MAC Protocols for Optimal Information Retrieval Pattern in Sensor Networks with Mobile Access

In signal ﬁeld reconstruction applications of sensor network, the locations where the measurements are retrieved from a ﬀ ect the reconstruction performance. In this paper, we consider the design of medium access control (MAC) protocols in sensor networks with mobile access for the desirable information retrieval pattern to minimize the reconstruction distortion. Taking both performance and implementation complexity into consideration, besides the optimal centralized scheduler, we propose three decentralized MAC protocols, namely, decentralized scheduling through carrier sensing, Aloha scheduling, and adaptive Aloha scheduling. Design parameters for the proposed protocols are optimized. Finally, performance comparison among these protocols is provided via simulations.


INTRODUCTION
In many applications, sensor networks operate in three phases: sensing, information retrieval, and information processing. As a typical example, in physical environmental monitoring, sensors first take measurements of the signal field at a particular time. The data are then collected from individual sensors to a central processing unit, where the signal field is finally reconstructed.
An appropriate network architecture for such applications is SEnsor Networks with Mobile Access (SENMA) [1,2]. As shown in Figure 1, SENMA consists of two types of nodes: low-power low-complexity sensors randomly deployed in a large quantity, and a few powerful mobile access This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. points communicating with the sensors. The use of mobile access points enables data collection from specific areas of the network.
We focus on the latter two operational phases in the SENMA architecture: information retrieval and processing, which are strongly coupled. To achieve the optimal performance of the sensor network, the two phases should be considered jointly. The key to information retrieval is medium access control (MAC) that regulates data retrieval from sensors to the access point. The main focus of this paper is to design MAC protocols for the optimal reconstruction of the signal field.
The MAC design for sensor network applications needs to take into account application-specific characteristics, for example, the correlation of the field, the randomness of the sensor locations, and the redundancy of the large-scale sensor deployment. The traditional MAC design criteria, such as throughput, fail to capture the characteristics of the specific Access point Sensor Figure 1: A 1D sensor network with a mobile access point. sensor application; a high-throughput MAC does not imply low reconstruction distortion. In this paper, we propose a new MAC design criterion for the field reconstruction application.
The new MAC design criterion is motivated by the need to collect data evenly across the field for a given throughput. If we have an infinitely dense network, the optimal data collection strategy is to retrieve samples from evenly spaced locations. For a finite density network considered in this work, however, there may not exist sensors in the desired locations. The optimal centralized scheduler, with the location information of all sensors, calculates the optimal location set and retrieves data from the optimal set to minimize the reconstruction distortion. Such optimal centralized scheduler comes with the substantial cost of sensor-location information gathering. Decentralized MAC protocols, on the other hand, require much less intervention from the mobile access point and bandwidth resources.
We consider a one-dimensional problem for simplicity, which can be extended to a two-dimensional setup. Taking both performance and implementation complexity into consideration, besides the optimal centralized scheduler, we propose three decentralized MAC protocols. We first propose a decentralized scheduler via carrier sensing, which, under the no-processing delay assumption, provides little performance loss compared to the performance of the optimal scheduler. Then, to simplify the implementation, we introduce a MAC scheme which uses Aloha-like random access within a resolution interval centered at the desired retrieval location. Finally, to improve the performance, we propose an adaptive Aloha scheduling scheme which adaptively chooses the desired retrieval locations based on the history of retrieved samples. Design parameters are optimized for the proposed schemes. The performance comparison under various sensor density conditions and packet collection sizes is also provided.
The problems on sensor network communications have attracted a growing research interest. In terms of medium access control, many MAC protocols have been proposed aiming at the special needs and requirements for both ad-hoc sensor networks [3,4,5,6] and sensor networks with mobile access [2]. Most of these proposed schemes only consider the MAC layer performance, that is, throughput. The effect of MAC for information retrieval on information processing is analyzed in [7,8] for infinite and finite sensor density networks, respectively, where the performance of the centralized scheduler and that of the decentralized random access are analyzed and compared.
The idea of using carrier sensing for energy-efficient transmission in sensor networks was first proposed in [9,10,11], where backoff delays are chosen as a function of the channel strength. The carrier sensing strategy presented here generalized that in [9,10,11] by using carrier sensing to distinguish nodes in different locations.

