activejNconv]=(cid:18)NconvN(cid:19)(cid:12)N(1(cid:0)(cid:12))Nconv(cid:0)N
`'e(cid:0)(cid:12)Nconv((cid:12)Nconv)N
`;
`N !
`and the average packet loss probability, givenNconv conversa-
`PL(Nconv)=NconvXN=0
`P[a given packet is lostjN]
`(cid:1)P[N activejNconv]:
`equations,Sbm, and thereforePL(Nconv), are functions ofp.
`p= 1.
`LetBs be the bandwidth required for a single TDMA channel
`(i.e., a single user),Bt=MBs be the total bandwidth in the
`system, andBc=Bt=C be the bandwidth per cell, ifC is
`the number of cells in a cluster (typically,C= 7). In CDMA
`Bc=Bt, since the frequency band is reused in all cells (i.e.,C =
`(of bandwidthBs) at his disposal. Note thatM has the meaning
`(cid:13)=NconvM;
`(24)
`whereNconv is the number of conversations that can be simul-
`On the MB link, ifNact=(cid:12)Nconv is the average number of
`Nact=(cid:12)Nconv=SmbZ:
`(25)
`
`C. Channel efficiency
`The major goal in systems of this sort is to maximize the
`number of admitted users, i.e., the system capacity.
`In this
`framework, we want to define an index to measure the spec-
`trum efficiency, in such a way that it is possible to compare the
`efficiency of systems with different bandwidths.
`
`(21)
`
`(22)
`
`(23)
`
`tions, is therefore expressed by:
`
`We remark that, although not explicitly indicated in the above
`
`However, as will be shown later, the best choice is very often
`
`and ALOHA, a wideband channel is provided to all users, and
`
`1), whereas in TDMA and FDMA each user has a single channel
`
`of “number of equivalent TDMA channels” in the system. We
`define channel efficiency by the quantity
`
`taneously supported by a base station.
`
`active conversations per cell, in steady-state we have that
`
`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`5
`
`(cid:13)=SbmZ(cid:12)M=Sbm(cid:23)(cid:12);
`(26)
`where(cid:23)=Z=M is the fraction of information bits in a packet.
`On the BM channel, for a given value ofSbm,Nconv is com-
`puted from (23), and(cid:13) is found from (24). On the other hand,
`reuse strategy, can support at mostBc=Bs=M=C conversa-
`TDMA is 1=C.
`presence of noise vanishes asPs decreases. Referring to the
`((cid:26);G)=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:27)2p
`2(cid:25)(cid:27)e(cid:0)(cid:22)0(cid:26)(cid:17)e(cid:0)(cid:24)(cid:0)J((cid:24);G(cid:26)2);
`P(n)s
`(27)
`(cid:22)0=bWR(cid:17)APT;
`(28)
`over(cid:26) can be solved in closed form only for(cid:17)= 4, to obtain
`S(n)mb(G)=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:25)(cid:27)r(cid:25)(cid:22)0
`2(cid:27)2p
`Ge(cid:18)2=2(cid:0)(cid:24)=2
`
`(cid:1)h
`(cid:16)p2(cid:22)0e(cid:0)(cid:24)=2(cid:0)(cid:18)(cid:17)(cid:0) F((cid:18))i; (29)
`(cid:18)=UGp
`(30)
`2(cid:22)0
`(x)=Zx(cid:0) dte(cid:0)t2
`2p
`(31)
`2(cid:25):
`(p;})=Z (cid:0) d(cid:24)e(cid:0)(cid:24)2
`2(cid:25)(cid:27)e(cid:0)(cid:22)0(cid:26)(cid:17)e(cid:0)(cid:24) 18Yi=1
`(cid:0)
`1(cid:0)pJi((cid:24);})(cid:1);(32)
`2(cid:27)2p
`P(n)s
`bandwidth isBt = 1.25 MHz, the vocoder bit rate is 8 kbps,
`corresponding to a bandwidthBc' 10 kHz, so that the number
`of equivalent TDMA channels isM = 128; the spread of the
`
`Therefore, the channel utilization is computed as
`
`in TDMA and FDMA, a base station, because of the frequency
`
`tions at a time, so that the attainable channel efficiency (24) in
`
`IV. PRESENCE OF NOISE
`
`In the previous analysis we neglected noise, which on the
`other hand is always present in a communication system. In a
`system like slotted ALOHA, heavily loaded and with moderate
`noise levels, the probability of having an interfering user much
`stronger than noise is very high. Therefore, the effect of the
`
`analysis carried out in [17], in the presence of noise the success
`probability (6) is replaced by
`
`where
`
`and (10) is modified accordingly. In particular, the integration
`
`where
`
`Similarly, (17) is replaced by
`
`and (19) is modified accordingly.
