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Showing papers in "IEEE Transactions on Communications in 1992"


Journal ArticleDOI
TL;DR: It is shown that although the problem of optimally establishing lightpaths is NP-complete, simple heuristics provide near optimal substitutes for several of the basic problems motivated by a lightpath-based architecture.
Abstract: An architectural approach that meets high bandwidth requirements by introducing a communication architecture based on lightpaths, optical transmission paths in the network, is introduced. Since lightpaths form the building block of the proposed architecture, its performance hinges on their efficient establishment and management. It is shown that although the problem of optimally establishing lightpaths is NP-complete, simple heuristics provide near optimal substitutes for several of the basic problems motivated by a lightpath-based architecture. >

1,308 citations


Journal ArticleDOI
TL;DR: The results suggest that on a Rayleigh channel, the standard trellis codes may not be the correct approach for improving the reliability of the communication channel.
Abstract: A suboptimal trellis coding approach based on the concept of combining a good convolutional code and bit interleavers is presented. The aim is to improve the reliability of digital radio communication over a fading channel. It is shown that over a Rayleigh channel and for a fixed code complexity the proposed system is superior to the baseline system. Its performance is analyzed using the generalized R/sub o/ and the upper bound on the bit error rate. The results suggest that on a Rayleigh channel, the standard trellis codes may not be the correct approach for improving the reliability of the communication channel. The discussion is restricted to a rate 2/3 coded system with 8-PSK modulation. >

1,074 citations


Journal ArticleDOI
J.M. Holtzman1
TL;DR: The author derives an accurate Gaussian approximation which is also computationally very simple, the 'standard approximation', which is not generally accurate enough for direct-sequence spread-spectrum multiple-access systems.
Abstract: The exact calculation of error probabilities for direct-sequence spread-spectrum multiple-access (DS/SSMA) systems has been addressed in the literature. The exact calculation is computationally difficult, so emphasis has been on approximations and bounds. One particularly attractive approximation is to just use a signal-to-noise ratio in a Gaussian approximation, the 'standard approximation'. Unfortunately, that approximation is not generally accurate enough. An improved Gaussian approximation with good accuracy has recently been presented. The author derives an accurate Gaussian approximation which is also computationally very simple. >

485 citations


Journal ArticleDOI
P. Balaban1, J. Salz1
TL;DR: A comprehensive theory for Nth-order space diversity reception combined with various equalization techniques in digital data transmission over frequency-selective fading channels is developed and is applied to optimize system parameters and to predict performance for QAM data transmission operating over a model for the mobile radio channel.
Abstract: A comprehensive theory for Nth-order space diversity reception combined with various equalization techniques in digital data transmission over frequency-selective fading channels is developed. The channels are characterized by N arbitrary impulse responses possessing random parameters as well as N additive Gaussian noise sources. Various combiner-equalizers that minimize the mean-squared error are determined. Formulas are presented for the attainable least-mean-squared errors and upper bounds on average probabilities of error. The theory is applied to optimize system parameters and to predict performance for QAM data transmission operating over a model for the mobile radio channel. For this model, estimates of average attainable error rates and outage probabilities are provided as functions of system parameters. In the channel models the uncoded data rates as well as Shannon capacity are regarded as random variables. >

397 citations


Journal ArticleDOI
TL;DR: The spectral-line regenerators can outperform both types of radiometers by a wide margin and are quantified in terms of receiver operating characteristics for several noise and interference environments and receiver collection times.
Abstract: The problem of detecting the presence of spread-spectrum phase-shift-keyed signals in variable noise and interference backgrounds is considered, and the performances of four detectors are evaluated and compared. The detectors include the optimum radiometer, the optimum modified radiometer that jointly estimates the noise level and detects the signal, and the maximum-SNR spectral-line regenerator for spectral-line frequencies equal to the chip rate and the doubled carrier frequency. It is concluded that the spectral-line regenerators can outperform both types of radiometers by a wide margin. The performance advantages are quantified in terms of receiver operating characteristics for several noise and interference environments and receiver collection times. >

