Power levels and packet lengths in random ... - Semantic Scholar
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002
Power Levels and Packet Lengths in Random Multiple Access Wei Luo, Member, IEEE, and Anthony Ephremides, Fellow, IEEE
Abstract—Multiple-power-level ALOHA has been proposed to take advantage of the capture phenomenon in order to improve the throughput of a multiple random access system. In this paper, we study the effect of the use of multiple transmission power levels and of the corresponding packet lengths on the system throughput and energy efficiency. We prove that the single-power-level system in which all transmit at the maximum allowable power level achieves both optimal throughput and energy usage efficiency under a condition on the decodability threshold value. Index Terms—ALOHA, capture, energy efficiency, random access.
I. INTRODUCTION
I
N distributed random access, users transmit packets in an uncoordinated way. Although collisions among packets of different users reduce throughput as well as waste energy and bandwidth, random access is an indispensable method in wireless networks because of its simplicity and low overhead. Distributed random access is commonly used in the control channel of the cellular system, and ad hoc networks, where users have little knowledge about whether the channel is used by others or not. In wireless networks, capture is possible since the receiver may be able to decode the packet with the highest power even when there are multiple packets arriving at the receiver at the same time. To take advantage of power capture, packets can be transmitted at multiple discrete power levels, and the packet received at the highest power may be captured (e.g., see [1]–[3]). In those systems with multiple power levels, there are power power levels. The transmitter randomly chooses one of the levels and transmits the packet at that level. Various power capture models have been used to analyze the system performance. The perfect power capture model is based on the assumption that if and only if a packet has higher power than all the other packets overlapping with it, the packet is captured (e.g., [1], [4]).
A more accurate model is based on the assumption that if and only if a packet’s signal-to-interference-plus-noise ratio (SINR) is greater than a certain decodability threshold, the packet is captured (e.g., [5]–[7]). We call this the SINR capture model. In all of the above works, packet lengths at different power levels are the same. Throughput is measured by the average number of successfully received packets per slot. Under the assumption of perfect power capture, the throughput is a function of offered traffic and the number of power levels as shown in Fig. 1. The derivation of the curves in Fig. 1 follows the method used in [1] and will be provided later. We can see from this figure that the use of multiple power levels can increase the system throughput. Similar conclusions are also drawn from the SINR capture model. However, the use of multiple power levels may have its side effect when energy efficiency is considered. Indeed, the use of multiple power levels increases the transmission energy per packet transmission. As energy efficiency is becoming a very important concern in wireless communication systems, we need to consider the tradeoff between increasing system throughput against improving system energy efficiency. In addition, we observe that for a packet that is sent at a higher power level, the length of the packet can be reduced since the can be met through transmission at correquired level of respondingly increased rate. By reducing the packet length, we can reduce the chance of packet collision. On the other hand, if all the users use the maximum possible power, the likelihood of packet overlap (and hence, collision) decreases, but the possibility of capture (with its beneficial effects on throughput) is eliminated. Thus, both the value of power level and the value of packet length are the variables that can be optimized to improve the system performance. The system performance that we consider in this paper is measured by the throughput and energy efficiency. A commonly used definition for energy efficiency is (e.g., [8]) energy efficiency
Manuscript received November 9, 1999; revised September 6, 2001. This work was prepared through collaborative participation in the Advanced Telecommunications and Information Distribution Research Program (ATIRP) Consortium sponsored by the U.S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL01-96-2-0002. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. W. Luo is with Bell Laboratories, Lucent Technologies, Holmdel, NJ 07733 USA (e-mail: [email protected]). A. Ephremides is with the Electrical Engineering Department and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]; [email protected]). Communicated by V. Anantharam, Associate Editor for Communication Networks. Publisher Item Identifier S 0018-9448(02)00049-4.
average number of correctly received packets per energy unit system throughput system average power consumption
where “system throughput” is the number of successfully received packets received per time unit, and “system average power consumption” is the total power consumption summed over all the users in the system averaged over time. For example, if there are two users in the system and each user’s power consumption is , then the system average power con. In random sumption is summed over two users, which is access, usually the higher the throughput the lower the energy
0018–9448/02$17.00 © 2002 IEEE
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Fig. 1. Throughput per packet slot versus offered traffic. A traditional model in which all packets are transmitted at the same packet lengths is used. Throughput is measured by the number of successfully received packets per slot.
efficiency because higher throughput usually implies larger number of packet transmissions per time unit, which leads to more collisions and lower energy efficiency. Hence, increase of both throughput and energy efficiency at the same time seems to be difficult if not impossible. An appropriate way to consider the system performance is to consider the system throughput under the constraint on the system average power consumption. In addition, in practical applications, packet peak power is also constrained. Therefore, in this paper, we consider the problem of maximizing the system throughput under the packet peak power constraint and system average power constraint. An attempt to consider the packet peak power constraint and system average power constraint in random access can be found in [9], where the concept of spread ALOHA is introduced. It is shown through Shannon theory that the capacity of low duty cycle ALOHA channel without capture cannot be exceeded by any other multiple-access methods given the same bandwidth and average power constraint. In this paper, we show the similar result that ALOHA without capture achieves the maximum achievable throughput through a different approach. The contribution of this paper is the analytical proof of Propositions 1 and 2, in which it states that in a single-receiver ALOHA system with power capture, the system maximum achievable throughput is maximized by the use of a single power level if the decodability threshold is greater than a certain value. In the proof, we derive the optimal setting of the power levels and packet levels by using a bounding method. In this bounding method, we set up an optimistic model whose throughput upper-bounds that of the real model. We prove that the maximum achievable throughput of the optimistic model is maximized by the use of a single power level. We also show that when a single power level is used, the real model has the
same throughput as that of the optimistic model. So for the real model, the maximum achievable throughput is also maximized by using the single power level. The organization of the paper is as follows. In Section II, the system model is described. In Section III, we consider uniform packet length with perfect power consumption and the derivation of the results shown in Fig. 1. In Section IV, we state the main results of this paper, outline the proofs, and discuss their implications. In Section V, we describe the optimistic model used in the proofs. Section VI concludes the paper. The mathematical details of the proofs are provided in the Appendix.
II. SYSTEM MODEL Suppose there are an infinite number of bufferless users in the system (see [10] for more discussions). Each user has a clock that is synchronized to a global clock so that slotted transmission can be achieved. Each user generates and transmits a packet at a certain power level independent of other users. We adopt the conventional assumption that the overall packet arrival pattern (including new arrivals and retransmissions) is Poisson (e.g., [1], [6], [11]). There are discrete power levels in the system. The values of power levels and packet lengths are determined at the system deployment and fixed once the system is in use. Time is slotted at each power level, and the slot durations can be different at different power levels. A packet transmitted at a power level has packet length equal to the slot duration at that power level, and transmission starts at the slot edge (see Fig. 2). Because the slot durations are different at different power levels, the time offset of the slots between different power levels cannot be controlled. Therefore, we do not have slot synchronization
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002
Fig. 2. Multiple power level random access. Within each power level, time is slotted. The packet is transmitted at the beginning of a time slot. No synchronization among different power levels is maintained.
between different power levels. If a packet is lost due to collision, the packet will be retransmitted at a later time. Each packet is transmitted or retransmitted on power level with probability for from to . The packet is involved in a collision when it has overlap with other packets either at the same power level or different power levels. We make the following assumptions. symbols. 1) Each packet contains 2) A packet can be received correctly if and only if a) during the whole transmission period, the packet has the highest power over all the other overlapping packets; b) during the whole transmission period, the packet’s SINR is always larger than a certain threshold . The SINR is defined as symbol energy to interference plus noise ratio, i.e. SINR
3) 4) 5) 6) 7)
(1)
are the transmission power and where , are the symbol duration of the packet, and interference and background noise energy in the output of the symbol matched filter. BPSK modulation scheme is used. Every packet is transmitted at constant power. The distances between the transmitters and the receiver are equal. . The maximum power that a packet can use is The system average power constraint is , for which (2)
where , , and are the offered traffic rate in packets per seconds, packet length, and power at the th power level, respectively, and is the number of power levels. 8) The offered traffic on the system (including newly arrivals and retransmissions) is Poisson.
