Showing papers in "IEEE Transactions on Wireless Communications in 2005"

Energy-constrained modulation optimization

Abstract: Wireless systems where the nodes operate on batteries so that energy consumption must be minimized while satisfying given throughput and delay requirements are considered. In this context, the best modulation strategy to minimize the total energy consumption required to send a given number of bits is analyzed. The total energy consumption includes both the transmission energy and the circuit energy consumption. For uncoded systems, by optimizing the transmission time and the modulation parameters, it is shown that up to 80% energy savings is achievable over nonoptimized systems. For coded systems, it is shown that the benefit of coding varies with the transmission distance and the underlying modulation schemes.

Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints

Abstract: Multiuser orthogonal frequency division multiplexing (MU-OFDM) is a promising technique for achieving high downlink capacities in future cellular and wireless local area network (LAN) systems. The sum capacity of MU-OFDM is maximized when each subchannel is assigned to the user with the best channel-to-noise ratio for that subchannel, with power subsequently distributed by water-filling. However, fairness among the users cannot generally be achieved with such a scheme. In this paper, a set of proportional fairness constraints is imposed to assure that each user can achieve a required data rate, as in a system with quality of service guarantees. Since the optimal solution to the constrained fairness problem is extremely computationally complex to obtain, a low-complexity suboptimal algorithm that separates subchannel allocation and power allocation is proposed. In the proposed algorithm, subchannel allocation is first performed by assuming an equal power distribution. An optimal power allocation algorithm then maximizes the sum capacity while maintaining proportional fairness. The proposed algorithm is shown to achieve about 95% of the optimal capacity in a two-user system, while reducing the complexity from exponential to linear in the number of subchannels. It is also shown that with the proposed resource allocation algorithm, the sum capacity is distributed more fairly and flexibly among users than the sum capacity maximization method.

Symbol error probabilities for general Cooperative links

Abstract: Cooperative diversity (CD) networks have been receiving a lot of attention recently as a distributed means of improving error performance and capacity. For sufficiently large signal-to-noise ratio (SNR), this paper derives the average symbol error probability (SEP) for analog forwarding CD links. The resulting expressions are general as they hold for an arbitrary number of cooperating branches, arbitrary number of cooperating hops per branch, and various channel fading models. Their simplicity provides valuable insights to the performance of CD networks and suggests means of optimizing them. Besides revealing the diversity, they clearly show from where this advantage comes from and prove that presence of diversity does not depend on the specific (e.g., Rayleigh) fading distribution. Finally, they explain how diversity is improved in multihop CD networks.

Cross-layer optimization for OFDM wireless networks-part I: theoretical framework

Abstract: In this paper, we provide a theoretical framework for cross-layer optimization for orthogonal frequency division multiplexing (OFDM) wireless networks. The utility is used in our study to build a bridge between the physical layer and the media access control (MAC) layer and to balance the efficiency and fairness of wireless resource allocation. We formulate the cross-layer optimization problem as one that maximizes the average utility of all active users subject to certain conditions, which are determined by adaptive resource allocation schemes. We present necessary and sufficient conditions for utility-based optimal subcarrier assignment and power allocation and discuss the convergence properties of optimization. Numerical results demonstrate a significant performance gain for the cross-layer optimization and the gain increases with the number of active users in the networks.

Autoregressive modeling for fading channel simulation

Abstract: Autoregressive stochastic models for the computer simulation of correlated Rayleigh fading processes are investigated. The unavoidable numerical difficulties inherent in this method are elucidated and a simple heuristic approach is adopted to enable the synthesis of accurately correlated, bandlimited Rayleigh variates. Startup procedures are presented, which allow autoregressive simulators to produce stationary channel gain samples from the first output sample. Performance comparisons are then made with popular fading generation techniques to demonstrate the merits of the approach. The general applicability of the method is demonstrated by examples involving the accurate synthesis of nonisotropic fading channel models.