377 citations


Journal ArticleDOI
TL;DR: An automatic modulation classification algorithm utilizing the statistical moments of the signal phase is developed and used to classify the modulation type of general M-ary PSK signals and is compared with the quasi-log-likelihood radio, square-law, and phase-based classifiers.
Abstract: An automatic modulation classification algorithm utilizing the statistical moments of the signal phase is developed and used to classify the modulation type of general M-ary PSK signals. It is shown that the nth moment (n even) of the phase of the signal is a monotonic increasing function of M. On the basis of this property, the authors formulate a general hypothesis test, develop a decision rule, and derive an analytic expression for the probability of misclassification. Two examples are given to demonstrate the performance of the algorithm. The algorithm is compared with the quasi-log-likelihood radio (qLLRC), square-law (SLC), and phase-based (PBC) classifiers. The algorithm is outperformed by qLLRC at low CNR but is comparable to SLC and is better than PBC. The qLLRC algorithm is only valid at CNR >

286 citations


Journal ArticleDOI
TL;DR: This work presents an exact expression for the pairwise error event probability of trellis-coded modulation (TCM) transmitted over Rayleigh-fading channels, which includes phase shift keying and multilevel quadrature amplitude modulation (QAM) codes, as well as coherent and partially coherent codes.
Abstract: This work presents an exact expression for the pairwise error event probability of trellis-coded modulation (TCM) transmitted over Rayleigh-fading channels. It includes phase shift keying (PSK) and multilevel quadrature amplitude modulation (QAM) codes, as well as coherent and partially coherent (e.g. differential, pilot tone, etc.) detection. Due to the form of the exact pairwise error event probabilities, this calculation technique cannot be used with the transfer function technique to obtain an upper (union) bound on the overall bit error probability. For this reason, the authors estimate the bit error probability by considering only a small number of short error events. Through simulations, they found that the estimation is usually very accurate at high signal-to-noise ratios but not as accurate at lower signal-to-noise ratios. They study several coded modulation schemes this way. Among the results are the fact that TCM provides significant improvement in the error floor when detected differentially, and an asymmetry in the pairwise error event probability for 16 QAM. >

275 citations


Journal ArticleDOI
TL;DR: A modified backpropagation-type algorithm is introduced for single-user and multiuser detection with near-optimum performance that could have applications in other classification and pattern recognition problems.
Abstract: Two simple structures employing multilayer perceptrons are proposed for demodulation of spread-spectrum signals in both synchronous and asynchronous Gaussian channels. The optimum receiver is used to benchmark the performance of the proposed receiver; in particular, it is proved to be instrumental in identifying the decision regions for the neural networks. The neutral networks are trained for the demodulation of signals via backpropagation-type algorithms. A modified backpropagation-type algorithm is introduced for single-user and multiuser detection with near-optimum performance that could have applications in other classification and pattern recognition problems. A comparative performance analysis of the three receivers, optimum, conventional, and the one employing neural networks, is carried out via Monte Carlo simulations. An importance sampling technique is employed to reduce the number of simulations necessary to evaluate the performance of these receivers in a multiuser environment. In examples given, the receiver significantly outperforms the conventional receiver. >

268 citations


Journal ArticleDOI
TL;DR: It is proved that the rate distortion limit for coding stereopairs cannot in general be achieved by a coder that first codes and decodes the right picture sequence independently of the left picture sequence, and then codes anddecodes theleft picture sequence given the decoded right picture sequences.
Abstract: Two fundamentally different techniques for compressing stereopairs are discussed. The first technique, called disparity-compensated transform-domain predictive coding, attempts to minimize the mean-square error between the original stereopair and the compressed stereopair. The second technique, called mixed-resolution coding, is a psychophysically justified technique that exploits known facts about human stereovision to code stereopairs in a subjectively acceptable manner. A method for assessing the quality of compressed stereopairs is also presented. It involves measuring the ability of an observer to perceive depth in coded stereopairs. It was found that observers generally perceived objects to be further away in compressed stereopairs than they did in originals. It is proved that the rate distortion limit for coding stereopairs cannot in general be achieved by a coder that first codes and decodes the right picture sequence independently of the left picture sequence, and then codes and decodes the left picture sequence given the decoded right picture sequence. >

