# Partial Interference and Its Performance Impact on Wireless Multiple Access Networks

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## Abstracts

To determine the capacity of *wireless multiple access networks*, the interference among the wireless links must be accurately modeled. In this paper, we formalize the notion of the partial interference phenomenon observed in many recent wireless measurement studies and establish analytical models with tractable solutions for various types of wireless multiple access networks. In particular, we characterize the stability region of IEEE 802.11 networks under partial interference with two potentially unsaturated links numerically. We also provide a closed-form solution for the stability region of slotted ALOHA networks under partial interference with two potentially unsaturated links and obtain a partial characterization of the boundary of the stability region for the general M-link case. Finally, we derive a closed-form approximated solution for the stability region for general M-link slotted ALOHA system under partial interference effects. Based on our results, we demonstrate that it is important to model the partial interference effects while analyzing wireless multiple access networks. This is because such considerations can result in not only significant quantitative differences in the predicted system capacity but also fundamental qualitative changes in the shape of the stability region of the systems.

### Keywords

Stability Region Slot Aloha Backoff Stage Admissible Region Successful Transmission Probability## 1. Introduction

In a *wireless network*, all stations communicate with each other through wireless links. A fundamental difference between a wireless network and its wired counterpart is that wireless links may *interfere* with each other, resulting in performance degradation. Therefore in the study of wireless networks, one important performance measure is the capacity of the network when the effects of interlink interference are considered.

In establishing the capacity of a wireless network, we have to predict whether the wireless links interfere with each other. Two most common interference models in the wireless networking literature, namely, for example, *protocol model* and *physical model* [1], were proposed to predict whether transmissions in a wireless network are successful. In these interference models, one key assumption is that interference is a *binary* phenomenon, that is, either the links mutually interfere with each other to result in *total* loss of throughput of a target link, or there is no link throughput degradation at all. In other words, these models exclude the possibility that interfering links can be active simultaneously and still realize their capacity partially. However, recent empirical studies [2, 3, 4, 5, 6] have shown that these binary interference models are not valid in practice. Instead, measurement results have confirmed that there is a *nonbinary* transitional region [2, 4] (also known as the *gray zone* in some literature [3]) for the successful packet reception rate (PRR) of a wireless link which changes from zero, that is, 100% lossy, to almost 100%, that is, perfectly reliable, as its signal-to-interference-plus-noise ratio (SINR) increases. These studies have indicated that the range of the transitional regional (in SINR) can exceed 10dB for various types of practical networks including IEEE 802.11a wireless mesh [3, 7] and other low-power multihop sensor networks [2, 4]. More importantly, measurement studies on large-scale wireless mesh testbeds [8, 9] found that a significant number of links in those testbeds were indeed operating at the SINR *transitional region*, that is, with intermediate level of PRR between zero and 100%. In this paper, we call this phenomenon *partial interference*. From the physical layer implementation perspective, the partial interference phenomenon can be viewed as a consequence/manifestation of the probabilistic nature of signal decoding in the receiver, its interaction with the well-known capture effect [10, 11], and the specific implementation of the frame reception and capture algorithms in individual chipsets [12].

While the phenomenon of partial interference in wireless networks has been widely observed as mentioned above, its incorporation in the performance modeling of such networks is still in its infancy. Most of the efforts in this direction so far ([2, 7, 12, 13]) have been limited to the characterization of the nonbinary transitional region in the PRR-versus-SINR curve based on measurement data [7, 12, 13] or some analytical means [2, 14]. However, once the PRR-versus-SINR curve is obtained, they only resort to simulations to evaluate the effects of partial interference on the system performance.

While the phenomenon of partial interference in wireless networks has been widely observed as mentioned above, its incorporation in the performance modeling of such networks is still in its infancy. Most of the efforts in this direction so far ([2, 7, 12, 13]) have been limited to the characterization of the nonbinary transitional region in the PRR-versus-SINR curve based on measurement data [7, 12, 13] or some analytical means [2, 14]. However, once the PRR-versus-SINR curve is obtained, they only resort to simulations to evaluate the effects of partial interference on the system performance.