Queuing with adaptive modulation and coding over wireless links: cross-Layer analysis and design

Abstract: Assuming there are always sufficient data waiting to be transmitted, adaptive modulation and coding (AMC) schemes at the physical layer have been traditionally designed separately from higher layers. However, this assumption is not always valid when queuing effects are taken into account at the data link layer. In this paper, we analyze the joint effects of finite-length queuing and AMC for transmissions over wireless links. We present a general analytical procedure, and derive the packet loss rate, the average throughput, and the average spectral efficiency (ASE) of AMC. Guided by our performance analysis, we introduce a cross-layer design, which optimizes the target packet error rate of AMC at the physical layer, to minimize thpacket loss rate and maximize the average throughput, when combined with a finite-length queue at the data link layer. Numerical results illustrate the dependence of system performance on various parameters, and quantify the performance gain due to cross-layer optimization. Our focus is on the single user case, but we also discuss briefly possible applications to multiuser scenarios.

Capacity of MIMO systems with antenna selection

Abstract: We consider the capacity of multiple-input multiple-output systems with reduced complexity. One link-end uses all available antennas, while the other chooses the L out of N antennas that maximize capacity. We derive an upper bound on the capacity that can be expressed as the sum of the logarithms of ordered chi-square-distributed variables. This bound is then evaluated analytically and compared to the results obtained by Monte Carlo simulations. Our results show that the achieved capacity is close to the capacity of a full-complexity system provided that L is at least as large as the number of antennas at the other link-end. For example, for L = 3, N = 8 antennas at the receiver and three antennas at the transmitter, the capacity of the reduced-complexity scheme is 20 bits/s/Hz compared to 23 bits/s/Hz of a full-complexity scheme. We also present a suboptimum antenna subset selection algorithm that has a complexity of N/sup 2/ compared to the optimum algorithm with a complexity of (N/sub L/).

Cross-layer optimization for OFDM wireless networks-part II: algorithm development

Abstract: We have established a theoretical framework for cross-layer optimization in orthogonal frequency division multiplexing (OFDM) wireless networks. In this paper, we focus on effective and practical algorithms for efficient and fair resource allocation in OFDM wireless networks. We have taken various conditions into account and developed a variety of efficient algorithms, including sorting-search dynamic subcarrier assignment, greedy bit loading, and power allocation, as well as objective aggregation algorithms. We have also modified those algorithms for a certain type of nonconcave utility functions. To further improve performance by exploiting time diversity, a low-pass time filter can be easily incorporated into all of the algorithms. Simulation results have confirmed that the utility-based cross-layer optimization can significantly enhance the system performance and guarantee fairness. The gains come from multiuser diversity, frequency diversity, as well as time diversity. The fairness is automatically achieved by the behavior of marginal utility functions.

On energy provisioning and relay node placement for wireless sensor networks

Abstract: Wireless sensor networks that operate on batteries have limited network lifetime. There have been extensive recent research efforts on how to design protocols and algorithms to prolong network lifetime. However, due to energy constraint, even under the most efficient protocols and algorithms, the network lifetime may still be unable to meet the mission's requirements. In this paper, we consider the energy provisioning (EP) problem for a two-tiered wireless sensor network. In addition to provisioning additional energy on the existing nodes, we also consider deploying relay nodes (RNs) into the network to mitigate network geometric deficiencies and prolong network lifetime. We formulate the joint problem of EP and RN placement (EP-RNP) into a mixed-integer nonlinear programming (MINLP) problem. Since an MINLP problem is NP-hard in general, and even state-of-the-art software and techniques are unable to offer satisfactory solutions, we develop a heuristic algorithm, called Smart Pairing and INtelligent Disc Search (SPINDS), to address this problem. We show a number of novel algorithmic design techniques in the design of SPINDS that effectively transform a complex MINLP problem into a linear programming (LP) problem without losing critical points in its search space. Through numerical results, we show that SPINDS offers a very attractive solution and some important insights to the EP-RNP problem.

Performance analysis of priority schemes for IEEE 802.11 and IEEE 802.11e wireless LANs

Abstract: The IEEE 802.11 distributed coordination function (DCF) enables fast installation with minimal management and maintenance costs, and is a very robust protocol for the best effort service in wireless medium. However, the current DCF is unsuitable for real-time applications. This paper studies backoff-based priority schemes for IEEE 802.11 and the emerging IEEE 802.11e standard by differentiating the minimum backoff window size, the backoff window-increasing factor, and the retransmission limit. An analytical model is proposed to derive saturation throughputs, saturation delays, and frame-dropping probabilities of different priority classes for all proposed priority schemes. Simulations are conducted to validate analytical results. The proposed priority schemes can be easily implemented, and the results from this paper are beneficial in designing good priority parameters.

