Bandwidth Utilization/Fidelity Tradeoffs in Predictive Filtering

In confirmation of earlier developed theory, this paper presents some very promising empirical results obtained for tradeoff between message bandwidth utilization versus error incurred in a predictive filtering method called predictive quantization. The results relate bandwidth utilization and error against quantum size for the same federation executing on DEVS/HLA (an HLA compliant distributed simulation environment) in Unix and NT networking platforms. The theoretical and empirical results so far indicate that predictive quantization can be very scaleable due to reduced local computation demands as well as having extremely favorable bandwidth requirement reduction/simulation fidelity tradeoffs. Since the DEVS formalism was employed to formally characterize predictive contract mechanisms.in a generic manner, application in distributed simulation environments, whether HLA-compliant or not, is facilitated. 1 This work was supported by Advance Simulation Technology Thrust (ASTT) DARPA Contract N6133997K-0007 1. Quantization In recent work we have developed a theory of predictive filtering which takes into account the global error due to error propagation [1, 2]. Previous analyses [3] of predictive contracts’ accuracy/performance tradeoffs have assumed that open loop analysis carries over to the closed loop case. Unfortunately, experience in numerical analysis suggests that the dynamics of feedback interaction may cause errors generated to grow without bound. The theory of quantized systems we developed provides conditions, based on a formulation of input sensitivity, under which homomorphic (error-free) quantization-based predictive filtering is possible. It shows how error can be generated if the conditions are violated and formulates a suitable concept of approximate homomorphism. We have verified the theory when applied to predictive quantization of arbitrary ordinary differential equation models. On this paper, we will present some very promising empirical results that we obtained for tradeoff between message bandwidth utilization (number of bits transmitted) versus error incurred. The results plot bandwidth utilization and error against quantum size for the same federation executing on DEVS/HLA (an HLA compliant distributed simulation environment) in three networking platforms: PC LAN, Unix Ethernet, and Unix WAN. The theoretical and empirical results so far indicate that predictive quantization can be very scalable due to reduced local computation demands as well as having extremely favorable bandwidth requirement reduction/simulation fidelity tradeoffs. The paper also shows how the DEVS formalism is employed to formally characterize predictive contract mechanisms in a generic manner. This allows application in distributed simulation environments, whether DEVS-based or not, whether HLA-compliant or not. The basic concept in quantization is illustrated in Figure 1. Rather than represent a continuous curve by points sampled at regular time intervals, the curve is represented by the crossings of an equal spaced set of boundaries, separated by a quantum size. In classical Dead Reckoning, quantization is applied to the error between a reduced order model and a high fidelity model of a federate[9]. However, a more fundamental approach is to allow any desired object attribute to be quantized. This has two advantages [5]. First, a wider space of possible algorithms is opened up for investigation, including more direct approaches that do not require federates to keep local models of other federates. Second, it enables us to clearly distinguish the global error incurred as a result of predictive filtering from the local error that may be used to, as in Dead Reckoning, to effect a change in local models exported to others. We’ll return to this point in a moment. Figure 1 Quantization The baseline mechanism for quantization, called nonpredictive quantization, is illustrated in Figure 2. We assume a sender federate is updating a receiver federate on a numerical, real-valued, state variable (dynamically changing attribute), V. In the non-predictive approach, a quantizer demon is applied to the sender’s output which checks for threshold (boundary) crossings whenever a change in V occurs. Only when such a crossing occurs, is a new value of V sent across to the receiver. We note that in this change-based filtering operation, the frequency of message updates may be substantially reduced, thereby reducing network traffic but potentially incurring error. The cost/benefit analysis between reduced traffic and increased error can be framed in terms of tradeoff curves as we shall soon discuss. Note, for future reference, that while message traffic is reduced, the size of messages sent is unaffected in this approach. The quantizer demon incurs some computation at the sender federate but this is relatively inexpensive. Such additional computation, does raise another issue – scalability, how fast does the additional computation required by a predictive filtering method grow with increasing simulation size. Also, the receiver must be prepared to handle updates arriving asynchronously. If 2 Several possible variations on this theme, concerning for example, whether we send the actual value of V, or a modification depending on the boundary crossing. Such details are beyond the scope of this report. synchronous updating was assumed in its design, this may require redesign – depending on the flexibility of the underlying simulation code, this may, or may not, be easy to achieve. In object