Assumption 1) implies that the packet length is proportional to the symbol duration. Assumption 2a) is based on the single receiver scenario. We assume that the receiver can only receive one packet at a time and is always locked to the packet with the highest power. This can be accomplished in a variety of ways that expend a negligible amount of additional power or bandwidth so as to enable the receiver to correctly read appropriate packet preambles. We do not enter into a detailed discussion of this as it pertains to the very nature of capture. Here we simply assume capture can be achieved without regard to the specific mechanism or signal structure. When the receiver is receiving a packet while another higher power packet is coming, the receiver will switch to lock onto the coming higher power packet and drop the reception of the ongoing packet. So the packet is captured only if it has the highest power during its overall transmission period. According to Assumption 2b), note that it is possible that multiple overlapping packets may have the SINR level higher than the threshold at the same time, but at most one of them can be captured due to Assumption 2a). (which usually occurs when spread specIn addition, if trum or a strong error-correction code is used), it is possible that two packets with equal power may both have SINR level higher than . However, in this paper, we focus on the sce, in which case if two or more packets are nario when received at equal power at the same time, none of them is captured. We do not consider the multiple-packet-reception capability in this paper. Other statistical capture models can be found in [12], [13]. In Assumption 3), we assume binary phase shift keying (BPSK) modulation just for simple explanation. We can extend our results to other modulation schemes such as quaternary phase shift keying (QPSK) and quadrature amplitude modulation (QAM). Assumption 5) makes the received power and transmission power interchangeable. If this condition is not satisfied, we need to transform the received power to transmission power by using different scaling coefficients. Assumption 6) puts an upper power limit on individual packet transmission. Assumption 7), on the other hand, puts an upper power limit is the enon the system wide power consumption. In (2), ergy expended on a packet transmitted at the th power level is the number of packets transmitted at the th power and is the average power consumplevel per second, thus tion at the th power level. The summation over all power levels, , is the overall system average power consumption. Assumption 8) is a conventional assumption to approximate the system with a large number of users, in which the packet length is far shorter than the packet retransmission interval. There can be multiple power levels so that the benefit of capture on the throughput can be obtained. As described before, the transmitter randomly chooses one power level before transmission. The value of the power level and packet length should be set so that capture of the packet is possible when there are overlapping packets at the lower levels. There might be multiple packets at the lower power levels. Thus, a packet at the highest power level is captured only if its signal power is large enough to combat the accumulation of interference from the multiple packets at the lower power levels. So whatever the values of power levels and packet lengths are, the packet at the
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at the same time. Therefore, the throughput per packet slot contributed on the th power level is
(3)
The throughput maximization problem given the offered traffic is formulated as follows:
maximize subject to Fig. 3.
Multiple-power-level random access with uniform packet lengths.
highest power level cannot be captured with 100% probability. But at the least, the power levels should be set so that the capture of a packet is guaranteed when there is a single interfering packet at a lower power level. Otherwise, if the separation of the adjacent power levels is too small to guarantee the capture of a packet at the higher power level, the use of more energy transmitting a packet at the higher power level does not bring any benefit but wastes energy. In Section V, where we set up a hypothetical and optimistic model for the multiple-power-level system, however, we assume that if the values of power levels guarantee the capture of a packet at a higher power level when there is a single interfering packet at the lower power level, then the capture of a packet is guaranteed irrespective of the number of interfering packets at lower power levels as long as there is no other packet transmitted at the same or higher power level. The hypothetical and optimistic model is used to compare with a single-power-level system, and we find that the single-power-level system yields higher throughput under the packet peak power constraint and the system average power constraint. Therefore, it is safe to say that the single-power-level system is better than any multiple-power-level system.
III. UNIFORM PACKET LENGTH If packet lengths on different power levels are same, i.e., , then it is possible that slots on different power levels can be synchronized as shown in Fig. 3. In this section, we assume that the packet is captured if there are no other packets at the higher or the same power level overlapping with it. The same assumptions are adopted in [1]. power levels . Assume Consider that each packet is transmitted at power level with probability . If the offered traffic (including newly arrived and retransmission packets) on the system is assumed to be Poisson with arrival rate , then the offered traffic at any power level is also . Obviously, . Poisson with arrival rate . We denote by the offered traffic vector A packet is received correctly on the th power level, if there is no other packet transmitted at a higher or the same power level
(4)
are the variables to be optiare constants. The numerical solutions for are depicted in Fig. 1. As we can see from the figure, the use of multiple power levels can increase system throughput if power consumption is not considered. In the next section, we take into consideration the peak and average power consumption. Instead of fixing the offered traffic rate , we will give an upper limit on the system average power consumption, which provides the constraint on the system offered traffic as we will see later.
where mized,
and
IV. OPTIMALITY OF SINGLE-POWER-LEVEL SYSTEM Based on the assumptions in Section II, we have the following propositions. , if the Proposition 1: Given a peak power constraint , decodability threshold in (1) is greater than achieves maximum possible a single power level throughput. In such a system, packets are transmitted at maxwith corresponding packet length imum allowable power . and Proposition 2: Given both peak-power constraint average-power constraint , if the decodability threshold in , then a single power level (1) is greater than achieves maximum possible throughput. In such a system, with packets are transmitted at maximum allowable power . packet lengths Note that if is large enough, Proposition 2 degenerates into , the average power Proposition 1. In fact, when constraint does not affect the optimal system performance. Also, note that we state that the single-power-level system can achieve the maximum possible throughput. The detailed proofs of the above propositions are given in the Appendixes. Here, we sketch the proofs. The proofs are based on the construction of an optimistic model, which overestimates the throughput of the multiple-power-level system, but provides an exact evaluation of the throughput for the single-power-level system, a degeneration of the multiple-power-level system. This optimistic model simplifies the analysis. We find that a singlepower-level system yields the maximum throughput under this
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optimistic model, and thus the single-power-level system is also optimal under the real model. We construct a system, in which the values of the power levels and packet lengths are chosen so that if there are two overlapping packets transmitted at two adjacent power levels, respectively, the SINR of the packet at the higher power level is greater than the decodability threshold. Therefore, the packet at the higher power level can be captured when there is only one overlapping packet at the next power level. But if there are more than one overlapping packets, the SINR of the packet at the highest power level may not be greater than the decodability threshold. Whether it can be captured or not depends on how much interference power is accumulated from overlapping packets at the lower power levels. This makes analysis difficult because we have to calculate the probability that the accumulated interference is below certain value. In the optimistic model, however, we simply assume that the packet received at the highest power level can be received correctly (captured) if there is no other packet overlapping with it at the same power level, regardless of how many overlapping packets are transmitted at the lower power levels. Based on this assumption, we can determine the values of the power levels and packet lengths, and also determine the maximum throughput of the multiple-power-level system. The estimation based on this optimistic assumption obviously yields an upper bound of the system performance. Because for the single-power-level system there is no capture, this optimistic model on capture yields an exact estimation of the throughput. Furthermore, the analysis shows that a single-power-level system achieves maximum throughput over all the multiple-power-level systems under the optimistic model. Therefore, the single-power-level system is optimal under the realistic model. The idea of the , we can proof is that for an -power-level system -power-level system which has always construct an as that of the -power-level the same parameters but has a higher throughput, system for and less energy consumption. This means that for any multiple-power-level system, we can reduce the lowest two power levels to a single power level so that throughput/power performance can be increased. By iteratively repeating this procedure, we obtain that a single power level which operates at the highest allowable packet power with the shortest possible packet length and achieves maximum throughput. To meet the average power constraint while achieving the maximum possible throughput, the offered traffic rate must be equal to the lesser of the two quantities: and . This indicates that to achieve optimal energy efficiency in a random access system, it is essential to apply admission control on the offered traffic. Note that in both propositions, there is the additional assumption that the . decodability threshold be greater than a fixed value Indeed, our proofs work only if this condition is satisfied. If is less than , use of a the decodability threshold single power level is no longer necessarily optimal. In this case, a multiple-power-level system, in which all packets are still transmitted at the maximum allowable power but with different packet lengths can achieve higher throughput than the single-power-level system. Although it seems counterintuitive,
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002
the packets transmitted at the lowest level should have the shortest allowable packet length, while the packets at higher levels should have longer length. The benefit from capture in this case seems to outweigh the adverse impact from larger collision probability that the longer packet lengths imply. However, this observation follows from the optimistic model and it may or may not carry any real significance. For a real multiple-power-level system, the throughput is actually less than that of the optimistic model used in our derivations. Thus, even when , it is still possible that a single power level is optimal. We simply cannot prove our claim for values . of less than V. THE OPTIMISTIC MODEL In this section, we describe the optimistic model that we use for proving the propositions. As we mentioned before, the optimistic model is based on two simplifying assumptions. • Among mutually overlapping packets, the packet transmitted at the highest power level is captured, if there is no other packet transmitted at the same level overlapping with it. This is assumed to be true regardless of how many packets are transmitted at the lower level. • To make the capture of the packet at higher power level possible, the value of power levels and packet lengths should satisfy the condition that if there are two overlapping packets at consecutive power levels the higher level packet’s SINR is no less than the decodability threshold . The first assumption greatly simplifies the throughput expression, and the second assumption simplifies the value setup of power levels and packet lengths, as we will see in the next two subsections. We should emphasize that the propositions stated in the preceding section hold for the real model, the optimistic model is used just to prove the propositions. These two assumptions are optimistic because the way we choose power level and packet length does not guarantee that a packet can be captured if there are more than one packets overlapping with it at the lower power levels. A. ALOHA With
