The de facto standard for wireless Internet. Interference Estimation in IEEE Networks

Transcription

1 Interference Estimation in IEEE Networks A KALMAN FILTER APPROACH FOR EVALUATING CONGESTION IN ERROR-PRONE LINKS ILENIA TINNIRELLO and GIUSEPPE BIANCHI The de facto standard for wireless Internet access is currently the IEEE Wireless Local Area Networking (WLAN) technology, also known to the general public as Wi-Fi (see Wi-Fi and the IEEE Standards ). Wi-Fi connectivity is integrated by default in every modern portable computer, laptop, and palmtop. Wi-Fi networks for wireless Internet connectivity are available in most airports, university campuses, offices, and homes as well as in many restaurants and coffee shops. Wi-Fi is extensively integrated in dedicated devices such as electric utilities and parking meters and even exploited in specialized applications such as garden hose sprinklers [3]. One of the key factors underlying the broad acceptance of Wi-Fi is the simplicity and robustness of the Medium Access Control (MAC) protocol. The MAC, called Distributed Coordination Function (DCF), belongs to the family of random access protocols, as detailed in Random Access in Wireless Local Area Networks. The protocol combines the listen before talk principle, which prevents a terminal from transmitting when the wireless channel is sensed busy, with a collision-avoidance technique that minimizes the probability of simultaneous transmission by multiple stations. These features allow DCF to work in the Digital Object Identifier 1.119/MCS FABIO BUCCIARELLI presence of interfering transmissions, a critical requirement for networks operating in unlicensed bands. Before transmitting, the carrier sense function reveals interfering sources by detecting channel activity. After transmitting, the missed reception of an explicit acknowledgment reveals a transmission failure due to interfering transmissions. In the first case, the protocol reacts to interference by delaying frame transmissions until the medium is sensed to be interference free. In the second case, the protocol attempts to retransmit the corrupted frame. Sources of interference affecting a given station may include not only other stations sharing the channel on the same network but also external noise, for example, from microwave ovens or overlapping networks. Station 3 IEEE CONTROL SYSTEMS MAGAZINE» APRIL X/1/$26. 21IEEE

2 Wi-Fi and the IEEE Standards Wi-Fi, which stands for wireless fidelity, is the name under which the general public refers to a specific wireless local area network (WLAN) technology promoted by the IEEE Working Group. IEEE launched the standardization activities for short/medium range (up to a couple of hundred meters) wireless communication technology operating in the unlicensed band at the beginning of the 199s. The first version of this standard, today called legacy, was released in 1997 and was characterized by a low transmission rate (up to 2 Mb/s). In 1999, two breakthrough specifications were released, specifically, the IEEE 82.11b standard, operating in the 2.4-GHz unlicensed band and targeted to support transmission rates up to 11 Mb/s, and the IEEE 82.11a standard, operating in the 5-GHz industrial, scientific, and medical frequency band and supporting up to 54 Mb/s transmission rates thanks to the usage of multicarrier orthogonal frequency division multiplexing (OFDM) [1]. Since the 5-GHz spectrum is tightly regulated in several countries, a further standard, the IEEE 82.11g, was released in 23 to bring the benefits of OFDM to the 2.4-GHz ISM unlicensed frequency band. The 82.11b, 82.11a, and 82.11g specifications define multiple modulation and coding schemes for frame transmissions, corresponding to multiple available transmission rates. The evolution of wireless LANs is still in progress. IEEE is currently working on a new higher rate standard, called 82.11n, targeted to exploit multiple-input, multiple-output (MIMO) antenna technologies for providing transmission rates up to five times faster than those achieved by current off-the-shelf 82.11a/g wireless cards. Although the standardization activities are still in progress, some vendors have already started to commercialize pre-82.11n products. Moreover, a study group, provisionally called Very High Throughput, is being launched by IEEE to explore the possibility of standardizing a faster version of 82.11, devised to reach up to 1 Gb/s operating at the 6-GHz frequency band. Although the terms Wi-Fi and IEEE are often used interchangeably, there is a subtle difference that justifies our consistent usage of the term IEEE instead of Wi-Fi. Indeed, Wi-Fi is not the name of an IEEE standard or of a group of standards. Rather, Wi-Fi is a certification that guarantees that an product from a vendor can interoperate with products of other vendors. Wi-Fi certification is issued by the Wi-Fi alliance, a nonprofit organization composed of 3 members from more than 2 countries as well as most industry leaders. Despite the rigorous testing methodology, Wi-Fi certification does not necessarily imply full conformance with the IEEE standard. Experimental results [2] show that some Wi-Fi certified products employ nonstandard contention parameters for improving their performance in comparison to competitors. interference is due to the channel access scheme, while noise interference is due to the physical medium. Therefore, station interference is defined as MAC interference and noise interference as physical layer (PHY) interference. Several solutions have been considered for maximizing the throughput of DCF networks in the presence of interference. In the standard IEEE protocol [1], collisionavoidance parameters depend on hardware-transmission technology called the PHY layer according to the Open System Interconnection (OSI) network model. Although this hardware cannot be modified, it is shown in [5] [7] that tuning the collision-avoidance parameters as a function of the number of interfering stations can reduce channel waste due to collisions and backoff expiration times. This finding has led to solutions for adaptively tuning the contention window [8] [11], as incorporated in a recent amendment to the standard [12]. A complementary approach to reducing transmission failures due to noise corruption is to dynamically select modulation and coding schemes according to