- Email: [email protected]
Generalized model for scheduling in MIMO multiple access systems: A cross-layer approach
ARTICLE IN PRESS
Signal Processing 86 (2006) 1834–1847 www.elsevier.com/locate/sigpro
Generalized model for scheduling in MIMO multiple access systems: A cross-layer approach$ Marc Realpa,, Ana I. Pe´rez-Neirab a
Telecommun. Technological Center of Catalonia (CTTC), Barcelona (Catalonia-Spain) b Technical University of Catalonia (UPC), Barcelona (Catalonia-Spain)
Received 3 January 2005; received in revised form 15 September 2005; accepted 15 September 2005 Available online 22 December 2005
Abstract Research on scheduling in MIMO multiple access systems is traditionally carried out from an information-theoretic point of view. Such information-theoretic approaches are feasible in homogeneous systems where, although channel fading is assumed, all terminals use the same modulation format, channel coding scheme, data transmission rate, packet lengths and no traffic issues are considered. In these systems, to select the set of terminals that maximize the instantaneous sum capacity leads to the highest system throughput. Unfortunately, in heterogeneous systems this assumption is not always true. Consequently, there is a need to evaluate the effects of such heterogeneity from a network point of view. In this paper, following a cross-layer approach, a generalized model for scheduling in MIMO multiple access is presented. Such model allows to evaluate the effects of heterogeneity on system throughput. Particularly, this generalized model accommodates any kind of heterogeneity from multiuser diversity due to channel fading to heterogeneity due to different data transmission rates. Different scheduling policies are analyzed under this generalized model and further applied to a sensor network. The results obtained are useful to achieve practical insight of such scheduling policies. r 2005 Elsevier B.V. All rights reserved. Keywords: Cross-layer; Scheduling; Multiuser diversity; MAC; MIMO
1. Introduction In an OSI layered packet oriented communications system, the medium access control (MAC) layer is the one in charge of controlling access of terminals to the medium in order to optimize system $
This work is supported by the Spanish government under TIC2002-04594-C02 and FIT-070000-2003-257 and by the IST European Commission under FP6-507572-WIDENS and FP6507525-NEUCOM and jointly financed by FEDER. Corresponding author. Tel.: +34 936 452 912; fax: +34 936 452 901. E-mail addresses: [email protected] (M. Realp), [email protected] (A.I. Pe´rez-Neira).
performance in terms of packet throughput, delay and jitter and to guarantee quality of service (QoS) requirements such as instantaneous packet error rate (PER). From a MAC layer point of view, the PHY layer has been typically modeled by the collision channel model. Then, a typical MAC protocol is designed considering that packets arrive error free at destination only when one terminal transmits. Otherwise, it is assumed that if there are more than one simultaneous transmissions, packets collide and are lost. Recently, as consequence of the great advances in signal processing techniques in the last two decades, MAC designers have realized that the classical
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.09.031
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
collision channel model is no longer an appropriate model for wireless communications. First, channel effects such as fading and noise are not considered and second, it does not consider that, with the use of advanced receivers, the successful reception of multiple simultaneous packets is possible. The reception of multiple simultaneous packets at the PHY layer is called multipacket reception (MPR). Such MPR capability of the receiver is achieved by means of applying some kind of diversity (code, frequency or space) and using a multiuser detector. In a multiuser detector, interference from other users or the so-called multi-access interference (MAI) can be reduced and then, some information can be recovered from the collision of packets. The use of multiuser detectors for interference cancelling in wireless networks or in digital subscriber line (DSL) systems has been widely studied [1,2]. Consequently, with the aim of going further in modelling layers and their interaction, the concept of cross layer (CL) design has appeared [3–5]. The idea is to achieve optimal performance in one layer of the communications system by allowing it to be aware of some parameters or characteristics of the others. A PHY layer model that accommodates the MPR capability of the receiver is called a MPR channel model. In general, the MPR channel model is based on the definition of a MPR matrix C. Each element of this matrix, cK;k is the probability of successfully receiving k packets when K packets have been sent. Assuming some statistical independency between packets and terminals, these probabilities can be obtained from the bit error rate (BER) and binomial distributions. Basically, cK;k is a measure of the trade-off between the number of simultaneous transmissions and the MAI created among them. Therefore, in a system with MPR capability, the optimal MAC strategy is no longer to avoid collisions as much as possible (as in the classical collision channel model) but instead, to control the number of simultaneous transmissions in order to keep MAI under an optimal threshold. Some recent articles in the literature present new MPR MAC protocols that make use of the MPR channel model [6–10]. The work in [6] is perhaps the first to introduce the concept of MPR matrix. Optimal transmission probabilities are obtained for an Aloha random access system under the MPR channel model. Besides, in [7], a cross-layer centralized approach is described. Assuming that each terminal has a packet with probability qi and considering the MPR receiver matrix, an optimal
1835
terminal access set aiming packet delay minimization and throughput maximization is obtained. However, with the use of the information on the MPR matrix only, the PHY–MAC cross-layer information reduces to a BER information exchange. In [8], the idea of [7] is used to show that the use of additional information such as the knowledge of the active terminals, can further improve system performance. An example of MPR MAC protocol for ad-hoc networks is presented in [9]. The MPR channel model as presented in [6–9] does not contemplate heterogeneity among terminals. Hence, we will call this model as homogeneous MPR channel model. In a packet oriented system, heterogeneity among terminals implies for example, different channel fadings, different transmission rates or different traffic sources. In [10], asymmetric traffic sources are considered. The stability region of a two-user Aloha random access system with MPR receivers is found. Furthermore, an inner bound for the asymmetrical N-user case is also given. Heterogeneity between terminals due to channel fading is commonly known as multiuser diversity (MUD). Although, channel fading was traditionally considered as a disadvantage rather than an advantage, advantages of exploiting multiuser diversity in multiple access channels were introduced in [11]. MUD is based on the idea that in multiple access channels if access is given to the terminal whose SNR is the greatest, then, the information-theoretic capacity is maximized. Hence, in a wireless system the channel state is estimated before transmitting user data and, according to such ‘‘instantaneous’’ channel state information, the best user’s data are scheduled. In [12], MUD scheduling is considered under fairness constraints and, in very slow fading channels, multiuser-diversity is created by means of dumb antennas. In [13], four MUD scheduling strategies are analyzed considering different criteria such as fairness and reduced feedback information. Such information–theoretic approaches are feasible in homogeneous systems where all terminals use the same modulation format, channel coding scheme, data transmission rate, and packet length. In homogeneous systems, the selection of the terminal with highest SNR leads to the highest system throughput. Unfortunately, in heterogeneous systems, one might find that the user with highest SNR does not lead to highest system throughput in general [14]. Once again, a cross-layer analysis becomes necessary.
ARTICLE IN PRESS 1836
M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
In [15], the idea of using the channel state information to vary the transmission probability in an Aloha random access systems was proposed. Although the system analysis was based on the collision channel, a cross-layer approach was intrinsically taken because multiuser diversity was combined to optimize network throughput and delay. Multiuser diversity in Aloha random access systems with MPR receivers is evaluated in [16]. Transmission probabilities are channel state dependant and the maximum stable throughput is obtained. However, the channel model presented in [16] is specific for random access mechanisms. Besides, does not accommodate other degrees of heterogeneity rather than multiuser diversity. Clearly, although random access mechanisms are good in terms of system simplicity, the randomness in the number of terminals accessing the channel implies that MAI cannot always be controlled. Therefore, instantaneous heterogeneity among terminals cannot be fully exploited and hard QoS guarantees such as instantaneous PER are not possible. Furthermore, although in a multiple access systems with multiple antennas different algorithms might be designed in order to obtain the optimal set of simultaneous transmitting/receiving antenna that maximizes the total information-theoretic channel capacity [17,18], to the best of our knowledge, a generalized MPR channel model for cross-layer scheduling in MIMO multiple access systems have not been yet presented in the literature. Only in [19,20] different cross-layer scheduling approaches for MIMO multiple access systems are presented. In [19], an average scheduling policy was considered. Heterogeneity due to multiuser diversity is averaged over the terminals and over the channel statistics and then, an optimal number of terminals that should be scheduled simultaneously is obtained. Contrary to [19], in [20], scheduling policies intended to fully exploit instantaneous multiuser diversity are studied. Emphasis on guaranteeing hard QoS is put in [20]. Consequently, one might rise the following questions. Is there any generalized MPR channel model to evaluate scheduling policies in MIMO multiple access systems? Apart from multiuser diversity, is it possible to consider other degrees of heterogeneity? Which kind of cross-layer scheduling policies can be used in heterogeneous systems? Is it possible to guarantee hard QoS? The work presented in this paper is intended to throw light on answers of these questions. Although focused on a
MIMO multiple access system with cross-layer scheduling policies, extrapolation to random access systems or to the broadcast channel is straightforward. This paper is structured as follows. In Section 2, the signal model for a MIMO multiple access system is presented. Then, in Section 3, a generalized MPR channel model is described with examples that particularize such generalized channel model to well-known channel models. Different scheduling strategies are shown in Section 4 and an application example is given in Section 5. The paper finalizes with simulations in Section 6 and conclusions in Section 7. 2. Signal model Let us consider a MIMO multiple access channel as in Fig. 1. N terminals (potentially sensors) with one antenna each wish to communicate with another terminal, namely receiver terminal (RT), provided with M antennas. Time axis is divided into slots and transmissions of different terminals are synchronized. Assume that a set k of simultaneous transmissions takes place and that the cardinality of k is jkj. Given the set k, the following is a model for this multiaccess antenna-array communication link: yk ¼ H k s k þ w k .
(1) T
In (1), sk ¼ ðs1 ; s2 ; . . . ; sjkj Þ is the transmitted symbol vector where si is the transmitted symbol from the ith terminal in the set k which is transmitted with a power P, Hk ¼ ðh1 ; h2 ; . . . ; hjkj Þ is M jkj flat–fading channel matrix where the scalar hj;i represents the fading suffered by the ith transmitter in the set k at the jth receiver antenna. Note that Hk is a matrix that gathers the columns in HMN corresponding to the terminals in the set k. The vector wk is a complex-valued, background Gaussian noise with zero mean and variance s2w . Previous to the transmission of packets, each terminal must receive access to the channel through a scheduling mechanism. Particularly, by means of the so-called feedback channel, the RT is able to schedule (or poll) the terminals that are allowed to access the channel. The feedback channel consists of a N bit feedback multicast sequence (FMS) where the ith position (FMSi ) indicates whether the ith terminal is allowed to transmit (FMSi ¼ 1) or not (FMSi ¼ 0). Instantaneous perfect feedback channel is assumed. As we will see, information conveyed
Fig. 1. MIMO multiple access channel with M receiving antennas and N transmitting terminals.