Power Levels
power levels . Assume Consider that each packet is transmitted at power level with probability . If the offered traffic (including newly arrived and retransmission packets) on the system is assumed to be Poisson with arrival rate , then the offered traffic at any power level is also . Obviously, . Poisson with arrival rate . We denote by the offered traffic vector Note that since the packet lengths at different power levels can be different, the unit for offered traffic is number of packets per second instead of number of packets per time slot. However, within each power level, the packets are still transmitted as in slotted ALOHA. So we can use the formula for slotted as the offered traffic ALOHA throughput [10] by using is the packet length, i.e., the slot duration, at per slot where level . We assume that if during the entire transmission time is of a packet at level , no other packet at level
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transmitted, then that packet is successfully received. We also assume, optimistically, that overlapping transmission of packets do not affect the reception of a packet at level at level , no matter how many of these may be. For a packet at level , no packets at other power levels can affect its decodability. So the throughput at that power level is packets per slot. Here, is the probability that no other packet at power level overlaps with the packet we are interested in. The slot duration is . The absolute throughput per time unit is
if it is the strongest packet at that time, the bit-error rate is sufficiently high to result in a packet reception error (collision). In this way, we will be able to account for excessive interference from packets at lower power levels. We assume that if one symbol in a packet is received incorrectly, the whole packet is received incorrectly. In other words, for a packet to be correctly received, every symbol must be correctly received. For any one symbol, we assume that it can be received correctly if and only if its SINR is greater than the decodability threshold. To be more precise, the symbol energy to interference-to-noise ratio should be larger than a threshold , that is,
For level , , a packet is successfully transmitted in a given time slot if it is the only packet transmitted at that level, , and if no which happens with probability packet at higher power levels overlaps with it. There may be several slots corresponding to power level that overlap with a given slot corresponding to power level . The total duration . Thereof each of those overlaps is greater than overlaps fore, the probability that no packet at level . with the packet at level is less than Because the transmission of packets at different power levels are independent, the probability that no packet at higher power levels overlaps with the packet of interest at level is less than . So, the throughput at level is bounded as follows:
(7) are the power and the symbol duration of the where and packet, respectively; denotes interference from overlapping is the background white noise, and is the SINR packets, decodability threshold. Now let us consider how the interference from overlapping packets affects SINR. If two packets, say 1 and 2, with different symbol durations, are overlapping, we need to calculate the interference after the matched filter for each packet. Assume and are the symbol durations of packets 1 and 2, respectively. We can write the received signal as
(8) (5) Note that this bound also holds for the case that all the power levels have the same packet lengths and transmission slots between different power levels are synchronized. The total throughput for the overall system is then bounded by
are amplitude, information symbols and where , respectively; symbol waveforms of the packet is background noise. For simplicity, we assume BPSK modulation and rectangular waveforms. It is easy to extend our analysis to more complex modulation schemes with little essential modification. Without loss of generality, we assume and
(6) , i.e., when a Note that strict equality holds in (6) when single power level is employed (that is, it reduces to the classical slotted ALOHA throughput formula). This point is very important in our proof. B. Power Levels and Packet Lengths In this section, we determine the values of power levels and packet lengths in the optimistic system. Those values have to satisfy a condition so that a packet overlapping with another packet one power level lower can be captured. We assume that there is a single receiver in the system. Furthermore, we assume that the receiver is able to select at any given time to decode the packet with the highest power. We also assume that after the packet is selected by the receiver, it can be passed to the corresponding matched filter and decoder. The packet’s SINR should be larger than a decodability threshold, otherwise even
The illustration of overlap symbols from two users is shown in Fig. 4. In this figure, we are looking at a symbol of packet 1 that is involved in collisions with packet 2 throughout the symbol duration. If , the number of symbols per packet is relatively large this will be the case for at least one symbol when a packet is involved in a collision, so we may do this without essential loss of generality. For the th symbol of packet 1 after the matched filter, the received energy of the desired signal is . The background white noise energy is . Interference from packet 2 after the matched filter is , and ) given by (see Fig. 4 for definition of
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002
Fig. 4.
Two interfering users. T
= (K + )T
k + )T
=( +
, where ;
; < 1, k = K , or k = K 0 1 (only possible when T
T
).
(9) and where in the second equality, we assume that . In fact, those properties can be extended to other modulation schemes in which the center of the constellation is the origin and average amplitude of the constellation is unit. , i.e., , it follows from When (9) that
(14)
(10)
It does not make sense to use a high power level unless capture is possible when a packet at such a power level is interfered with by a packet at a lower power level. Our SINR calculation above must shows that this must imply that the power level and necessarily satisfy the condtions
, i.e., , the values of , , When and in (9) can be either one of the two possibilities:
(15)
1) 2)
and and
while for the requires
.
th (i.e., the lowest) level
, decodability simply
It then follows from (9) that (16)
(11) Now suppose packet 1 and packet 2 overlap with each other. symbols, Recall that we assume that in each packet there are and , and let , i.e., , and . It follows that the symbol SINR of packet 1 is
(12) When
, i.e.,
, then using (10), we have (13)
and when
, using (11), we have
When more interfering packets are present, even if (15) is satisfied, a single highest level packet may not be received correctly because accumulation of interference from multiple lower level packets precludes correct reception. However, we assume an optimistic model in which if (20) and (21) are satisfied, the packet can be received correctly as long as no other packet transmitted at higher or equal power levels overlaps with it (regardless of how many lower level packets are transmitted at the same time). The reason we use (15) and (16) is their relative simplicity. They provide an optimistic model which upper-bounds the performance of the real system. We will show later that the maximum value of the upper bound of the optimistic capture system of (15) and (16) can be actually achieved by a real single-power-level system. Thus, we will prove that the system that achieves that maximum is indeed the optimum actual system despite the apparent largesse of our assumption regarding capture. C. System Power Constraints We assume that no packet can exceed the peak power con. Furthermore, to capture the finiteness of energy straint
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These characteristics are used in the proof of the two propositions. 1) Among mutually overlapping packets, the packet transmitted at the highest power level is captured, if there is no other packet transmitted at the same level overlapping with it. This is assumed to be true regardless of how many packets are transmitted at the lower level. The total throughput of the multiple-power-level system is given optimistically by (19)
Fig. 5. Maximum throughput as a function of average system power constraint.
we also specify an average power constraint for the entire system. Although this construction does not capture the finiteness of battery life at each transmitter it provides a cap to energy expenditures system-wide. The overall system power consumption for the ALOHA system with power levels is given by
This follows from (6) by always assuming equality instead of “ ” in (6). Again, note that for a single-powerlevel system, equality holds in (6), which means (19) holds in the real single-power-level system even without the optimistic assumption. 2) To make the capture of the packet at higher power level possible, the value of power levels and packet lengths should satisfy the condition that if there are two overlapping packets at consecutive power levels the higher level packet’s SINR is no less than the decodability threshold . Optimistically, we assume that if the following is satisfied then the only packet at the highest power level is sure to be captured
(17) (20)
We can also express the throughput per “energy” unit as (18) depends on the offered traffic Note that the quantity through the resulting throughput as well as through the denominator . Suppose that, for a given constraint value , we can determine the maximum achievable throughput . Then, can be thought of as a function of . We call it the throughput is initially a power function. It is rather obvious that nondecreasing function. But throughput cannot keep increasing with . At some point, the throughput must saturate at a maximum value. Increasing beyond that point is not going to help increase the maximum stable throughput since the peak power constraint will kick in and will not permit further increases by making the packets shorter. Thus, the available additional power can only be consumed by means of more packet transmissions, which will start causing increasing collisions, thereby reducing the throughput. In Fig. 5, we show the throughput power functions for the optimal single-power-level system. If we look at and from this figure, we can the relationship between decreases as increases. Since we hope to insee that crease the throughput as well as energy efficiency, it is more appropriate to consider the throughput power function instead alone. of energy efficiency D. Summary of the Optimistic Model The optimistic model is discussed in the previous subsections. We summarize the characteristics of the optimistic model here.
(21) In a single-power-level system, there is no packet capture. Therefore, only (21) is required. These characteristics in the optimistic model are optimistic only in the multiple-power-level system. In the single-powerlevel system, these assumptions become realistic. VI. CONCLUSION In this paper, we examined the throughput and energy efficiency performance of multiple-power-level random-access systems. We described the relationship among the power constraints, the packet lengths, and the throughput. We found that a single maximum power level achieves the best throughput–power performance under a condition on the decodability threshold value. APPENDIX A. Proof of Proposition 1 We first prove some useful lemmas. is maximized at . Lemma 1: The function In addition, the function is monotonically increasing in the in, and monotonically decreasing in the interval terval .