the channel quality perceived by stations. In fact, for a given interfering noise, the probability of correctly decoding the frame depends on the modulation alphabet and coding redundancy, which in turns affect the data-transmission rate. Commercial Wi-Fi adaptors sometimes implement automatic rate selection algorithms, also known as link adaptation algorithms. The choice of the rate selection algorithm depends on the latency and buffering capabilities of the hardware, which needs to store modulation and coding descriptors for each packet [13]. The feasibility of rate adaptation, however, depends on the availability of MAC and PHY measurements, which are not always practicable or reliable. On the one hand, the IEEE MAC provides no direct method for retrieving information on network congestion and number of competing stations. On the other, the station hardware has no mechanism for distinguishing frame errors due to low signal-to-noise ratio (SNR) from collisions. The lack of direct mechanisms for detecting collisionand channel-induced errors is addressed by techniques that distinguish collisions indirectly by means of special control frames, acknowledgment monitoring, or tracking of consecutive failures [14] [16]. However, most of these methods have drawbacks in terms of overhead or accuracy of the channel error-rate estimator. In this article, we describe a technique for distinguishing and quantifying MAC and PHY interference in error-prone networks. This technique, which generalizes the framework discussed in [17], is fully distributed, allowing each station to estimate interference individually. The APRIL 21 «IEEE CONTROL SYSTEMS MAGAZINE 31

3 Random Access in Wireless Local Area Networks The majority of the WLAN technologies, including the most representative one, namely IEEE 82.11, are characterized by two properties. First, transmissions occur over a single, shared, wireless communication channel. The use of a single channel is different from most wireless cellular technologies (for example, GSM), where communication occurs simultaneously in independent wireless channels deployed in different time slots and frequency bands. Second, WLANs may operate in scenarios, called independent basic service set or ad hoc mode according to the IEEE terminology, where no pre-established network infrastructure is deployed and any pair of stations can exchange data. The ad-hoc mode is not possible in wireless cellular networks where communication always occurs between a terminal and a central device called a base station. As a consequence, a fundamental part of every WLAN standard is the specification of the medium access control operation. The medium access control (MAC) is a sublayer of the data link layer specified in the seven-layer OSI model (layer 2). The MAC sublayer provides addressing and channel-access control mechanisms that make it possible for several terminals or network nodes to communicate within a multipoint network. In WLANs, communication occurs through the exchange of elementary data units (packets) called MAC frames. The MAC function s role is to guarantee that MAC frames are transmitted by different terminals at different time instants, hence without colliding with each other. Among the several MAC approaches that have been considered, the family of random access protocols has acquired a prominent role in WLANs. Random access was considered in packet radio networks almost 4 years ago. In 197, the University of Hawaii, under the leadership of Norman Abramson, developed ALOHAnet, the world s first packet radio computer communication network. The MAC protocol employed was called ALOHA. Its idea is simple, namely, when a terminal has a packet to deliver, it blindly transmits the packet over the shared radio channel. Since this operation causes collisions whenever the transmission overlaps with an ongoing transmission originated by a different terminal, in case of error the packet must be retransmitted at a later, randomly chosen, period of time. Despite its simplicity, the ALOHA protocol proved that a completely random and decentralized operation could achieve a nonzero throughput. Indeed, the ALOHA protocol can reach a maximum throughput, defined as the percentage of time the channel is used for successful transmissions, of 18.4% [4]. The maximum achievable throughput is further doubled if the channel time is subdivided into discrete intervals called slots and transmissions are forced to occur inside a given slot. The MAC protocols employed in IEEE WLANs comprise a distributed coordination function (DCF). Although DCF is based on random access and distributed operation, this protocol improves the basic ALOHA approach in the following major aspects. In wireless networks, the sender cannot determine whether a transmission is successful. Therefore, in DCF, the receiver immediately replies with an acknowledgment message whenever it successfully receives a MAC frame. Hence, the lack of an acknowledgment informs the sender that the current frame has either collided or has been corrupted by channel noise and thus that the frame must be retransmitted at a later time. Second, DCF uses a technique called carrier sense multiple access (CSMA). Developed by Bob Metcalfe in 1973 for the Ethernet wired LAN, this technique requires every station to listen to the channel before transmitting. A MAC frame is transmitted only if the channel is sensed idle. This rule prevents the transmission from interfering with other frames being delivered in the same time by other stations. Third, DCF prevents collisions by trying to minimize the probability that two stations start their frame transmission simultaneously. For this purpose, a station that senses the channel is busy not only defers transmission until the channel is idle again but continues to wait for an additional random delay called backoff. The range in which the backoff delay is extracted is called the contention window. Since all the other competing stations follow the same rule, the chance that two stations select the same time for their frame transmission is reduced. This operation is called collision avoidance. estimator is based on an extended Kalman filter coupled to a mechanism for revealing abrupt changes in state. The network state is a vector of two components, representing PHY interference, expressed in terms of channel-error