SYSTEM MODEL AND MAC DESIGN OBJECTIVE
In this section, we introduce the system model and the signal field reconstruction distortion measure, which leads to a simple MAC design objective.

Signal field model
Consider a one-dimensional field of unit length, denoted by be the source of interest in A at a particular time. We assume that the spatial dynamic of S(x) is a homogeneous Gaussian random field given by the following linear stochastic differential equation: where is the stationary solution of (1). The random field modeled in (1) is essentially a diffusion process which is often used to model many physical phenomena of interest. Being homogeneous in A, S(x) has the autocorrelation for x 0 < x 1 , which is only a function of the distance between the two points x 1 and x 0 .

Sensor network model
We assume that sensors in A are deployed randomly, and their distribution forms a one-dimensional homogeneous spatial Poisson field with local density ρ sensors/unit area. That is, in a length-l interval, the number of sensors N(l) is a Poisson random variable with distribution and the numbers of sensors in any two disjoint intervals are independent. To avoid the boundary effect, we assume that there is a sensor at each of the two boundary points x = 0 and x = 1. Let N denote the number of sensors in the field excluding the two boundary points. Denote x N = [x 1 , x 2 , . . . , x N ] T the sensor locations, where 0 < x 1 < x 2 < · · · < x N < 1. After its deployment, each sensor obtains its own location information through some positioning method. At a prearranged time, all sensors measure their local signals, forming a snapshot of the signal field. The measurement of a sensor at location x is given by where Z(x) is zero mean, spatially white Gaussian measurement noise with variance σ 2 Z , and is independent of S(x). Each sensor stores its local measurement along with its location information in the form of a packet for future data collection.

The multiple-access channel
When the mobile access point is ready for data collection, sensors transmit their measurement packets to the access point through a common wireless channel. We assume slotted transmission in a collision channel, that is, a packet is correctly received if and only if no other users attempt transmission. To retrieve measurement packets from the field through a collision channel, some form of MAC is needed. In this paper, we propose and discuss four MAC protocols, with different performance and complexity trade-off, to optimize the reconstruction performance.
In each time slot, sensors compete for the channel use. The channel output may be a collision, an empty slot, or a data packet that contains the measurement and the location of the sensor. We assume that the access point uses m time slots to retrieve measurement data and refer to m as the packet collection size. Let q i , 1 ≤ i ≤ m, denote the sample location of the ith channel outcome if a packet is successfully received. Otherwise, let q i = ∅. Let q = [q 1 , q 2 , . . . , q m ] T denote the output location vector. To avoid the boundary effect for signal reconstruction, we assume that, in addition to the m retrieval attempts, the two boundary measurements are also retrieved by the mobile access point.

Information processing and performance measure
After the information retrieval, we reconstruct the original signal field based on the received data samples. Let K denote the number of q i 's not equal to ∅ in q, excluding the two boundary points. Let r K = [r 1 , r 2 , . . . , r K ] T , r 1 ≤ r 2 ≤ · · · ≤ r K , be the ordered sample location vector constructed from q by ordering the non-∅ elements. For convenience, let r 0 = 0 and r K+1 = 1. We estimate S(x) at location x using its two immediate neighbor samples by the MMSE smoothing, that is, for r i < x < r i+1 , 0 ≤ i ≤ K,  Given q, we define the maximum field reconstruction distortion as he maximum mean-square estimation error in A, The expected maximum distortion of the signal reconstruction in m collection time slots is then given bȳ where the expectation is taken over the output location vector q.

MAC design objective
Our objective is to design MAC protocols that result in the smallest signal field reconstruction distortion for a fixed number of retrieval slots. From [7,8], we have shown that the maximum distortion is determined only by the maximum distance between two adjacent data samples, The maximum distortion in (6) is a monotonically increasing function of d max . Thus, a smaller E{d max } indicates a smaller reconstruction distortion. Our objective now is to design MAC for the minimum E{d max }.