Design and analysis of an MST-based topology control algorithm

Abstract: In this paper, we present a minimum spanning tree (MST)-based algorithm, called local minimum spanning tree (LMST), for topology control in wireless multihop networks. In this algorithm, each node builds its LMST independently and only keeps on-tree nodes that are one-hop away as its neighbors in the final topology. We analytically prove several important properties of LMST: 1) the topology derived under LMST preserves the network connectivity; 2) the node degree of any node in the resulting topology is bounded by 6; and 3) the topology can be transformed into one with bidirectional links (without impairing the network connectivity) after removal of all unidirectional links. Simulation results show that LMST can increase the network capacity as well as reduce the energy consumption.

ICI mitigation for pilot-aided OFDM mobile systems

Abstract: Orthogonal frequency-division multiplexing (OFDM) is robust against frequency selective fading due to the increase of the symbol duration. However, for mobile applications channel time-variations in one OFDM symbol introduce intercarrier-interference (ICI) which degrades the performance. This becomes more severe as mobile speed, carrier frequency or OFDM symbol duration increases. As delay spread increases, symbol duration should also increase in order to maintain a near-constant channel in every frequency subband. Also, due to the high demand for bandwidth, there is a trend toward higher carrier frequencies. Therefore, to have an acceptable reception quality for the applications that experience high delay and Doppler spread, there is a need for ICI mitigation within one OFDM symbol. We introduce two new methods to mitigate ICI in an OFDM system with coherent channel estimation. Both methods use a piece-wise linear model to approximate channel time-variations. The first method extracts channel time-variations information from the cyclic prefix. The second method estimates these variations using the next symbol. We find a closed-form expression for the improvement in average signal-to-interference ratio (SIR) when our mitigation methods are applied for a narrowband time-variant channel. Finally, our simulation results show how these methods would improve the performance in a highly time-variant environment with high delay spread.

Nonline-of-sight error mitigation in mobile location

Abstract: The location of mobile terminals has received considerable attention in the recent years. The performance of mobile location systems is limited by errors primarily caused by nonline-of-sight (NLOS) propagation conditions. We investigate the NLOS error identification and correction techniques for mobile user location in wireless cellular systems. Based on how much a priori knowledge of the NLOS error is available, two NLOS mitigation algorithms are proposed. Simulation results demonstrate that with the prior information database, the location estimate can be obtained with good accuracy even in severe NLOS propagation conditions.

A QRD-M/Kalman filter-based detection and channel estimation algorithm for MIMO-OFDM systems

Abstract: The use of multiple transmit/receive antennas forming a multiple-input multiple-output (MIMO) system can significantly enhance channel capacity. This paper considers a V-BLAST-type combination of orthogonal frequency-division multiplexing (OFDM) with MIMO (MIMO-OFDM) for enhanced spectral efficiency and multiuser downlink throughput. A new joint data detection and channel estimation algorithm for MIMO-OFDM is proposed which combines the QRD-M algorithm and Kalman filter. The individual channels between antenna elements are tracked using a Kalman filter, and the QRD-M algorithm uses a limited tree search to approximate the maximum-likelihood detector. A closed-form symbol-error rate, conditioned on a static channel realization, is presented for the M=1 case with QPSK modulation. An adaptive complexity QRD-M algorithm (AC-QRD-M) is also considered which assigns different values of M to each subcarrier according to its estimated received power. A rule for choosing M using subcarrier powers is obtained using a kernel density estimate combined with the Lloyd-Max algorithm.

How well can the IEEE 802.11 wireless LAN support quality of service

Abstract: This paper studies an important problem in the IEEE 802.11 distributed coordination function (DCF)-based wireless local area network (WLAN): how well can the network support quality of service (QoS). Specifically, this paper analyzes the network's performance in terms of maximum protocol capacity or throughput, delay, and packet loss rate. Although the performance of the 802.11 protocol, such as throughput or delay, has been extensively studied in the saturated case, it is demonstrated that maximum protocol capacity can only be achieved in the nonsaturated case and is almost independent of the number of active nodes. By analyzing packet delay, consisting of medium access control (MAC) service time and waiting time, accurate estimates were derived for delay and delay variation when the throughput increases from zero to the maximum value. Packet loss rate is also given for the nonsaturated case. Furthermore, it is shown that the channel busyness ratio provides precise and robust information about the current network status, which can be utilized to facilitate QoS provisioning. The authors have conducted a comprehensive simulation study to verify their analytical results and to tune the 802.11 to work at the optimal point with maximum throughput and low delay and packet loss rate. The simulation results show that by controlling the total traffic rate, the original 802.11 protocol can support strict QoS requirements, such as those required by voice over Internet protocol (VoIP) or streaming video, and at the same time achieve high channel utilization.