rate, and MAC interference. Two distinct state models are considered. When PHY interference can be assumed to be constant for all stations, network congestion is expressed by the number of competing terminals as in [17]. Conversely, in the absence of information on PHY interference, congestion is expressed in terms of the collision probability perceived by individual stations. As explained below, the former approach provides faster tracking of PHY interference dynamics. The latter is suitable for more general application scenarios. This article is organized as follows. First, we review the IEEE DCF and its equivalent persistent model. Second, we derive the measurement model and the relationship between the number of competing stations, the packet collision probability, and the frame error rate. We then define and evaluate two run-time estimation techniques for uniform and heterogeneous interference conditions. A final section summarizes the conclusions. See Acronyms for a reference for the acronyms used throughout the article. 32 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 21

4 THE DISTRIBUTED COORDINATION FUNCTION Stations in a wireless IEEE network are set up to communicate either indirectly through a central node, called an access point, or directly to the other stations. In both cases, the access to the shared medium is managed by the DCF. In DCF, a station with a data packet to be transmitted monitors channel activity. If the channel is idle for a period called distributed interframe space (DIFS), the station transmits. The receiving station signals the successful packet reception by replying with an acknowledgment frame (ACK). The ACK is transmitted at the end of the datapacket reception, and within a period of time called short interframe space (SIFS). The SIFS period is shorter than the DIFS period to avoid allowing other stations (waiting for an idle DIFS time) to access the channel and interfere with the ACK transmission. If the transmitting station does not receive the ACK within a specified ACK timeout, or if the station detects the transmission of a different packet over the channel, then the station reschedules the transmission of the packet after a random interval of time, using the backoff rules described below. DCF employs a technique called collision avoidance to reduce the probability that two or more competing stations simultaneously transmit and hence cause packet corruption. Whenever the channel is sensed busy, either at the instant of time a station starts to monitor the channel activity or during a DIFS, the station continues to monitor the channel until it is idle for a DIFS. An extended interframe space (EIFS) is considered instead of a DIFS period, when the channel-sensing function reveals a corrupted frame on the channel. Since decoding a corrupted frame does not imply that the frame receiver cannot decode the frame, the EIFS period is equal to an ACK timeout for avoiding interference with ACK transmissions. After the expiration of the DIFS (or EIFS) period from the last channel activity, rather than immediately transmitting, the station schedules the packet transmission after a random time interval called backoff. In addition, a random backoff is also extracted between two consecutive packet transmissions, even if the medium is sensed idle during the DIFS. This technique prevents a single station from capturing the channel for an indefinite period of time. For efficiency reasons, DCF employs a discrete-time backoff scale. The time immediately following an idle DIFS is slotted. Stations are allowed to transmit only at the beginning of each slot time because the random backoff time is given by an integer number of slot times. This number is chosen uniformly in the range (, v21), where v is called the contention window, and follows an exponentially truncated increment law. On the first attempt at transmission, v is set to the value CW min, called the minimum contention window. After each unsuccessful transmission, v is doubled, up to the maximum value CW max 5 2 m CW min. The values for CW min and CW max depend on the specific PHY layer technology considered. For the most common case of the 82.11b PHY [1], CW min 5 32 and CW max Acronyms ACK ARF CW DCF DIFS EIFS MAC OSI PHY SIFS SNR Wi-Fi WLAN The backoff time counter is decremented as long as the channel is sensed idle, frozen when a transmission is detected on the channel, and reactivated when the channel is again sensed idle for a DIFS. The station transmits when the backoff counter reaches zero. Figure 1 illustrates this operation, where stations A and B share the same wireless channel. After a packet is transmitted over the channel, station A waits for a DIFS. Station A then sets the backuounter to nine. We assume that station B receives its first packet at the time indicated by the arrow in the figure. After a DIFS, station B transmits its packet during T h P h M Legend: m = Backoff for Station A: b = 9 Station A Station B Station A DIFS Payload MPDU Packet h P = PHY Header h M = MAC Header Acknowledgment Automatic rate fallback Contention window Distributed coordination function Distributed interframe space Extended interframe space Medium access control Open system interconnection Physical layer Short interframe space Signal-to-noise ratio Wireless fi delity Wireless local area network T SIFS ACK SIFS DIFS Slot Time T = Packet Transmission Time m = Intertransmission Time FIGURE 1 Example of the distributed coordination function (DCF) operation. Stations A and B share the same wireless channel. During the backoff countdown for station A, station B receives its first packet at the time indicated by the arrow. After a distributed interframe space (DIFS), the station transmits its packet, including the MAC packet data unit (MPDU), while station A freezes its backoff counter to 4. Each packet is acknowledged after a short interframe space (SIFS). APRIL 21 «IEEE CONTROL SYSTEMS MAGAZINE 33