M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
ARTICLE IN PRESS 1837
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1838
in the FMS sequence might be determined by means of different scheduling policies. Besides, different cross-layer information will be used throughout the paper in order to improve PHY and MAC layers performance. In Fig. 1, such additional cross-layer information is shown. In the next sections we will explain why the MAC layer allows the PHY layer to know the set of terminals that are scheduled and why the PHY layer provides information to the MAC layer regarding the receiver architecture, the set of active transmitters and the channel state H: Of course, PHY and MAC layers exchange packets sent and received at the RT, i.e., FMS packets and data packets. 3. Generalized MPR channel model Consider a system as described in Section 2. Then, given a channel realization H (which in turn defines Hk ) and a set k of simultaneous transmitters, the post-detection instantaneous SNIR (gi ) for the ith transmitting terminal in the set k is defined by g gi ¼ ai ðHk Þ, 2
(2)
P=s2w
where g ¼ denotes the average SNR at the reception. The noise enhancement factor ai ðHk Þ can be seen as a PHY layer quality measurement that accounts for both, channel fading and MAI through receiver implementation. Define the set aðHk Þ ¼ fai ðHk Þ : i 2 kg and the set aH ¼ faðHk Þ : k Pf1; . . . ; Ngg where Pf1; . . . ; Ng defines the set of partitions of f1; . . . ; Ng. From (2), we can obtain a value for the instantaneous BER, by using the approximate expression presented in [21]: g BERaðHk Þ ðiÞ ’ C 1 exp C 2 ai ðHk Þ , (3) 2 where constants C 1 and C 2 are modulation dependant. For instance, for a QPSK, C 1 ¼ 0:2 and C 2 ¼ 7=ð23:8 þ 1Þ. Further, consider that packets in the system are Pl bits long. Then, the instantaneous packet success rate (PSR) for the ith transmission in the set k or, equivalently, the probability of successfully receiving a packet from the i th transmitter in k is defined by r X Pl PSRaðHk Þ ðiÞ ¼ ðBERaðHk Þ ðiÞÞm m m¼0 ð1 BERaðHk Þ ðiÞÞPl m .
ð4Þ
In (4), we have considered the use of harddecision decoding of perfect linear binary block codes where r is the number of correctable errors in a packet. However, expression (4) could be modified to consider other codification techniques as shown in [14] or further simplified in the case that we consider no channel coding and short packets (as it could be the case in sensor networks). Then, for example, PSR could also be computed as PSRaðHk Þ ðiÞ ¼ ð1 BERaðHk Þ ðiÞÞPl .
(5)
The important idea behind (4) and (5) is to assume a PSR function that depends on Hk , or equivalently, on the noise enhancement set aðHk Þ. However, we could further consider terminals using different modulation formats, channel coding schemes, transmission rates or packet lengths, leading to different PSRaðHk Þ ðiÞ. Then, the average number of packets successfully received per timeslot when all terminals in k transmit is X C aH ðkÞ ¼ RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ, (6) i2k
where RðiÞ is the transmission data rate of the ith terminal expressed in packets/slot. For simplicity assume RðiÞ 2 N. Note that different RðiÞ among terminals implies also different PSRaðHk Þ ðiÞ among them. Either because the use of different modulation formats (in systems using link adaptation) or because the use of different channel bandwidths (in multicarrier systems). At this point, it is worth remarking that C aH ðkÞ depends on the channel realization H through the corresponding noise enhancement sets aðHk Þ. Particularly, C aH ðkÞ gives a measure of the tradeoff between the number of simultaneous transmissions and the MAI. Clearly, the higher the number of terminals in k, i.e. the higher the jkj, the higher the C aH ðkÞ. In the opposite direction, usually, higher the number of terminals in k, higher the amount of MAI and consequently, lower the C aH ðkÞ. How C aH ðkÞ depends on k is given by the set aH . Using the definition of FMS given in Section 2, let K ¼ fi : FMSi ¼ 1g define the set of terminals scheduled at a given timeslot and, as before, k K define the set of active terminals, i.e., the set of terminals that simultaneously transmit a packet. Then, for a given K, the average number of packets
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1839
3.1. The collision channel
successfully received per timeslot is X X ZaH ðKÞ ¼ pKk RðiÞPSRaðHk Þ ðiÞ
where pKk denotes the probability that all terminals in k transmit when access to the channel is given to the terminals in K. Obviously, if kD / K, then, pKk ¼ 0. Assume the channel process to vary in an i.i.d. manner according to some stationary distribution pH where H is the set of all possible states of H. The channel coherence time is T and hence, channels hold their state for timeslots of length T with transitions occurring at slot boundaries t ¼ nT. Define a scheduling policy SH such that
The collision channel is the classical channel model that has been used for decades to model the channel in the design of MAC and network protocols for communication systems. The basic principle of this channel model is that if two or more packets are sent through the radio channel simultaneously these packets collide and consequently, information is lost. This model could be used, for instance, in the case that the RT of the system in Fig. 1 had only one antenna (M ¼ 1). Clearly, this model implies that when there is more than one simultaneous transmission, the SNIR is 0. Then, the ith element in aðHk Þ is ( ai ðhk Þ for k 2 f1; . . . ; Ng; ai ðHk Þ ¼ (10) 0 for jkj41
SH
and
kK
¼
X
i2k
pKk C aH ðkÞ ðpackets=slotÞ,
ð7Þ
kK
¼
(
2N
sH 2 ½0; 1
:
X
) sH ðKÞ ¼ 1; H 2 H .
KPf1;...;Ng
ð8Þ Note that sH is a vector with entries sH ðKÞ defining the proportional amount of time a set K of terminals is scheduled for a given channel realization H, or equivalently, the probability that a set K of terminals is scheduled for a given channel realization H. Since K Pf1; . . . ; Ng there are 2N different possible sets. The total system throughput is computed as X X ZðSH Þ ¼ pH sH ðKÞZaH H
KPf1;...;Ng
ðKÞ ðpackets=slotÞ.
ð9Þ
The probability pKk in (7) might be either constant or a function of the traffic characteristics of the terminals, input rates and departure rates. For example, pKk could represent the stationary probabilities of a Markov chain that models the evolution of terminals’ buffer size. Note that in this case, our model implies that the channel coherence time, T, must be long enough for the system to reach its stationary regime at each channel state. In the next subsections, we will present some examples to show that with the proper definition of the noise enhancement sets aðHk Þ, the general MPR channel model presented in this section can be particularized to model the multiple access channel in any kind of communications system.
C aH ðkÞ ( ¼
R PSRaðhk Þ
for k 2 f1; . . . ; Ng
0
for jkj41 ðpackets=slotÞ.
ð11Þ
3.2. The orthogonal channel The orthogonal channel is the opposite to the collision channel. The basic principle of this channel model is that the channel can be viewed as a set of jkj parallel channels where terminals do not interfere with one other. This means that the noise enhancement suffered by the ith terminal in k, only depends on its own channel realization hi . Then, the ith element in aðHk Þ is ai ðHk Þ ¼ ai ðhi Þ and X C aH ðkÞ ¼ RðiÞPSRaðhi Þ ðiÞ ðpackets=slotÞ. i2k
3.3. The homogeneous MPR channel Recently, with the aim of constructing a better channel model for wireless systems with MPR receivers, the homogeneous MPR model has appeared as a simple interesting model. Basically, this homogeneous model simplifies channel and interference among users by considering that all users suffer from the same channel fading and hence, the MAI is only dependant on the number of simultaneous transmissions jkj, and not on the transmitters
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1840
in k. Consequently, as the MAI does not depend on the channel, ai ðHk Þ ¼ ajkj . Then PSRaðHk Þ ðiÞ ¼ PSRajkj ðiÞ and, assuming that all terminals use the same modulation format, channel coding scheme, transmission rate R, and packet length, we have C aH ðkÞ ¼ C ajkj ðjkjÞ ¼ jkjR PSRajkj ðpackets=slotÞ. (12) 3.4. The zero forcing MPR channel The channel models presented above are somewhat simplifications of the real channel model in a wireless communication system provided with multiple antennas. Let us particularize the generalized MPR channel model to the system presented in Fig. 1 with a zero forcing (ZF) receiver. Assume then that during transmission of packets, a ZF beamforming is performed at the RT by means of the M antennas and that the channel matrix H is completely known at the RT. The following considerations have to be done: (a) Since access to the channel of terminals in the set K is given through the FMS sequence, if no additional information is provided, at the RT the set of k K terminals that simultaneously transmit is unknown. Then, the ZF receiver at the RT will null all the interferences in the set K by considering the channel matrix HK . (b) Due to the constraints on the ZF receiver, successful reception of packets is only possible when jKjpM. Then, ai ðHK Þ can be easily shown to be given by 8 2 > < for jKjpM; 1 H ai ðHK Þ ¼ ½ðHK HK Þ ii (13) > :0 for jKj4M 1 with ½ðHH K HK Þ ii defining the ith element of the 1 diagonal of ðHH K HK Þ . Then, C H ðkÞ depends on the set K through
C aH ðkÞ ¼ C aH ðK; kÞ X ¼ RðiÞPSRaðHK Þ ðiÞ ðpackets=slotÞ.
ð14Þ
i2k
Improving ZF performance via cross-layer: Clearly, from (13), the ZF performance is drastically
reduced in the case that not all the expected transmitting terminals do, in fact, transmit a packet, i.e., when kiK. This problem can be solved by introducing previous to the ZF, an additional stage that detects the set of k active transmissions. This is fully described in [22] and references therein. Notice that, as shown in Fig. 1, cross-layer information regarding the terminals in K would facilitate the detection of terminals in k. We will assume that the set of active transmissions k K is detected without error. Then, given H, 8 2 > < for jkjpN; 1 H (15) ai ðHk Þ ¼ ½ðHk Hk Þ ii > :0 for jkj4N and C aH ðkÞ is as in (6). 4. Scheduling policies At each timeslot, the FMS sequence is constructed in order to schedule (or poll) a set of terminals. The process that decides the set K of terminals to be scheduled is governed by the scheduling policy SH (see expression (8)). Such scheduling policy determines the proportional amount of time that every set of terminals has to be served or equivalently, the probability of every set of terminals to be scheduled. In this section we will focus on analyzing three different scheduling policies. First, we will present the optimal scheduling policy, based on maximizing the average number of successfully received packets at each channel state. Second, we will analyze a so called QoS scheduling policy that selects terminals following a minimum SNIR criterion and finally, we will concentrate on the less optimal but simplest policy, called the average scheduling policy. This policy is based on averaging the MAI over the channel and hence, models the channel as an homogeneous MPR channel model. Although this is not the main motivation of the section, comments on fairness are also presented. 4.1. The optimal scheduling policy If we aim system throughput maximization, the optimal scheduling policy Sop H is such that Sop H ¼ arg maxðZðSH ÞÞ. SH
(16)
Obviously, the maximum is achieved if, at each channel state, the average number of successfully
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
received packets is maximized. Then, Sop H
(
¼
sop H
2 ½0; 1
2M
:
sop H ðKÞ ¼ 1
if K ¼ Kop ; H 2 H
sop H ðKÞ ¼ 0
if KaKop ; H 2 H
) ,
ð17Þ where Kop ¼ arg
that now the main system requirement is to guarantee all transmissions to be over a desired SNIR threshold, or equivalently, over a ath (see expression (2)). Then, our scheduling policy is determined by such parameter ath as SQoS H ¼
max
ðZaH ðKÞÞ.