oriented designs, with flexible time management, this would not be a major issue. To summarize, the characteristics of non-predictive quantized filtering are: • Sender federate generates fixed (or variable) time step outputs. • Quantizer demon is applied to sender output. • This reduces the number of messages sent (although not their size). • The quantizer incurs some computation at the sender’s federate. • The sender’s model computation is unaffected. In sum, this approach is relatively easy to apply and requires minimal restructuring of the federates’ state computation processes. However, it can incur loss of accuracy due to the receiver’s diminished state updates and this may propagate in a global error due to feedback between sender and receiver as we will discuss. The DEVS/HLA environment supports this approach through its development of quantizer objects and their automatic interfacing to the HLA subscribe/publish data distribution service. However, the concept is generic and can be employed in any distributed simulation environment. Figure 2 Non-Predictive Quantization A more efficient form of quantization is predictive quantization, as illustrated in Figure 3. Here the sender employs a model to predict the next boundary crossing and time it will occur given its current state. As we will show, such computation can be quite inexpensive depending on the model used. This approach is inherently discrete-event based since the sender waits until the predicted next event (boundary crossing) time before sending its output and effecting its state change. Note that the federate need not be computing state changes during this waiting period, thus gaining computational advantage over non-predictive quantization. Since the next boundary crossing is either one above or one below the last recorded boundary, the sender need not send the full floating point (double word) value to the receiver. Indeed, assume that the receiver keeps track of the last boundary and knows the quantum size. Then only one bit of information is required – for example, a +1 indicates adding the quantum to the last boundary, while –1 indicates a similar subtraction. Thus, not only the number of messages but also the message size – in other words, total number of bits transmitted – can be significantly reduced in this approach. As shown in Figure 3, the stream emitted by the sender must somehow convey the boundary crossing times. Whether this incurs additional network bandwidth depends on the context. In discrete event logical time simulations, messages can be easily time stamped. This usually involves no additional cost, as all messages are time stamped to enable strong synchronization in the accepted distributed simulation protocols. In real time simulations, preservation of the order and time spacing between messages may provide the required information without time stamping. This form of messaging is often assumed in classical DIS (Distributed Interactive Simulation) training exercises. HLA provides the flexibility to choose among logical and real-time time management schemes. To summarize the characteristics of predictive quantized filtering are: • The sender employs a model to predict successive boundary crossings. • It sends a one-bit message at crossings − whether the next higher or next lower boundary has been reached. • The main advantage over non-predictive quantization is that both number of messages and their size can be reduced A second advantage, is that if simple predictive models are used, discrete event prediction can also greatly reduce the sender’s state transition computation execution time and frequency. The messages must convey the time of boundary crossing. In discrete event logical time simulations, messages can be time stamped with usually, no additional cost, as this is the background approach. In real time, preservation of the order and spacing between messages may provide the required information without time stamping. Figure 3 Predictive Quantization A detailed theoretical and empirical study of the advantages of predictive quantization over nonpredictive quantization is provided in [1, 4]. Here we will briefly review some salient elements of this study. 1.1 Generic Predictive Quantization Methods First, we note that the model employed for predictive quantization can be very simple. An approach that is fully generic for differential equation systems is illustrated in Figure 4. An ordinary differential equation system (ODE) consists of a finite number of integrators connected by instantaneous derivative functions to each other and to the external interface. A straightforward mapping of such a network on to a distributed equivalent is as follows: Figure 4 Mapping Differential Equation Systems directly to Distributed Discrete Event Form Each derivative function is mapped to a persistent function element and each integrator is mapped to a predictive quantization equivalent while preserving the interconnection topology. Basically, a persistent function element receives event inputs produces the output of a derivative function after any one of the inputs changes. The predictive quantization integrator (PCI) is basically linear extrapolation as illustrated in Figure 5. The time to next boundary crossing is basically the quantum size divided by the input (derivative). The boundary is predicted either to be one up or one down according to the sign of the derivative. We note that when an input event is received, the state is updated using the old input before recalculating the predicted crossing. This provides an important correction for error