243 citations


Journal ArticleDOI
TL;DR: The authors provide an analysis of the performance of optical orthogonal codes in an optical code division multiple access (CDMA) network by considering the probability distribution of the interference patterns and it is shown that the actual performance is close to a previous estimate.
Abstract: The authors provide an analysis of the performance of optical orthogonal codes in an optical code division multiple access (CDMA) network by considering the probability distribution of the interference patterns. It is shown that the actual performance is close to a previous estimate. A less structured temporal code in which the code words are allowed to overlap at two pulse positions is also considered. The bit error probability for this class of codes is obtained for two cases: with and without optical hard-limiting at the receivers. It is shown that this code may increase the number of users in the network considerably without a significant loss in the performance. >

227 citations


Journal ArticleDOI
TL;DR: A rigorous analysis of digital coherent optical modulation schemes using the state of polarization as the modulating parameter obtains the exact performance of all the polarization-based modulation schemes proposed in the literature so far, including a differential demodulation scheme, named DPOLSK, which does not require either electrooptic or electronic polarization tracking.
Abstract: A rigorous analysis of digital coherent optical modulation schemes using the state of polarization as the modulating parameter is presented. The analysis obtains the exact performance of all the polarization-based modulation schemes proposed in the literature so far, including a differential demodulation scheme, named DPOLSK, which does not require either electrooptic or electronic polarization tracking. Preliminary results involving multilevel transmission schemes based on the state of polarization are introduced. A spectral analysis of POLSK signals is also proposed. >

Journal ArticleDOI
TL;DR: It is shown that a generalized capture model can be used to estimate q/sub i/'s for a simplified example system which uses noncoherent frequency shift keying modulation, and the dependence of the capture probability for a test mobile on its distance from the base station is obtained.
Abstract: The probability q/sub i/ of successful reception in a nonfading mobile radio channel with i contending mobiles transmitting to a central base station is studied for a number of different capture and spatial distribution models. It is shown that a generalized capture model can be used to estimate q/sub i/'s for a simplified example system which uses noncoherent frequency shift keying modulation. This model can be applied to other systems as well. An example of the use of the q/sub i/'s in the throughput evaluation of a finite population slotted ALOHA system is given. In most practical systems, the mobiles cannot get arbitrarily close to the base station. The effect of this constraint on q/sub i/ is examined. Finally, the dependence of the capture probability for a test mobile on its distance from the base station is obtained. >

Journal ArticleDOI
TL;DR: A discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gaussian noise that is linear in both the modulation and the demodulation, and is free from the effects of error propagation.
Abstract: A discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gaussian noise. Multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete Fourier transform (DFT) code vectors, and high gain coset or trellis codes. Symbol blocks from an inverse DFT (IDFT) are cyclically extended to generate ISI-free channel-output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels. Asymptotic performance of this system is derived, and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample. This discrete multiple tone technique is linear in both the modulation and the demodulation, and is free from the effects of error propagation that often afflict systems employing bandwidth-optimized decision feedback plus coset codes. >

Journal ArticleDOI
TL;DR: Packet throughput figures are obtained for direct sequence spread spectrum multiple access (DS/SSMA) slotted ALOHA radio systems where all users employ random signature sequences from bit-to-bit within all transmitted packets using an improved Gaussian approximation technique.
Abstract: Packet throughput figures are obtained for direct sequence spread spectrum multiple access (DS/SSMA) slotted ALOHA radio systems where all users employ random signature sequences from bit-to-bit within all transmitted packets. These calculations use an improved Gaussian approximation technique which gives accurate bit error probabilities and also incorporates the effect of bit-to-bit error dependence within each packet in the multiaccess interference environment. Numerical results are given for packets which employ varying amounts of block error control, and a comparison is made with results obtained by other methods which ignore the effects of bit-to-bit error dependence within each packet in the multiaccess interference environment. Numerical results are given for packets which employ varying amount of block error control, and a comparison is made with results obtained by other methods which ignore the effects of bit-to-bit error and/or employ less-accurate Gaussian approximations to the probability of data bit error. Maximum throughput per unit bandwidth figures are calculated which compare favorably to similar figures for narrowband signaling techniques. >