`
`V. NUMERICAL RESULTS
`
`A. Absence of noise
`We apply the above theory to the same system shown in [4],
`whose parameters are here reported for convenience. The total
`
`PETITIONERS EX. 1016 PAGE 5
`
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`F
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`IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 43, N. 4, NOV. 1994
`
`shadowing is(cid:27) = 8 dB, the propagation loss factor(cid:17)= 4, and
`the thresholdb = 6 dB. The voice activity is(cid:12) = 37.5%, the
`maximum backoffK = 10, the packet length is 16 bytes of
`the packet overhead, the effective number of slots in a frame,Z,
`is less thanM : this reduction is to be taken into account for a fair
`comparison. If we denote by(cid:23) the fraction of information bits in
`a packet ((cid:23) = 0.73 in the present case), we haveZ=(cid:23)M= 93.
`In Fig. 1, the packet loss probability (16) vs.(cid:16) is plotted. The
`maximum sustainable traffic under the constraintPL(cid:20) 0:01
`is the solution ofPL(G)= 0:01, as discussed in Section 3.1,
`and, in the present case, is equal toG0' 0:49. Fig. 2 shows
`the relationship betweenG andSmb: the maximum attainable
`throughput is computed asS0=Smb(G0), and turns out to be
`loss probability,PL. We remark that the values ofG0 andS0 are
`Fig. 3 shows the average delay,mD, (obtained averaging (14)
`over(cid:26)), vs. the throughput,Smb: the curve does not take into
`account the dropping mechanism (in fact,mD increases beyond
`Z= 93, which should be an obvious upper bound). The actual
`Fig. 3: for low delay, they almost coincide, whereas whenmD
`diverges the actual average delay tends to the valueZ= 93.
`Note that, when the throughput isSmb' 0:2, the average delay
`Z= 93 slots.
`With the above parameters, we obtainNconv' 49, and(cid:13)'
`0:38.
`If the bandwidth is increased, orK is decreased, the
`comparison, the channel utilization,(cid:13), reported in the rightmost
`utilization in TDMA, as discussed in Section 3.3, is 1=7' 0:143,
`In Fig. 4, the probability of having a successful slot,Sbm, is
`plotted vs.p, for the same system parameters. We note that the
`best performance (Sbm = 0.328) is obtained forp = 1, i.e., when
`a packet is lost is plotted vs.Nconv in Fig. 5 for some values
`ofSbm, withZ= 93. The curves in Fig. 5 are parametrized by
`Sbm, instead ofp, because in this case they are independent of
`the propagation parameters,(cid:17) and(cid:27), which, on the other hand,
`affect the relationship betweenp andSbm.
`Table II reportsNconv for different values ofM . The reported
`channel utilization,(cid:13), as in Table I, is the parameter to be con-
`results forM = 128, and compares them to those reported in
`
`information (16 ms of speech) plus 6 bytes of overhead. Due to
`
`A. Mobile-to-Base link
`
`about 0.2. From Figs. 1 and 2, the values of the maximum
`sustainable traffic and the corresponding maximum attainable
`throughput can be evaluated for any value of the required packet
`
`computed with reference to a worst-case situation, i.e., when the
`intended user is located at the cell boundary. This, of course, is
`not always the case, and, on the average, the experienced delay
`is substantially less than the maximum tolerable (i.e., a frame).
`
`curve of the average delay is of course upper bounded by that in
`
`is about 12 slots, much smaller than the maximum tolerable, i.e.,
`
`throughput is increased, as shown in Table I. Recall that, for a
`
`column, is the relevant parameter. Note also that the channel
`
`significantly smaller than the values of Table I.
`
`B. Base-to-Mobile link
`
`the base stations transmit in every slot. The probability (23) that
`
`sidered for a fair comparison. Table III summarizes the above
`
`6
`
`(cid:14)
`on the propagation loss factor,(cid:17). As(cid:17) decreases, the average
`formance of the system is dramatically reduced as(cid:17) decreases.
`PRN [20]:A=(cid:0)108 dB,PT = 0 dBW,W=(cid:0)144 dBW. By
`computingS(n)mb andS(n)bm , we verify our assumptions about the
`presence of noise. Let us define the averageSNR (due only to
`thermal noise) at the boundary of the cell,SNR0, as:
`SNR0=APTWR4
`=b(cid:22)0
`(33)
`:
`With the above