5 the time slot corresponding to the fifth backoff slot for station A. After sensing that the channel is busy, station A freezes its backoff time to four and decrements the counter again, only when the channel is sensed idle again for a DIFS. We omit the description of other DCF features that are not relevant for understanding the next sections of the article. A complete description of DCF can be found in [1]. DCF Model Although DCF is based on random access, DCF is accurately modeled in [6] as a persistent slotted access protocol. When all stations are permanently in contention, that is, when all stations permanently have at least one frame in the transmission buffer, the interval between the end of activity on the channel and the next access is equivalent to an integer number of idle backoff slots. Considering a nonuniform scale for slots, in which each slot represents either an idle backoff slot or an activity period plus the final DIFS time, the behavior of the protocol can be summarized by a single parameter t, representing the probability that a tagged station accesses the channel during a generic channel slot. When failure rates are the same for all stations, the access protocol is fair, and t is equal for all stations. Figure 2 shows an example of actual DCF times and equivalent model slots. In the persistent model [6], frame transmission times, ACK times, and DIFS intervals are included in a single model slot. Figure 2 also shows several specific channel events (observation slots, successful slots), described in the next section. Assuming that the probability of a collision does not depend on either the number of collisions in the past or the time since the last activity on the channel, t can be derived as a function of CW min and CW max. Data Packet Model Time Observation Slots (b = 19, c busy = 4) Channel Activity Successful Slots (s = 5) SIFS DIFS ACK Data Packet Actual Time... EIFS Channel Activity FIGURE 2 Distributed channel-access operations and equivalent slotted model. Frame transmission times, short interframe space, acknowledgment (ACK) times, and distributed interframe space (DIFS) or extended interframe space (EIFS) intervals are all included in a single model slot. For the station whose transmissions are marked in red, model slots are classified as observation slots or successful slots. Observation slots are model slots in which the given station does not transmit, while successful slots are model slots in which a successful transmission is performed. MEASUREMENT MODEL We consider a second-order model of the network, whose state components represent congestion (measured by the number n of contending stations or the probability of collisions) and channel-error rate. No parameter is directly accessible to the stations monitoring the state of the channel. Not even the access point knows the number of stations trying to access the channel in a specific interval of time. In addition, no station can know whether an ACK timeout is due to a collision or to corruption of the channel. We therefore need to define a system of measurements that allows us to estimate the network state indirectly. The carrier-sense feature of the access protocol obliges all contending stations to continuously monitor channel state. This monitoring can be used to indirectly probe external interference and internal congestion on the network. A first measurement that can be derived from channel observations is the probability of transmission failure. Each station counts the number t of transmissions and the number a of ACK timeouts [16]. Retransmission probability p r can be measured as pr r 5 a/t, where the prime indicates a measurement sample of the actual value p r. When external interference is the same for all stations, the failure rate is also the same. To increase the size of the sample used for the measurement, we can consider all successful transmissions over the channel. To identify its own frames, each station has to decode MAC headers for all frames received. In this setting, the retry bit in the MAC header can be used to count the total number r of retransmissions as well as the total number s of successful frames. In this case, the retransmission probability is given by pr r 5 (r 1 a) / (s 1 t). A second parameter available from channel observations is. Given that stations have no way of classifying their own transmission failures, each station performs the measurement of without considering the outcomes of its own transmissions. However, all other slots on the channel can be monitored. Using the monitoring data, stations classify idle slots as potential successes and busy slots as potential collisions [17], [16]. The reasoning is that an idle slot indicates a posteriori that it is possible to successfully transmit a frame in that slot. Conversely, a busy slot indicates a posteriori that it is not possible to transmit a frame in that slot without causing a collision. Thus, the probability of a collision can be obtained by counting the number c busy of busy slots and dividing by the total number b of observation slots to obtain pr c 5 c busy /b. Figure 2 illustrates how a tagged station (the station whose transmissions are shown in red) can count observation and successful slots. When the contending stations operate with saturated transmission buffers, it is possible to relate the measurements just described to a network state with two components. For convenience, exponential backoff parameters are expressed as W and m, where W 5 CW min and CW max 5 2 m CW min. Consequently, m 5 log 2 (CW max /CW min ). Let be the actual conditional collision probability, namely, the 34 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 21

6 probability that a packet transmitted over the channel collides because of two or more simultaneous channel accesses. Let p r be the actual probability that a frame fails and requires retransmission. Note that this probability depends both on the probability of collision and the probability of channel error given by p r 5 1 (1 2 ). (1) Equivalently, the channel-error rate is the function of and p r given by 5 p r 2. (2) 1 2 When the channel-error rate is uniform for all stations, retransmission probability p r is also uniform, and the DCF protocol is fair in terms of channel-access probability. Let t be the equivalent channel access probability, that is, the probability that a station transmits in a given, randomly chosen slot. It is shown in [6] that the probability t can be expressed as the function of p r given by t(p r ) 5 2(1 2 2p r ) (1 2 2p r )(W 1 1) 1 p r W(1 2 (2p r ) m ), (3) and the probability can be expressed as the function of t and n given by 5 12(1 2 t) n21. (4) By substituting t, as expressed by (3), into (4), and solving the resulting