Imagine a game of hide and seek with an energy-limited hider and an observation rate-limited seeker. We first show that hider energy expenditure is a function of the process covariance and thus, unsurprisingly, that Gaussian motion processes maximize hider entropy, a proxy for seeker effort. We formulate the optimization in terms of the process position (or velocity) spectral density. If the seeker knows when the hider has looked, the optimal motion strategy is a satisfyingly simple renewal process – after an observation, pick a velocity at random and proceed until the next observation where the process renews. However, if the seeker is oblivious to seeker attentions the problem is more complicated. While the power spectral density of the motion process is necessarily bandlimited owing to the energy constraint, a general closed form eluded us. So, we consider a few special cases for both stationary motion processes and a class of non-stationary motion processes.

The past decade has produced a large body of work on communication channels which use chemicals to communicate. Some work uses a finest grain model wherein the arrival times of individual emitted particles convey information. Others consider a related particle intensity system where the time is binned at the transmitter/receiver and the number of particles released and counted conveys information. Still others consider a macroscopic model that uses an Avogradrian number of particles and thus concentration as the information carrier. However, given the myriad emission, carrier transport and uptake/sensing methods studied, it has been difficult to precisely relate timing, intensity and concentration results. Here we attempt a partial unification through a simple upper bound applicable to any finite-mean first-passage time transport mechanism under an assumption of constant particle emission intensity λ̅. Our result is expressed in terms of three quantities: the particle emission rate λ̅, the average particle uptake rate μ, and the entropy of the first-passage time distribution, h(D).

Recent years have shown a rapid increase in the amount of study devoted to communication systems where molecules are information carriers. The reasons for such interest are varied, from seeking to understand the ubiquity of molecular communication in biology to the search for communication methods in media where electromagnetic and acoustic methods are inappropriate, to exploring the energy efficiency of methods where some delivery latency can be allowed. With this tutorial on recent discrete molecular communication research, we seek to organize the work into broad categories and thence under the umbrella of what can be called “inscribed matter communication,” where information is conveyed through assemblage, release, and capture of matter as opposed to the transmission of photons or phonons. We will begin by considering discrete passage of molecules between senders and receivers, and argue that matter emission and detection, even at Avogadrian levels, are a subset of the discrete problem, all with a focus on point-to-point communication. In this way, we hope to contextualize current work within a larger fundamental framework, illuminate the hard boundaries of what is known, and then stimulate further research on this fascinating topic.

We consider data transfer between a chemical vapor emitter and photoionization detectors (PIDs) under constant velocity gas flow in a constrained environment (flow tube). We show that the system, though stochastic owing to flow unsteadiness, is (on average) linear and then characterize the channel using a Karhunen-Loeve (KL) expansion. We measure bit error rates using straightforward detection methods in the KL-produced signal space. Data rates of 20bps are easily achieved at ≈ 10-3error rate and we suspect that similarly low error rates are achievable at much higher bit rates using suitable adaptive equalization methods.

Computation that can exploit the Avogadrian numbers of molecules in heterogeneous solutions, and the even larger number of potential interactions among these molecules, is a tantalizing dream. However, the lack of precise specificity/control of chemical interactions can be at odds with the dream. In this paper, we show how relatively simple chemistry can be used to produce a ubiquitous computational primitive (the multiply-accumulate or MAC operation) that forms the basis for a single-layer neural network called a perceptron. A chemical perceptron can be realized using distinct mixtures as inputs and different reagents as operations to produce the results of the perceptron MAC operation, that can be read out perhaps using simple indicators such as pH or fluorescence. With a moderately large chemical library, the number of potential inputs can be Avogadrian so that reagent addition implicitly performs a concomitantly large number of MAC operations in parallel.

This work introduces the particle-intensity channel (PIC) as a model for molecular communication systems and characterizes the properties of the optimal input distribution and the capacity limits for this system. In the PIC, the transmitter encodes information, in symbols of a given duration, based on the number of particles released, and the receiver detects and decodes the message based on the number of particles detected during the symbol interval. In this channel, the transmitter may be unable to control precisely the number of particles released, and the receiver may not detect all the particles that arrive. We demonstrate that the optimal input distribution for this channel always has mass points at zero and the maximum number of particles that can be released. We then consider diffusive particle transport, derive the capacity expression when the input distribution is binary, and show conditions under which the binary input is capacity-achieving. In particular, we demonstrate that when the transmitter cannot generate particles at a high rate, the optimal input distribution is binary.