KPf1;...;Ng
(18)
In the case that maximum in (18) is not unique, Kop will be the set with minimum cardinality among all the sets that maximize ZaH ðKÞ. Then, the total transmitted power would be minimized. If all possible channel states were known beforehand, the optimal values Kop could be computed offline. However, the set of channel states H is not typically known beforehand. Therefore, expression (18) implies an online exhaustive search for each channel realization. Furthermore, the optimal scheduling policy maximizes system throughput by exploiting the instantaneous heterogeneity among terminals to the maximum. Hence, giving the possibility to present unfairness. If fairness was desired, scheduling probabilities in (17) should be modified accordingly at the expense of loosing in throughput performance. It is worth remarking that one should observe some similarities with the Multiuser Diversity strategies in single-user detectors presented by [11]. Particularly, here we have presented the Multiuser Diversity scheduling for a generalized MPR channel. Note that in the case that the RT does not have MPR capabilities, the optimal scheduling policy reduces to the Multiuser Diversity strategies in single-user detectors. Nevertheless, the main difference is that, in MPR channels, many simultaneous transmissions might occur and therefore, the scheduler have to choose the best compromise between number of simultaneous transmissions and MAI. Furthermore, optimization is performed by maximizing throughput rather than maximizing capacity. Hence, a cross-layer approach. 4.2. The QoS scheduling policy The scheduling policy presented above is optimal in the sense that system throughput is maximized. However, as it was highlighted, such strategy might lead to high computational costs. Furthermore, no QoS requirements are considered. Let us assume
1841
8 < :
sQoS H
2N
2 ½0; 1
:
QoS sQoS H ðKÞ ¼ PHK
9 if K 2 KQoS H ;H 2 H=
sQoS H ðKÞ ¼ 0
; if KeKQoS H ;H 2 H
,
ð19Þ where KQoS is the set that gathers all sets of H terminals that guarantee that transmissions of all terminals in each set are over ath at channel state H. That is KQoS ¼ fI : I Pf1; . . . ; Ng H s:t ai ðHI ÞXath 8i 2 Ig. If for every H 2 H, the set K 2 randomly chosen: PQoS HK ¼
1 jKQoS H j
.
ð20Þ KQoS H
is
(21)
Expression (21) says that all sets in KQoS H have the same probability to be scheduled. However, if we would like to select the set with largest cardinality, i.e., to schedule at each timeslot the maximum number of terminals that are over ath , then, 8 jIj; < 1 if jKj ¼ max QoS I2KH ¼ (22) PQoS HK : 0 otherwise: We could also have a look at the optimal QoS requirement, aop th that maximizes throughput. That is QoS aop th ¼ arg maxðZðSH ÞÞ. ath
(23)
The system throughput will be clearly affected by PQoS HK , i.e., by the algorithm implemented at the RT to select the set of terminals K 2 KQoS H . Since typically ath cannot be tuned and is a parameter fixed by upper or lower layers to achieve, for instance, application layer requirements or to determine a received signal strength indicator (RSSI), the main motivation of computing aop th is to evaluate the difference between the throughput obtained when using a QoS scheduling policy with ath ¼ aop th and the throughput obtained by means of the optimal scheduling policy. Such comparison would give a qualitative measure of the efficiency of
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1842
the selection algorithm used to select the set of terminals K 2 KQoS H .
Rayleigh fading channel and performance of the different scheduling policies is presented.
4.3. The average scheduling policy
5.1. The network
The main motivation for this average scheduling policy is to simplify the complexity of the both strategies presented before at the expense of experiencing a lower throughput performance. Basically, what is intended is to schedule, at any slot, the same number of terminals rather than, as in the optimal scheduling policy, to schedule the optimal set of terminals by performing an exhaustive search. To this end, the average scheduling policy is such that
Assume that our problem is the information retrieval of a large network of sensors. For that purpose, consider transmitting terminals depicted in Fig. 1 to be sensors. By means of the FMS sequence, sensors are synchronized and scheduled (or polled) periodically. We further assume our sensors to be environmental sensors that send real-time information related to the temperature, the sea level or rain intensity of a given area. For instance, assume that a group of temperature sensors are spread over an area and we want to monitor the real-time temperature of the area. Hence, the RT terminal is in charge of scheduling the sensors periodically in order to get as much real-time information as possible. We further assume that when a sensor is scheduled, this sensor only sends information if temperature is over a designed threshold. One main property of this network is that old information is not necessary and only information regarding the current temperature is needed. This allows both, to keep sensors very simple because buffering of information is not necessary and to save energy in the sense that sensors only transmit information upon request.
Sav ¼
8 > > < > > :
sav 2 ½0; 12
M
:
sðKÞ ¼
1
N jKj
sðKÞ ¼ 0
if jKj ¼ K
av
9 > > =
if jKj ¼ K
av
> > ;
,
(24) where K av
8 > > <
19 > > X X B 1 C= C ¼ min arg maxB p Z ðIÞ H aH A> . > jKj @ N > > : ; jKj IPf1;...;Ng H 0
s:t: jIj¼jKj
(25) av
In (25), K determines the optimal set cardinality that maximizes the system throughput averaged over all sets with same cardinality. Therefore, the average scheduling policy determines that, with equal probability, sets with such optimal cardinality are scheduled. Note that (24) ensures fairness among terminals. However, if priority among terminals was desired, different probabilities in (24) could be designed subject to X sðIÞ ¼ 1. (26) IPf1;...;Ng s:t: jIj¼Kav
An example in Section 5 shows that the average scheduling policy models the channel as an homogeneous MPR channel model. 5. An Example: the sensor network The motivation of this section is to give practical insight of the theory presented in previous sections. First, a network of sensors is presented as a case of study and a traffic model for this network is given. After that, the ZF MPR channel model is particularized when the entries of H correspond to a
5.2. The traffic model Given a set of sensors K, the probability of a set of sensors k K to simultaneously transmit information is only a function of environmental characteristics. Assume that at a given timeslot n 1, the probability of a sensor i to transmit a packet, i.e., the probability that temperature is over the designed threshold is pi ðn 1Þ. Then, at timeslot n, the probability of this terminal to transmit a packet is updated as pi ðnÞ ¼ bIðn 1Þ þ ð1 bÞpi ðn 1Þ,
(27)
where b 2 ½0; 1 is a parameter of design and Iðn 1Þ 2 ½0; 1 is a function of the information collected in previous slots. For instance, if a sensor that is scheduled does not transmit any information, the probability of this sensor to transmit data is decreased. On the other hand, when information of a sensor is received, the transmitting probability is updated regarding to this information. Furthermore, cross-layer information could also
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
be exploited in this case. Imagine the situation where a sensor transmits data but this is lost, if information of active users is passed from the PHY layer to the MAC layer (as shown in Fig. 1) the MAC layer knows that the temperature sensed by this sensor is at least over a given threshold and then, pi ðn 1Þ could be estimated accordingly. Following notation in Section 3: Y Y pKk ðnÞ ¼ pi ðnÞ ð1 pi ðnÞÞ. (28) i2k
i2K iek
1843
otherwise. Expression (9) can be rewritten as X X X op Þ ¼ p s ðKÞ pKk ZðSop H H H H
X
kK
KPf1;...;Ng
RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ
or equivalently X ZðSop HÞ ¼
X
pKk
KPf1;...;Ng kK
X
RðiÞ
X
i2k
pH sop H ðKÞ
H
PSRaðHk Þ ðiÞ ðpackets=slotÞ.
Furthermore, it makes sense to consider that, if sensors are in the same area, the information collected is a group of samples of the temperature in the area. Hence, all sensors will evolve according to this collected information and the traffic model can be simplified by defining pjKjjkj as the probability that jkj terminals transmit when jKj terminals are scheduled. That is ! jKj pjKjjkj ðnÞ ¼ ð pðnÞÞjkj ð1 pðnÞÞjKjjkj , (29) jkj
ð31Þ
i2k
ð32Þ
The optimal scheduling policy takes scheduling decisions according to ZaH ðKÞ (see expression (17)). Then, saying that sop H ðKÞ depends on the channel state H is equivalent to saying that sop H ðKÞ depends op on aH and therefore, sop ðKÞ s aH ðKÞ. If we H assume that elements in aðHk Þ are i.i.d distributed according to f jkj ðaÞ, expression (32) is rewritten as Z 1 X X X Þ ¼ p RðiÞ sop ZðSop Kk aH ðKÞ H KPf1;...;Ng kK
i2k
0
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð33Þ
where pðnÞ is updated as in (27) but in this case, Iðn 1Þ is related to the information received from all the sensors. For simplicity, in what follows, we will consider that pjKjjkj ðnÞ is not updated and then, pjKjjkj ðnÞ ¼ pjKjjkj . 5.3. The channel model We consider that an improved ZF receiver is used at the RT and that expression (15) applies for the computation of different values of ai ðHk Þ. Therefore, if we consider a Rayleigh fading channel, i.e., the entries of H are independent and identically distributed complex Gaussian random variables with zero mean and unit variance, ai ðHk Þ is a weightened Chi-Square distributed variable with 2ðM jkj þ 1Þ degrees of freedom and probability density function (p.d.f.) f jkj ðaÞ ¼
an=21 expð a2Þ 2n=2 Gðn2Þ
,
(30)
where n ¼ 2ðM jkj þ 1Þ and Gðn=2Þ stands for Gamma function.
5.5. The QoS scheduling policy In general, many different selection algorithms could be applied to construct a FMS sequence such that all terminals with FMSi ¼ 1 have a guaranteed minimum SNIR. Since finding the optimal selection algorithm is beyond the scope of this paper, let us describe a simple selection algorithm, called random selection algorithm: (1) Set FMS ¼ f1; . . . ; 1g and K ¼ fi 2 f1; . . . ; Ng : FMSi ¼ 1g. (2) Obtain H ¼ ½. . . ; hi ; . . .. (3) According to H, compute ai ðHK Þ for i 2 K. (4) If minimum ai ðHK Þ is over ath . Go to step 6. (5) Else, set FMSi ¼ 0 i being the index corresponding to a terminal randomly chosen from those in K. Set K ¼ fi 2 f1; . . . ; Ng : FMSi ¼ 1g and go to step 3. (6) Send FMS.
5.4. The optimal scheduling policy
According to this random selection algorithm 0 1 jKjþ1 QoS Y NðjJj; K Þ 1 @ H A , PQoS 1 (34) HK ¼
At each channel realization H, a FMS sequence is built such that FMSi ¼ 1 if i 2 Kop and FMSi ¼ 0
where NðjJj; KQoS H Þ is the number of sets of cardinality jJj in KQoS H . Note that the random
jJj¼N
N jJj
N jKj
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1844
selection of terminals creates independency among two successive trials. Hence, PQoS HK is the probability mass function of a geometric random variable where the probability of the desired outcome to N occur at the ith trial is Nði; KQoS H Þ=ð i Þ because any QoS of the Nði; KH Þ sets can be selected. However, at the last trial the probability of success is 1=ðNi Þ because only one set among those with cardinality i is the desired outcome.
Then
(39) where
X
X
RjKjjkj ¼
X
2 RðiÞ4
jJPf1;...;Ng i2k s:t: jJj¼jKj s:t: jjj¼jk
Z
jKj1 Y
!
ath
f jJj ðaÞ da 0
jJj¼N
3R 1 f ðaÞ da PSRaðH Þ ðiÞf jkj ðaÞ da jKj a 5 ath R k th 1 N jKj a f jkj ðaÞ da jKj jkj R1
kK
KPf1;...;Ng
pjKjjkj RjKjjkj ðpackets=slotÞ,
jKj¼1 jkj¼1
Expression (9) can be rewritten as X X X QoS ZðSQoS Þ ¼ p s ðKÞ pKk H H H H
jKj N X X
ZðSQoS H Þ ¼
RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ
th
i2k
ð40Þ
ð35Þ or equivalently ZðSQoS H Þ ¼
X
X
pKk
X
KPf1;...;Ng kK
RðiÞ
X
i2k
and in the case that there was no heterogeneity in terms of modulation format, channel coding scheme, transmission rate R, and packet length
pH sQoS H ðKÞ
RjKjjkj
H
PSRaðHk Þ ðiÞ ðpackets=slotÞ.