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Proof: It follows trivially from the form of the derivative (22)
as that of the -power-level system . The parameters on the th power level (lowest power level in ) are as follows:
(27) Lemma 2: If
,
, then (23)
and
It can be seen that
is monotonically decreasing. Proof:
(28)
(24) where the first inequality follows from the fact that the second inequality follows from the fact that is monotonically decreasing. It is trivial to see that .
, and . So
For the system , it is trivial to see that the capture criterion (20) is satisfied. Next, we need to compare the throughputs of the two systems. Because the throughput for the highest th power levels are the same for both system and , we only need to compare the throughput of ’s lowest power level with the throughput of ’s two lowest power levels. Following (19), the throughput of ’s lowest power level is
Now we provide the proof of Proposition 1 as follows. Proof: Suppose there are
power levels
(29)
We can construct a new system with power levels, which is under the same peak power constraint. The parameters of the highest power levels are the same for th power level the two systems. The parameters on the of the new system are properly chosen such that the throughput is higher than that of the -power-level system according to the optimistic model we described in the paper. We will show . Therefore, that such construction can always be done if we can always increase the throughput by reducing the number of power levels. Consequently, the optimal system (in the sense of maximizing the throughput) is the one that only has a single power level. Note that our optimistic model overestimates the . For , the optimistic model cointhroughput for cides with reality. The optimal single-power-level system yields greater throughput than the overestimate of the throughput of the multiple-power-level system. So the single-power-level system is also optimal in reality. Now we start to show that construction of the new system power levels is always possible. The capture and with throughput formulas are all based on the optimistic model we described before. For the -power-level system (say, system ), recall from (20) and (21) that
The throughput of ’s two lowest power levels is
(30) where the inequality follows from (28). Our objective is to show that (29) and (30), it suffices to show that
. From
(31) (we will consider the case First, suppose later). Then (25) becomes (25) (26) The new system is constructed as follows. The parameters of the highest power levels are the same
(32)
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From the first inequality of (32), we obtain
(39) where the inequality follows from Lemma 1. Note that the last and in . term in (39) is monotonically decreasing in From (37) and (38), Inequality (39) becomes
From the second inequality of (32)
Notice that because
, we have (33)
Consider the term on the left-hand side of (31) (40) Let (40) that
. By using Lemma 2, we obtain from
(34) where the second inequality follows from Lemma 1, the third , and the fourth inequality follows from the fact that inequality follows from (33). Thus, we prove (31) and hence . the proposition, for the case , inequalities in (25) become If
(41) . Thus, we which establishes (31) for the case proved that the optimal system should have a single power level. The throughput of the single power level system is (42)
(35) Therefore,
. We can see that the where equality holds if and only if value of the throughput increases as decreases. But cannot be arbitrarily small. In order to maintain the requested SINR . Therefore, level, we must have (43)
(36)
So the maximum achievable throughput is achieved when
So (37) This completes the proof of Proposition 1. and from the second inequality of (35) B. Proof of Proposition 2 (38) Now, let us consider the term on the left-hand side of (31)
We also need to introduce several lemmas before we proceed to the proof. Lemma 3: The function . terval Proof: In the interval
is convex in the in-
(44) So the function
is convex.
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From Lemma 3, if
and
Thus, in both cases we have
then (45)
(52)
This property will be used to prove the following lemma. Lemma 4: If
,
,
From (50), it follows that
, and we let (46) (47)
, then
and
(48) Proof: Let
The proof of the lemma is completed.
, and (49)
We now prove Proposition 2. Proof: The idea of the proof is similar to that of Proposition 1. Suppose there are power levels in a system, say system . As before, we can prove that a new power levels can be constructed under the system with same power constraints with higher throughput. The construction of the new system is similar to that of the proof of Proposiof the highest power tion 1. The parameters levels are the same for the two systems. Assume that the average power used by the two lowest levels in system is
The term on the left-hand side of (48) can be rewritten as
(53) (50) Consider first the case that is convex, we have tion
. Because the func-
For the system , the parameters of the lowest level are
th power
(54)
The average power at the lowest level of system
is (55)
Now consider the case that . By Lemma 1 is obvious that
. Let
. It
It follows that
(51) , The first inequality follows monotonicity of the function and the second inequality follows the fact that the function is . convex in the interval
highest levels are the Because the average power of the same for the two systems and the average power of the last power level of is not greater than that of the last two power levels of according to (55), the total average power of the system is not greater than that of the system . Therefore, if system satisfies the average power constraint, system also satisfies the average power constraint. We now show that the throughput of is greater than that of . The throughput of the highest levels are the same for the two systems. So we only need to show that the throughput th power level of the system is greater than the of the throughput of the two lowest power levels of the system . The throughput of the lowest level of the system is
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As in (30), the throughput of ’s two lowest power level is
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In addition,
(62) Let
be a positive number such that (63)
It is trivial to see that
. So from (62), we have (64)
Now let us return to the objective inequality (57). The term on the left-hand side is (56) The objective is to show that suffices to show that
. Then it
(57) , then following (54). In that If case, inequality (57) reduces to (31) which is already proven in Proposition 1. Therefore, we only need to prove (57) for the case . , If Again, we consider first the case , it follows that (33) is still true, i.e.,
where the second inequality follows from (64), and the third inequality follows from Lemma 4. Thus, we prove the inequality . in (57) for the case , following the same reasoning as in the proof If of Proposition 1, we observe (37) and (38). That is,
(58)
(65)
In addition, the capture criterion (32) holds. For reader’s convenience, we repeat that criterion here
In addition, because of (53), we have
(66) and (59) Let
where we use the second inequality in (35) and (36). Then, the left-hand side of (57) becomes
(60) , it follows that Because we assume , and can also see from (54) that It also follows from (53) that
(67)
. We .
(61)
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ACKNOWLEDGMENT The authors wish to thank V. Anantharam for discussions and comments leading to improvement of presentation and quality of this paper. where the first equality follows from the assumption that , and the first inequality follows from (66); the second inequality follows from (67); the third inequality follows from ; and the last inequality follows the assumption that . from the assumption that So (57) is proven. We know that the optimal system should have a single power level. For a single-power-level system under the peak and average power constraints, we must have
(68) which implies
(69) The throughput is then given by By Lemma 1, if achieved at throughput is achieved at
, the maximum throughput is ; otherwise, the maximum . In short (70)
and the maximum achievable throughput is achieved when (71) This completes the proof of Proposition 2.
REFERENCES [1] J. J. Metzner, “On improving utilization in ALOHA networks,” IEEE Trans. Commun., vol. COM-24, pp. 447–448, Apr. 1976. [2] J. P. Linnartz, Narrowband Land-Mobile Radio Networks. Norwood, MA: Artech House, 1993. [3] C. T. Lau and C. Leung, “Capture models for mobile packet radio networks,” IEEE Trans. Commun., vol. 40, pp. 917–925, May 1992. [4] N. Abramson, “The throughput of packet broadcasting channels,” IEEE Trans. Commun., vol. COM-25, pp. 117–128, Jan. 1977. [5] R. O. LaMaire, A. Krishna, and M. Zorzi, “Optimization of capture in mulitple access radio systems with Rayleigh fading and random power levels,” in Multiaccess, Mobility and Teletraffic for Personal Communications, B. Jabbari, P. Godlewski, and X. Lagrange, Eds. Boston, MA: Kluwer, 1996, pp. 321–336. [6] I. Cidon, J. Kodesh, and M. Sidi, “Erasure, capture, and random power level selection in multipleaccess systems,” IEEE Trans. Commun., vol. 36, pp. 263–271, Mar. 1988. [7] C. C. Lee, “Random signal levels for channel access in packet broadcast networks,” IEEE J. Select. Areas Commun., vol. 5, pp. 1026–1034, July 1987. [8] Y. W. Leung, “Mean power consumption of artificial power capture in wireless networks,” IEEE Trans. Commun., vol. 45, pp. 957–964, Aug. 1997. [9] N. Abramson, “VSAT data networks,” Proc. IEEE, vol. 78, no. 7, pp. 1267–1274, July 1990. [10] D. Bertsekas and R. Gallager, Data Network, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1992. [11] L. Kleinrock and F. Tobagi, “Packet switch in radio channels: Part I—Carrier sense multiple-access modes and their throughput-delay characteristics,” IEEE Trans. Commun., vol. COM-23, pp. 1400–1416, Dec. 1975. [12] B. Hajek, A. Krishna, and R. O. LaMaire, “On the capture probability for a large number of stations,” IEEE Trans. Commun., vol. 45, pp. 254–260, Feb. 1997. [13] H. Zhou and R. H. Deng, “Capture model for mobile radio slotted ALOHA systems,” Proc. Inst. Elec. Eng.–Commun., vol. 145, no. 2, pp. 91–97, Apr. 1998.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002
Power Levels and Packet Lengths in Random Multiple Access Wei Luo, Member, IEEE, and Anthony Ephremides, Fellow, IEEE
Abstract—Multiple-power-level ALOHA has been proposed to take advantage of the capture phenomenon in order to improve the throughput of a multiple random access system. In this paper, we study the effect of the use of multiple transmission power levels and of the corresponding packet lengths on the system throughput and energy efficiency. We prove that the single-power-level system in which all transmit at the maximum allowable power level achieves both optimal throughput and energy usage efficiency under a condition on the decodability threshold value. Index Terms—ALOHA, capture, energy efficiency, random access.