Consider observation of a system with initial state x(0) through some signal r(t) corrupted by white noise of spectral height N0. When the system is cast in state-space form and the observations projected onto the relevant orthonormal bases, completely unbidden, two well-known wireless communications tropes emerge: a colored noise channel and a multi-access channel wherein elements of system state are associated with different “signatures” defined by the system. That is, from a mathematical perspective, the system could communicate to the observer in a well-understood way. Taking these tropes at face value, we investigate the efficiency of conveying a state vector x(0) through classical estimation to that wherein a “demon” manipulates an identical initially-at-rest system so as to communicate x(0) to the observer on successive epochs (channel uses). An energy constraint on the initial state E[|x(0)|2]=ε is assumed, and the demon's signaling efforts over the ensemble of epochs are constrained similarly. In all cases, the demon conveys the x(0) with less error - by orders of magnitude for moderate signal to noise ratio ε/N0. Furthermore, the demon scenario results in some number of reliably-conveyed bits of information and imposes crisp limits on relative uncertainty of different state element estimates. In fact, the form of these limits is identical to that of the quantum mechanical Uncertainty Principle (although there is no requirement of a momentum-position analog). Nonetheless, the appearance of these tropes raises the question of whether communication and information theory have something deeper to say about physical interactions and the cacophony of system voices in conversation.

As system sizes shrink, the usual macroscopic methods of communication using electromagnetic and acoustic waves become increasingly less efficient owing to mismatches between realizable antenna sizes and the propagation characteristics of the medium. Thus, at the scale of microns and below, communication methods which utilize molecular messengers become attractive, a notion supported by the ubiquity of molecular signaling in biological systems, sometimes using identical molecules (tokens) and sometimes using tokens with embedded payloads such as, for instance, m-RNA. Here we consider a wide range of molecular signaling techniques used by biological systems, and by applying simple information-theoretic concepts seek to develop an outerbound model which distills the plethora of channel details to (1) timing, and (2) molecular “packet” payloads as the information-bearing agents. We find that both bits/joule and bits/sec/joule efficiencies are greatly increased by using tokens with only 1-bit payloads and that overall information carriage efficiency (and consequently, bit rate) is best served by using tokens with large payloads.

As system sizes shrink to the nanoscale, the usual macroscopic methods of communication using electromagnetic and acoustic waves become increasingly difficult and energy-inefficient owing to, essentially, a mismatch between realizable antenna sizes and the propagation characteristics of the medium. Thus, at the scale of microns and below, communication methods which utilize molecular messengers become increasingly attractive, a notion supported by the ubiquity of molecular signaling in biological systems, usually with identical molecules. In a large portion of previous work, time-varying signal molecule/token concentration is used as the observable and various analyses performed. However, from an information-theoretic standpoint, concentration masks the underlying process which consists, fundamentally, of signal token emission, transit through some medium, and reception. We build here on previous work to establish machinery which allows upper bounds to be derived on the identical token timing channel. We then consider the special case of exponential token transit times.

The articles in this special issue focus on the technologies and applications that support molecular, biological, and multiscale communication.

A navigation filter combines measurements from sensors currently available on vehicles - Global Positioning System (GPS), inertial measurement unit, inertial measurement unit (IMU), camera, and light detection and ranging (lidar) - for achieving lane-level positioning in environments where stand-alone GPS can suffer or fail. Measurements from the camera and lidar are used in two lane-detection systems, and the calculated lateral distance (to the lane markings) estimates of both lane-detection systems are compared with centimeter-level truth to show decimeter-level accuracy. The navigation filter uses the lateral distance measurements from the lidar- and camera-based systems with a known waypoint-based map to provide global measurements for use in a GPS/Inertial Navigation System (INS) system. Experimental results show that the inclusion of lateral distance measurements and a height constraint from the map creates a fully observable system even with only two satellite observations and, as such, greatly enhances the robustness of the integrated system over GPS/INS alone. Various scenarios are presented, which affect the navigation filter, including satellite geometry, number of satellites, and loss of lateral distance measurements from the camera and lidar systems.