Linear field and circular field
The above 1D field model with two boundary points is referred to as the linear field ( Figure 2). Another filed of interest is the circular field which is a circle with unit circumference ( Figure 3). As in the linear field, sensors in the circular field are deployedaccording to Poisson distribution with density ρ sensors/unit length; see (3). The location of each sensor on the circular field is described by its angle θ, 0 ≤ θ < 2π, as shown in Figure 3. Alternatively, the location can also be de- denote the sensor locations where N is the number of sensors in the field. 1 Similar to the linear field, let q = [q 1 , q 2 , . . . , q m ] T denote the output location vector, where q i , 1 ≤ i ≤ m, is the sample location of the ith channel outcome if a packet is successfully received in the ith slot, or q i = ∅ otherwise. Let K be the number of non-∅ elements in q and let r K = [r 1 , r 2 , . . . , r K ] T be the ordered sample location vector constructed from q by ordering the non-∅ elements, with r 1 being the smallest. For convenience, let r K+1 = 1+r 1 . The maximum distance for the circular field is defined as To avoid ambiguity, define d max to be 1 if only one sample is retrieved, or 2 if none is retrieved. Since we are not working in the extremely low-density regime, the probability of retrieving only one or no sample is small. Besides the vector form as in (9) and (10), the input parameters of d max (q) for both fields also take other forms in this paper for the ease of presentation. The MAC design objective for the circular field is also to minimize E{d max }.

Optimal centralized scheduling
Assume that the location information x N of all sensors is available to the mobile access point. Also assume that the mobile access point is able to activate individual nodes for data transmission. The mobile access point is then able to precompute the optimal set of m locations and to activate only those sensors. This results in the minimum d max , and therefore, the best performance. The performance under this scheduler can be used as a benchmark for performance comparison. For a given sensor location realization x N and a fixed m, the optimal d max is The optimal set of sensor locations are those that produce d * max , and the mobile access point activates these sensors one at a time to avoid collision.
The optimization problem (11) can be solved by a brute force search. To reduce the computational complexity, we propose an efficient algorithm for the linear field, Algorithm 1. It first looks for an initial set of locations and The search scheme consists of three steps.
Step 1. Location initialization. A set of m sensor locations is chosen from x N as the initial set, (q (0) 1 , . . . , q (0) m ). The d max of the chosen set is assigned to d (0) max . Let i = 0.
Step 2. Within interval (0, d (i) max ), find the sensor location closest to d (i) max and assign it to q (i+1) j+1 be the sensor location closest to the right boundary of the interval; if , the algorithm ends and is the optimal set of locations for the given x N and m. We select the initial set as follows. Choose q (0) i to be the sensor location that is closest to i/(m + 1), 1 ≤ i ≤ m, and let the corresponding d max be d (0) max .
Algorithm 1 the corresponding d max . Based on this d max , it looks for another set of locations resulting in a smaller d max . Iteratively, d max converges to its minimum value in finite steps.
In each iteration, d (i) max is strictly decreasing. Algorithm 1 stops only when d (i) max has reached its minimum value. For a field with finite sensors, the possible values of d max is finite. Therefore, Algorithm 1 finds the optimal locations in finite steps.
Next, we consider the circular field. Algorithm 1 can be adapted to solve the optimization of (11) by converting the circular field to the linear field. For the ease of discussion, for a given x N , let Suppose that x i is included in the optimal set, 1 ≤ i ≤ N. Then we break the circle at point x i , and (x i+1 , . . . , x N+i−1 ) are sensor locations in the linear field with x i and x N+i being the two boundary points. The other m − 1 points that minimize d max under the assumption that x i is selected can be solved by Algorithm 1. 2 Exhausting all x i gives the global optimal d * max . To shorten the search time, use the smallest d max obtained in previous runs of Algorithm 1 as the initialization value d (0) max for the new search with a new x i . It can be shown that exhausting x 1 ≤ x i < x 1 + d max is enough, where d max is any value greater than or equal to the global minimum d * max . The initialization value d (0) max for the current x i can be used as d max for the exhaustion stopping criterion.
The centralized scheme gives the best performance under the condition that all sensor location information is available to the mobile access point. However, the bandwidth required for sensors reporting their locations is prohibitively large, especially for large-scale sensor networks. Decentralized schemes that do not require the knowledge of sensor locations at the mobile access point are desirable. Nonetheless, the centralized scheme gives the best possible performance and serves as a benchmark.