ð36Þ
As for the optimal scheduling policy, the elements in aðHk Þ are i.i.d distributed according to f jkj ðaÞ. Furthermore, due to the random selection of terminals, at each iteration of the algorithm there is no a-posteriory information related to aðHk Þ. Then, the a-posteriory p.d.f. is equal to f jkj ðaÞ. Therefore, considering (34), X pH sQoS H ðKÞ H jKj1 Y
¼
jJj¼N
Z
!R1
ath
f jJj ðaÞ da 0
ath
f jKj ðaÞ da . N jKj
¼
X
X
X
2
RðiÞ4
i2k
R1
ath
R1
jKj1 Y jJj¼N
Z
!
ath
¼ jkjR R1
ath
jJj¼N
!Z f jJj ðaÞ da
0
#
1
f jKj ðaÞ da ath
PSRaðHk Þ f jkj ðaÞ da R1 . a f jkj ðaÞ da
ð41Þ
th
Note that, as described in Section 4.2, expression (39) allows us to evaluate the throughput efficiency of the random selection algorithm by computing throughput when ath ¼ aop th . 5.6. The average scheduling policy
pKk
KPf1;...;Ng kK
jKj1 Y Z ath
ð37Þ
Hence, applying conditional probabilities ZðSQoS H Þ
"
f jJj ðaÞ da 0
3 f jKj ðaÞ da 5 N jKj
At each timeslot, the FMS sequence is constructed such that the number of scheduled terminals is K av where K av is computed as in (25). To compute K av , ZaH ðKÞ is averaged over the channel and over all sets of cardinality jKj. Then, the channel model is approximated to an homogeneous channel model. Applying (24) to (9) X X X 1 ZðSH Þ ¼ N pKk RðiÞ K av KPf1;...;Ng kK s:t: jKj¼K av
X
i2k
pH PSRaðHk Þ ðiÞ ðpackets=slotÞ.
ð42Þ
H
PSRaðH Þ ðiÞf jkj ðaÞ da R k ðpackets=slotÞ. ð38Þ 1 jKj f ðaÞ da jkj a jkj
ath
th
The last term in (42) is an average of PSRaðHk Þ ðiÞ over all channel states. Then, if ai ðHk Þ is distributed according to pjkj ðaÞ, expression (42) is
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
rewritten as
Throughput Vs. Packet Transmission Probability 3.5
ZðSH Þ K av X
X
1
X
X
pKj N i2j K av KPf1;...;Ng jkj¼1 jK s:t: jjj¼jkj s:t: jKj¼K av Z 1
RðiÞ
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð43Þ
0
Operating ZðSH Þ ¼
K av X
pK av jkj
K av jkj
jkj¼1
Z
1
X
X
0
2.5 2 1.5 1 0.5
RðiÞ
0 0.2
jKPf1;...;Ng i2j s:t: jKj¼K av s:t: jjj¼jkj
1
M=1 M=2 M=4
3 Throughput (packets/slot)
¼
1845
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð44Þ
Throughput Vs. SNIR threshold 1.5
pK av jkj RK av jkj ðpackets=slotÞ,
jkj¼1
where ( K av ¼ min arg max jKj
jKj X
!) pjKjjkj RjKjjkj
p=0.7 p=1
(45)
.
(46)
jkj¼1
In the case that there was no heterogeneity in terms of modulation format, channel coding scheme, transmission rate R, and packet length, Z 1 PSRaðHk Þ f jkj ðaÞ da. (47) RjKjjkj ¼ jkjR 0
Throughput (packets/slot)
ZðSH Þ ¼
1
Fig. 2. Throughput vs. p obtained when using the optimal scheduling policy and M ¼ 1; 2 and 4.
Then, the throughput under an average scheduling policy is K av X
0.5 0.7 Packet Transmission Probability (p)
1.25 1 0.75 0.5 0.25 0
0
5
10 15 SNIR Threshold (dB)
20
6. Simulations
Fig. 3. Throughput vs. SNIR threshold when the QoS scheduling s used M ¼ 4 and p ¼ ½0:7 1.
A network as described in Section 5 is simulated. The number of sensors in the network is 10 (N ¼ 10) and it has been considered that the RT is provided with M ¼ 1; 2 and 4 antennas. The MPR receiver is an improved MPR ZF receiver and the channel behaves as a Rayleigh fading channel. The traffic model considered is the same as in Section 5 where the probability of one terminal to transmit a packet p is changed from 0 to 1. In Fig. 2, the throughput performance of the optimal scheduling policy as a function of p is shown. It can be observed that when p ¼ 1 the optimal scheduler exploits multiuser diversity and can always schedule sensors with orthogonal
channels. Hence, the throughput obtained is M times the throughput obtained with M ¼ 1. However, if the probability of transmitting a packet decreases, the number of terminals simultaneously transmitting a packet also decreases and hence, heterogeneity among terminals (multiuser diversity) cannot fully be exploited. In Fig. 3, it is shown that the dependence on ath of the QoS scheduling policy with a random selection algorithm when p ¼ 0:7 and p ¼ 1. In both cases M ¼ 4. It is observed that it is possible to find an aop th that maximizes throughput. In Fig. 4, also for M ¼ 4 and, p ¼ 0:5, p ¼ 0:7 and p ¼ 1, it is shown the
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1846
Throughput Vs. Set Cardinality 1.4
p=0.5 p=0.7 p=1
Throughput (pasckets/slot)
1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4
1
2
3
4
Set Cardinality (|K|) Fig. 4. Throughput vs. set cardinality when the average scheduling policy is used M ¼ 4 and p ¼ ½0:5 0:7 1.
throughput obtained when the average scheduling policy is used. We observe that there is an optimal number of terminals K av that maximize the average throughput. Reasonably, in order to maintain the optimal trade-off between the number of simultaneous transmissions and the MAI, the value of K av increases as p decreases. We observe that both, the QoS scheduling policy and the average scheduling policy lead to lower throughput results compared to those obtained with the optimal scheduling policy. Interestingly enough, due to the randomness introduced in the random selection algorithm, the QoS scheduling policy with random selection algorithm leads to the same throughput as the one obtained with the average scheduling policy. However, the QoS scheduling policy guarantees hard QoS requirements. Probably, a selection algorithm as described by expression (22) would lead to better throughput results. 7. Conclusions The existence of heterogeneity among terminals in wireless networks and the need to evaluate the effects of such heterogeneity at the MAC and upper layers have motivated the generalized model for scheduling in MIMO multiple access systems presented in this paper. Following a cross-layer approach, a generalized MPR channel model that allows to evaluate system throughput is developed. Particularly, this generalized model accommodates any kind of heterogeneity from multiuser diversity
due to channel fading to heterogeneity due to different data transmission rates. Hence, some examples have been given to show that, although the model is specially designed to accommodate receivers with MPR capabilities, with a proper definition of the noise enhancement factor, it also accommodates any other channel model as for instance the collision channel model or the orthogonal channel model. Different scheduling policies have been analyzed under this generalized model. First, it has been investigated that an optimal scheduling policy based on an exhaustive search gives the maximum achievable throughput. Second, hard QoS requirements are achieved by means of the QoS scheduling policy. Furthermore, since many different scheduling policies might guarantee QoS requirements, we have provided a mechanism to evaluate how far is the QoS scheduling policy from the optimal one. Finally, the last scheduling policy investigated is the average scheduling policy. This policy provides a very simple way of scheduling by averaging the effects of heterogeneity. All these scheduling policies have been particularized for a sensor network and have been evaluated in terms of the statistics of the noise enhancement factor. The results obtained have been useful to achieve practical insight of such scheduling policies.
Acknowledgements The authors of this paper would like to specially thank Xavier Mestre Ph.D. for his valuable help.
References [1] K.-W. Cheong, W.-J. Choi, J.M. Cioffi, Multiuser soft interference canceler via iterative decoding for DSL applications, IEEE. J. Selec. Areas Comm. 20 (2) (February 2002). [2] S. Verdu´, Multiuser Detection, Cambridge University Press, Cambridge, 1998. [3] L. Tong, Q. Zhao, G. Mergen, Multipacket reception in random access wireless networks: from signal processing to optimal medium access control, Proc. IEEE Comm. Magazine (November 2001) 108–112. [4] C. Anton-Haro, M.A. Lagunas, Array processing techniques for wireless: a cross-layer perspective, in: International Forum on Future Mobile Telecommunications and ChinaEU Post Conference on Beyond 3G, 2002. [5] IEEE Signal Processing Magazine, 21(5) (September 2004). [6] S. Ghez, S. Verdu´, S.C. Schwartz, Optimal decentralized control in the random access multipacket channel, IEEE Trans. Automat. Control 34 (11) (November 1989).
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847 [7] Q. Zhao, L. Tong, The dynamic queue protocol for spread spectrum random access networks, in: Military Communications Conference, 2, 2001, pp. 1024–1028. [8] M. Realp, A.I. Pe´rez-Neira, PHY-MAC dialogue with multi-packet reception, in: ETSI Workshop on Broadband Wireless Ad-Hoc Networks and Services, 2002. [9] M. Realp, A.I. Pe´rez-Neira, LAMAN: load adaptable MAC for Ad-hoc networks, EURASIP J. Appl. Signal Process. Special Issue on Cross Layer Design for Communications and Signal Processing Systems 2 (February 2005) 116–128. [10] V. Naware, G. Mergen, L. Tong, Stability and delay of finite user slotted ALOHA with multipacket reception, IEEE Trans. Inform. Theory 51 (7) (July 2006). [11] R. Knopp, P.A. Humlet, Information Capacity and Power Control in Single-Cell Multiuser Communications, in: IEEE International Conference on Communications (ICC), June 1995. [12] P. Viswanath, D.N.C. Tse, R. Laroia, Opportunistic beamforming using dumb antennas, IEEE Trans. Inform. Theory 48 (6) (June 2002). [13] F. Flore´n, O. Edfors, B.-A. Molin, Throughput analysis of three multiuser diversity schemes, in: Proceedings of the Vehicular Technology and Communications (VTC), April 2003. [14] M. Realp, A.I. Pe´rez-Neira, C. Mecklenbra¨uker, Cross-layer approach to multi-user diversity in heterogeneous wireless systems, in: IEEE International Conference on Communications (ICC 2005), May 2005.
1847
[15] X. Qin, R. Berry, Exploiting multiuser diversity in wireless ALOHA networks, in: Proceedings of the Allerton Conference Communication, Control and Computing, October 2001, pp. 793–794. [16] S. Adireddy, L. Tong, Exploiting decentralized channel state information for random access, IEEE Trans. Inform. Theory 51 (2) (February 2005) 537–561. [17] M. Gharavi-Alkhansari, A.B. Gershman, Fast antenna subset selection in MIMO systems, Trans. Signal Proc. 52 (2) (February 2004). [18] A. Gorokhov, Antenna selection algorithms for MEA transmission systems, in: International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2002. [19] M. Realp, A.I. Pe´rez-Neira, Generalized multipacket reception model for heterogeneous networks, in: 12th European Signal Processing Conference (EUSIPCO 2004), September 2004. [20] M. Realp, A.I. Pe´rez-Neira, Cross-layer MAC scheduling for multiple antenna systems, in: IEEE Global Telecommunications Conference (Globecom 2004), December 2004. [21] S.T. Chung, A.J. Goldsmith, Degrees of freedom in adaptive modulation: a unified view, IEEE Trans. Comm. 49 (9) (September 2001). [22] B. Chen, L. Tong, Traffic-aided multiuser detection for random-access CDMA networks, IEEE Trans. Signal Proc. 49 (7) (July 2001).