As system sizes shrink to the nanoscale, the usual macroscopic methods of communication using electromagnetic and acoustic waves become increasingly difficult owing to, essentially, a mismatch between realizable antenna sizes and the propagation characteristics of the medium. Thus, at the scale of microns and below, communication methods which utilize molecular messengers become increasingly attractive, a notion supported by the ubiquity of molecular signaling in biological systems, usually with identical molecules. In a large portion of previous work, time-varying signal molecule/token concentration is used as the observable and various analyses performed. However, from an information-theoretic standpoint, concentration masks the underlying process which consists, fundamentally, of signal token emission, diffusion through some medium, and reception. In this paper we establish a lower bound on identical token signaling with energy constraints and thereby indirectly provide max-min bounds on concentration-based signaling rates.

GPS receiver performance can suffer in difficult environments such as urban canyons and heavy foliage. Inertial sensors provide information between GPS updates and can enhance the position solution in a GPS/INS architecture. Additional information from safety sensors already on the vehicle, such as lane departure warning (LDW) sensors, can enhance the navigation solution further by constraining inertial errors even in the presence of GPS errors. This paper outlines a scalable navigation solution that can use a combination of GPS, reduced inertial sensors, full inertial data, vehicle CAN data, and vision sensors, depending on what data is available in difficult environments. Data was collected in Detroit, Michigan in a diverse mix of environments that includes heavy foliage, highway, and downtown areas, in proportions representative of what is expected in typical driving. Validation of the approach consists of both a qualitative analysis of the resulting trajectories overlaid on a map of the area and quantitative comparison of the trajectories produced by the proposed system and the reference system.

In this short paper we describe an approach and primary results for obtaining bounds on mutual information between release times and capture times for a set of M identical quanta traveling from a source to a target. The first-passage times are assumed independent and identically distributed and the launch times are constrained. One major application of area is intercellular molecular signaling in biological systems whereby a cell (or group of cells) must deliver some message (such as developmental instructions) over distance with reasonable certainty to another cell (or group of cells). Another application area is communication between components of nano/molecular computers. However, the model can also be applied to any communication systems wherein indistinguishable signals have random transit latencies.

We derive the maximum mutual information for an additive exponential noise (AEN) channel with a peak input constraint. We find that the optimizing input density is mixed (with singularities) similar to previous results for AEN channels with a mean input constraint. Likewise, the maximum mutual information takes a similar form, though obviously the maximum for the peak constraint is smaller than for the corresponding mean-constrained channel. This model is inspired by multiple biological phenomena and processes which can be abstracted as follows: inscribed matter is sent by an emitter, moves through a medium, and arrives eventually at its destination receptor. The inscribed matter can convey information in a variety of ways such as the number of signaling quanta - molecules, macromolecular complexes, organelles, cells and tissues - that are emitted as well as the detailed pattern of their release. However, rather than focus on a general class of emitter-receptor systems or a particular exemplar of biomedical importance, our ultimate goal is to provide bounds on the potential efficacy of timed-release signaling for any system which emits identical signaling quanta. That is, we seek to apply one of the most potent aspects of information theory to biological signaling - mechanism blindness - in the hopes of gaining insights applicable to diverse systems that span a wide range of spatiotemporal scales.

In this paper we present application of interference avoidance in the context of a general multiple access vector channel model. We show that this monotonically increases sum capacity, and discuss algorithms for code division multiple access (CDMA) codeword optimization based on this procedure. A greedy interference avoidance algorithm for multiaccess vector channels is presented in the paper, for which we discuss convergence to a class of codeword ensembles that satisfy a simultaneous water filling solution and maximize sum capacity. Numerical results obtained from simulations that corroborate our analytical results are also presented.

In this paper we investigate properties of simultaneous water filling for a wireless system with two mutually interfering transmitters and receivers with non-cooperative coding strategies. This is slightly different from the traditional interference channel problem which assumes that transmitters cooperate in their respective coding strategies, and that interference cancellation can be performed at the receivers. In this noncooperative setup, greedy capacity optimization by individual transmitters through various algorithms leads to simultaneous water filling fixed points where the spectrum of the transmit covariance matrix of one user water fills over the spectrum of its corresponding interference-plus-noise covariance matrix, and in our paper we study the properties of these fixed points. We show that at a simultaneous water filling point the eigenvectors of transmit covariance matrices at each receiver are aligned, and identify three regimes which correspond to simultaneous water filling that depend on the interference gains: a) complete spectral overlap, b) partial spectral overlap, and c) spectral segregation. These imply that the transmit covariance matrices will be white in regions of both overlap and segregation, but not necessarily white overall. We also consider performance as a function of interference gain and show that complete spectral overlap is a strongly suboptimal solution over a wide range of gains. Overall, our results suggest that for strong mutual interference, an effort should be made to do joint decoding over receivers since such collaboration can provide large capacity increases. For moderate interference, distributed and/or centralized conflict resolution algorithms would be most effective since more complex collaborative methods do not afford much improvement and strictly greedy methods such as water filling perform poorly, while for weak interference a laissez faire approach seems reasonable