In this paper, our focus is to develop *analytical* models with tractable solutions for various types of wireless multiple access networks which can accurately capture the performance impact of partial interference. Via analytical and numerical results throughout this paper, we demonstrate that it is important to model the partial interference effects while analyzing wireless multiple access networks. This is because such considerations can result in not only significant quantitative differences in the predicted system capacity but also fundamental qualitative changes in the shape of the stability region of the systems (e.g., from a concave to a convex region).

To quantify the impact of interference on multiple access networks, we propose an analytical framework to characterize partial interference for two representative types of multiple access wireless networks, namely, the IEEE 802.11 Wireless LANs and the classical slotted ALOHA networks. For IEEE 802.11 Wireless LANs, we extend the single-channel Markov model in [15] to take into account the *unsaturated* traffic conditions, the SINR attained at the receivers, and the modulation scheme employed. These modifications result in a *partial interference region*, which cannot be captured by the binary interference models used in previous works. We also find out the *stability (admissible) region* of IEEE 802.11 networks with two interfering, potentially *unsaturated* links numerically. For slotted ALOHA networks, we extend the model in [16] to derive the exact stability region of slotted ALOHA with two links while considering partial interference. We show that as the link separation increases, the stability region obtained expands *gradually* under partial interference, as in the case of 802.11.

Despite the simplicity of slotted ALOHA, characterizing its exact stability region with *unsaturated* links is extremely difficult and has remained to be a key *open* problem for decades when there are more than two, potentially unsaturated links in the system [16, 17, 18, 19, 20, 21, 22, 23]. However, by extending the FRASA (*Feedback Retransmission Approximation for Slotted ALOHA*) approach [24] to model the partial interference effects, we obtain a *closed-form* approximation for the exact stability region for *any* number of links.

- (1)
After reviewing related work in Section 2, we formalize the notion of partial interference in Section 3 and then demonstrate its significant performance impact on different types of wireless networks via various examples and their analytical/numerical results throughout the rest of the paper. As an illustration, we show in Section 4 that, by considering partial interference effects while scheduling traffic in a wireless network of regular topology, the gain in

*network capacity across unit cut*can be as high as 67%. - (2)
In Section 5, we establish a model to analyze the effects of partial interference on the throughput of IEEE 802.11 networks with

*unsaturated*links. Our approach enables one to compute numerically the stability region of any 2-link 802.11 system under*unsaturated*traffic conditions. - (3)
In Section 6, we investigate the effects of partial interference on the capacity of a slotted ALOHA system with

*unsaturated*links by (i) establishing the*exact*stability region in closed-form for the 2-link case and (ii) providing a closed-form,*partial*characterization of the stability region of the general M-link case. - (4)
In Section 7, we extend the FRASA approach in [24] to yield a

*closed-form approximation*for the stability region of the general M-link slotted ALOHA system while considering partial interference effects. The capacity region derived by our approximation and the corresponding simulation results are provided for some sample cases. Again, this is to demonstrate the potential qualitative and quantitative differences in the system capacity region when the effect of partial interference is taken into account. We then conclude the paper in Section 8.

## 2. Related Work

In [1], two interference models, called the *protocol model* and the *physical model*, were introduced. The protocol model states that a transmission is successful if the corresponding receiver is located inside the transmission range of the transmitter, and all other active transmitters are located outside the interference range of the receiver. In the physical model, the transmissions from other transmitters are considered as noise, and a transmission is successful if the SINR attained at the receiver exceeds a certain threshold. Based on these models, the capacities of a multihop wireless network under random and optimal node placement were derived.

In [5], the authors measured the interference among links in a single-channel, static 802.11 multihop wireless network. They measured the interference between pairs of links by the *link interference ratio* and observed that this ratio exhibited a continuum between 0 and 1. In [6], two interfering links were set up in a wireless network with multiple partially overlapped channels to measure TCP and UDP throughputs of an individual link. It was found that the throughputs increased smoothly when the separation between the links increased. The throughputs increased more rapidly as the channel separation between the links increased. Such nonbinary transitional region in the link throughput (or PRR equivalently) as the receiver SINR varies has also been observed by numerous measurement studies including [2, 3, 4]. These experimental results all confirmed that the binary assumption in the protocol or physical interference models are not valid in practice.