Journal ArticleDOI
TL;DR: It is shown that parallel architectures fall somewhat short of ideal speedups in practice, but they should still enable current CMOS technologies to go well beyond 1 Gb/s data rates.
Abstract: The use of VLSI technology to speed up cyclic redundancy checking (CRC) circuits used for error detection in telecommunications systems is investigated By generalizing the analysis of a parallel prototype, performance is estimated over a wide range of external constraints and design choices It is shown that parallel architectures fall somewhat short of ideal speedups in practice, but they should still enable current CMOS technologies to go well beyond 1 Gb/s data rates >

Journal ArticleDOI
Se-Hyun Oh1, Dong-Wan Tcha1
TL;DR: Exact incremental algorithms which efficiently solve the proposed problems are devised based on the properties of the blocking probabilities of new calls and handoff calls and are appropriate for the fair allocation of frequency channels among cells.
Abstract: Dimensioning procedures for prioritized channel assignment in a cellular radio network are considered. Under the cutoff priority discipline, the prioritized channel assignment procedures for a single cell and multicell system are formulated as nonlinear discrete capacity allocation problems. Exact incremental algorithms which efficiently solve the proposed problems are devised. They are based on the properties of the blocking probabilities of new calls and handoff calls. Given the number of available frequency channels together with the arrival rates and the grade of service (GOS) for both types of calls in each cell, algorithm SP1 generates an optimal channel assignment which ensures priority for handoff calls. Given the arrival rates and distinct GOSs for new and handoff calls, algorithm SP2 finds the minimum number of channels required in each cell. Algorithm MP extends algorithm SP1 to a multicell system and provides the prioritized channel assignment for all calls in the system. The algorithms are very fast and are appropriate for the fair allocation of frequency channels among cells. >

Journal ArticleDOI
TL;DR: Results show that the maximum throughput achievable with hot-potato routing can be as low as 25% of that for store-and-forward routing, and that the relative degradation increases as the number of nodes grows larger.
Abstract: The achievable aggregate capacity for a variant of the basic multihop approach in which minimum distance store-and-forward routing is replaced by a hot-potato routing algorithm is determined. With hot-potato routing, all packets simultaneously arriving at a given node and not intended for reception at that node are immediately placed onto the outbound links leaving that node; if two or more packets contend for the same outgoing link to achieve a minimum distance routing, then all but one will be misrouted to links which produce longer paths to the eventual destination. Attention is confined to the development of an analytical methodology for finding the probability distribution of the number of hops with hot potato routing for symmetric networks under uniform traffic load. Results show that the maximum throughput achievable with hot-potato routing can be as low as 25% of that for store-and-forward routing, and that the relative degradation increases as the number of nodes grows larger. This implies that the link speed up needed to produce a significant overall capacity advantage with hot potato should be at least a factor of 10. >

Journal ArticleDOI
TL;DR: It is shown that multiple-symbol differential detection is a very effective strategy for eliminating the irreducible error floor associated with a conventional differential detector.
Abstract: The error performance of multiple-symbol differential detection of uncoded QPSK signals transmitted over correlated Rayleigh fading channels is studied. The optimal detector is presented, along with an exact expression for the corresponding pairwise error event probability. It is shown that multiple-symbol differential detection is a very effective strategy for eliminating the irreducible error floor associated with a conventional differential detector. In all of the cases investigated, a detector with an observation interval as small as two symbols is sufficient for this purpose. It is also found that the error performance of a multiple-symbol differential detector is not sensitive to the mismatch between the decoding metric and the channel fading statistics. >

Journal ArticleDOI
TL;DR: A differentially coherent detection scheme with improved bit error rate (BER) performance is presented for differentially encoded binary and quaternary phase shift keying (PSK) modulation.
Abstract: A differentially coherent detection scheme with improved bit error rate (BER) performance is presented for differentially encoded binary and quaternary phase shift keying (PSK) modulation. The improvement is based on using L symbol detectors with delays of 1, 2, . . ., L symbol periods and on feeding back detected PSK symbols. Exact formulas for the bit error probability are derived for the case that correct symbols are fed back. The effect of symbol errors in the feedback path on the BER is determined by computer simulations. >