I. INTRODUCTION
I
N distributed random access, users transmit packets in an uncoordinated way. Although collisions among packets of different users reduce throughput as well as waste energy and bandwidth, random access is an indispensable method in wireless networks because of its simplicity and low overhead. Distributed random access is commonly used in the control channel of the cellular system, and ad hoc networks, where users have little knowledge about whether the channel is used by others or not. In wireless networks, capture is possible since the receiver may be able to decode the packet with the highest power even when there are multiple packets arriving at the receiver at the same time. To take advantage of power capture, packets can be transmitted at multiple discrete power levels, and the packet received at the highest power may be captured (e.g., see [1]–[3]). In those systems with multiple power levels, there are power power levels. The transmitter randomly chooses one of the levels and transmits the packet at that level. Various power capture models have been used to analyze the system performance. The perfect power capture model is based on the assumption that if and only if a packet has higher power than all the other packets overlapping with it, the packet is captured (e.g., [1], [4]).
A more accurate model is based on the assumption that if and only if a packet’s signal-to-interference-plus-noise ratio (SINR) is greater than a certain decodability threshold, the packet is captured (e.g., [5]–[7]). We call this the SINR capture model. In all of the above works, packet lengths at different power levels are the same. Throughput is measured by the average number of successfully received packets per slot. Under the assumption of perfect power capture, the throughput is a function of offered traffic and the number of power levels as shown in Fig. 1. The derivation of the curves in Fig. 1 follows the method used in [1] and will be provided later. We can see from this figure that the use of multiple power levels can increase the system throughput. Similar conclusions are also drawn from the SINR capture model. However, the use of multiple power levels may have its side effect when energy efficiency is considered. Indeed, the use of multiple power levels increases the transmission energy per packet transmission. As energy efficiency is becoming a very important concern in wireless communication systems, we need to consider the tradeoff between increasing system throughput against improving system energy efficiency. In addition, we observe that for a packet that is sent at a higher power level, the length of the packet can be reduced since the can be met through transmission at correquired level of respondingly increased rate. By reducing the packet length, we can reduce the chance of packet collision. On the other hand, if all the users use the maximum possible power, the likelihood of packet overlap (and hence, collision) decreases, but the possibility of capture (with its beneficial effects on throughput) is eliminated. Thus, both the value of power level and the value of packet length are the variables that can be optimized to improve the system performance. The system performance that we consider in this paper is measured by the throughput and energy efficiency. A commonly used definition for energy efficiency is (e.g., [8]) energy efficiency
Manuscript received November 9, 1999; revised September 6, 2001. This work was prepared through collaborative participation in the Advanced Telecommunications and Information Distribution Research Program (ATIRP) Consortium sponsored by the U.S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL01-96-2-0002. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. W. Luo is with Bell Laboratories, Lucent Technologies, Holmdel, NJ 07733 USA (e-mail: [email protected]). A. Ephremides is with the Electrical Engineering Department and the Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]; [email protected]). Communicated by V. Anantharam, Associate Editor for Communication Networks. Publisher Item Identifier S 0018-9448(02)00049-4.
average number of correctly received packets per energy unit system throughput system average power consumption
where “system throughput” is the number of successfully received packets received per time unit, and “system average power consumption” is the total power consumption summed over all the users in the system averaged over time. For example, if there are two users in the system and each user’s power consumption is , then the system average power con. In random sumption is summed over two users, which is access, usually the higher the throughput the lower the energy
0018–9448/02$17.00 © 2002 IEEE
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Fig. 1. Throughput per packet slot versus offered traffic. A traditional model in which all packets are transmitted at the same packet lengths is used. Throughput is measured by the number of successfully received packets per slot.
efficiency because higher throughput usually implies larger number of packet transmissions per time unit, which leads to more collisions and lower energy efficiency. Hence, increase of both throughput and energy efficiency at the same time seems to be difficult if not impossible. An appropriate way to consider the system performance is to consider the system throughput under the constraint on the system average power consumption. In addition, in practical applications, packet peak power is also constrained. Therefore, in this paper, we consider the problem of maximizing the system throughput under the packet peak power constraint and system average power constraint. An attempt to consider the packet peak power constraint and system average power constraint in random access can be found in [9], where the concept of spread ALOHA is introduced. It is shown through Shannon theory that the capacity of low duty cycle ALOHA channel without capture cannot be exceeded by any other multiple-access methods given the same bandwidth and average power constraint. In this paper, we show the similar result that ALOHA without capture achieves the maximum achievable throughput through a different approach. The contribution of this paper is the analytical proof of Propositions 1 and 2, in which it states that in a single-receiver ALOHA system with power capture, the system maximum achievable throughput is maximized by the use of a single power level if the decodability threshold is greater than a certain value. In the proof, we derive the optimal setting of the power levels and packet levels by using a bounding method. In this bounding method, we set up an optimistic model whose throughput upper-bounds that of the real model. We prove that the maximum achievable throughput of the optimistic model is maximized by the use of a single power level. We also show that when a single power level is used, the real model has the
same throughput as that of the optimistic model. So for the real model, the maximum achievable throughput is also maximized by using the single power level. The organization of the paper is as follows. In Section II, the system model is described. In Section III, we consider uniform packet length with perfect power consumption and the derivation of the results shown in Fig. 1. In Section IV, we state the main results of this paper, outline the proofs, and discuss their implications. In Section V, we describe the optimistic model used in the proofs. Section VI concludes the paper. The mathematical details of the proofs are provided in the Appendix.
II. SYSTEM MODEL Suppose there are an infinite number of bufferless users in the system (see [10] for more discussions). Each user has a clock that is synchronized to a global clock so that slotted transmission can be achieved. Each user generates and transmits a packet at a certain power level independent of other users. We adopt the conventional assumption that the overall packet arrival pattern (including new arrivals and retransmissions) is Poisson (e.g., [1], [6], [11]). There are discrete power levels in the system. The values of power levels and packet lengths are determined at the system deployment and fixed once the system is in use. Time is slotted at each power level, and the slot durations can be different at different power levels. A packet transmitted at a power level has packet length equal to the slot duration at that power level, and transmission starts at the slot edge (see Fig. 2). Because the slot durations are different at different power levels, the time offset of the slots between different power levels cannot be controlled. Therefore, we do not have slot synchronization
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Fig. 2. Multiple power level random access. Within each power level, time is slotted. The packet is transmitted at the beginning of a time slot. No synchronization among different power levels is maintained.
between different power levels. If a packet is lost due to collision, the packet will be retransmitted at a later time. Each packet is transmitted or retransmitted on power level with probability for from to . The packet is involved in a collision when it has overlap with other packets either at the same power level or different power levels. We make the following assumptions. symbols. 1) Each packet contains 2) A packet can be received correctly if and only if a) during the whole transmission period, the packet has the highest power over all the other overlapping packets; b) during the whole transmission period, the packet’s SINR is always larger than a certain threshold . The SINR is defined as symbol energy to interference plus noise ratio, i.e. SINR
3) 4) 5) 6) 7)
(1)
are the transmission power and where , are the symbol duration of the packet, and interference and background noise energy in the output of the symbol matched filter. BPSK modulation scheme is used. Every packet is transmitted at constant power. The distances between the transmitters and the receiver are equal. . The maximum power that a packet can use is The system average power constraint is , for which (2)
where , , and are the offered traffic rate in packets per seconds, packet length, and power at the th power level, respectively, and is the number of power levels. 8) The offered traffic on the system (including newly arrivals and retransmissions) is Poisson.