There has been some analytical work on finding the relationship between the SINR attained at a receiver and the throughput (or PRR equivalently) achieved by the corresponding wireless link. In [14], a methodology for estimating the packet error rate in the *affected wireless network* due to the interference from the *interfering wireless network* was presented. The throughput of the affected wireless network was found to increase continuously with the SINR attained at the corresponding receiver, which increased with the separation between the networks. Similarly, [2] derived expressions for the PRR as a function of distance, radio channel parameters, and the modulation/encoding scheme used by the radio. However, they did not provide analytical model on how the PRR function would impact the performance of the corresponding networks.

In [25], the throughput achieved by an M-link IEEE 802.11 network under physical layer capture was derived. While their analysis can be viewed as another case study of the effects of the partial interference over 802.11 networks, their approach only works for the case where all of the links are always saturated, that is, with infinite backlog at the transmitter side. In contrast, the approach proposed in Section 5 of this paper can handle unsaturated links and has provided explicit numerical solutions for the stability region of the 2-link case.

The study of the stability region of Open image in new window -user infinite-buffer slotted ALOHA was initiated by the study in [17] decades before and is still an ongoing research. The authors in [17] obtained the exact stability region when Open image in new window under the collision channel (i.e., binary interference) model. References [18, 19] used *stochastic dominance* and derived the same result as in [17] for the case of Open image in new window .

For general Open image in new window , there were attempts to find the exact stability region, but there was only limited success. Reference [21] established the boundary of the stability region, but it involves stationary joint queue statistics, which still do not have closed form to date. Instead, many researchers focused on finding bounds on the stability region for general Open image in new window . Reference [17] obtained separate sufficient and necessary conditions for stability. References [18, 19] derived tighter bounds on the stability region by using stochastic dominance in different ways. Reference [22] introduced *instability rank* and used it to improve the bounds on the stability region. However, the bounds in [18, 22] are not always applicable. Also, the bounds obtained may not be piecewise linear.

With the advances in multiuser detection, researchers also studied this problem with the *multipacket reception* (MPR) model. Reference [23] studied this problem in the infinite-user, single-buffer, and symmetric MPR case. Reference [16] considered the problem with finite users and infinite buffer. They obtained the boundary for the asymmetric MPR case with two users, and also the inner bound on the stability region for general Open image in new window .

## 3. Partial Interference—Basic Idea

As an illustration to the methodology in [14], assume the underlying modulation scheme used is binary phase shift keying (BPSK). The distance between the transmitter and the receiver and that between the interferer and the receiver are Open image in new window and Open image in new window meters, respectively. The transmission power of the transmitter and the interferer are Open image in new window and Open image in new window watts, respectively.

In general, Open image in new window depends on the BER, which, in turn, is a function of the SINR at the receiver as well as the specific modulation scheme being used. While we use BPSK as an example here, the actual expression for Open image in new window under other modulation schemes can be readily derived as shown in [2, Table Open image in new window ].

We observe that if the value we assign to Open image in new window is too large (or the threshold distance is too large), we underestimate the throughput that the links can achieve. On the other hand, if Open image in new window is too small (or the threshold distance is too small), we introduce excessive interference into the network. In other words, it is difficult to use a single threshold to describe accurately the relationship between interference and throughput of each link in a network.

## 4. Capacity Gain When Partial Interference Is Considered

In this section, we demonstrate that there is a gain in system capacity when the effect of partial interference is considered. We consider one variation of the Manhattan network [27], that is, a network consisting of a rectangular grid extending to infinity in both dimensions. The horizontal and vertical separations between neighboring stations are denoted by Open image in new window and Open image in new window , respectively.

Under infinite transmitter backlog, the packet-level capacity of each link, that is, the maximum packet reception rate without interference, is denoted by Open image in new window . We assume that differential binary phase shift keying (DBPSK) is employed and a packet consists of Open image in new window bits. We use the two-ray ground reflection model (1) as in previous section to model the path loss. To apply the physical model, we let the SINR threshold Open image in new window be the case that the packet error rate is Open image in new window , that is, Open image in new window , where Open image in new window is the bit error rate of DBPSK [26]. We let Open image in new window and Open image in new window , therefore the SINR requirement is Open image in new window . Assuming that there is no interferer, this SINR requirement is met when the length of a link is smaller than 493 meters.