Signal Processing 86 (2006) 1834–1847 www.elsevier.com/locate/sigpro
Generalized model for scheduling in MIMO multiple access systems: A cross-layer approach$ Marc Realpa,, Ana I. Pe´rez-Neirab a
Telecommun. Technological Center of Catalonia (CTTC), Barcelona (Catalonia-Spain) b Technical University of Catalonia (UPC), Barcelona (Catalonia-Spain)
Received 3 January 2005; received in revised form 15 September 2005; accepted 15 September 2005 Available online 22 December 2005
Abstract Research on scheduling in MIMO multiple access systems is traditionally carried out from an information-theoretic point of view. Such information-theoretic approaches are feasible in homogeneous systems where, although channel fading is assumed, all terminals use the same modulation format, channel coding scheme, data transmission rate, packet lengths and no traffic issues are considered. In these systems, to select the set of terminals that maximize the instantaneous sum capacity leads to the highest system throughput. Unfortunately, in heterogeneous systems this assumption is not always true. Consequently, there is a need to evaluate the effects of such heterogeneity from a network point of view. In this paper, following a cross-layer approach, a generalized model for scheduling in MIMO multiple access is presented. Such model allows to evaluate the effects of heterogeneity on system throughput. Particularly, this generalized model accommodates any kind of heterogeneity from multiuser diversity due to channel fading to heterogeneity due to different data transmission rates. Different scheduling policies are analyzed under this generalized model and further applied to a sensor network. The results obtained are useful to achieve practical insight of such scheduling policies. r 2005 Elsevier B.V. All rights reserved. Keywords: Cross-layer; Scheduling; Multiuser diversity; MAC; MIMO
1. Introduction In an OSI layered packet oriented communications system, the medium access control (MAC) layer is the one in charge of controlling access of terminals to the medium in order to optimize system $
This work is supported by the Spanish government under TIC2002-04594-C02 and FIT-070000-2003-257 and by the IST European Commission under FP6-507572-WIDENS and FP6507525-NEUCOM and jointly financed by FEDER. Corresponding author. Tel.: +34 936 452 912; fax: +34 936 452 901. E-mail addresses: [email protected] (M. Realp), [email protected] (A.I. Pe´rez-Neira).
performance in terms of packet throughput, delay and jitter and to guarantee quality of service (QoS) requirements such as instantaneous packet error rate (PER). From a MAC layer point of view, the PHY layer has been typically modeled by the collision channel model. Then, a typical MAC protocol is designed considering that packets arrive error free at destination only when one terminal transmits. Otherwise, it is assumed that if there are more than one simultaneous transmissions, packets collide and are lost. Recently, as consequence of the great advances in signal processing techniques in the last two decades, MAC designers have realized that the classical
0165-1684/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2005.09.031
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
collision channel model is no longer an appropriate model for wireless communications. First, channel effects such as fading and noise are not considered and second, it does not consider that, with the use of advanced receivers, the successful reception of multiple simultaneous packets is possible. The reception of multiple simultaneous packets at the PHY layer is called multipacket reception (MPR). Such MPR capability of the receiver is achieved by means of applying some kind of diversity (code, frequency or space) and using a multiuser detector. In a multiuser detector, interference from other users or the so-called multi-access interference (MAI) can be reduced and then, some information can be recovered from the collision of packets. The use of multiuser detectors for interference cancelling in wireless networks or in digital subscriber line (DSL) systems has been widely studied [1,2]. Consequently, with the aim of going further in modelling layers and their interaction, the concept of cross layer (CL) design has appeared [3–5]. The idea is to achieve optimal performance in one layer of the communications system by allowing it to be aware of some parameters or characteristics of the others. A PHY layer model that accommodates the MPR capability of the receiver is called a MPR channel model. In general, the MPR channel model is based on the definition of a MPR matrix C. Each element of this matrix, cK;k is the probability of successfully receiving k packets when K packets have been sent. Assuming some statistical independency between packets and terminals, these probabilities can be obtained from the bit error rate (BER) and binomial distributions. Basically, cK;k is a measure of the trade-off between the number of simultaneous transmissions and the MAI created among them. Therefore, in a system with MPR capability, the optimal MAC strategy is no longer to avoid collisions as much as possible (as in the classical collision channel model) but instead, to control the number of simultaneous transmissions in order to keep MAI under an optimal threshold. Some recent articles in the literature present new MPR MAC protocols that make use of the MPR channel model [6–10]. The work in [6] is perhaps the first to introduce the concept of MPR matrix. Optimal transmission probabilities are obtained for an Aloha random access system under the MPR channel model. Besides, in [7], a cross-layer centralized approach is described. Assuming that each terminal has a packet with probability qi and considering the MPR receiver matrix, an optimal
1835
terminal access set aiming packet delay minimization and throughput maximization is obtained. However, with the use of the information on the MPR matrix only, the PHY–MAC cross-layer information reduces to a BER information exchange. In [8], the idea of [7] is used to show that the use of additional information such as the knowledge of the active terminals, can further improve system performance. An example of MPR MAC protocol for ad-hoc networks is presented in [9]. The MPR channel model as presented in [6–9] does not contemplate heterogeneity among terminals. Hence, we will call this model as homogeneous MPR channel model. In a packet oriented system, heterogeneity among terminals implies for example, different channel fadings, different transmission rates or different traffic sources. In [10], asymmetric traffic sources are considered. The stability region of a two-user Aloha random access system with MPR receivers is found. Furthermore, an inner bound for the asymmetrical N-user case is also given. Heterogeneity between terminals due to channel fading is commonly known as multiuser diversity (MUD). Although, channel fading was traditionally considered as a disadvantage rather than an advantage, advantages of exploiting multiuser diversity in multiple access channels were introduced in [11]. MUD is based on the idea that in multiple access channels if access is given to the terminal whose SNR is the greatest, then, the information-theoretic capacity is maximized. Hence, in a wireless system the channel state is estimated before transmitting user data and, according to such ‘‘instantaneous’’ channel state information, the best user’s data are scheduled. In [12], MUD scheduling is considered under fairness constraints and, in very slow fading channels, multiuser-diversity is created by means of dumb antennas. In [13], four MUD scheduling strategies are analyzed considering different criteria such as fairness and reduced feedback information. Such information–theoretic approaches are feasible in homogeneous systems where all terminals use the same modulation format, channel coding scheme, data transmission rate, and packet length. In homogeneous systems, the selection of the terminal with highest SNR leads to the highest system throughput. Unfortunately, in heterogeneous systems, one might find that the user with highest SNR does not lead to highest system throughput in general [14]. Once again, a cross-layer analysis becomes necessary.
ARTICLE IN PRESS 1836
M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
In [15], the idea of using the channel state information to vary the transmission probability in an Aloha random access systems was proposed. Although the system analysis was based on the collision channel, a cross-layer approach was intrinsically taken because multiuser diversity was combined to optimize network throughput and delay. Multiuser diversity in Aloha random access systems with MPR receivers is evaluated in [16]. Transmission probabilities are channel state dependant and the maximum stable throughput is obtained. However, the channel model presented in [16] is specific for random access mechanisms. Besides, does not accommodate other degrees of heterogeneity rather than multiuser diversity. Clearly, although random access mechanisms are good in terms of system simplicity, the randomness in the number of terminals accessing the channel implies that MAI cannot always be controlled. Therefore, instantaneous heterogeneity among terminals cannot be fully exploited and hard QoS guarantees such as instantaneous PER are not possible. Furthermore, although in a multiple access systems with multiple antennas different algorithms might be designed in order to obtain the optimal set of simultaneous transmitting/receiving antenna that maximizes the total information-theoretic channel capacity [17,18], to the best of our knowledge, a generalized MPR channel model for cross-layer scheduling in MIMO multiple access systems have not been yet presented in the literature. Only in [19,20] different cross-layer scheduling approaches for MIMO multiple access systems are presented. In [19], an average scheduling policy was considered. Heterogeneity due to multiuser diversity is averaged over the terminals and over the channel statistics and then, an optimal number of terminals that should be scheduled simultaneously is obtained. Contrary to [19], in [20], scheduling policies intended to fully exploit instantaneous multiuser diversity are studied. Emphasis on guaranteeing hard QoS is put in [20]. Consequently, one might rise the following questions. Is there any generalized MPR channel model to evaluate scheduling policies in MIMO multiple access systems? Apart from multiuser diversity, is it possible to consider other degrees of heterogeneity? Which kind of cross-layer scheduling policies can be used in heterogeneous systems? Is it possible to guarantee hard QoS? The work presented in this paper is intended to throw light on answers of these questions. Although focused on a
MIMO multiple access system with cross-layer scheduling policies, extrapolation to random access systems or to the broadcast channel is straightforward. This paper is structured as follows. In Section 2, the signal model for a MIMO multiple access system is presented. Then, in Section 3, a generalized MPR channel model is described with examples that particularize such generalized channel model to well-known channel models. Different scheduling strategies are shown in Section 4 and an application example is given in Section 5. The paper finalizes with simulations in Section 6 and conclusions in Section 7. 2. Signal model Let us consider a MIMO multiple access channel as in Fig. 1. N terminals (potentially sensors) with one antenna each wish to communicate with another terminal, namely receiver terminal (RT), provided with M antennas. Time axis is divided into slots and transmissions of different terminals are synchronized. Assume that a set k of simultaneous transmissions takes place and that the cardinality of k is jkj. Given the set k, the following is a model for this multiaccess antenna-array communication link: yk ¼ H k s k þ w k .
(1) T
In (1), sk ¼ ðs1 ; s2 ; . . . ; sjkj Þ is the transmitted symbol vector where si is the transmitted symbol from the ith terminal in the set k which is transmitted with a power P, Hk ¼ ðh1 ; h2 ; . . . ; hjkj Þ is M jkj flat–fading channel matrix where the scalar hj;i represents the fading suffered by the ith transmitter in the set k at the jth receiver antenna. Note that Hk is a matrix that gathers the columns in HMN corresponding to the terminals in the set k. The vector wk is a complex-valued, background Gaussian noise with zero mean and variance s2w . Previous to the transmission of packets, each terminal must receive access to the channel through a scheduling mechanism. Particularly, by means of the so-called feedback channel, the RT is able to schedule (or poll) the terminals that are allowed to access the channel. The feedback channel consists of a N bit feedback multicast sequence (FMS) where the ith position (FMSi ) indicates whether the ith terminal is allowed to transmit (FMSi ¼ 1) or not (FMSi ¼ 0). Instantaneous perfect feedback channel is assumed. As we will see, information conveyed
Fig. 1. MIMO multiple access channel with M receiving antennas and N transmitting terminals.