Assumption 1) implies that the packet length is proportional to the symbol duration. Assumption 2a) is based on the single receiver scenario. We assume that the receiver can only receive one packet at a time and is always locked to the packet with the highest power. This can be accomplished in a variety of ways that expend a negligible amount of additional power or bandwidth so as to enable the receiver to correctly read appropriate packet preambles. We do not enter into a detailed discussion of this as it pertains to the very nature of capture. Here we simply assume capture can be achieved without regard to the specific mechanism or signal structure. When the receiver is receiving a packet while another higher power packet is coming, the receiver will switch to lock onto the coming higher power packet and drop the reception of the ongoing packet. So the packet is captured only if it has the highest power during its overall transmission period. According to Assumption 2b), note that it is possible that multiple overlapping packets may have the SINR level higher than the threshold at the same time, but at most one of them can be captured due to Assumption 2a). (which usually occurs when spread specIn addition, if trum or a strong error-correction code is used), it is possible that two packets with equal power may both have SINR level higher than . However, in this paper, we focus on the sce, in which case if two or more packets are nario when received at equal power at the same time, none of them is captured. We do not consider the multiple-packet-reception capability in this paper. Other statistical capture models can be found in [12], [13]. In Assumption 3), we assume binary phase shift keying (BPSK) modulation just for simple explanation. We can extend our results to other modulation schemes such as quaternary phase shift keying (QPSK) and quadrature amplitude modulation (QAM). Assumption 5) makes the received power and transmission power interchangeable. If this condition is not satisfied, we need to transform the received power to transmission power by using different scaling coefficients. Assumption 6) puts an upper power limit on individual packet transmission. Assumption 7), on the other hand, puts an upper power limit is the enon the system wide power consumption. In (2), ergy expended on a packet transmitted at the th power level is the number of packets transmitted at the th power and is the average power consumplevel per second, thus tion at the th power level. The summation over all power levels, , is the overall system average power consumption. Assumption 8) is a conventional assumption to approximate the system with a large number of users, in which the packet length is far shorter than the packet retransmission interval. There can be multiple power levels so that the benefit of capture on the throughput can be obtained. As described before, the transmitter randomly chooses one power level before transmission. The value of the power level and packet length should be set so that capture of the packet is possible when there are overlapping packets at the lower levels. There might be multiple packets at the lower power levels. Thus, a packet at the highest power level is captured only if its signal power is large enough to combat the accumulation of interference from the multiple packets at the lower power levels. So whatever the values of power levels and packet lengths are, the packet at the
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at the same time. Therefore, the throughput per packet slot contributed on the th power level is
(3)
The throughput maximization problem given the offered traffic is formulated as follows:
maximize subject to Fig. 3.
Multiple-power-level random access with uniform packet lengths.
highest power level cannot be captured with 100% probability. But at the least, the power levels should be set so that the capture of a packet is guaranteed when there is a single interfering packet at a lower power level. Otherwise, if the separation of the adjacent power levels is too small to guarantee the capture of a packet at the higher power level, the use of more energy transmitting a packet at the higher power level does not bring any benefit but wastes energy. In Section V, where we set up a hypothetical and optimistic model for the multiple-power-level system, however, we assume that if the values of power levels guarantee the capture of a packet at a higher power level when there is a single interfering packet at the lower power level, then the capture of a packet is guaranteed irrespective of the number of interfering packets at lower power levels as long as there is no other packet transmitted at the same or higher power level. The hypothetical and optimistic model is used to compare with a single-power-level system, and we find that the single-power-level system yields higher throughput under the packet peak power constraint and the system average power constraint. Therefore, it is safe to say that the single-power-level system is better than any multiple-power-level system.
III. UNIFORM PACKET LENGTH If packet lengths on different power levels are same, i.e., , then it is possible that slots on different power levels can be synchronized as shown in Fig. 3. In this section, we assume that the packet is captured if there are no other packets at the higher or the same power level overlapping with it. The same assumptions are adopted in [1]. power levels . Assume Consider that each packet is transmitted at power level with probability . If the offered traffic (including newly arrived and retransmission packets) on the system is assumed to be Poisson with arrival rate , then the offered traffic at any power level is also . Obviously, . Poisson with arrival rate . We denote by the offered traffic vector A packet is received correctly on the th power level, if there is no other packet transmitted at a higher or the same power level
(4)
are the variables to be optiare constants. The numerical solutions for are depicted in Fig. 1. As we can see from the figure, the use of multiple power levels can increase system throughput if power consumption is not considered. In the next section, we take into consideration the peak and average power consumption. Instead of fixing the offered traffic rate , we will give an upper limit on the system average power consumption, which provides the constraint on the system offered traffic as we will see later.
where mized,
and
IV. OPTIMALITY OF SINGLE-POWER-LEVEL SYSTEM Based on the assumptions in Section II, we have the following propositions. , if the Proposition 1: Given a peak power constraint , decodability threshold in (1) is greater than achieves maximum possible a single power level throughput. In such a system, packets are transmitted at maxwith corresponding packet length imum allowable power . and Proposition 2: Given both peak-power constraint average-power constraint , if the decodability threshold in , then a single power level (1) is greater than achieves maximum possible throughput. In such a system, with packets are transmitted at maximum allowable power . packet lengths Note that if is large enough, Proposition 2 degenerates into , the average power Proposition 1. In fact, when constraint does not affect the optimal system performance. Also, note that we state that the single-power-level system can achieve the maximum possible throughput. The detailed proofs of the above propositions are given in the Appendixes. Here, we sketch the proofs. The proofs are based on the construction of an optimistic model, which overestimates the throughput of the multiple-power-level system, but provides an exact evaluation of the throughput for the single-power-level system, a degeneration of the multiple-power-level system. This optimistic model simplifies the analysis. We find that a singlepower-level system yields the maximum throughput under this
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optimistic model, and thus the single-power-level system is also optimal under the real model. We construct a system, in which the values of the power levels and packet lengths are chosen so that if there are two overlapping packets transmitted at two adjacent power levels, respectively, the SINR of the packet at the higher power level is greater than the decodability threshold. Therefore, the packet at the higher power level can be captured when there is only one overlapping packet at the next power level. But if there are more than one overlapping packets, the SINR of the packet at the highest power level may not be greater than the decodability threshold. Whether it can be captured or not depends on how much interference power is accumulated from overlapping packets at the lower power levels. This makes analysis difficult because we have to calculate the probability that the accumulated interference is below certain value. In the optimistic model, however, we simply assume that the packet received at the highest power level can be received correctly (captured) if there is no other packet overlapping with it at the same power level, regardless of how many overlapping packets are transmitted at the lower power levels. Based on this assumption, we can determine the values of the power levels and packet lengths, and also determine the maximum throughput of the multiple-power-level system. The estimation based on this optimistic assumption obviously yields an upper bound of the system performance. Because for the single-power-level system there is no capture, this optimistic model on capture yields an exact estimation of the throughput. Furthermore, the analysis shows that a single-power-level system achieves maximum throughput over all the multiple-power-level systems under the optimistic model. Therefore, the single-power-level system is optimal under the realistic model. The idea of the , we can proof is that for an -power-level system -power-level system which has always construct an as that of the -power-level the same parameters but has a higher throughput, system for and less energy consumption. This means that for any multiple-power-level system, we can reduce the lowest two power levels to a single power level so that throughput/power performance can be increased. By iteratively repeating this procedure, we obtain that a single power level which operates at the highest allowable packet power with the shortest possible packet length and achieves maximum throughput. To meet the average power constraint while achieving the maximum possible throughput, the offered traffic rate must be equal to the lesser of the two quantities: and . This indicates that to achieve optimal energy efficiency in a random access system, it is essential to apply admission control on the offered traffic. Note that in both propositions, there is the additional assumption that the . decodability threshold be greater than a fixed value Indeed, our proofs work only if this condition is satisfied. If is less than , use of a the decodability threshold single power level is no longer necessarily optimal. In this case, a multiple-power-level system, in which all packets are still transmitted at the maximum allowable power but with different packet lengths can achieve higher throughput than the single-power-level system. Although it seems counterintuitive,
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the packets transmitted at the lowest level should have the shortest allowable packet length, while the packets at higher levels should have longer length. The benefit from capture in this case seems to outweigh the adverse impact from larger collision probability that the longer packet lengths imply. However, this observation follows from the optimistic model and it may or may not carry any real significance. For a real multiple-power-level system, the throughput is actually less than that of the optimistic model used in our derivations. Thus, even when , it is still possible that a single power level is optimal. We simply cannot prove our claim for values . of less than V. THE OPTIMISTIC MODEL In this section, we describe the optimistic model that we use for proving the propositions. As we mentioned before, the optimistic model is based on two simplifying assumptions. • Among mutually overlapping packets, the packet transmitted at the highest power level is captured, if there is no other packet transmitted at the same level overlapping with it. This is assumed to be true regardless of how many packets are transmitted at the lower level. • To make the capture of the packet at higher power level possible, the value of power levels and packet lengths should satisfy the condition that if there are two overlapping packets at consecutive power levels the higher level packet’s SINR is no less than the decodability threshold . The first assumption greatly simplifies the throughput expression, and the second assumption simplifies the value setup of power levels and packet lengths, as we will see in the next two subsections. We should emphasize that the propositions stated in the preceding section hold for the real model, the optimistic model is used just to prove the propositions. These two assumptions are optimistic because the way we choose power level and packet length does not guarantee that a packet can be captured if there are more than one packets overlapping with it at the lower power levels. A. ALOHA With