*capacity across unit cut*Open image in new window as the performance metric, where Open image in new window is the ratio of the horizontal separation to the vertical separation. It is a measure on how much traffic we can send through a cut in a network on average while physically packing the links towards each other. Consider the SINR attained at the receiver marked with the blue circle, which has the position assigned as the origin in the Cartesian coordinate plane. We assume that all stations transmit with power Open image in new window , and each station has a background noise power of Open image in new window . The SINR is defined by Open image in new window , where Open image in new window is the received power from the intended transmitter and Open image in new window is the power received from all interferers. The packet-level capacity achieved by each link, that is, the successful packet reception rate at the receiver, is Open image in new window under our partial interference model. On the other hand, under the physical interference model, Open image in new window if Open image in new window and Open image in new window otherwise. A

*cut*Open image in new window in the network is an infinitely long horizontal line. Let Open image in new window be the set of all active transmitters such that Open image in new window intersects the link used by Open image in new window . We divide Open image in new window into segments Open image in new window , Open image in new window , where

and Open image in new window is the Euclidean norm. Then the length Open image in new window of the cut occupied by an active transmitter is the length of Open image in new window , and the capacity across unit cut is therefore Open image in new window , where Open image in new window is the fraction of time that a link is active.

Capacity gain in the modified Manhattan network with different lengths of links.

% increase | |||||
---|---|---|---|---|---|

350 | 3.02 | 0.2365 Open image in new window | 2.55 | 0.2671 Open image in new window | 12.93% |

400 | 3.48 | 0.1796 Open image in new window | 2.73 | 0.2163 Open image in new window | 20.45% |

450 | 5.58 | 0.0996 Open image in new window | 3.06 | 0.1661 Open image in new window | 66.82% |

If we allow partial interference, the active transmitters can be packed more closely. When Open image in new window decreases, more spatial reuse is allowed. The increase in the density of active transmitters outweighs the degradation in capacity, so there is an increase in the capacity across unit cut. However, if Open image in new window decreases further, interference will be the dominant factor in determining the capacity across unit cut. Therefore, the capacity across unit cut drops, and there exists Open image in new window for the optimal performance under partial interference. This behavior is depicted by the blue solid line in Figure 3. The optimal value of Open image in new window under partial interference is Open image in new window , and the capacity across unit cut is Open image in new window bits per second per kilometer. There is a percentage increase of 66.82% in the capacity across unit cut when the effect of partial interference is considered. Similar results are shown in Table 1 and Figure 3 for Open image in new window meters. The percentage increase is larger when the links are longer, but the capacity achieved by each link reduces. We can view Open image in new window as the carrier sensing range in the modified Manhattan network with the scheduling pattern in Figure 2, as it is the smallest horizontal separation allowed by the physical model. We observe that if the length of the links increases, the carrier sensing range needs to be increased in a larger proportion. Also, this carrier sensing range is much larger than double of the length of the links, which is the usual convention used in defining the relationship between carrier sensing range and transmission range.

## 5. Partial Interference in 802.11

In this section, we study partial interference in 802.11 networks, the prevalent wireless random access networks. We present an analytical framework to characterize partial interference in a single-channel wireless network under unsaturated traffic conditions, which uses 802.11b with basic access scheme and DBPSK. We show that there is a partial interference region, in which the throughput of each link increases continuously with the separation between the links in the network. As a first attempt to relate the capacity-finding problem in wireless random access networks to the stability region of such networks, we derive the admissible (stability) region of an 802.11 network with two potentially unsaturated links numerically.

### 5.1. The 802.11 Model

We present our framework to characterize partial interference in a wireless network with random access protocols. In this framework, we derive the *transmission probabilities* Open image in new window and the *packet corruption probabilities* Open image in new window of the links in the network. Open image in new window is the probability that a station transmits in a randomly chosen slot, while Open image in new window is the probability that a packet is received with error.

- (i)
The network consists of two links Open image in new window and Open image in new window , where Open image in new window and Open image in new window denote the transmitter and the receiver of the links, respectively, Open image in new window .

- (ii)
There are a constant

*buffer nonempty probability*Open image in new window that the transmission buffer of Open image in new window is nonempty and a constant*channel idle probability*Open image in new window that Open image in new window senses the channel to be idle, Open image in new window . - (iii)
Open image in new window transmits with power Open image in new window , and the background noise power at Open image in new window is Open image in new window , Open image in new window .