equation for n, we obtain n 5 f(, p r ) log(1 2 ) log(1 2t(p r )). (5) Equations (5) and (2) explicitly associate (n, ) with collision and retransmission probabilities as well as the known, constant backoff parameters m and W. Given that all stations can independently measure probabilities and p r by monitoring channel activity, they can also estimate the two components of the network state. Figure 3 compares simulation results with results from (5). The graph plots the number of contending stations n versus the conditional collision probability, for various values of. The simulation results are obtained using object-oriented, event-driven simulator software written in C11. The results are cross validated against the NS2 simulation platform [19]. Each simulation run lasts 1 s. To reach saturation conditions, the load is set to be higher than per-station throughput. A 1-s warmup time is added at the beginning of the simulation. Each simulation point is obtained by taking n and as constant and measuring the resulting conditional collision probability pr c. Parameter values are summarized in Table 1. In all cases the difference between simulation and analytical results is less than 3%. Number of Stations n = =.1 =.3 = Conditional Collision Probability FIGURE 3 Number of stations versus conditional collision probability for various values of the channel-error probability. The analytical relation is shown with dashes. Simulation results are shown using symbols. Each simulation point is obtained by taking n and as constant and measuring the resulting conditional collision probability pr c. The results indicate the accuracy of the relationship between the collision probability and the number of contending stations. RUN-TIME ESTIMATION OF NETWORK STATE WITH UNIFORM INTERFERENCE When the channel-error rate is fixed for all stations, DCF protocol is fair. The fairness assumption is realistic if all contending stations transmit with comparable powers to a common receiver, such as an access point, that experiences PHY interference due to noise or external networks. To estimate (n, ) at run time, all that is required is an algorithm that allows each station to make independent estimates using its own observations of interference on the channel. During each time interval I, each monitoring station counts the number c busy (I) of busy slots, the number b(i) of observation slots, the number r(i) of retransmitted frames, and the number s(i) of successful frames. Stations then use these values to measure pr c and pr r. Measurements can be smoothed at run time by applying an autoregressive TABLE 1 Packet format and parameter values used in the simulations. These simulation parameters refer to the 82.11b PHY layer. Packet payload MAC header ACK length PHY header Channel bit rate Propagation delay SIFS DIFS ASK_timeout Slot time (s) 15 bytes 224 bits 112 bits 1 PHY header 192 ms 11 Mb/s 1 ms 1 ms 5 ms 344 ms 2 ms APRIL 21 «IEEE CONTROL SYSTEMS MAGAZINE 35

7 filter and then processed to produce the time-dependent network-state estimates n^ (k) log(1 2 p^ c (k)) log(1 2t(p^, (6) r(k))) p^ e(k) 5 p^ r (k) 2 p^ c (k), (7) 1 2 p^ c(k) where p^ c(k) and p^ r(k) are the smoothed measurements, and n^ (k) and p^ e(k) are estimates of the network state at time k. However, the autoregressive filters suffer from a tradeoff between accuracy and tracking capability, and estimates of the network state may be biased by the nonlinearity between measurements and state. To avoid this difficulty, we use the extended Kalman filter [17], which handles nonlinear dynamics and exploits previously unused information, such as variance of measurements and knowledge about the updating laws for the network state. The Kalman filter requires a model of the network state. This model must specify the law governing the update of network state as well as a measurement model for the relationship between the measurements and the state. The filter is updated at discrete steps k, corresponding to the time intervals I at which new measurement samples become available. The network state is represented by the number n(k) of stations in the network and by the channel-error probability (k) at a discrete time k. In the absence of further information about traffic and channel models, we assume that network state evolves according to the law n(k) 5 n(k 2 1) 1 w n (k), (8) (k) 5 (k 2 1) 1 w e (k), (9) where the state noise random variables w n (k) and w e (k) describe changes in the number of stations that have become active or inactive as well as changes in PHY interference over the previous time interval. Obviously, whenever a traffic model or an interference model is available, (8), (9) can be modified accordingly. To obtain the measurement model, we invert (5) and (2) to determine the relationships n 5 f(, p r ) and 5 g(, p r ) between state variables and measurements. Substituting (1) into (5), we obtain an implicit relationship between n, and. Inverting this relationship yields 5 h(n, ). The complete inverse relationship between state and measurements can thus be expressed as (k) 5 h(n(k), (k)) 1 v c (k), (1) p r (k) 5 (k) 1 (1 2 (k))h(n(k), (k)) 1 v e (k), (11) where the random variables v c (k) and v e (k) account for the measurement noise. Equations (8) (11) describe the state model for the system. Given the state model, the extended Kalman filter can be applied [2]. The linearized sensitivity matrix H, relating variations in network state to variations in measurements at time k, is given by 'h(n, ) 'n (12 ) 'h(n, ) 'n 'h(n, ) ' 12 h(n, ) 1 (12 ) 'h(n, ). ' (12) At each time k, we compute the matrix H for n 5 n^ (k 2 1) and 5 p^ e(k 2 1), where n^ (k 2 1) and p^ e(k 2 1) represent the estimated network state at the previous step. Numerical simulations with n() 5 1 and () 5 show that, when initial values for the error variance matrix are high, the initial state is not critical. Noise Statistics To complete the design of the filter, it is necessary to specify statistics for noise. Measurement noise depends on the size of the time interval I. Given b observation slots during I, that is, b slots where monitoring stations do not transmit, and an unknown conditional collision probability, the measurement r (k) is a random variable with distribution Probapr c (k) 5 x b b 5 ab x b x (1 2 ) b2x, (13) where x [ (, b). The mean value and variance pr c (k) are given by and (1 2 ) /b, respectively. Similarly, given s successful frames, t transmissions performed by the monitoring station during I, and the actual, unknown retransmission probability p r, it follows that the measurement p rr (k) is a random variable with distribution Probapr r (k) 5 x s 1 t b 5 as 1 t x bp r x (1 2 p r ) s1t2x, (14) where x [ (, s 1 t) The mean value and variance of pr r (k) are given by p r and p r (1 2 p r ) / (s 1 t), respectively. Assuming that the randomness of the measurements pr c and pr r is due to an additive measurement noise, that is, pr c 5 1 v c, and pr r 5 p r 1 v r, the measurement noise v c and v r are binomial random variables, whose mean value is zero and whose variance depends on the network state and observation interval I. We can thus use the estimated network state and observation samples b(k) and s(k) 1 t(k) at time k to dynamically tune the measurement variances Var3v c (k)4 5 p^ c (k)(12p^ c (k)), (15) b(k) Var3v r (k)4 5 p^ r (k)(12p^ r (k)), (16) s(k) 1 t(k) 36 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 21

8 where p^ c(k) and p^ r(k) are the expected measurements at time k. We consider v c and v r as independent. Numerical simulations confirm the lack of significant correlation between measurement samples pr c and pr r. When no model for the evolution of network state is available, state noise is harder to model. In cases where the number of competing stations and the channel-error rate appear to be constant, the noise variances Var3w n (k)4 and Var3w e (k)4 must be set to zero or low values. In cases where the network state changes, the noise variances must be set to high values, allowing the filter to react promptly. The state model does not require that w n (k) and w e (k) be stationary processes. State noise variance can thus be tuned dynamically, depending on whether the monitoring station has detected changes in network state. Several statistical tests are capable of revealing changes in the state [21]. For example, we consider two independent change detection filters, based on the CUSUM (cumulative summary) test. We then apply these filters to innovation components z c 5 pr c 2 p^ c and z r 5 pr r 2 p^ r. For convenience, we replace the innovation processes z c and z r with two related processes i c and i r, representing the original innovation processes normalized by their respective standard deviations. At time k, the cumulative sums g 1 (k) and g 2 (k) are constructed from each generic innovation process i(k) as g 1 (k) 5 max(, g 1 (k 2 1)1 i(k)2v), (17) g 2 (k) 5 min(, g 2 (k 2 1) 2 i(k)1v). (18) In (17), (18), the drift parameter v is a filter design parameter. As v decreases, the sensitivity of the test results to fluctuations in the process i(k) increases. The initial conditions are given by g 1 () 5 and g 2 () 5. When the level of network interference changes, the magnitude of the sums g 1 (k) and g 2 (k) tends to increase or decrease without bounds. For example, suppose that a new station becomes active. In this event, the collision probability p^ c and the retransmission probability p^ r predicted from the current estimate of the network state are, on average, lower than the measured values. In this way, the mean values for i c and i r are positive. As soon as the mean value becomes greater than v, the innovation sums from the two CUSUM filters begin to diverge. Conversely, with increased external interference, the predicted value for p^ r becomes lower than the current value, whereas the predicted value of p^ c becomes higher. On average, an increase in channel errors leads to an enlargement of the contention window, reducing the real probability of collisions. The cumulative innovation sum for the i c filter decreases without bound, and the innovation sum for i r increases in the same way. To avoid further complications, the two CUSUM filters are not correlated. An alarm is generated whenever at least one of the cumulative sums is higher than the alarm threshold h. After an alarm, all CUSUM sums are restored to zero. As h increases, the probability of false alarms decreases, and the time taken to detect a change in the network state increases. The techniques described in [21] make it possible to automatically choose parameter values to achieve a specified false-alarm rate or change detection delay. Alarms from the CUSUM tests are used to adaptively set variances Q n (k) and Q e (k) for the state-noise variables w n and w e according to the following rules:» When no alarm is raised at time k, Q n 5 and Q e 5. Under these conditions, the measurements z c and z r can be used to increase the accuracy of the previous estimate.» In the event of an alarm, Q n and Q e are set to values that are sufficiently large to produce a noise impulse in the state update equation. This impulse at time k allows the Kalman filter to move away from its previous estimate and converge. As discussed in the next section, the values of these parameters have only a marginal impact on estimator performance. Performance Evaluation We consider an infrastructure network in which the contending stations transmit toward the access point and the measurement interval I is set to.5 s. The first-order state model in [17] assumes that all frame errors are due to collisions. Under this hypothesis, the network interference is represented by the MAC interference only. Even neglecting the PHY interference, the collision-induced error rate can be correctly measured by monitoring only potential collisions as described earlier. Despite a correct measurement methodology, the first-order model does not work in the presence of channel-induced errors. Figure 4 plots the temporal behavior of the filter working on first- and secondorder state models for a scenario with ten competing stations and a fixed channel-error rate of.3. As Figure 4 Number of Stations Second-Order Kalman n First-Order Kalman n Second-Order Kalman FIGURE 4 Estimating first- and second-order network states. The plot refers to a scenario with ten competing stations and a fixed channel-error rate of.3. The filter underestimates the actual network contention in the case of a first-order state model. By contrast, in the case of a second-order state model, the filter correctly tracks both the number of contending stations and the channelerror rate Probability APRIL 21 «IEEE CONTROL SYSTEMS MAGAZINE 37