Journal ArticleDOI
TL;DR: An effective bandwidth control algorithm and its calculated performance are described, and the bandwidth control is shown to successfully improve network transmission efficiency with only a slight increase in processing load compared to the fixed bandwidth scheme.
Abstract: The virtual path concept has several valuable features to construct an economical and efficient asynchronous transfer mode (ATM) network. One of them is bandwidth control which affords transmission efficiency improvement through statistical sharing of capacity. An effective bandwidth control algorithm and its calculated performance are described. Network performance with the algorithm is evaluated, and the bandwidth control is shown to successfully improve network transmission efficiency with only a slight increase in processing load compared to the fixed bandwidth scheme. A method is also proposed to equalize call loss probability for each virtual path. The effectiveness of the method is demonstrated by analysis. >

Journal ArticleDOI
TL;DR: A growable switch architecture is presented that is based on three key principles: a generalized knockout principle exploits the statistical behaviour of packet arrivals and thereby reduces the interConnect complexity, output queuing yields the best possible delay/throughput performance, and distributed intelligence in routing packets through the interconnect fabric eliminates internal path conflicts.
Abstract: The problem of designing a large high-performance, broadband packet of ATM (asynchronous transfer mode) switch is discussed. Ways to construct arbitrarily large switches out of modest-size packet switches without sacrificing overall delay/throughput performance are presented. A growable switch architecture is presented that is based on three key principles: a generalized knockout principle exploits the statistical behaviour of packet arrivals and thereby reduces the interconnect complexity, output queuing yields the best possible delay/throughput performance, and distributed intelligence in routing packets through the interconnect fabric eliminates internal path conflicts. Features of the architecture include the guarantee of first-in-first-out packet sequence, broadcast and multicast capabilities, and compatibility with variable-length packets, which avoids the need for packet-size standardization. As a broadband ISDN example, a 2048*2048 configuration with building blocks of 42*16 packet switch modules and 128*128 interconnect modules, both of which fall within existing hardware capabilities, is presented. >

Journal ArticleDOI
TL;DR: Using a trace of address references, the author compares the efficiency of several different hashing functions such as cyclic redundancy checking polynomials, Fletcher checksum, folding of address octets using the exclusive-OR operation, and bit extraction from the address.
Abstract: Using a trace of address references, the author compares the efficiency of several different hashing functions such as cyclic redundancy checking polynomials, Fletcher checksum, folding of address octets using the exclusive-OR operation, and bit extraction from the address. Guidelines are provided for determining the size of hash masks required to achieve a specified level of performance. >

Journal ArticleDOI
TL;DR: In this paper, the authors propose an algorithm for assignment of multiple minimum-length UIO sequences for each state of the finite-state-machine (FSM) specification, which is polynomial in the number of states and transitions.
Abstract: Automatic generation of conformance test sequences for communication protocols by means of unique input/output (UIO) sequences is addressed. It is shown that if multiple minimum-length UIO sequences are computed for each state of the finite-state-machine (FSM) specification, then the length of the resulting test sequence is significantly reduced without an appreciable increase in the time needed to compute the sequence. An algorithm for assignment of the multiple UIO sequences is given. This algorithm, which is based on network flow, is polynomial in the number of states and transitions of the FSM and is effective in reducing the overall length of the test sequence. >

Journal ArticleDOI
S.J. Howard1, K. Pahlavan
TL;DR: Based on frequency domain measurements in the 0.9-1.1-GHz band, an autoregressive model for the frequency response of the indoor radio channel is introduced and it is shown that a second-order process is sufficient to represent the important statistical characteristics of the channel both in the frequency domain and the time domain.
Abstract: Based on frequency domain measurements in the 0.9-1.1-GHz band, an autoregressive model for the frequency response of the indoor radio channel is introduced. It is shown that a second-order process is sufficient to represent the important statistical characteristics of the channel both in the frequency domain and the time domain where each pole identifies the arrival of a cluster of paths. A comparison is made between the statistical characteristics of the empirical data and of the channel responses regenerated from the second-order AR processes. Four methods to regenerate the indoor radio channel responses from a second-order AR model are proposed. The accuracy of the methods is examined by comparing the cumulative distribution functions of the RMS delay spread and the 3-dB width of the frequency correlation function with that of the measurements performed in global, local, and mixed indoor radio propagation experiments. >