- (iv)
Channel defects like shadowing and fading are neglected, and a generic path loss model Open image in new window is used to model the wireless channel, where Open image in new window is the propagation distance, Open image in new window is the path loss exponent, and Open image in new window is a constant.

- (v)
The interference from other transmitters plus the receiver background noise is assumed to be Gaussian distributed.

- (vi)
All bits in a packet must be received correctly for correct reception of the packet.

- (vii)
The size of an acknowledgement is much smaller than that of the payload, so the bit errors on acknowledgement are negligible.

We follow the approach as in [15], using a discrete-time Markov chain to model the 802.11 Distributed Coordination Function (DCF) and obtain the transmission probability of a station. An ordered pair Open image in new window is used to denote the state of the Markov chain, where Open image in new window represents the backoff stage and Open image in new window is the current backoff counter value. In stage Open image in new window , Open image in new window is in the range Open image in new window , where Open image in new window is the contention window size in stage Open image in new window . Open image in new window is the maximum number of backoff stages. However, there are some discrepancies between the model in [15] and the actual behavior of 802.11 DCF. First, the model assumes that a station retransmits indefinitely until the packet is successfully transmitted. This assumption is inconsistent with 802.11 basic access scheme. Also, the model does not account for the unsaturated traffic conditions, which is the scenario appeared in practical situations.

The details of the Markov chain and the derivation of this equation can be found in [31].

The packet corruption probability is calculated according to the modulation scheme used in the PHY layer, the distance between the transmitter and the receiver, and the existence of nearby interferer(s). For a fixed *carrier sensing threshold* Open image in new window , we differentiate into two cases, whether both transmitters can sense the transmission of each other or not.

where Open image in new window , Open image in new window , and Open image in new window represent the number of bits in the PHY header, the MAC header, and the payload, respectively.

The channel idle probability is defined as follows. If Open image in new window can sense the transmission of Open image in new window , then Open image in new window will consider the channel to be idle whenever Open image in new window is inactive, that is, Open image in new window ; otherwise Open image in new window always senses the channel to be idle and Open image in new window .

*offered load*and the

*carried load,*respectively. We calculate Open image in new window by

Similarly, we can obtain three equations for link Open image in new window . With these six equations we can solve for the variables Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window by Newton's method [32] and obtain the loadings of these two links by (14).

### 5.2. Some Analytical Results

Parameters used for the analytical results.

192 bits | 272 bits | ||
---|---|---|---|

7 | 5 | ||

9020 Open image in new window s | 9020 Open image in new window s | ||

32 | |||

24.5 dBm | Open image in new window 88 dBm | ||

1 | 1.5 m |

In the following we attempt to find the maximum carried loads of each link in various scenarios. One observation from solving the system of equations in Section 5.1 is that the carried load will be smaller than the offered load when the offered load is too large. This corresponds to the instability of 802.11 observed in previous works (e.g., [15]). Therefore, we use binary search to find the maximum carried load under stable conditions. Initially, the search range for the offered load is between 0 and 1 Mbps. We choose the midpoint of the search range to be the offered load and solve the system of equations. If the resultant carried load is the same as the offered load, the offered load can be increased, and the next search range will be the upper half of the original one. Otherwise, the offered load results in instability, and the next search range will be the lower half of the original one. This procedure is repeated until the search range is sufficiently small.

Consider the curve corresponding to the carrier sensing threshold of −78 dBm in Figure 6(c), which is a common value used in NS-2 simulation and the practical value used in Orinoco wireless LAN card. The corresponding carrier sensing range is 550 meters, which is in line with the carrier sensing range used in practice. In our model, we assume that carrier sensing works when the separation is within the carrier sensing range and fails otherwise and use two different sets of equations to model the system in these situations. Therefore there is an abrupt change in the aggregate throughput when the separation equals the carrier sensing range. If there is no carrier sensing in the system, the aggregate throughput will reduce to zero smoothly when the link separation reduces.