Decentralized scheduling through carrier sensing
In practice, the sensor location information may not be available at the mobile access point. Each senor only knows its own location. In this case, in order to retrieve data with the desired pattern and in a decentralized fashion, we propose decentralized scheduling through carrier sensing. We assume that each sensor has a transmission coverage radius R. Since the propagation delay is relatively small as compared to the slot length, we assume perfect carrier sensing with no propagation delay within radius R, that is, a sensor's transmission is detected immediately by other sensors within distance R.
In the proposed protocol, sensor transmissions are scheduled through carrier sensing, where the distances of sensors from the desired locations are used in the backoff scheme. The backoff time of a sensor is a function of the distance from the sensor to the desired location. A similar idea of using carrier sensing for decentralized transmission was first proposed in [9,10,11], where the channel state information was used in the backoff function of the carrier sensing scheme for opportunistic transmission.
Protocol. In each time slot, a segment of length R is activated. Sensors within the activated region compete for the channel use. Let p j denote the center of the jth segment, 1 ≤ j ≤ m. Each sensor within the activated segment computes its distance to p j , that is, if x i is within the activated segment, its distance is d i, j = |x i − p j | for the linear field, or d i, j = min(|x i − p j |, 1 − |x i − p j |) for the circular field. The activated sensors then choose their respective backoff time based on a backoff function τ(d), which maps the distance to a backoff time. A sensor listens to the channel during its backoff time. If it detects a transmission before its backoff timer expires, the sensor will not transmit in this time slot. Otherwise, the sensor transmits its measurement sample packet immediately when its timer expires. The function τ(d) is designed to be strictly increasing; therefore, if there are sensors in the activated region, only the sensor closest to the center of the activated segment will be received successfully in this time slot. An example of τ(d) is given in Figure 4. The activation sequence is deterministic in the sense that it does not change based on the previous data collection results.
Where the activation segments should be centered is a design issue. As the next lemma shows, for the circular field, the segments should be separated evenly. Lemma 1. Consider the circular field. Suppose that in the ith time slot, 1 ≤ i ≤ m, the length-L segment centered at p i , 0 ≤ p i < 1, is activated to compete for the collision channel use. Suppose that these segments do not overlap. Let q i , 0 ≤ q i < 1, be the outcome location in the ith slot if a packet is successfully received, or q i = ∅ otherwise. Define the relative outcome where the conditions in (12) are to deal with the coordinate transition around θ = 0 or θ = 2π on the circular field. If b i 's are independent and identically distributed (i.i.d.), then evenly spaced segments produce the minimum E{d max } for the circular field.
For the proof, see Appendix A. For the linear field, however, evenly spaced activation segment sequence is not optimal because of the asymmetry introduced by the two boundary points. Nonetheless, evenly spaced segment sequence has good performance for large m and ρ since the boundary effect is negligible in this scenario. We will use the evenly spaced segment sequence p i = i/(m + 1), 1 ≤ i ≤ m, for the linear field in the simulations.
The carrier sensing protocol has high throughput because, if there are nodes within an activation segment, the packet closest to the center will be successfully received with probability one.