We present the application of greedy interference avoidance methods to codeword optimization in the uplink of a code division multiple access (CDMA) system in which the channel between a given user and the base station receiver is assumed known and stable for the duration of the transmission. Repeated application of greedy interference avoidance monotonically increases sum capacity and yields an optimal codeword ensemble that satisfies a simultaneous water-filling distribution. However, algorithms for codeword optimization based on the greedy interference avoidance procedure are in general different from water-filling schemes. We illustrate the algorithms with examples and look at properties of optimal codeword ensembles.

For multiple transmitters sending independent data to a single receiver, by modulating their data symbols with fixed length codewords, the problem of optimizing these codewords to maximize capacity has been addressed by S. Verdu (Proc. 24th Allerton Conf. on Comm., Control and Computing, 1989) and P. Viswanath and V. Anantharam (IEEE Trans. Inf. Theory, vol. 45, no. 6, pp. 1984-1991, 1999). This paper considers an analogous scenario when the information sent by the transmitters is correlated. The optimal codeword set and power allocation which minimizes TMSE (total mean square error) at the receiver under a total power constraint have been derived. The equivalence between TMSE and sum capacity is also shown, in the sense that minimizing the former corresponds to maximizing the latter.

We consider a communication system with multiple independent user-base pairings in a white Gaussian noise environment, and for which a simultaneous water filling condition is satisfied by users at their respective bases. We focus on the weak mutual interference case for which the simultaneous water filling solution is unique and corresponds to complete user overlap in signal space. We show that when users at other bases are treated as Gaussian noise, simple separation of users in signal space usually offers better performance than simultaneous water filling, and present a distributed algorithm which separates users in signal space.

Extremely large-scale wireless networks of interconnected mobile devices are inevitable in the near future. Almost all these varied devices are likely to require some form of Internet access. The uncertainty associated with wireless mobile networks produces unique challenges to achieving seamless integration with the Internet while provisioning end-to-end quality of service (QoS). In particular, the uncertainty in wireless channels, and in network topology, due to node mobility, can bedevil protocols more suited to a "classical" Internet structure. Therefore, new protocols have to be designed that must be: (i) robust against the uncertainty in traffic load, host mobility, resource availability and wireless link characteristics; (ii) adaptive to the network dynamics, thus learning and prediction become integral components in the design methodology; (iii) intrinsically on-line so as to make real-time decisions based on temporal and spatial information. In order to cope with the uncertainty, we propose an overarching theoretical framework to represent relevant network information in terms of underlying entropies, entropy rates and their inter-relationships. We demonstrate how to apply information theoretic learning and prediction tools for collecting and disseminating network state information that can be used for robust and adaptive protocol design. Specifically, we investigate the applicability of this novel framework in designing optimal mobility tracking and resource management, while coping with uncertainty in traffic load, topology control and routing.

Mobility, rather than being a liability, can be an asset. If delay constraints are loose, it is possible for a given packet to observe many different network topologies as nodes move relative one another, and these different topologies can be treated as diversity. Opportunistic strategies can exploit these large scale changes in the channel quality to decrease transmit power at the expense of greater delay. We study the tradeoff between mobility, transmit power and delay and along the way develop simple greedy (packet-based) threshold rules for packet transmission.

We consider a wireless system with base stations which collaborate, and derive bounds on sum capacity and total squared correlation for uniform channels between users and bases. The correspondence also investigates structural properties which must be satisfied by user transmit covariance matrices at the optimal sum capacity/total squared correlation (TSC) point, and shows that for multibase systems, maximizing sum capacity and minimizing TSC are, in general, not equivalent problems.