Quasi-orthogonal STBC with minimum decoding complexity

Abstract: In this paper, we consider a quasi-orthogonal (QO) space-time block code (STBC) with minimum decoding complexity (MDC-QO-STBC). We formulate its algebraic structure and propose a systematic method for its construction. We show that a maximum-likelihood (ML) decoder for this MDC-QO-STBC, for any number of transmit antennas, only requires the joint detection of two real symbols. Assuming the use of a square or rectangular quadratic-amplitude modulation (QAM) or multiple phase-shift keying (MPSK) modulation for this MDC-QO-STBC, we also obtain the optimum constellation rotation angle, in order to achieve full diversity and optimum coding gain. We show that the maximum achievable code rate of these MDC-QO-STBC is 1 for three and four antennas and 3/4 for five to eight antennas. We also show that the proposed MDC-QO-STBC has several desirable properties, such as a more even power distribution among antennas and better scalability in adjusting the number of transmit antennas, compared with the coordinate interleaved orthogonal design (CIOD) and asymmetric CIOD (ACIOD) codes. For the case of an odd number of transmit antennas, MDC-QO-STBC also has better decoding performance than CIOD.

Quantifying the power loss when transmit beamforming relies on finite-rate feedback

Abstract: Transmit beamforming achieves optimal performance in systems with multiple transmit antennas and a single receive antenna from both the capacity and the received signal-to-noise ratio (SNR) perspectives but ideally requires perfect channel knowledge at the transmitter. In practical systems where the feedback link can only convey a finite number of bits, transmit beamformer designs have been extensively investigated using either the outage probability or the average SNR as the figure of merit. In this paper, we study the symbol error rate (SER) for transmit beamforming with finite-rate feedback in a multi-input single-output setting. We derive an SER lower bound that is tight for good beamformer designs. Comparing this bound with the SER corresponding to the ideal case, we quantify the power loss due to the finite-rate constraint across the entire SNR range.

Space-time decoding with imperfect channel estimation

Abstract: Under the assumption of a frequency-flat slow Rayleigh fading channel with multiple transmit and receive antennas, we examine the effects of imperfect estimation of the channel parameters on error probability when known pilot symbols are transmitted among information data. Three different receivers are considered. The first one derives an estimate of the channel [by using either a maximum-likelihood (ML) or a minimum mean square error (MMSE) criterion], and then uses this estimate in the same metric that would be applied if the channel were perfectly known. The second receiver derives again an estimate of the channel, but uses the ML metric conditioned on the channel estimate. Our last receiver simultaneously processes the pilot and data symbols received. Simulation results are exhibited, showing that only a relatively small percentage of the transmitted frame need be allocated to pilot symbols in order to experience an acceptable degradation of error probability due to imperfect channel knowledge. Algorithms for the recursive calculation of the decision metric of the last two receivers are also developed for application to sequential decoding of trellis space-time codes.

On non-WSSUS wireless fading channels

Abstract: We propose a novel framework for the statistical characterization of fading dispersive channels that do not satisfy the assumption of wide-sense stationary uncorrelated scattering (WSSUS). The local scattering function (LSF) and the channel correlation function (CCF) are introduced and shown to characterize, respectively, the mean power and the correlation of non-WSSUS scatterers. Furthermore, the practically important class of doubly underspread (DU) channels is introduced, and it is shown that for DU channels, the LSF has numerous useful properties. The practical relevance of our approach is illustrated via application examples and numerical experiments.

Alamouti-type space-time coding for free-space optical communication with direct detection

Abstract: A modification of the Alamouti code originally proposed for RF wireless applications is described that allows it to be applied in scenarios such as free-space optical communication with direct detection where unipolar modulations like pulse-position modulation and on-off keying are traditionally used to convey the information. The modification of the code and associated decision metric is such as to maintain all of the desirable properties of the original scheme.