M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
ARTICLE IN PRESS 1837
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1838
in the FMS sequence might be determined by means of different scheduling policies. Besides, different cross-layer information will be used throughout the paper in order to improve PHY and MAC layers performance. In Fig. 1, such additional cross-layer information is shown. In the next sections we will explain why the MAC layer allows the PHY layer to know the set of terminals that are scheduled and why the PHY layer provides information to the MAC layer regarding the receiver architecture, the set of active transmitters and the channel state H: Of course, PHY and MAC layers exchange packets sent and received at the RT, i.e., FMS packets and data packets. 3. Generalized MPR channel model Consider a system as described in Section 2. Then, given a channel realization H (which in turn defines Hk ) and a set k of simultaneous transmitters, the post-detection instantaneous SNIR (gi ) for the ith transmitting terminal in the set k is defined by g gi ¼ ai ðHk Þ, 2
(2)
P=s2w
where g ¼ denotes the average SNR at the reception. The noise enhancement factor ai ðHk Þ can be seen as a PHY layer quality measurement that accounts for both, channel fading and MAI through receiver implementation. Define the set aðHk Þ ¼ fai ðHk Þ : i 2 kg and the set aH ¼ faðHk Þ : k Pf1; . . . ; Ngg where Pf1; . . . ; Ng defines the set of partitions of f1; . . . ; Ng. From (2), we can obtain a value for the instantaneous BER, by using the approximate expression presented in [21]: g BERaðHk Þ ðiÞ ’ C 1 exp C 2 ai ðHk Þ , (3) 2 where constants C 1 and C 2 are modulation dependant. For instance, for a QPSK, C 1 ¼ 0:2 and C 2 ¼ 7=ð23:8 þ 1Þ. Further, consider that packets in the system are Pl bits long. Then, the instantaneous packet success rate (PSR) for the ith transmission in the set k or, equivalently, the probability of successfully receiving a packet from the i th transmitter in k is defined by r X Pl PSRaðHk Þ ðiÞ ¼ ðBERaðHk Þ ðiÞÞm m m¼0 ð1 BERaðHk Þ ðiÞÞPl m .
ð4Þ
In (4), we have considered the use of harddecision decoding of perfect linear binary block codes where r is the number of correctable errors in a packet. However, expression (4) could be modified to consider other codification techniques as shown in [14] or further simplified in the case that we consider no channel coding and short packets (as it could be the case in sensor networks). Then, for example, PSR could also be computed as PSRaðHk Þ ðiÞ ¼ ð1 BERaðHk Þ ðiÞÞPl .
(5)
The important idea behind (4) and (5) is to assume a PSR function that depends on Hk , or equivalently, on the noise enhancement set aðHk Þ. However, we could further consider terminals using different modulation formats, channel coding schemes, transmission rates or packet lengths, leading to different PSRaðHk Þ ðiÞ. Then, the average number of packets successfully received per timeslot when all terminals in k transmit is X C aH ðkÞ ¼ RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ, (6) i2k
where RðiÞ is the transmission data rate of the ith terminal expressed in packets/slot. For simplicity assume RðiÞ 2 N. Note that different RðiÞ among terminals implies also different PSRaðHk Þ ðiÞ among them. Either because the use of different modulation formats (in systems using link adaptation) or because the use of different channel bandwidths (in multicarrier systems). At this point, it is worth remarking that C aH ðkÞ depends on the channel realization H through the corresponding noise enhancement sets aðHk Þ. Particularly, C aH ðkÞ gives a measure of the tradeoff between the number of simultaneous transmissions and the MAI. Clearly, the higher the number of terminals in k, i.e. the higher the jkj, the higher the C aH ðkÞ. In the opposite direction, usually, higher the number of terminals in k, higher the amount of MAI and consequently, lower the C aH ðkÞ. How C aH ðkÞ depends on k is given by the set aH . Using the definition of FMS given in Section 2, let K ¼ fi : FMSi ¼ 1g define the set of terminals scheduled at a given timeslot and, as before, k K define the set of active terminals, i.e., the set of terminals that simultaneously transmit a packet. Then, for a given K, the average number of packets
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1839
3.1. The collision channel
successfully received per timeslot is X X ZaH ðKÞ ¼ pKk RðiÞPSRaðHk Þ ðiÞ
where pKk denotes the probability that all terminals in k transmit when access to the channel is given to the terminals in K. Obviously, if kD / K, then, pKk ¼ 0. Assume the channel process to vary in an i.i.d. manner according to some stationary distribution pH where H is the set of all possible states of H. The channel coherence time is T and hence, channels hold their state for timeslots of length T with transitions occurring at slot boundaries t ¼ nT. Define a scheduling policy SH such that
The collision channel is the classical channel model that has been used for decades to model the channel in the design of MAC and network protocols for communication systems. The basic principle of this channel model is that if two or more packets are sent through the radio channel simultaneously these packets collide and consequently, information is lost. This model could be used, for instance, in the case that the RT of the system in Fig. 1 had only one antenna (M ¼ 1). Clearly, this model implies that when there is more than one simultaneous transmission, the SNIR is 0. Then, the ith element in aðHk Þ is ( ai ðhk Þ for k 2 f1; . . . ; Ng; ai ðHk Þ ¼ (10) 0 for jkj41
SH
and
kK
¼
X
i2k
pKk C aH ðkÞ ðpackets=slotÞ,
ð7Þ
kK
¼
(
2N
sH 2 ½0; 1
:
X
) sH ðKÞ ¼ 1; H 2 H .
KPf1;...;Ng
ð8Þ Note that sH is a vector with entries sH ðKÞ defining the proportional amount of time a set K of terminals is scheduled for a given channel realization H, or equivalently, the probability that a set K of terminals is scheduled for a given channel realization H. Since K Pf1; . . . ; Ng there are 2N different possible sets. The total system throughput is computed as X X ZðSH Þ ¼ pH sH ðKÞZaH H
KPf1;...;Ng
ðKÞ ðpackets=slotÞ.
ð9Þ
The probability pKk in (7) might be either constant or a function of the traffic characteristics of the terminals, input rates and departure rates. For example, pKk could represent the stationary probabilities of a Markov chain that models the evolution of terminals’ buffer size. Note that in this case, our model implies that the channel coherence time, T, must be long enough for the system to reach its stationary regime at each channel state. In the next subsections, we will present some examples to show that with the proper definition of the noise enhancement sets aðHk Þ, the general MPR channel model presented in this section can be particularized to model the multiple access channel in any kind of communications system.
C aH ðkÞ ( ¼
R PSRaðhk Þ
for k 2 f1; . . . ; Ng
0
for jkj41 ðpackets=slotÞ.
ð11Þ
3.2. The orthogonal channel The orthogonal channel is the opposite to the collision channel. The basic principle of this channel model is that the channel can be viewed as a set of jkj parallel channels where terminals do not interfere with one other. This means that the noise enhancement suffered by the ith terminal in k, only depends on its own channel realization hi . Then, the ith element in aðHk Þ is ai ðHk Þ ¼ ai ðhi Þ and X C aH ðkÞ ¼ RðiÞPSRaðhi Þ ðiÞ ðpackets=slotÞ. i2k
3.3. The homogeneous MPR channel Recently, with the aim of constructing a better channel model for wireless systems with MPR receivers, the homogeneous MPR model has appeared as a simple interesting model. Basically, this homogeneous model simplifies channel and interference among users by considering that all users suffer from the same channel fading and hence, the MAI is only dependant on the number of simultaneous transmissions jkj, and not on the transmitters
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1840
in k. Consequently, as the MAI does not depend on the channel, ai ðHk Þ ¼ ajkj . Then PSRaðHk Þ ðiÞ ¼ PSRajkj ðiÞ and, assuming that all terminals use the same modulation format, channel coding scheme, transmission rate R, and packet length, we have C aH ðkÞ ¼ C ajkj ðjkjÞ ¼ jkjR PSRajkj ðpackets=slotÞ. (12) 3.4. The zero forcing MPR channel The channel models presented above are somewhat simplifications of the real channel model in a wireless communication system provided with multiple antennas. Let us particularize the generalized MPR channel model to the system presented in Fig. 1 with a zero forcing (ZF) receiver. Assume then that during transmission of packets, a ZF beamforming is performed at the RT by means of the M antennas and that the channel matrix H is completely known at the RT. The following considerations have to be done: (a) Since access to the channel of terminals in the set K is given through the FMS sequence, if no additional information is provided, at the RT the set of k K terminals that simultaneously transmit is unknown. Then, the ZF receiver at the RT will null all the interferences in the set K by considering the channel matrix HK . (b) Due to the constraints on the ZF receiver, successful reception of packets is only possible when jKjpM. Then, ai ðHK Þ can be easily shown to be given by 8 2 > < for jKjpM; 1 H ai ðHK Þ ¼ ½ðHK HK Þ ii (13) > :0 for jKj4M 1 with ½ðHH K HK Þ ii defining the ith element of the 1 diagonal of ðHH K HK Þ . Then, C H ðkÞ depends on the set K through
C aH ðkÞ ¼ C aH ðK; kÞ X ¼ RðiÞPSRaðHK Þ ðiÞ ðpackets=slotÞ.
ð14Þ
i2k
Improving ZF performance via cross-layer: Clearly, from (13), the ZF performance is drastically
reduced in the case that not all the expected transmitting terminals do, in fact, transmit a packet, i.e., when kiK. This problem can be solved by introducing previous to the ZF, an additional stage that detects the set of k active transmissions. This is fully described in [22] and references therein. Notice that, as shown in Fig. 1, cross-layer information regarding the terminals in K would facilitate the detection of terminals in k. We will assume that the set of active transmissions k K is detected without error. Then, given H, 8 2 > < for jkjpN; 1 H (15) ai ðHk Þ ¼ ½ðHk Hk Þ ii > :0 for jkj4N and C aH ðkÞ is as in (6). 4. Scheduling policies At each timeslot, the FMS sequence is constructed in order to schedule (or poll) a set of terminals. The process that decides the set K of terminals to be scheduled is governed by the scheduling policy SH (see expression (8)). Such scheduling policy determines the proportional amount of time that every set of terminals has to be served or equivalently, the probability of every set of terminals to be scheduled. In this section we will focus on analyzing three different scheduling policies. First, we will present the optimal scheduling policy, based on maximizing the average number of successfully received packets at each channel state. Second, we will analyze a so called QoS scheduling policy that selects terminals following a minimum SNIR criterion and finally, we will concentrate on the less optimal but simplest policy, called the average scheduling policy. This policy is based on averaging the MAI over the channel and hence, models the channel as an homogeneous MPR channel model. Although this is not the main motivation of the section, comments on fairness are also presented. 4.1. The optimal scheduling policy If we aim system throughput maximization, the optimal scheduling policy Sop H is such that Sop H ¼ arg maxðZðSH ÞÞ. SH
(16)
Obviously, the maximum is achieved if, at each channel state, the average number of successfully
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
received packets is maximized. Then, Sop H
(
¼
sop H
2 ½0; 1
2M
:
sop H ðKÞ ¼ 1
if K ¼ Kop ; H 2 H
sop H ðKÞ ¼ 0
if KaKop ; H 2 H
) ,
ð17Þ where Kop ¼ arg
that now the main system requirement is to guarantee all transmissions to be over a desired SNIR threshold, or equivalently, over a ath (see expression (2)). Then, our scheduling policy is determined by such parameter ath as SQoS H ¼
max
ðZaH ðKÞÞ.
KPf1;...;Ng
(18)
In the case that maximum in (18) is not unique, Kop will be the set with minimum cardinality among all the sets that maximize ZaH ðKÞ. Then, the total transmitted power would be minimized. If all possible channel states were known beforehand, the optimal values Kop could be computed offline. However, the set of channel states H is not typically known beforehand. Therefore, expression (18) implies an online exhaustive search for each channel realization. Furthermore, the optimal scheduling policy maximizes system throughput by exploiting the instantaneous heterogeneity among terminals to the maximum. Hence, giving the possibility to present unfairness. If fairness was desired, scheduling probabilities in (17) should be modified accordingly at the expense of loosing in throughput performance. It is worth remarking that one should observe some similarities with the Multiuser Diversity strategies in single-user detectors presented by [11]. Particularly, here we have presented the Multiuser Diversity scheduling for a generalized MPR channel. Note that in the case that the RT does not have MPR capabilities, the optimal scheduling policy reduces to the Multiuser Diversity strategies in single-user detectors. Nevertheless, the main difference is that, in MPR channels, many simultaneous transmissions might occur and therefore, the scheduler have to choose the best compromise between number of simultaneous transmissions and MAI. Furthermore, optimization is performed by maximizing throughput rather than maximizing capacity. Hence, a cross-layer approach. 4.2. The QoS scheduling policy The scheduling policy presented above is optimal in the sense that system throughput is maximized. However, as it was highlighted, such strategy might lead to high computational costs. Furthermore, no QoS requirements are considered. Let us assume
1841
8 < :
sQoS H
2N
2 ½0; 1
:
QoS sQoS H ðKÞ ¼ PHK
9 if K 2 KQoS H ;H 2 H=
sQoS H ðKÞ ¼ 0
; if KeKQoS H ;H 2 H
,
ð19Þ where KQoS is the set that gathers all sets of H terminals that guarantee that transmissions of all terminals in each set are over ath at channel state H. That is KQoS ¼ fI : I Pf1; . . . ; Ng H s:t ai ðHI ÞXath 8i 2 Ig. If for every H 2 H, the set K 2 randomly chosen: PQoS HK ¼
1 jKQoS H j
.