The curve in Figure 6(c) can be divided into three parts according to the link separation Open image in new window . When Open image in new window meters, both transmitters are in the carrier sensing range of each other. As a result, at most only one transmitter is active at a time. If Open image in new window meters, the transmitters are unaware of the existence of each other, and they contend for the wireless channel as if there were no interferers nearby. When Open image in new window meters, the separation is so large that there will not be any interference between the links. When Open image in new window lies between 550 and 800 meters, the aggregate throughput of the links increases smoothly as Open image in new window increases. We label this range of Open image in new window as the *partial interference region*. The existence of this partial interference region suggests that the interference models proposed by [1] that a single threshold can represent the interference relationship in wireless networks may be overly simplified.

The width of this partial interference region depends on the carrier sensing threshold Open image in new window used. Smaller Open image in new window , for example, −80 dBm, results in a narrower partial interference region as in Figure 6(d). Simultaneous transmissions are allowed only for the links separated far enough, and the throughput is suppressed significantly. For larger Open image in new window , for example, −75 and −70 dBm, more *spatial reuse* is allowed, and the width of the partial interference region is larger, as shown in Figures 6(a)-6(b). However, excessive interference is introduced for larger Open image in new window , so there is a reduction in the aggregate throughput.

### 5.3. Admissible (Stability) Region

*admissible (stability) region*predicted from our model. The admissible region includes all flow vectors Open image in new window such that if Open image in new window is located inside the admissible region, then a flow of Open image in new window can be allocated on and achieved by Open image in new window , Open image in new window . We use the same settings as above and choose the carrier sensing threshold to be −78 dBm. The link separations are chosen to be 500, 600, and 900 meters for illustrative purposes, because they correspond to different shapes of the admissible region. Figure 8 shows the admissible region for these three link separations. The link separation of 500 meters represents that the links are in mutual interference and the admissible region has a triangular shape. When the links are separated by 900 meters, the links do not interfere with each other, and the admissible region is rectangular. For the link separation of 600 meters, partial interference exists and the admissible region becomes convex.

Although we are able to compute the admissible region for a two-link 802.11 network numerically, the closed-form expression for the admissible region is unknown. Also, for an 802.11 network with two links, we have to solve a system of six nonlinear equations to compute the admissible region. When the number of links in the network grows, the number of equations involved will increase, and the system of equations will be more difficult to solve. Therefore the computation of the admissible region of general 802.11 networks seems to be forbiddingly intractable.

## 6. Partial Interference in Slotted ALOHA

In order to obtain insights in the stability region of general 802.11 networks, in this section, we study the stability of slotted ALOHA, which is a simpler random access protocol, under the assumptions of finite links and infinite buffer.

### 6.1. The Finite-Link Slotted ALOHA Model

Let Open image in new window be the set of links in the slotted ALOHA system. Time is slotted. The following assumptions apply to all links Open image in new window . Let Open image in new window and Open image in new window be the transmitter and the receiver of link Open image in new window , respectively. Open image in new window has an infinite buffer. The packet arrival process at Open image in new window is Bernoulli with mean Open image in new window and is independent of the arrivals at other transmitters. Open image in new window attempts a *virtual transmission* with probability Open image in new window , that is, if its buffer is nonempty, Open image in new window attempts an *actual transmission* with probability Open image in new window ; otherwise, Open image in new window always remains silent. Also define Open image in new window .

In the system, each time slot is just enough for transmission of one packet. Packets are assumed to have equal lengths. We assume that transmission results are independent in each slot. For Open image in new window , let Open image in new window be the probability that the transmission on link Open image in new window is successful when Open image in new window is the set of active transmitters. Open image in new window depends on the SINR at the receiver and the modulation scheme used. We also assume that the transmitters know immediately the transmission results, so that the transmitters remove successfully transmitted packets and retain only those unsuccessful ones.

We let Open image in new window be the queue length in Open image in new window at the beginning of slot Open image in new window and use an Open image in new window -dimensional Markov chain Open image in new window to represent the queue lengths in all transmitters. We denote by Open image in new window the number of packets arrived at Open image in new window in slot Open image in new window and Open image in new window the number of packets successfully transmitted in slot Open image in new window by Open image in new window when Open image in new window . Then Open image in new window , where Open image in new window is used to account for the case that there is no packet transmitted when Open image in new window . We use the definition of stability in [16, 21, 22].

Definition 1.

*-*dimensional stochastic process Open image in new window is stable if for Open image in new window the following holds:

then the process is *substable*. The process is *unstable* if it is neither stable nor substable.