Consider a wireless system with multiple and independent user-base pairings over some region. Each user tries to greedily optimize its performance, and eventually a simultaneous water-filling fixed point is reached. Here we seek to analytically understand properties of different water-filling fixed points. In so doing we show that water-filling alone does not generally result in optimum resource sharing and in some cases is a poor solution. We close by suggesting dynamic strategies for performance enhancement.

We investigate practical methods of distributed interference avoidance where users iteratively adapt their codewords in response to global feedback from the receiver. In turn, the receiver adaptively tracks user codewords and offers a reasonable alternative to feeding back codewords. We introduce variants of standard interference avoidance procedures which produce more easily tracked incremental codewords and study the response of the system to abrupt changes in the interference background as might be encountered in a practical system. Furthermore, the methods we propose are strongly reminiscent of adaptive equalization for which a large body of knowledge and hardware expertise exist.

We present the application of interference avoidance methods to multiuser systems with multiple inputs and multiple outputs. A general signal space formulation is used which makes the approach applicable to any MIMO system model regardless of the choice of basis functions. Information is transmitted via multicode CDMA where symbols that comprise the data frame from a given user are "spread" over the available dimensions using a precoding matrix. Optimal precoding matrices that maximize signal-to-interference-plus-noise ratio for all symbols/users are then obtained by application of distributed greedy interference avoidance methods. Numerical simulations have been performed and the signal-to-noise ratio distribution for receiver antennas and complementary cumulative distribution functions for sum capacity with optimal precoding matrices are also presented.

Motivated by the emergence of programmable radios, we seek to understand a new class of communication system where pairs of transmitters and receivers can adapt their modulation/demodulation method in the presence of interference to achieve better performance. Using signal to interference ratio as a metric and a general signal space approach, we present a class of iterative distributed algorithms for synchronous systems which results in an ensemble of optimal waveforms for multiple users connected to a common receiver (or colocated independent receivers). That is, the waveform ensemble meets the Welch (1974) bound with equality and, therefore, achieves minimum average interference over the ensemble of signature waveforms. We derive fixed points for a number of scenarios, provide examples, look at ensemble stability under user addition and deletion as well as provide a simplistic comparison to synchronous code-division multiple-access. We close with suggestions for future work.

Interference avoidance has been shown to reduce total square correlation (TSC) for given ensembles of user signature waveforms (codewords) in a synchronous code-division multiple-access (CDMA) system. In all experiments we have conducted, sequential application of interference avoidance produces an optimal codeword set when starting from randomly chosen initial codewords. Here we provide the first formal proof of convergence to optimal codeword ensembles for greedy interference avoidance algorithms augmented by a technique called "class warfare" whereby users which reside in more heavily loaded areas of the signal space purposely interfere with (attack) the reception of users in less crowded areas. Coordination of deliberate interference by a complete class of aggrieved user is also sometimes necessary. Such "attacks" and subsequent codeword adjustment by attacked users are shown to strictly decrease TSC. Along the way we also show using linear algebra and a variant of stochastic ordering, equivalence between minimization of TSC and maximization of sum capacity.

Programmable radios offer a new perspective on wireless communications since the modulation method is no longer fixed. Adaptive methods where user signatures and corresponding receiver filters are iteratively adapted can be used to improve performance. However, since codeword adjustments must be fed back to the transmitter, compact representation of codewords is extremely important. This issue is important for systems which employ interference avoidance since as opposed to current CDMA systems where uniform-amplitude codeword chips are used, interference avoidance employs real-valued "chips"-real-valued coefficients for a set of orthonormal basis functions of the signal space used by the transmitter and receiver. The paper represents a simple investigation of how codeword quantization affects the performance of interference avoidance algorithms. Results indicate that using 4-5 bits per chip for codeword representation is sufficient to maintain performance close to optimal values.

The infostations wireless data network architecture features discontinuous coverage and ultra-high radio rates for burst transfers of information between base and mobile. It has been shown previously that the infostations architecture can greatly increase the capacity of wireless data systems at the expense of increased delivery delay. In this work we explicitly consider the multiple access issue on both the uplink and downlink-to and from an infostation respectively-and sharing of fixed network links for transporting information to and from infostations. We find that in order to maximize throughput, an infostation radio link should not be shared among users. Furthermore, this sole use paradigm is echoed in the fixed network which transports information to and from infostations. In order to minimize average delay the fixed link to any given infostation should serve users sequentially, as opposed to in a shared manner.