ð20Þ KQoS H
is
(21)
Expression (21) says that all sets in KQoS H have the same probability to be scheduled. However, if we would like to select the set with largest cardinality, i.e., to schedule at each timeslot the maximum number of terminals that are over ath , then, 8 jIj; < 1 if jKj ¼ max QoS I2KH ¼ (22) PQoS HK : 0 otherwise: We could also have a look at the optimal QoS requirement, aop th that maximizes throughput. That is QoS aop th ¼ arg maxðZðSH ÞÞ. ath
(23)
The system throughput will be clearly affected by PQoS HK , i.e., by the algorithm implemented at the RT to select the set of terminals K 2 KQoS H . Since typically ath cannot be tuned and is a parameter fixed by upper or lower layers to achieve, for instance, application layer requirements or to determine a received signal strength indicator (RSSI), the main motivation of computing aop th is to evaluate the difference between the throughput obtained when using a QoS scheduling policy with ath ¼ aop th and the throughput obtained by means of the optimal scheduling policy. Such comparison would give a qualitative measure of the efficiency of
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1842
the selection algorithm used to select the set of terminals K 2 KQoS H .
Rayleigh fading channel and performance of the different scheduling policies is presented.
4.3. The average scheduling policy
5.1. The network
The main motivation for this average scheduling policy is to simplify the complexity of the both strategies presented before at the expense of experiencing a lower throughput performance. Basically, what is intended is to schedule, at any slot, the same number of terminals rather than, as in the optimal scheduling policy, to schedule the optimal set of terminals by performing an exhaustive search. To this end, the average scheduling policy is such that
Assume that our problem is the information retrieval of a large network of sensors. For that purpose, consider transmitting terminals depicted in Fig. 1 to be sensors. By means of the FMS sequence, sensors are synchronized and scheduled (or polled) periodically. We further assume our sensors to be environmental sensors that send real-time information related to the temperature, the sea level or rain intensity of a given area. For instance, assume that a group of temperature sensors are spread over an area and we want to monitor the real-time temperature of the area. Hence, the RT terminal is in charge of scheduling the sensors periodically in order to get as much real-time information as possible. We further assume that when a sensor is scheduled, this sensor only sends information if temperature is over a designed threshold. One main property of this network is that old information is not necessary and only information regarding the current temperature is needed. This allows both, to keep sensors very simple because buffering of information is not necessary and to save energy in the sense that sensors only transmit information upon request.
Sav ¼
8 > > < > > :
sav 2 ½0; 12
M
:
sðKÞ ¼
1
N jKj
sðKÞ ¼ 0
if jKj ¼ K
av
9 > > =
if jKj ¼ K
av
> > ;
,
(24) where K av
8 > > <
19 > > X X B 1 C= C ¼ min arg maxB p Z ðIÞ H aH A> . > jKj @ N > > : ; jKj IPf1;...;Ng H 0
s:t: jIj¼jKj
(25) av
In (25), K determines the optimal set cardinality that maximizes the system throughput averaged over all sets with same cardinality. Therefore, the average scheduling policy determines that, with equal probability, sets with such optimal cardinality are scheduled. Note that (24) ensures fairness among terminals. However, if priority among terminals was desired, different probabilities in (24) could be designed subject to X sðIÞ ¼ 1. (26) IPf1;...;Ng s:t: jIj¼Kav
An example in Section 5 shows that the average scheduling policy models the channel as an homogeneous MPR channel model. 5. An Example: the sensor network The motivation of this section is to give practical insight of the theory presented in previous sections. First, a network of sensors is presented as a case of study and a traffic model for this network is given. After that, the ZF MPR channel model is particularized when the entries of H correspond to a
5.2. The traffic model Given a set of sensors K, the probability of a set of sensors k K to simultaneously transmit information is only a function of environmental characteristics. Assume that at a given timeslot n 1, the probability of a sensor i to transmit a packet, i.e., the probability that temperature is over the designed threshold is pi ðn 1Þ. Then, at timeslot n, the probability of this terminal to transmit a packet is updated as pi ðnÞ ¼ bIðn 1Þ þ ð1 bÞpi ðn 1Þ,
(27)
where b 2 ½0; 1 is a parameter of design and Iðn 1Þ 2 ½0; 1 is a function of the information collected in previous slots. For instance, if a sensor that is scheduled does not transmit any information, the probability of this sensor to transmit data is decreased. On the other hand, when information of a sensor is received, the transmitting probability is updated regarding to this information. Furthermore, cross-layer information could also
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
be exploited in this case. Imagine the situation where a sensor transmits data but this is lost, if information of active users is passed from the PHY layer to the MAC layer (as shown in Fig. 1) the MAC layer knows that the temperature sensed by this sensor is at least over a given threshold and then, pi ðn 1Þ could be estimated accordingly. Following notation in Section 3: Y Y pKk ðnÞ ¼ pi ðnÞ ð1 pi ðnÞÞ. (28) i2k
i2K iek
1843
otherwise. Expression (9) can be rewritten as X X X op Þ ¼ p s ðKÞ pKk ZðSop H H H H
X
kK
KPf1;...;Ng
RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ
or equivalently X ZðSop HÞ ¼
X
pKk
KPf1;...;Ng kK
X
RðiÞ
X
i2k
pH sop H ðKÞ
H
PSRaðHk Þ ðiÞ ðpackets=slotÞ.
Furthermore, it makes sense to consider that, if sensors are in the same area, the information collected is a group of samples of the temperature in the area. Hence, all sensors will evolve according to this collected information and the traffic model can be simplified by defining pjKjjkj as the probability that jkj terminals transmit when jKj terminals are scheduled. That is ! jKj pjKjjkj ðnÞ ¼ ð pðnÞÞjkj ð1 pðnÞÞjKjjkj , (29) jkj
ð31Þ
i2k
ð32Þ
The optimal scheduling policy takes scheduling decisions according to ZaH ðKÞ (see expression (17)). Then, saying that sop H ðKÞ depends on the channel state H is equivalent to saying that sop H ðKÞ depends op on aH and therefore, sop ðKÞ s aH ðKÞ. If we H assume that elements in aðHk Þ are i.i.d distributed according to f jkj ðaÞ, expression (32) is rewritten as Z 1 X X X Þ ¼ p RðiÞ sop ZðSop Kk aH ðKÞ H KPf1;...;Ng kK
i2k
0
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð33Þ
where pðnÞ is updated as in (27) but in this case, Iðn 1Þ is related to the information received from all the sensors. For simplicity, in what follows, we will consider that pjKjjkj ðnÞ is not updated and then, pjKjjkj ðnÞ ¼ pjKjjkj . 5.3. The channel model We consider that an improved ZF receiver is used at the RT and that expression (15) applies for the computation of different values of ai ðHk Þ. Therefore, if we consider a Rayleigh fading channel, i.e., the entries of H are independent and identically distributed complex Gaussian random variables with zero mean and unit variance, ai ðHk Þ is a weightened Chi-Square distributed variable with 2ðM jkj þ 1Þ degrees of freedom and probability density function (p.d.f.) f jkj ðaÞ ¼
an=21 expð a2Þ 2n=2 Gðn2Þ
,
(30)
where n ¼ 2ðM jkj þ 1Þ and Gðn=2Þ stands for Gamma function.
5.5. The QoS scheduling policy In general, many different selection algorithms could be applied to construct a FMS sequence such that all terminals with FMSi ¼ 1 have a guaranteed minimum SNIR. Since finding the optimal selection algorithm is beyond the scope of this paper, let us describe a simple selection algorithm, called random selection algorithm: (1) Set FMS ¼ f1; . . . ; 1g and K ¼ fi 2 f1; . . . ; Ng : FMSi ¼ 1g. (2) Obtain H ¼ ½. . . ; hi ; . . .. (3) According to H, compute ai ðHK Þ for i 2 K. (4) If minimum ai ðHK Þ is over ath . Go to step 6. (5) Else, set FMSi ¼ 0 i being the index corresponding to a terminal randomly chosen from those in K. Set K ¼ fi 2 f1; . . . ; Ng : FMSi ¼ 1g and go to step 3. (6) Send FMS.
5.4. The optimal scheduling policy
According to this random selection algorithm 0 1 jKjþ1 QoS Y NðjJj; K Þ 1 @ H A , PQoS 1 (34) HK ¼
At each channel realization H, a FMS sequence is built such that FMSi ¼ 1 if i 2 Kop and FMSi ¼ 0
where NðjJj; KQoS H Þ is the number of sets of cardinality jJj in KQoS H . Note that the random
jJj¼N
N jJj
N jKj
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1844
selection of terminals creates independency among two successive trials. Hence, PQoS HK is the probability mass function of a geometric random variable where the probability of the desired outcome to N occur at the ith trial is Nði; KQoS H Þ=ð i Þ because any QoS of the Nði; KH Þ sets can be selected. However, at the last trial the probability of success is 1=ðNi Þ because only one set among those with cardinality i is the desired outcome.
Then
(39) where
X
X
RjKjjkj ¼
X
2 RðiÞ4
jJPf1;...;Ng i2k s:t: jJj¼jKj s:t: jjj¼jk
Z
jKj1 Y
!
ath
f jJj ðaÞ da 0
jJj¼N
3R 1 f ðaÞ da PSRaðH Þ ðiÞf jkj ðaÞ da jKj a 5 ath R k th 1 N jKj a f jkj ðaÞ da jKj jkj R1
kK
KPf1;...;Ng
pjKjjkj RjKjjkj ðpackets=slotÞ,
jKj¼1 jkj¼1
Expression (9) can be rewritten as X X X QoS ZðSQoS Þ ¼ p s ðKÞ pKk H H H H
jKj N X X
ZðSQoS H Þ ¼
RðiÞPSRaðHk Þ ðiÞ ðpackets=slotÞ
th
i2k
ð40Þ
ð35Þ or equivalently ZðSQoS H Þ ¼
X
X
pKk
X
KPf1;...;Ng kK
RðiÞ
X
i2k
and in the case that there was no heterogeneity in terms of modulation format, channel coding scheme, transmission rate R, and packet length
pH sQoS H ðKÞ
RjKjjkj
H
PSRaðHk Þ ðiÞ ðpackets=slotÞ.