The stability problem of slotted ALOHA we consider here is to determine whether the slotted ALOHA system with the set of links Open image in new window is stable given the parameters Open image in new window and Open image in new window . We use the result from [34]. On the assumption that the arrival and the service processes of a queue are stationary, the queue is stable if the average arrival rate is less than the average service rate, and the queue is unstable if the average arrival rate is larger than the average service rate. We also define the slotted ALOHA system to be stable when all queues in the system are stable.

### 6.2. Stability Region of 2-Link Slotted ALOHA under Partial Interference

*bit*error rate when the SINR is Open image in new window . In particular, if DBPSK is used in the physical layer, Open image in new window . Under binary interference, we let the SINR threshold Open image in new window be the case that the

*packet*error rate is Open image in new window , that is, Open image in new window . Consider Open image in new window . When only Open image in new window is active, the SINR attained at Open image in new window is Open image in new window , and

Similarly, we can derive expressions for Open image in new window and Open image in new window under binary and partial interference.

*dominant system*of the original system Open image in new window , with Open image in new window being the

*persistent*set. The transmitters of the links in this set transmit dummy packets when they decide to transmit but do not have packets queued in their buffer. The remaining transmitters behave identically as those in Open image in new window . We first consider the dominant system Open image in new window . In this dominant system, the successful transmission probability of link 2 is Open image in new window . For link 1, the queue in Open image in new window is empty with probability Open image in new window ; in this case the successful transmission probability is Open image in new window ; otherwise, the successful transmission probability is Open image in new window . Hence, the average successful transmission probability of link 1 is

The union of these two regions constitutes the inner bound on the stability region of the original system Open image in new window .

The reason for the union of these two regions to be the outer bound on the stability region follows from the indistinguishability argument [16, 18]. Consider the dominant system Open image in new window . With a particular initial condition on the length of the queues, if the queue in Open image in new window is unstable, it is equivalent to the case that the queue in Open image in new window never empties with nonzero probability. Then Open image in new window and Open image in new window will be indistinguishable, in the sense that the packets transmitted from Open image in new window in Open image in new window are always real packets and Open image in new window is also unstable. Therefore, the union of the regions defined by (29) and (30) is the *exact* stability region for Open image in new window .

### 6.3. Some Illustrations

Parameters used for the analytical results.

24.5 dBm | Open image in new window 88 dBm | ||
---|---|---|---|

1 | 1.5 m | ||

0.001 | 450 m |

*switch*between the collision channel and the orthogonal channel.

Figure 9(b) shows the corresponding results under partial interference. When the link separation is small, the amount of interference is so large that partial interference degenerates to the collision channel. As the link separation increases, the stability region expands gradually and changes from nonconvex to convex. At another extreme, when the links are sufficiently far apart, partial interference is identical to the orthogonal channel. Therefore, partial interference can be viewed as a generalization of binary interference that it *interpolates* the transition from the collision channel to the orthogonal channel. Notice that the results here are similar to the case in 802.11, therefore our results should be applicable to networks with practical random access protocols like 802.11.

### 6.4. Partial Characterization of the Stability Region for the M-Link Case

to be a *corner point* corresponding to the case that Open image in new window is the set of persistent links. Notice that RHS of (35) is zero when Open image in new window . Then we obtain the following theorem.

Theorem 2.

All corner points lie on the boundary of the stability region.

Proof.

Refer to Appendix A.

By using stochastic dominance and the indistinguishability argument, we obtain the following theorem.

Theorem 3.

Let Open image in new window , Open image in new window be two corner points such that Open image in new window . Then the line segment joining these two points lies on the boundary of the stability region. This line segment represents the case that Open image in new window is the set of persistent links while Open image in new window is the only nonempty nonpersistent link in the system.

Proof.

Refer to Appendix B.

## 7. Stability Region of the General M-Link Slotted ALOHA System under Partial Interference

Theorems 2 and 3 cover all cases with zero or one nonempty nonpersistent link in an M-link system, respectively. However, if there are at least two *nonempty unsaturated* links, the stationary joint queue statistics must be involved in calculating the boundary. Unless we are able to compute the stationary joint queue statistics in closed-form, we are unable to solve the capacity-finding problem, even assuming one of the simplest random access protocol, that is, slotted ALOHA. In fact, the characterization of the exact stability region of a general M-link slotted ALOHA system with *nonpersistent* links has remained to be a key *open* problem for decades when Open image in new window [16, 17, 18, 19, 20, 21, 22, 23] even under the simplified binary interference model.