We consider a new mobility management scheme-the adaptive location area tracking scheme-in which each mobile performs a location registration as it crosses the boundary of its current personal location area and is assigned a new location area. The size and shape of the new location area depend on the mobile's mobility and call characteristics in its previous location area. The objective is to minimize the combined average signaling cost of both paging and registration activities for each individual mobile user. We model the mobility and incoming call traffic of an individual mobile user using Brownian motion with the drift process and Poisson arrival process. Under the assumption of a one-dimensional cellular network environment, we investigate the effects of user mobility parameters such as average movement speed, location uncertainty and mean call arrival rate on the size and shape of individual location areas. This study reveals that, besides size, the shape of the location areas also plays an important role in signaling cost reduction. Performance analysis shows that this scheme offers a cost reduction up to 50% as compared to a previously proposed scheme.

We examine several methods to allocate radio frequency channels across a cellular network. In this paper we consider centralized channel assignment strategies that utilize global network traffic information. The channel assignment methods considered are linear programming, quadratic programming, and a heuristic algorithm to assign resources given the network traffic loading. Each method attempts to assign resources, under frequency reuse constraints, to achieve a desired target performance measure for the network. The network performance measures used in this paper are the average and maximum probability of blocking across the network. We compare the performance of a linear highway network and a planar network using our assignment methods to those of uniform fixed channel allocation (FCA) for non-uniform network traffic loading. We show that under non-uniform traffic loading a better grade-of-service can be provided to network subscribers than by fixed channel assignment.>

The minimum necessary aggregate link capacity in a telecommunication network is directly proportional to the mean distance between nodes. The mean internodal distance is therefore an important network characteristic. It is shown that most network topologies, including those constructed at random, display mean internodal distances comparable to those of many carefully designed networks. Thus, careful selection of network topology to minimize the mean internodal distance may be important in only the most sensitive applications. Furthermore, even in such sensitive applications, an almost randomly chosen network topology may be the best choice.>

Networks have been found through simulated annealing with mean internodal distances lower than any previously reported for a given number of nodes N with a maximum of p outgoing links per node. These networks form the closest known approximations to Moore networks. However, the improvements in mean internodal distance obtained are relatively small (a few percent) and this improvement decreases rapidly with increasing p. The relation between average nodal switching complexity and mean internodal distance is briefly explored through simulated annealing. Initial sharp decreases in mean internodal distance can be obtained through moderate increases in switching complexity. However, further decreases require much larger increases in average complexity.>

Many time-multiplex switching systems require that the incoming traffic be scheduled to avoid conflict at the switch output (two or more users converging simultaneously upon a single output). Optimal scheduling provides a means to assign traffic on demand such that either blocking probability is minimized (unbuffered system) or packet waiting time is minimized (buffered system). However, computation of an optimal schedule for switches of a reasonable size (i.e. N=100) may require many seconds or even minutes, whereas the traffic demand may vary much more rapidly. Since the computation time varies as O(N/sup 2/), the problem becomes readily intractable for large N. This computational bottleneck is overcome by using a scheduling algorithm which is run on a simple special-purpose parallel computer (cellular automaton). A schedule is produced in O(N) time if signal propagation time in the automaton is considered negligible, and therefore increases in computation speed by several orders of magnitude should be possible; the time to compute a schedule for a 1000-input switch would be measured in milliseconds rather than minutes.>

In a time-multiplex switching system, the incoming traffic must be scheduled to avoid conflict at the switch output (two or more users converging simultaneously upon a single output). Two scheduling algorithms, random scheduling and optimal scheduling, are explored in this paper. Random scheduling is computationally simple, whereas optimal scheduling is currently very difficult. We have found, using a traffic model appropriate for circuit switched traffic that increases of typically 10 to 15 percent in offered load can be obtained through optimal scheduling (as compared to the much simpler random scheduling algorithm). The improvement is a function of the number of time slots (or circuits) per time-multiplexed frame, and falls to zero for both very small and very large frame sizes. Thus, in many circuit switching applications, providing a computationally expensive optimal schedule may not be warranted. This conclusion has important ramifications for both electronic and emerging photonic switching systems since it reduces the importance of the costly design feature of optimal scheduling.