Power Levels
power levels . Assume Consider that each packet is transmitted at power level with probability . If the offered traffic (including newly arrived and retransmission packets) on the system is assumed to be Poisson with arrival rate , then the offered traffic at any power level is also . Obviously, . Poisson with arrival rate . We denote by the offered traffic vector Note that since the packet lengths at different power levels can be different, the unit for offered traffic is number of packets per second instead of number of packets per time slot. However, within each power level, the packets are still transmitted as in slotted ALOHA. So we can use the formula for slotted as the offered traffic ALOHA throughput [10] by using is the packet length, i.e., the slot duration, at per slot where level . We assume that if during the entire transmission time is of a packet at level , no other packet at level
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transmitted, then that packet is successfully received. We also assume, optimistically, that overlapping transmission of packets do not affect the reception of a packet at level at level , no matter how many of these may be. For a packet at level , no packets at other power levels can affect its decodability. So the throughput at that power level is packets per slot. Here, is the probability that no other packet at power level overlaps with the packet we are interested in. The slot duration is . The absolute throughput per time unit is
if it is the strongest packet at that time, the bit-error rate is sufficiently high to result in a packet reception error (collision). In this way, we will be able to account for excessive interference from packets at lower power levels. We assume that if one symbol in a packet is received incorrectly, the whole packet is received incorrectly. In other words, for a packet to be correctly received, every symbol must be correctly received. For any one symbol, we assume that it can be received correctly if and only if its SINR is greater than the decodability threshold. To be more precise, the symbol energy to interference-to-noise ratio should be larger than a threshold , that is,
For level , , a packet is successfully transmitted in a given time slot if it is the only packet transmitted at that level, , and if no which happens with probability packet at higher power levels overlaps with it. There may be several slots corresponding to power level that overlap with a given slot corresponding to power level . The total duration . Thereof each of those overlaps is greater than overlaps fore, the probability that no packet at level . with the packet at level is less than Because the transmission of packets at different power levels are independent, the probability that no packet at higher power levels overlaps with the packet of interest at level is less than . So, the throughput at level is bounded as follows:
(7) are the power and the symbol duration of the where and packet, respectively; denotes interference from overlapping is the background white noise, and is the SINR packets, decodability threshold. Now let us consider how the interference from overlapping packets affects SINR. If two packets, say 1 and 2, with different symbol durations, are overlapping, we need to calculate the interference after the matched filter for each packet. Assume and are the symbol durations of packets 1 and 2, respectively. We can write the received signal as
(8) (5) Note that this bound also holds for the case that all the power levels have the same packet lengths and transmission slots between different power levels are synchronized. The total throughput for the overall system is then bounded by
are amplitude, information symbols and where , respectively; symbol waveforms of the packet is background noise. For simplicity, we assume BPSK modulation and rectangular waveforms. It is easy to extend our analysis to more complex modulation schemes with little essential modification. Without loss of generality, we assume and
(6) , i.e., when a Note that strict equality holds in (6) when single power level is employed (that is, it reduces to the classical slotted ALOHA throughput formula). This point is very important in our proof. B. Power Levels and Packet Lengths In this section, we determine the values of power levels and packet lengths in the optimistic system. Those values have to satisfy a condition so that a packet overlapping with another packet one power level lower can be captured. We assume that there is a single receiver in the system. Furthermore, we assume that the receiver is able to select at any given time to decode the packet with the highest power. We also assume that after the packet is selected by the receiver, it can be passed to the corresponding matched filter and decoder. The packet’s SINR should be larger than a decodability threshold, otherwise even
The illustration of overlap symbols from two users is shown in Fig. 4. In this figure, we are looking at a symbol of packet 1 that is involved in collisions with packet 2 throughout the symbol duration. If , the number of symbols per packet is relatively large this will be the case for at least one symbol when a packet is involved in a collision, so we may do this without essential loss of generality. For the th symbol of packet 1 after the matched filter, the received energy of the desired signal is . The background white noise energy is . Interference from packet 2 after the matched filter is , and ) given by (see Fig. 4 for definition of
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Fig. 4.
Two interfering users. T
= (K + )T
k + )T
=( +
, where ;
; < 1, k = K , or k = K 0 1 (only possible when T
T
).
(9) and where in the second equality, we assume that . In fact, those properties can be extended to other modulation schemes in which the center of the constellation is the origin and average amplitude of the constellation is unit. , i.e., , it follows from When (9) that
(14)
(10)
It does not make sense to use a high power level unless capture is possible when a packet at such a power level is interfered with by a packet at a lower power level. Our SINR calculation above must shows that this must imply that the power level and necessarily satisfy the condtions
, i.e., , the values of , , When and in (9) can be either one of the two possibilities:
(15)
1) 2)
and and
while for the requires
.
th (i.e., the lowest) level
, decodability simply
It then follows from (9) that (16)
(11) Now suppose packet 1 and packet 2 overlap with each other. symbols, Recall that we assume that in each packet there are and , and let , i.e., , and . It follows that the symbol SINR of packet 1 is
(12) When
, i.e.,
, then using (10), we have (13)
and when
, using (11), we have
When more interfering packets are present, even if (15) is satisfied, a single highest level packet may not be received correctly because accumulation of interference from multiple lower level packets precludes correct reception. However, we assume an optimistic model in which if (20) and (21) are satisfied, the packet can be received correctly as long as no other packet transmitted at higher or equal power levels overlaps with it (regardless of how many lower level packets are transmitted at the same time). The reason we use (15) and (16) is their relative simplicity. They provide an optimistic model which upper-bounds the performance of the real system. We will show later that the maximum value of the upper bound of the optimistic capture system of (15) and (16) can be actually achieved by a real single-power-level system. Thus, we will prove that the system that achieves that maximum is indeed the optimum actual system despite the apparent largesse of our assumption regarding capture. C. System Power Constraints We assume that no packet can exceed the peak power con. Furthermore, to capture the finiteness of energy straint
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These characteristics are used in the proof of the two propositions. 1) Among mutually overlapping packets, the packet transmitted at the highest power level is captured, if there is no other packet transmitted at the same level overlapping with it. This is assumed to be true regardless of how many packets are transmitted at the lower level. The total throughput of the multiple-power-level system is given optimistically by (19)
Fig. 5. Maximum throughput as a function of average system power constraint.
we also specify an average power constraint for the entire system. Although this construction does not capture the finiteness of battery life at each transmitter it provides a cap to energy expenditures system-wide. The overall system power consumption for the ALOHA system with power levels is given by
This follows from (6) by always assuming equality instead of “ ” in (6). Again, note that for a single-powerlevel system, equality holds in (6), which means (19) holds in the real single-power-level system even without the optimistic assumption. 2) To make the capture of the packet at higher power level possible, the value of power levels and packet lengths should satisfy the condition that if there are two overlapping packets at consecutive power levels the higher level packet’s SINR is no less than the decodability threshold . Optimistically, we assume that if the following is satisfied then the only packet at the highest power level is sure to be captured
(17) (20)
We can also express the throughput per “energy” unit as (18) depends on the offered traffic Note that the quantity through the resulting throughput as well as through the denominator . Suppose that, for a given constraint value , we can determine the maximum achievable throughput . Then, can be thought of as a function of . We call it the throughput is initially a power function. It is rather obvious that nondecreasing function. But throughput cannot keep increasing with . At some point, the throughput must saturate at a maximum value. Increasing beyond that point is not going to help increase the maximum stable throughput since the peak power constraint will kick in and will not permit further increases by making the packets shorter. Thus, the available additional power can only be consumed by means of more packet transmissions, which will start causing increasing collisions, thereby reducing the throughput. In Fig. 5, we show the throughput power functions for the optimal single-power-level system. If we look at and from this figure, we can the relationship between decreases as increases. Since we hope to insee that crease the throughput as well as energy efficiency, it is more appropriate to consider the throughput power function instead alone. of energy efficiency D. Summary of the Optimistic Model The optimistic model is discussed in the previous subsections. We summarize the characteristics of the optimistic model here.
(21) In a single-power-level system, there is no packet capture. Therefore, only (21) is required. These characteristics in the optimistic model are optimistic only in the multiple-power-level system. In the single-powerlevel system, these assumptions become realistic. VI. CONCLUSION In this paper, we examined the throughput and energy efficiency performance of multiple-power-level random-access systems. We described the relationship among the power constraints, the packet lengths, and the throughput. We found that a single maximum power level achieves the best throughput–power performance under a condition on the decodability threshold value. APPENDIX A. Proof of Proposition 1 We first prove some useful lemmas. is maximized at . Lemma 1: The function In addition, the function is monotonically increasing in the in, and monotonically decreasing in the interval terval .