To overcome the problem caused by the stationary joint queue statistics, we have introduced the FRASA (Feedback Retransmission Approximation for Slotted Aloha) model in [24] to obtain a *closed-form approximation* for the stability region of a general M-link slotted ALOHA system under binary interference (i.e., simple collision channel) assumptions. We refer the readers to [24, 31] for a detailed description of the FRASA approach. In the following subsection, we extend the model and analysis in [24] to cover the partial interference case. We remark that our results here automatically apply to the binary interference case if we allow Open image in new window to be either zero or one only by comparing the corresponding SINR, that is, Open image in new window , against a predefined threshold Open image in new window , as illustrated in (32) and (33).

### 7.1. FRASA under Partial Interference

*reduced FRASA system*, in which we let Open image in new window of the links have fixed aggregate arrival rates and the remaining link is assumed with infinite backlog. Take Open image in new window to be the link with infinite backlog and denote this reduced FRASA system by Open image in new window . Let Open image in new window be the aggregate arrival rate of link Open image in new window , where Open image in new window is between zero and one. Hence, link Open image in new window is active with probability Open image in new window , while for Open image in new window , link Open image in new window is active with probability Open image in new window . Let Open image in new window and Open image in new window . Introduce the following notations:

with Open image in new window and Open image in new window is between zero and one for all Open image in new window . Open image in new window is the successful transmission probability vector under partial interference.

With this parametric form, we obtain the stability region of FRASA under partial interference as in the following theorem.

Theorem 4.

with Open image in new window and Open image in new window is between zero and one for all Open image in new window . Then Open image in new window , the stability region of FRASA under partial interference, is enclosed by Open image in new window , Open image in new window in the positive orthant.

Proof.

Refer to Appendix C.

### 7.2. Simulation Results

*qualitative*changes in the shapes of the cross-sections of the stability region from concave to convex when the more realistic partial interference phenomenon is considered. This reinforces our argument that it is important to model the partial interference effects while analyzing the performance of wireless multiple access protocols.

## 8. Conclusion

In this paper, we have introduced the notion of partial interference in wireless multiple access systems and illustrated the potential gain in system capacity when the effect of partial interference is taken into account. In particular, we characterize the stability region of IEEE 802.11 networks under partial interference with two potentially unsaturated links numerically. We also derive the stability region of slotted ALOHA networks under partial interference with two links analytically and obtain a partial characterization of the boundary of the stability region for the case of more than two, potentially unsaturated links in a slotted ALOHA system. By extending the FRASA model, we provide a closed-form approximation for the stability region for general M-link slotted ALOHA system under partial interference effects. Our analyses demonstrate that partial interference considerations can result in not only significant quantitative differences in the predicted system capacity but also fundamental qualitative changes in the shape of the stability region of the systems. This highlights the need of capturing the partial interference effects instead of adopting the conventional, overly simplified binary interference models while analyzing wireless MAC protocols.

## Appendices

### A. Proof of Theorem 2

Therefore Open image in new window lies on the boundary.

### B. Proof of Theorem 3

The necessity follows directly from the indistinguishability argument. We observe that Open image in new window varies linearly with Open image in new window on the boundary, Open image in new window . It is trivial that Open image in new window and Open image in new window correspond to Open image in new window and Open image in new window , respectively.

### C. Proof of Theorem 4

where we assume that Open image in new window , which is in general true because the probability of successful transmission is larger when there are less interferers. This implies that the stability region of FRASA obtained by assuming all links in Open image in new window in persistent conditions is contained inside the stability region of FRASA obtained by assuming all links in Open image in new window in persistent conditions. Hence, to obtain the boundary of stability region of FRASA under partial interference, we only have to consider the case that only one link is persistent. Then we can use the parametric form (40) to obtain the boundary when Open image in new window . By repeating over all possible values of Open image in new window , we get the desired result.

## Notes

### Acknowledgments

The material in this paper was presented in part at the IEEE International Conference on Communications 2007, Glasgow, Scotland, June, 2007 and the 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Athens, Greece, September, 2007.

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