Security analysis and enhancements of 3GPP authentication and key agreement protocol

Abstract: This paper analyzes the authentication and key agreement protocol adopted by Universal Mobile Telecommunication System (UMTS), an emerging standard for third-generation (3G) wireless communications. The protocol, known as 3GPP AKA, is based on the security framework in GSM and provides significant enhancement to address and correct real and perceived weaknesses in GSM and other wireless communication systems. In this paper, we first show that the 3GPP AKA protocol is vulnerable to a variant of the so-called false base station attack. The vulnerability allows an adversary to redirect user traffic from one network to another. It also allows an adversary to use authentication vectors corrupted from one network to impersonate all other networks. Moreover, we demonstrate that the use of synchronization between a mobile station and its home network incurs considerable difficulty for the normal operation of 3GPP AKA. To address such security problems in the current 3GPP AKA, we then present a new authentication and key agreement protocol which defeats redirection attack and drastically lowers the impact of network corruption. The protocol, called AP-AKA, also eliminates the need of synchronization between a mobile station and its home network. AP-AKA specifies a sequence of six flows. Dependent on the execution environment, entities in the protocol have the flexibility of adaptively selecting flows for execution, which helps to optimize the efficiency of AP-AKA both in the home network and in foreign networks.

SLM and PTS peak-power reduction of OFDM signals without side information

Abstract: Selected mapping (SLM) and partial transmit sequence (PTS) are well-known techniques for peak-power reduction in orthogonal frequency-division multiplexing (OFDM). We derive a simplified maximum likelihood (ML) decoder for SLM and PTS that operates without side information. This decoder exploits the fact that the modulation symbols belong to a given constellation and that the multiple signals generated by the PTS or SLM processes are widely different in a Hamming distance sense. Pairwise error probability (PEP) analysis suggests how SLM and PTS vectors should be chosen. The decoder performs well over additive white Gaussian noise (AWGN) channels, fading channels, and amplifier nonlinearities.

Compensation of IQ imbalance and phase noise in OFDM systems

Abstract: Nowadays, a lot of effort is spent on developing inexpensive orthogonal frequency-division multiplexing (OFDM) receivers. Especially, zero intermediate frequency (zero-IF) receivers are very appealing, because they avoid costly IF filters. However, zero-IF front-ends also introduce significant additional front-end distortion, such as IQ imbalance. Moreover, zero-IF does not solve the phase noise problem. Unfortunately, OFDM is very sensitive to the receiver nonidealities IQ imbalance and phase noise. Therefore, we developed a new estimation/compensation scheme to jointly combat the IQ imbalance and phase noise at baseband. In this letter, we describe the algorithms and present the performance results. Our compensation scheme eliminates the IQ imbalance based on one OFDM symbol and performs well in the presence of phase noise. The compensation scheme has a fast convergence and a small residual degradation: even for large IQ imbalance, the overall system performance for an OFDM-wireless local area network (WLAN) case study is within 0.6 dB of the optimal case. As such, our approach greatly relaxes the mismatch specifications and thus enables low-cost zero-IF receivers.

Frequency offset and I/Q imbalance compensation for direct-conversion receivers

Abstract: Two types of wireless system imperfections, namely, frequency offset and in-phase/quadrature (I/Q) imbalance, are addressed in this paper. The I/Q imbalance in radio frequency (RF) direct-conversion not only introduces an unwanted in-band image interference but also degrades the accuracy of carrier estimation. Toward this end, we propose a pilot-based scheme for both frequency offset and I/Q imbalance compensation at the baseband. A low-cost nonlinear least squares (NLS) frequency estimator robust to the I/Q imbalance is developed. Also derived is an I/Q imbalance compensation structure that consists of two stages: a finite impulse response (FIR) filter that compensates the frequency dependent imbalance; and an asymmetric phase compensator that corrects the frequency independent error. The compensation coefficients are optimized by exploiting the phase rotation embedded in the pilot symbols. Both computer simulations and experiment results verify the effectiveness of the proposed scheme in various I/Q imbalance scenarios.