ð36Þ
As for the optimal scheduling policy, the elements in aðHk Þ are i.i.d distributed according to f jkj ðaÞ. Furthermore, due to the random selection of terminals, at each iteration of the algorithm there is no a-posteriory information related to aðHk Þ. Then, the a-posteriory p.d.f. is equal to f jkj ðaÞ. Therefore, considering (34), X pH sQoS H ðKÞ H jKj1 Y
¼
jJj¼N
Z
!R1
ath
f jJj ðaÞ da 0
ath
f jKj ðaÞ da . N jKj
¼
X
X
X
2
RðiÞ4
i2k
R1
ath
R1
jKj1 Y jJj¼N
Z
!
ath
¼ jkjR R1
ath
jJj¼N
!Z f jJj ðaÞ da
0
#
1
f jKj ðaÞ da ath
PSRaðHk Þ f jkj ðaÞ da R1 . a f jkj ðaÞ da
ð41Þ
th
Note that, as described in Section 4.2, expression (39) allows us to evaluate the throughput efficiency of the random selection algorithm by computing throughput when ath ¼ aop th . 5.6. The average scheduling policy
pKk
KPf1;...;Ng kK
jKj1 Y Z ath
ð37Þ
Hence, applying conditional probabilities ZðSQoS H Þ
"
f jJj ðaÞ da 0
3 f jKj ðaÞ da 5 N jKj
At each timeslot, the FMS sequence is constructed such that the number of scheduled terminals is K av where K av is computed as in (25). To compute K av , ZaH ðKÞ is averaged over the channel and over all sets of cardinality jKj. Then, the channel model is approximated to an homogeneous channel model. Applying (24) to (9) X X X 1 ZðSH Þ ¼ N pKk RðiÞ K av KPf1;...;Ng kK s:t: jKj¼K av
X
i2k
pH PSRaðHk Þ ðiÞ ðpackets=slotÞ.
ð42Þ
H
PSRaðH Þ ðiÞf jkj ðaÞ da R k ðpackets=slotÞ. ð38Þ 1 jKj f ðaÞ da jkj a jkj
ath
th
The last term in (42) is an average of PSRaðHk Þ ðiÞ over all channel states. Then, if ai ðHk Þ is distributed according to pjkj ðaÞ, expression (42) is
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
rewritten as
Throughput Vs. Packet Transmission Probability 3.5
ZðSH Þ K av X
X
1
X
X
pKj N i2j K av KPf1;...;Ng jkj¼1 jK s:t: jjj¼jkj s:t: jKj¼K av Z 1
RðiÞ
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð43Þ
0
Operating ZðSH Þ ¼
K av X
pK av jkj
K av jkj
jkj¼1
Z
1
X
X
0
2.5 2 1.5 1 0.5
RðiÞ
0 0.2
jKPf1;...;Ng i2j s:t: jKj¼K av s:t: jjj¼jkj
1
M=1 M=2 M=4
3 Throughput (packets/slot)
¼
1845
PSRaðHk Þ ðiÞf jkj ðaÞ da ðpackets=slotÞ. ð44Þ
Throughput Vs. SNIR threshold 1.5
pK av jkj RK av jkj ðpackets=slotÞ,
jkj¼1
where ( K av ¼ min arg max jKj
jKj X
!) pjKjjkj RjKjjkj
p=0.7 p=1
(45)
.
(46)
jkj¼1
In the case that there was no heterogeneity in terms of modulation format, channel coding scheme, transmission rate R, and packet length, Z 1 PSRaðHk Þ f jkj ðaÞ da. (47) RjKjjkj ¼ jkjR 0
Throughput (packets/slot)
ZðSH Þ ¼
1
Fig. 2. Throughput vs. p obtained when using the optimal scheduling policy and M ¼ 1; 2 and 4.
Then, the throughput under an average scheduling policy is K av X
0.5 0.7 Packet Transmission Probability (p)
1.25 1 0.75 0.5 0.25 0
0
5
10 15 SNIR Threshold (dB)
20
6. Simulations
Fig. 3. Throughput vs. SNIR threshold when the QoS scheduling s used M ¼ 4 and p ¼ ½0:7 1.
A network as described in Section 5 is simulated. The number of sensors in the network is 10 (N ¼ 10) and it has been considered that the RT is provided with M ¼ 1; 2 and 4 antennas. The MPR receiver is an improved MPR ZF receiver and the channel behaves as a Rayleigh fading channel. The traffic model considered is the same as in Section 5 where the probability of one terminal to transmit a packet p is changed from 0 to 1. In Fig. 2, the throughput performance of the optimal scheduling policy as a function of p is shown. It can be observed that when p ¼ 1 the optimal scheduler exploits multiuser diversity and can always schedule sensors with orthogonal
channels. Hence, the throughput obtained is M times the throughput obtained with M ¼ 1. However, if the probability of transmitting a packet decreases, the number of terminals simultaneously transmitting a packet also decreases and hence, heterogeneity among terminals (multiuser diversity) cannot fully be exploited. In Fig. 3, it is shown that the dependence on ath of the QoS scheduling policy with a random selection algorithm when p ¼ 0:7 and p ¼ 1. In both cases M ¼ 4. It is observed that it is possible to find an aop th that maximizes throughput. In Fig. 4, also for M ¼ 4 and, p ¼ 0:5, p ¼ 0:7 and p ¼ 1, it is shown the
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847
1846
Throughput Vs. Set Cardinality 1.4
p=0.5 p=0.7 p=1
Throughput (pasckets/slot)
1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4
1
2
3
4
Set Cardinality (|K|) Fig. 4. Throughput vs. set cardinality when the average scheduling policy is used M ¼ 4 and p ¼ ½0:5 0:7 1.
throughput obtained when the average scheduling policy is used. We observe that there is an optimal number of terminals K av that maximize the average throughput. Reasonably, in order to maintain the optimal trade-off between the number of simultaneous transmissions and the MAI, the value of K av increases as p decreases. We observe that both, the QoS scheduling policy and the average scheduling policy lead to lower throughput results compared to those obtained with the optimal scheduling policy. Interestingly enough, due to the randomness introduced in the random selection algorithm, the QoS scheduling policy with random selection algorithm leads to the same throughput as the one obtained with the average scheduling policy. However, the QoS scheduling policy guarantees hard QoS requirements. Probably, a selection algorithm as described by expression (22) would lead to better throughput results. 7. Conclusions The existence of heterogeneity among terminals in wireless networks and the need to evaluate the effects of such heterogeneity at the MAC and upper layers have motivated the generalized model for scheduling in MIMO multiple access systems presented in this paper. Following a cross-layer approach, a generalized MPR channel model that allows to evaluate system throughput is developed. Particularly, this generalized model accommodates any kind of heterogeneity from multiuser diversity
due to channel fading to heterogeneity due to different data transmission rates. Hence, some examples have been given to show that, although the model is specially designed to accommodate receivers with MPR capabilities, with a proper definition of the noise enhancement factor, it also accommodates any other channel model as for instance the collision channel model or the orthogonal channel model. Different scheduling policies have been analyzed under this generalized model. First, it has been investigated that an optimal scheduling policy based on an exhaustive search gives the maximum achievable throughput. Second, hard QoS requirements are achieved by means of the QoS scheduling policy. Furthermore, since many different scheduling policies might guarantee QoS requirements, we have provided a mechanism to evaluate how far is the QoS scheduling policy from the optimal one. Finally, the last scheduling policy investigated is the average scheduling policy. This policy provides a very simple way of scheduling by averaging the effects of heterogeneity. All these scheduling policies have been particularized for a sensor network and have been evaluated in terms of the statistics of the noise enhancement factor. The results obtained have been useful to achieve practical insight of such scheduling policies.
Acknowledgements The authors of this paper would like to specially thank Xavier Mestre Ph.D. for his valuable help.
References [1] K.-W. Cheong, W.-J. Choi, J.M. Cioffi, Multiuser soft interference canceler via iterative decoding for DSL applications, IEEE. J. Selec. Areas Comm. 20 (2) (February 2002). [2] S. Verdu´, Multiuser Detection, Cambridge University Press, Cambridge, 1998. [3] L. Tong, Q. Zhao, G. Mergen, Multipacket reception in random access wireless networks: from signal processing to optimal medium access control, Proc. IEEE Comm. Magazine (November 2001) 108–112. [4] C. Anton-Haro, M.A. Lagunas, Array processing techniques for wireless: a cross-layer perspective, in: International Forum on Future Mobile Telecommunications and ChinaEU Post Conference on Beyond 3G, 2002. [5] IEEE Signal Processing Magazine, 21(5) (September 2004). [6] S. Ghez, S. Verdu´, S.C. Schwartz, Optimal decentralized control in the random access multipacket channel, IEEE Trans. Automat. Control 34 (11) (November 1989).
ARTICLE IN PRESS M. Realp, A.I. Pe´rez-Neira / Signal Processing 86 (2006) 1834–1847 [7] Q. Zhao, L. Tong, The dynamic queue protocol for spread spectrum random access networks, in: Military Communications Conference, 2, 2001, pp. 1024–1028. [8] M. Realp, A.I. Pe´rez-Neira, PHY-MAC dialogue with multi-packet reception, in: ETSI Workshop on Broadband Wireless Ad-Hoc Networks and Services, 2002. [9] M. Realp, A.I. Pe´rez-Neira, LAMAN: load adaptable MAC for Ad-hoc networks, EURASIP J. Appl. Signal Process. Special Issue on Cross Layer Design for Communications and Signal Processing Systems 2 (February 2005) 116–128. [10] V. Naware, G. Mergen, L. Tong, Stability and delay of finite user slotted ALOHA with multipacket reception, IEEE Trans. Inform. Theory 51 (7) (July 2006). [11] R. Knopp, P.A. Humlet, Information Capacity and Power Control in Single-Cell Multiuser Communications, in: IEEE International Conference on Communications (ICC), June 1995. [12] P. Viswanath, D.N.C. Tse, R. Laroia, Opportunistic beamforming using dumb antennas, IEEE Trans. Inform. Theory 48 (6) (June 2002). [13] F. Flore´n, O. Edfors, B.-A. Molin, Throughput analysis of three multiuser diversity schemes, in: Proceedings of the Vehicular Technology and Communications (VTC), April 2003. [14] M. Realp, A.I. Pe´rez-Neira, C. Mecklenbra¨uker, Cross-layer approach to multi-user diversity in heterogeneous wireless systems, in: IEEE International Conference on Communications (ICC 2005), May 2005.
1847
[15] X. Qin, R. Berry, Exploiting multiuser diversity in wireless ALOHA networks, in: Proceedings of the Allerton Conference Communication, Control and Computing, October 2001, pp. 793–794. [16] S. Adireddy, L. Tong, Exploiting decentralized channel state information for random access, IEEE Trans. Inform. Theory 51 (2) (February 2005) 537–561. [17] M. Gharavi-Alkhansari, A.B. Gershman, Fast antenna subset selection in MIMO systems, Trans. Signal Proc. 52 (2) (February 2004). [18] A. Gorokhov, Antenna selection algorithms for MEA transmission systems, in: International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2002. [19] M. Realp, A.I. Pe´rez-Neira, Generalized multipacket reception model for heterogeneous networks, in: 12th European Signal Processing Conference (EUSIPCO 2004), September 2004. [20] M. Realp, A.I. Pe´rez-Neira, Cross-layer MAC scheduling for multiple antenna systems, in: IEEE Global Telecommunications Conference (Globecom 2004), December 2004. [21] S.T. Chung, A.J. Goldsmith, Degrees of freedom in adaptive modulation: a unified view, IEEE Trans. Comm. 49 (9) (September 2001). [22] B. Chen, L. Tong, Traffic-aided multiuser detection for random-access CDMA networks, IEEE Trans. Signal Proc. 49 (7) (July 2001).