Aloha scheduling
The carrier sensing scheme requires additional hardware for the carrier sensing functionality. In addition, the synchronization and timing requirements are strict for the carrier sensing mechanism. Next, we present a cost-efficient protocol for sensor sample collection.
Protocol. Select a sequence of m nonoverlapping lengthsegments as the activation sequence. Activate one segment in the activation sequence every time slot. Sensors within the activated region transmit their packet independently with probability P. The activation sequence is deterministic in the 0 p 1 p 2 p 3 1 Figure 5: Aloha scheme on the linear field. A sequence of lengthsegments is activated sequentially. The sensors within the activated range transmit with probability P.
sense that it does not depend on the data collection results. Figure 5 illustrates the Aloha scheme on the linear field. In the Aloha protocol, the segment length , the transmission probability P, and the center locations of the activation segments are optimization parameters. Lemma 2. For both the linear and the circular fields, the optimal transmission probability P is one and the optimal segment length is strictly less than 1/ρ.
For the proof, see Appendix B. It can be shown that the result of Lemma 2 also holds in a more general setup where the transmission probability within the activation region is a function of the distance from the sensor to the center of the activation region. An intuitive way to explain Lemma 2 is that, for the same throughput, the smaller the activation interval length is, the more precise the outcome location can be. Therefore, the data collection outcomes for a smaller activation interval are closer to evenly spaced center locations, producing a smaller E{d max }. Letting P = 1 gives the smallest activation interval length for a given throughput. The result about can be explained as follows. Shortening the activation length has two effects on E{d max }: one is that it gives a lower throughput if the length is less than or equal to 1/ρ, which is a negative effect; the other is that it produces a more precise outcome location control, a positive effect. Although (P = 1, = 1/ρ) gives the maximum throughput for Aloha, when is shortened a little, the throughput only decreases a little because the derivative of the throughput with respect to is zero at = 1/ρ. Thus the negative effect is small. The positive effect from the more precise location control favors an activation length strictly shorter than 1/ρ, meaning that the optimal throughput is strictly less than 1/e. Nonetheless, the gain by selecting a length shorter than 1/ρ is small for dense sensor networks. We will use = 1/ρ in the simulations.
As shown in Lemma 1, for the circular field, evenly spaced center locations of the activation segments are optimal. As mentioned in the carrier sensing protocol, for the linear field, evenly spaced activation segments are not optimal. Nonetheless, evenly spaced segments have good performance for large m and ρ, and we will use evenly spaced activation segments in the simulations for the linear field.

Adaptive Aloha scheduling
The carrier sensing and Aloha scheduling protocols presented previously are deterministic scheduling since the center location of each activation segment does not change according to previous data collection outcomes. In deterministic scheduling, the activation location information may be preset to sensors before their deployment, eliminating the The algorithm tries to break the maximum distance by placing the next polling interval at the center of the two received data sample locations whose distance is d max .
need to broadcast the location information from the mobile access point and saving some hardware cost. Another approach is to let the mobile access point decide the next activation location on the fly, based on previous data collection results. Allowing the activation sequence to adapt to previous data collection results may give better performance. Next we present an adaptive scheduling for Aloha.
Protocol. The basic activation strategy is similar to the Aloha protocol. The mobile access point activates an interval of length = 1/ρ in each time slot; the sensors within the range transmit with probability P = 1. The difference is that, in the adaptive version, the locations of the activation intervals depend on the previous data collection results, which is described as follows.
After obtaining a new packet, the access point checks all the previous received data and finds the two adjacent sample locations that have the maximum distance. The access point then locates the next polling interval in the middle of these two samples locations (see Figure 6 for the linear field case). If an empty slot occurs, the access point then activates the length-interval adjacent (either left or right) to the previous empty intervals until a success or collision occurs. If a collision occurs, the access point resolves the collision by splitting the collision interval until a packet is successfully received (similar to the splitting algorithms [12]). If a packet is received successfully, the access point recalculates and tries to break the new d max of the received samples within the remaining time slots. The algorithm keeps running until it uses up the m time slots.
The above protocol works in an environment where the mobile access point can communicate to the whole field from one location, for example, high-altitude airplanes or satellites. There are other types of adaptive scheduling schemes. For example, we can also adapt the activation sequence on a carrier sensing scheduling setup. However, as will be shown in the simulations section, the gain of adapting activation sequence on a carrier sensing setup is small because the performance of the carrier sensing scheduling is already close to that of the optimal centralized scheduling.