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Proof: It follows trivially from the form of the derivative (22)
as that of the -power-level system . The parameters on the th power level (lowest power level in ) are as follows:
(27) Lemma 2: If
,
, then (23)
and
It can be seen that
is monotonically decreasing. Proof:
(28)
(24) where the first inequality follows from the fact that the second inequality follows from the fact that is monotonically decreasing. It is trivial to see that .
, and . So
For the system , it is trivial to see that the capture criterion (20) is satisfied. Next, we need to compare the throughputs of the two systems. Because the throughput for the highest th power levels are the same for both system and , we only need to compare the throughput of ’s lowest power level with the throughput of ’s two lowest power levels. Following (19), the throughput of ’s lowest power level is
Now we provide the proof of Proposition 1 as follows. Proof: Suppose there are
power levels
(29)
We can construct a new system with power levels, which is under the same peak power constraint. The parameters of the highest power levels are the same for th power level the two systems. The parameters on the of the new system are properly chosen such that the throughput is higher than that of the -power-level system according to the optimistic model we described in the paper. We will show . Therefore, that such construction can always be done if we can always increase the throughput by reducing the number of power levels. Consequently, the optimal system (in the sense of maximizing the throughput) is the one that only has a single power level. Note that our optimistic model overestimates the . For , the optimistic model cointhroughput for cides with reality. The optimal single-power-level system yields greater throughput than the overestimate of the throughput of the multiple-power-level system. So the single-power-level system is also optimal in reality. Now we start to show that construction of the new system power levels is always possible. The capture and with throughput formulas are all based on the optimistic model we described before. For the -power-level system (say, system ), recall from (20) and (21) that
The throughput of ’s two lowest power levels is
(30) where the inequality follows from (28). Our objective is to show that (29) and (30), it suffices to show that
. From
(31) (we will consider the case First, suppose later). Then (25) becomes (25) (26) The new system is constructed as follows. The parameters of the highest power levels are the same
(32)
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From the first inequality of (32), we obtain
(39) where the inequality follows from Lemma 1. Note that the last and in . term in (39) is monotonically decreasing in From (37) and (38), Inequality (39) becomes
From the second inequality of (32)
Notice that because
, we have (33)
Consider the term on the left-hand side of (31) (40) Let (40) that
. By using Lemma 2, we obtain from
(34) where the second inequality follows from Lemma 1, the third , and the fourth inequality follows from the fact that inequality follows from (33). Thus, we prove (31) and hence . the proposition, for the case , inequalities in (25) become If
(41) . Thus, we which establishes (31) for the case proved that the optimal system should have a single power level. The throughput of the single power level system is (42)
(35) Therefore,
. We can see that the where equality holds if and only if value of the throughput increases as decreases. But cannot be arbitrarily small. In order to maintain the requested SINR . Therefore, level, we must have (43)
(36)
So the maximum achievable throughput is achieved when
So (37) This completes the proof of Proposition 1. and from the second inequality of (35) B. Proof of Proposition 2 (38) Now, let us consider the term on the left-hand side of (31)
We also need to introduce several lemmas before we proceed to the proof. Lemma 3: The function . terval Proof: In the interval
is convex in the in-
(44) So the function
is convex.
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From Lemma 3, if
and
Thus, in both cases we have
then (45)
(52)
This property will be used to prove the following lemma. Lemma 4: If
,
,
From (50), it follows that
, and we let (46) (47)
, then
and
(48) Proof: Let
The proof of the lemma is completed.
, and (49)
We now prove Proposition 2. Proof: The idea of the proof is similar to that of Proposition 1. Suppose there are power levels in a system, say system . As before, we can prove that a new power levels can be constructed under the system with same power constraints with higher throughput. The construction of the new system is similar to that of the proof of Proposiof the highest power tion 1. The parameters levels are the same for the two systems. Assume that the average power used by the two lowest levels in system is
The term on the left-hand side of (48) can be rewritten as
(53) (50) Consider first the case that is convex, we have tion
. Because the func-
For the system , the parameters of the lowest level are
th power
(54)
The average power at the lowest level of system
is (55)
Now consider the case that . By Lemma 1 is obvious that
. Let
. It
It follows that
(51) , The first inequality follows monotonicity of the function and the second inequality follows the fact that the function is . convex in the interval
highest levels are the Because the average power of the same for the two systems and the average power of the last power level of is not greater than that of the last two power levels of according to (55), the total average power of the system is not greater than that of the system . Therefore, if system satisfies the average power constraint, system also satisfies the average power constraint. We now show that the throughput of is greater than that of . The throughput of the highest levels are the same for the two systems. So we only need to show that the throughput th power level of the system is greater than the of the throughput of the two lowest power levels of the system . The throughput of the lowest level of the system is
LUO AND EPHREMIDES: POWER LEVELS AND PACKET LENGTHS IN RANDOM MULTIPLE ACCESS
As in (30), the throughput of ’s two lowest power level is
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In addition,
(62) Let
be a positive number such that (63)
It is trivial to see that
. So from (62), we have (64)
Now let us return to the objective inequality (57). The term on the left-hand side is (56) The objective is to show that suffices to show that
. Then it
(57) , then following (54). In that If case, inequality (57) reduces to (31) which is already proven in Proposition 1. Therefore, we only need to prove (57) for the case . , If Again, we consider first the case , it follows that (33) is still true, i.e.,
where the second inequality follows from (64), and the third inequality follows from Lemma 4. Thus, we prove the inequality . in (57) for the case , following the same reasoning as in the proof If of Proposition 1, we observe (37) and (38). That is,
(58)
(65)
In addition, the capture criterion (32) holds. For reader’s convenience, we repeat that criterion here
In addition, because of (53), we have
(66) and (59) Let
where we use the second inequality in (35) and (36). Then, the left-hand side of (57) becomes
(60) , it follows that Because we assume , and can also see from (54) that It also follows from (53) that
(67)
. We .
(61)
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ACKNOWLEDGMENT The authors wish to thank V. Anantharam for discussions and comments leading to improvement of presentation and quality of this paper. where the first equality follows from the assumption that , and the first inequality follows from (66); the second inequality follows from (67); the third inequality follows from ; and the last inequality follows the assumption that . from the assumption that So (57) is proven. We know that the optimal system should have a single power level. For a single-power-level system under the peak and average power constraints, we must have
(68) which implies
(69) The throughput is then given by By Lemma 1, if achieved at throughput is achieved at
, the maximum throughput is ; otherwise, the maximum . In short (70)
and the maximum achievable throughput is achieved when (71) This completes the proof of Proposition 2.
REFERENCES [1] J. J. Metzner, “On improving utilization in ALOHA networks,” IEEE Trans. Commun., vol. COM-24, pp. 447–448, Apr. 1976. [2] J. P. Linnartz, Narrowband Land-Mobile Radio Networks. Norwood, MA: Artech House, 1993. [3] C. T. Lau and C. Leung, “Capture models for mobile packet radio networks,” IEEE Trans. Commun., vol. 40, pp. 917–925, May 1992. [4] N. Abramson, “The throughput of packet broadcasting channels,” IEEE Trans. Commun., vol. COM-25, pp. 117–128, Jan. 1977. [5] R. O. LaMaire, A. Krishna, and M. Zorzi, “Optimization of capture in mulitple access radio systems with Rayleigh fading and random power levels,” in Multiaccess, Mobility and Teletraffic for Personal Communications, B. Jabbari, P. Godlewski, and X. Lagrange, Eds. Boston, MA: Kluwer, 1996, pp. 321–336. [6] I. Cidon, J. Kodesh, and M. Sidi, “Erasure, capture, and random power level selection in multipleaccess systems,” IEEE Trans. Commun., vol. 36, pp. 263–271, Mar. 1988. [7] C. C. Lee, “Random signal levels for channel access in packet broadcast networks,” IEEE J. Select. Areas Commun., vol. 5, pp. 1026–1034, July 1987. [8] Y. W. Leung, “Mean power consumption of artificial power capture in wireless networks,” IEEE Trans. Commun., vol. 45, pp. 957–964, Aug. 1997. [9] N. Abramson, “VSAT data networks,” Proc. IEEE, vol. 78, no. 7, pp. 1267–1274, July 1990. [10] D. Bertsekas and R. Gallager, Data Network, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1992. [11] L. Kleinrock and F. Tobagi, “Packet switch in radio channels: Part I—Carrier sense multiple-access modes and their throughput-delay characteristics,” IEEE Trans. Commun., vol. COM-23, pp. 1400–1416, Dec. 1975. [12] B. Hajek, A. Krishna, and R. O. LaMaire, “On the capture probability for a large number of stations,” IEEE Trans. Commun., vol. 45, pp. 254–260, Feb. 1997. [13] H. Zhou and R. H. Deng, “Capture model for mobile radio slotted ALOHA systems,” Proc. Inst. Elec. Eng.–Commun., vol. 145, no. 2, pp. 91–97, Apr. 1998.