BER of OFDM systems impaired by carrier frequency offset in multipath fading channels

Abstract: We introduce an analytical approach to evaluate the error probability of orthogonal frequency-division-multiplexing (OFDM) systems subject to carrier frequency offset (CFO) in frequency-selective channels, characterized by Rayleigh or Rician fading. By properly exploiting the Gaussian approximation of the intercarrier interference (ICI), we show that the bit-error rate (BER) for an uncoded OFDM system with quadrature amplitude modulation (QAM) can be expressed by the sum of a few integrals, whose number depends on the constellation size. Each integral can be evaluated numerically, or, in Rayleigh fading, by using a series expansion that involves generalized hypergeometric functions. Simulation results illustrate that the theoretical analysis is quite accurate, especially for Rayleigh channels, and also with nonlinear amplifiers.

Iterative interference cancellation and channel estimation for mobile OFDM

Abstract: In mobile reception, the reliability of orthogonal frequency division multiplexing (OFDM) is limited because of the time-varying nature of the channel. This causes intercarrier interference (ICI) and increases inaccuracies in channel tracking. We model the ICI using derivatives of the channel amplitude. This allows us to design a relatively simple receiver scheme that iteratively cancels the ICI. The design of the canceler aims at maximizing the signal-to-noise-plus-ICI ratio at the detector input. We also propose a new channel estimator, and we show that it achieves reliable mobile reception in practical situations that are relevant to terrestrial Digital Video Broadcasting (DVB-T). Extensive simulations for a receiver with one or two antennas show that a small number of iterations between ICI cancellation and channel estimation allow a reliable reception at vehicle speeds above 100 km/h.

Optimal training for MIMO frequency-selective fading channels

Abstract: High data rates give rise to frequency-selective propagation effects. Space-time multiplexing and/or coding offer attractive means of combating fading and boosting capacity of multi-antenna communications. As the number of antennas increases, channel estimation becomes challenging because the number of unknowns increases, and the power is split at the transmitter. Optimal training sequences have been designed for flat-fading multi-antenna systems or for frequency-selective single transmit antenna systems. We design a low-complexity optimal training scheme for block transmissions over frequency-selective channels with multiple antennas. The optimality in designing our training schemes consists of maximizing a lower bound on the ergodic (average) capacity that is shown to be equivalent to minimizing the mean square error of the linear channel estimator. Simulation results confirm our theoretical analysis that applies to both single- and multicarrier transmissions.

Performance of cellular networks with relays and centralized scheduling

Abstract: Future cellular wireless networks could include multihop transmission through relays. We propose a centralized downlink scheduling scheme in a cellular network with a small number of relays. The scheduling scheme has the property that it guarantees stability of the user queues for the largest set of arrival rates. We obtain throughput results by simulation for various scenarios and study the effect of number of relays, relay transmit power relative to the base station (BS) power, and the effect of distributing a given total power between the BS and different numbers of relays. We also present results for the case without channel fading to determine what fraction of the throughput gain is achieved from diversity reception. We find that, with four relays deployed in each sector, it is possible to achieve significant throughput gain including the signaling overhead.

Fusion of censored decisions in wireless sensor networks

Abstract: Sensor censoring has been introduced for reduced communication rate in a decentralized detection system where decisions made at peripheral nodes need to be communicated to a fusion center. In this letter, the fusion of decisions from censoring sensors transmitted over wireless fading channels is investigated. The knowledge of fading channels, either in the form of instantaneous channel envelopes or the fading statistics, is integrated in the optimum and suboptimum fusion rule design. The sensor censoring and the ensuing fusion rule design have two major advantages compared with the previous work. 1) Communication overhead is dramatically reduced. 2) It allows incoherent detection, hence, the phase information of transmission channels is no longer required. As such, it is particularly suitable for wireless sensor network applications with severe resource constraints.

On routing in random Rayleigh fading networks

Abstract: This paper addresses the routing problem for large wireless networks of randomly distributed nodes with Rayleigh fading channels. First, we establish that the distances between neighboring nodes in a Poisson point process follow a generalized Rayleigh distribution. Based on this result, it is then shown that, given an end-to-end packet delivery probability (as a quality of service requirement), the energy benefits of routing over many short hops are significantly smaller than for deterministic network models that are based on the geometric disk abstraction. If the permissible delay for short-hop routing and long-hop routing is the same, it turns out that routing over fewer but longer hops may even outperform nearest-neighbor routing, in particular for high end-to-end delivery probabilities.