SIMULATIONS
In this section, we compare the performance of the MAC protocols proposed in the last section through simulations.  shown. For the circular field, similar results are observed. Sensors are randomly deployed according to the Poisson distribution with density ρ. For convenience, we name these MAC protocols as follows.
We use the d max found using π 2 as the initial maximum distance for the iteration algorithm in π 1 . The search stops after 1-2 iterations typically. In the comparison, we use E{d max } as the performance metric. Figures 7 and 8 plot E{d max } versus m for sensor density ρ = 40 and 200, respectively. The expectation of d max in the figures is averaged over 100 000 realizations of the Poisson sensor field. As expected, as m increases, the number of data samples received at the mobile access point increases, and thus E{d max } decreases. We see that there is little performance loss by using π 2 . Notice that, when m is larger than ρ (Figure 7), under π 1 and π 2 , data from all sensors can be retrieved with a high probability. Therefore, the performance gap for the two protocols diminishes. The performance under π 3 is worse than other schemes even when m is larger than ρ. This is because, under π 3 , some scheduled intervals do not have data packets received successfully due to either collision or void of sensors. Unlike π 3 , the location of each activation interval of π 4 is adapted to the previous data collection outcomes. When m is large, it has enough slots to search for intervals within which sensors exist and to resolve collision, therefore avoiding the problem in π 3 . From Figure 7, we see that, when m is large, the performance under π 4 is as good as the optimal case.  Figures 9 and 10 plot E{d max } versus ρ for packet collection size m = 10 and 50, respectively. As expected, as ρ increases, the density of the sensor field increases, and the received data locations are closer to the desired locations, resulting in a sample pattern closer to evenly spaced. Therefore, E{d max } converges to the minimum value as ρ increases. Again, we see that the performance under π 2 closely follows the optimal one. As ρ increases, we see the performance gap between the two Aloha schemes and π 1 increases. The performance loss under π 3 is mainly due to its lower throughput than that of π 1 and π 2 , which limits the number of received samples. We observe that there is a significant performance improvement of π 4 over π 3 by adaptively optimizing the retrieval pattern based on the retrieval history.

CONCLUSION
To reconstruct the signal field using sensor networks, the locations of the retrieved data affect the signal field reconstruction performance. In this paper, we design MAC protocols to obtain the desired data retrieval pattern. We propose a new MAC design criterion that takes into account the application characteristics of the signal field reconstruction. Taking both performance and implementation complexity into consideration, besides the optimal centralized scheduler, we propose three decentralized MAC protocols. We have shown that, for the carrier sensing and Aloha scheduling schemes, evenly spaced activation intervals are optimal for the circular field. For the Aloha scheduling in both the linear field and the circular field, the optimal transmission probability is one and the optimal activation interval length is strictly smaller than 1/ρ, resulting in a throughput strictly less than 1/e. Our simulations show that using the decentralized scheduling through carrier sensing results in little performance loss compared to the performance of the optimal scheduler. For the two Aloha schemes, by exploring the history of retrieved data locations, adaptive Aloha provides a significant performance gain over the simple Aloha scheme.