Journal ArticleDOI
TL;DR: A novel mathematical method is presented to express the outage probability for a desired radio signal received from a mobile transmitter in the presence of multiple interfering signals with combined log-normal and Rayleigh fading, useful in determining the spectrum efficiency and performance of radio networks for high-capacity cellular telephony, two-way paging, packet radio and other mobile data networks.
Abstract: A novel mathematical method is presented to express the outage probability for a desired radio signal received from a mobile transmitter in the presence of multiple interfering signals with combined log-normal and Rayleigh fading. In contrast to previously reported analyses, this exact method avoids approximation of the PDF of the received powers of the various signals. This is useful in determining the spectrum efficiency and performance of (interference-limited) radio networks for high-capacity cellular telephony, two-way paging, packet radio and other mobile data networks. >

Journal ArticleDOI
P. Balaban1, J. Salz1
TL;DR: The probability distributions of the data rates that can be supported by optimum receiver structures as well as the distribution of the Shannon capacity are studied and the dependences among the important system parameters are exhibited graphically for several illustrative examples including QPSK.
Abstract: For Pt.I, see ibid., vol.40, no.5, p.885-94 (1992). The probability distributions of the data rates that can be supported by optimum receiver structures as well as the distribution of the Shannon capacity are studied. The dependences among the important system parameters are exhibited graphically for several illustrative examples including QPSK. At outage probabilities >

Journal ArticleDOI
TL;DR: Call admission control in ATM networks without monitoring network load is proposed and a cell loss probability standard is guaranteed to be satisfied under this control without assumptions of a cell arrival process.
Abstract: Call admission control in ATM networks without monitoring network load is proposed. Traffic parameters specified by users are employed to obtain the upper bound of cell loss probability, thereby determining call admission. A cell loss probability standard is guaranteed to be satisfied under this control without assumptions of a cell arrival process. Implementation of this control to quickly evaluate cell loss probability after acceptance of a new call is discussed. The result is also applicable to ATM network dimensioning based on traffic parameters. >

Journal ArticleDOI
TL;DR: It is demonstrated that via a simple enhancement of using a 2 T s (instead of T s) delay in the second stage of the encoder and first stage ofThe decoder, the performance degradation can be significantly reduced.
Abstract: A variety of schemes for performing differential detection in environments characterized by frequency offset are discussed. All of the schemes involve encoding the input phase information as a second-order difference and performing an analogous second-order differential detection at the receiver. Because of the back-to-back differential detection operations at the receiver, The performance of most of the schemes is considerably degraded relative to that of first-order differential detection schemes. However, the latter are quite sensitive to frequency offset and, in many instances, cannot be used at all. It is demonstrated that via a simple enhancement of using a 2 T s (instead of T s) delay in the second stage of the encoder and first stage of the decoder, the performance degradation can be significantly reduced. This result is significant in view of the fact that it comes without any penalty in implementation complexity. >

Journal ArticleDOI
TL;DR: It is shown that the hyperbolic frequency hop codes possess nearly ideal characteristics for use in both types of system, specifically, in multiple-access communications the codes achieve minimum error probability, while in radar and sonar systems the codes have at most two hits in their auto- and cross-ambiguity function.
Abstract: The problem of constructing frequency hop codes for use in multiuser communication systems such as multiple-access spread-spectrum communications and multiuser radar and sonar systems is addressed. Previous frequency hopping techniques are reviewed. The construction of a new family of frequency hopping codes called hyperbolic frequency hop codes is given. The concepts of multiple-access spread-spectrum communication systems and multiuser radar and sonar systems are reviewed, and it is shown that the hyperbolic frequency hop codes possess nearly ideal characteristics for use in both types of system. Specifically, in multiple-access communications the codes achieve minimum error probability, while in radar and sonar systems the codes have at most two hits in their auto- and cross-ambiguity function. Examples of address assignment for multiple-access communications systems and radar and sonar auto- and cross-ambiguity functions are also given. >

Journal ArticleDOI
TL;DR: The approximations show that deflection routing performs remarkably well in hypercube networks, for small as well as large networks and for the whole range from light to heavy load.
Abstract: An approximate analysis of the transient and steady state behavior of deflection routing in hypercube networks is presented, under a uniform traffic model. In deflection routing congestion causes packets admitted to the network to be temporarily misrouted rather than buffered or dropped. The approximations show that deflection routing performs remarkably well in hypercube networks, for small as well as large networks and for the whole range from light to heavy load. Simulations suggest that the approximations are quite accurate. >