9 shows, in the case of a first-order model, the filter estimates the value of n as approximately five, an underestimate of actual network contention. By contrast, in the case of a second-order model, the filter correctly tracks both the number of contending stations and the channel-error rate. In this way, the filter generates an output that is suitable for use by link adaptation algorithms. The reason for the estimation errors of the first-order model is that it fails to model channel-induced errors. In the absence of such a model, the filter Number of Stations Second-Order Kalman n.2 First-Order Kalman n Second-Order Kalman FIGURE 5 Estimating first- and second-order network states. The plot refers to a scenario with ten competing stations and a fixed channel-error rate of zero. Under these conditions, the first- and second-order state models provide similar results. Number of Stations Kalman Actual Alarm ARMA FIGURE 6 Kalman filter tracking with dynamic load. The plot refers to a scenario in which the number of stations (1, 2, 3, 5, 1, 25, 14) changes step-wise, and the channel-error rate is assigned a fixed value of.2. When the filters detect a change in network state and raise an alarm, the variance in state noise is set to (Q n, Q e )5(5,.5). When no alarms are active, the variance is set to (, ). Alarms from the change-detection filter are plotted as small impulses on the horizontal axis. For comparison, the plot also r r shows estimates in which the measurements and p r are filtered using two ARMA filters with a large filter memory corresponding to a Probability incorrectly infers that the decrease in collision probability (due to the increased size of the contention window) is due to reduced contention. As far as the second-order model is concerned, estimated values for n are slightly lower than the actual number of stations, while those for are unbiased. The underestimation of the number of stations depends on minor differences between the simulator access algorithm and the model of the protocol used by the Kalman filter. In the model, interference is generated locally at the access point, which acts as the common receiver for all contending stations. Thus, while all stations receive a corrupted frame in case of collision, only the access point receives a corrupted frame in case of channel-induced interference. In the first case, the stations originating a collision resume their backoff process after an ACK timeout from the end of the channel activity, while the remaining stations resume their backoff process after an EIFS period. Therefore, all stations are simultaneously in contention for the next channel access. In the second case, the sender of a frame that has been corrupted by PHY interference must wait until the end of the ACK timeout before resuming the access protocol. The remaining stations wait for less time, since they wait for a single DIFS. As a result, only n 2 1 stations contend on the channel immediately after the transmission of the corrupted frame. The reduced contention level corresponds to a lower collision probability, which in turns leads to a lower estimation of the number of stations. As expected, this effect is large when the channel-error rate is high. Figure 5 confirms that with 5 the estimation error disappears. Under these conditions the first- and second-order state models provide similar results. Figure 6 demonstrates the effectiveness of the Kalman filter with a dynamic load. In this scenario we use a normalized drift parameter set to.75. The threshold for the two change-detection filters is set to ten. When the filters detect a change in network state and raise an alarm, the variance in state noise is set to (Q n, Q e ) 5 (5,.5). When no alarms are active, the variance is set to (, ). Figure 6 shows a scenario in which the number of stations (1, 2, 3, 5, 1, 25, 14) changes step-wise, and the channel-error rate is assigned a fixed value of.2. Alarms from the changedetection filter are shown in the figure as small impulses on the horizontal axis. Although this scenario is not realistic, the results show that the estimation technique can effectively track abrupt variations in the network state. The estimation errors are higher than 1% only for a few seconds after the filter reset. The results also show that the values of parameters (Q n, Q e ) are not critical for filter performance. When the number of stations on the network changes from ten to 25 (simulation time 35 s) and from 25 to 14 (simulation time 45 s,) the value of Q n (Q n 5 5) is smaller than the load variations. Given, however, that the change detector raises a second alarm a few milliseconds after the first alarm, the filter is able to 38 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 21

10 track the changes in the network state. For purposes of comparison, Figure 6 shows estimates in which measurements pr c and pr r are filtered using two autoregressive moving average (ARMA) filters with a large filter memory corresponding to a5.95. The filter memory produces an estimation error lower than 1% with a response time on the order of tens of seconds. The limitations of ARMA filters are evident when retransmission probability is measured using only transmission outcomes for the station making the measurement, that is, pr r 5 a/t. Under these conditions, the variance of samples of p r for the same time interval I is much higher. Furthermore, the nonlinear relationship between measurements and network state can produce strongly biased estimates. Figure 7 compares estimates from the Kalman filter with estimates from two ARMA filters, under conditions where p r has high variance. The filter ARMA1 uses a5.95 to smooth measurement samples of pr c and pr r, whereas the filter ARMA2 uses a5.95 for pr c and a5.995 for pr r. It can be seen from Figure 7 that neither filter has acceptable performance. While ARMA1 produces estimates of n that are three times higher than the actual number of contending stations, ARMA2 is ineffective in tracking the system dynamics. Regardless of the specific setting of a, the problem with autoregressive filters is that the filter memory is fixed and state changes cannot be revealed. In contrast, the high measurement variance produces no dramatic change in the performance of the Kalman filter. Indeed, the Kalman filter can account for