A. PROOF OF LEMMA 1
We first define four operations on integers or real numbers. Let i and j be two integers. Define i⊕ j to be equal to i+ j+km, where k is the integer such that . Let x 1 and x 2 be two real numbers. Define x 1 ⊕x 2 to be equal to x 1 +x 2 +k, where k is the integer such that 0 ≤ x 1 +x 2 +k < 1. Let x 1 x 2 x 1 ⊕ (−x 2 ). For convenience, extend the operations and on real numbers to include the symbol ∅. Let x 1 and x 2 be real numbers or the symbol ∅. Define x 1 ⊕x 2 and x 1 x 2 to be ∅ if either x 1 or x 2 is equal to ∅.
It can be verified that the inverse function of (12) is given by The average d max when p is the center location vector is given by where p ⊕ b is the vector with p i ⊕ b i as the ith entry. Without loss of generality, assume that p is an ordered vector with p 1 being the smallest. Letp be an equally spaced location vector on the circular field. Without loss of generality, let The proof is concluded if we show that, for all p, Therefore, the left-hand side of (A.3) can be expressed as Hence, it suffices to show that for any b and p, For a given b with one or no non-∅ element, by definition, d max is equal to 1 or 2, respectively, for both p andp. Therefore, (A.6) holds.
Let L(i, j) be the set of indices between i and j counterclockwise, 1 ≤ i, j ≤ m and i = j, that is, L(i, j) = {l : i < l < j} if i < j, or {l : i < l ≤ m, or 1 ≤ l < j} if i > j. For a given b with at least two non-∅ entries, search d max among the output locationsp ⊕ b on the circular field. Suppose that d max occurs from the ith point to the jth point counterclockwise, that is, b i , b j = ∅, b l = ∅ for l ∈ L(i, j), and is at least as large as the distance from where (A.8) holds because p j k p i k > L > b j − b i , and (A.9) holds because m−1 k=0 (p i k⊕l p i k⊕l 1 ) is equal to the circumference of the circular field, which is one.

B. PROOF OF LEMMA 2
We prove Lemma 2 for the linear field. The proof for the circular field is basically the same except that extra care should be taken for coordinate transitions around location x = 0 or x = 1. Consider a more general scheme which does not require that each activation segment has the same length and transmission probability. Let p i , P i , and i denote the center, the transmission probability, and the length of the ith activation segment, respectively, 1 ≤ i ≤ m. Let q i be the outcome location of the ith channel competition, or q i = ∅ if no sample packet is received successfully in the ith time slot, due to either collision or no transmission. The throughput of the ith time slot is s i P r q i = ∅ = i P i ρe − iPi ρ . (B.10) Given a packet is received successfully in the ith time slot, the location q i is uniformly distributed, p q i |q i = ∅ = 1 i 1 pi− i /2≤qi≤pi+ i /2 , (B.11) where 1 A is the indicator function. Let q = [q 1 , . . . , q m ] T . Since the activation segments do not overlap, q i 's are independent. Let q /i denote the length-(m−1) vector constructed by taking out q i from q. The expected d max (q) is given by (B.12) Suppose that (˜ i ,P i ) give the same throughput as ( i , P i ), that is,˜ iPi ρe −˜ iPi ρ = s i . And suppose that˜ i < i . We will show that if ( i , P i ) are replaced by (˜ i ,P i ) while other parameters remain the same, then E{d max (q)} decreases. Since the throughput s i remains the same, the first term of (B.12) remains the same. If we can show that, for all q /i and for − i /2 ≤ a ≤ i /2, d max q /i , q i = p i + a + d max q /i , q i = p i − a ≥ d max q /i , q i = p i +˜ i i a + d max q /i , q i = p i −˜ i i a , (B.13) then we have shown that the second term of (B.12) decreases. Therefore, we have proved that, with the same throughput, the shorter the activation length, the better the performance. Hence, the optimal P i is 1 and the optimal i is less than or equal to 1/ρ for all i because these conditions in Aloha give the shortest activation length for a given throughput.
Next we prove (B.13). Let length-m vectors q ,q, andq be functions of q given q i = ∅: q j =q j =q j = q j for j = i, q i = 2p i −q i ,q i = p i +˜ i / i (q i − p i ), andq i = p i −˜ i / i (q i − p i ) ( Figure 11). Equivalently, we are proving that d max (q) + d max (q ) ≥ d max (q) + d max (q ) (B.14) for all q with q i = ∅, or equivalently, for allq withq i = ∅. We first define three terms for the ease of discussion. d max (q) is said to be associated with q i if q i is one of the endpoints that produces d max given q as the outcome location vector. d max (q) is said to be associated with q i to the inside if d max (q) is associated with q i and the center p i is between the two endpoints of d max . d max (q) is said to be associated with q i to the outside if d max (q) is associated with q i and the center p i is d max ( q), d max ( q ) q i−1 q i p i q i q i+1 q i+2 q i q i Figure 11: Case 1.
not between the two endpoints of d max . We prove (B.14) by verifying all possible cases.