variance in the measurements and can weight measurements of pr c and pr r according to their accuracy. Finally, Figure 8 shows simulation results with ten competing stations, in which the channel-error rate changes dynamically. Figure 8 shows that the estimator can track changes in the value of without making major changes to its estimate of n even when its state model assumes that the channel-error rate is constant. RUN-TIME ESTIMATION OF THE NETWORK STATE WITH HETEROGENEOUS INTERFERENCE In the presence of nonuniform interference, it is possible to consider an alternative model of MAC and PHY interference state. In this case, the channel-error rate is station dependent, and the DCF protocol is no longer fair. Given that each station has a different channel-access probability t, the collision probability is also station dependent and can no longer be related to the number n of contending stations. We thus represent interference state at time k as the probabilities of collision- and channel-induced error. The evolution of the network state can be represented by * (k) 5 * (k 2 1) 1 w c (k), (19) (k) 5 (k 2 1) 1 w e (k), (2) Number of Stations Kalman Actual ARMA1 ARMA FIGURE 7 Kalman filter and ARMA filter tracking with high-variance in measurements of p r. The filter ARMA1 uses a for both pr c and pr r while the filter ARMA2 uses a 5.95, for pr c and a for pr r. Neither ARMA filter has acceptable performance. While ARMA1 produces estimates of n that are three times higher than the actual number of contending stations, ARMA2 is ineffective in tracking system dynamics. By contrast, the high measurement variance produces no dramatic change in the performance of the Kalman filter. Number of Stations Actual Kalman Actual n Kalman n FIGURE 8 Kalman filter tracking capability in dynamic channel conditions. The plot refers to a scenario with ten competing stations, in which the channel-error rate changes dynamically, following the dashed curve. The estimator can track changes in the value of while maintaining errors on the estimates of n below 1%, even when its state model includes no information for the evolution of the network state. where the random variable w c (k) indirectly accounts for stations that have become active or inactive over the previous time interval, and p * c represents network congestion in terms of the collision probability. Relationships between measurements and state variables are simplified as (k) 5 * (k) 1 v c (k), (21) 1.8 Probability APRIL 21 «IEEE CONTROL SYSTEMS MAGAZINE 39

11 Station B 1 p r = (A + r)/(t + S) p r = A/T Probability 1 Station A (a) 2 1 Actual Kalman ARMA1 (b) 2 Probability (a) 2 1 Actual Kalman ARMA1 (b) 2 FIGURE 9 Interference estimation with a heterogeneous channelerror rate. This plot shows both estimates of and for two stations A and B. Station A experiences the high error rate (A) 5.565, whereas station B experiences the low error rate (B) Despite the high variance of the measurements pr e, the Kalman estimate errors are lower than 1% after a few seconds. Conversely, the ARMA filter with a smoothing factor of.95 provides negligible errors on estimates of, but significant errors (higher than 1% for the station B case) on estimates of. FIGURE 1 Interference estimation with a uniform channel-error rate. This plot compares estimates of and made with the measurement model defined for (a) the uniform interference case, with estimates based on (b) the general interference model. The Kalman filter errors are lower than 1% in a few seconds with both measurement models, with a slower variance reduction of the estimate of for the general interference model. By contrast, the ARMA filter works adequately with only the uniform interference model. p r (k) 5 (k) 1 (1 2 (k)) * 1 v e (k), (22) where the random variables v c (k) and v e (k), which account for the measurement noise, have binomial distributions. In the measurement model (21), (22), the linearized sensitivity matrix H, which relates variations in the network state to variations in the measurements at time k, is given by 1 c d. (23) We now focus on the estimation of interference for a single target station. Given that retransmission probabilities for target and other stations are independent, measurements of p r must be computed as the number a of ACK timeouts over the number t of transmission attempts, that is, pr r 5 a/t. This computation implies that, for each time interval I, the variance of pr r is higher than the variance for the model with heterogeneous interference. Samples of pr c are computed as described in the previous section. Thus, Var3v c (k)4 5 p^ c (k)(12p^ c (k)), (24) b(k) Var3v r (k)4 5 p^ r (k)(12p^ r (k)), (25) t(k) where b(k) is the number of observation slots, and t(k) is the number of target station transmissions for the kth time interval I. In this model, as in (17), (18), we use two parallel CUSUM filters to detect changes in the network state by applying them to both innovation processes z c and the z r Given that in this case the congestion state is represented as a probability, the variance in state noise Q c is set to.5 after each alarm. Performance Evaluation We now consider an infrastructure network in which the contending stations transmit toward the access point. To account for heterogeneous channel conditions, we assume that each transmitter experiences a different channel-error rate. In the simulation shown in Figure 9, each station is assigned a different value for, with the values chosen uniformly in the range [, 1]. The total number of contending stations is set to ten. Figure 9 shows the estimate of interference for two stations A and B. Station A experiences the high error rate (A) 5.565, whereas station B experiences the low error rate (B) Measurement samples from the two stations are collected every.5 s. Figure 9 shows that despite the high variance of the samples of pr e, the Kalman filter errors on the estimates of are lower than 1% after a few seconds. Conversely, the ARMA filter with a smoothing factor of.95 requires more than 2 s to reduce the errors on the estimates of under 1% and provides biased estimates of. In particular, estimates of for station A are much higher than the actual values. The ARMA filters are also slower than the Kalman filters in responding to changes in the network state. Estimates of 4 IEEE CONTROL SYSTEMS MAGAZINE» APRIL 21