IEEE TRANSACTIONS ON COMPUTERS, MARCH 1969

Reviews of Books and Papers in the Computer FieldDONALD L. EPLEY, Reviews Editor

D. W. FiFE, A. I. RuBIN, R. A. SHORT, H. S. STONEAssistant Reviews Editors

Please address your comments and suggestions to the Reviews Editor: Donald L. Epley,Department of Electrical Engineering, University of Iowa, Iowa City, Iowa 52240

A. SIMULATION

R69-5 The Statistics of Discrete-Event Simulation-G. S. Fishmanand P. J. Kiviat (Simulation, vol. 10, pp. 185-195, April 1968).

This paper discusses the problems of simulating systems in whichsome random behavior exists. The title refers to "discrete-eventsimulation," but this remains undefined and unemphasized through-out most of the paper. (The expression refers to svstems in whichevents, such as customer arrivals, occur at a countable set of pointson the continuous time axis-this as opposed to a continuous-eventsimulation in which important properties of the system must beaccounted for over a continuum in a given time interval.) The authorstrace the elements of a typical experiment and identify the statisticalproblems along with a discussion of their possible solutions. Theyisolate a number of frequently overlooked considerations in therandom input process, particularly the independence of adjacentsamples. Similarly, in estimating system performance, the problemof correlated outputs must be solved. In short, in any simulation, theinput data and the system structure must be understood and ex-amined carefully.

Random number generators are discussed (without defining someof the quantities used, unfortunately), and the important conclusionis referred to, namely, that currently there is no better way forgenerating pseudorandom numbers than that of a simple multipli-cative congruence. The authors correctly state that it is importantfor any simulation to verify the independence of its input samples,but they neglect to tell us if the currently popular simulation lan-guages (e.g., GPSS, SIMSCRIPT, SIMULA) provide for this.

The use of analytical models and/or simulation of subsystems isrecommended as significant tools in reducing the complexity of largesystems simulations. An example of a simple queueing system is con-sidered and the question of simulation detail is discussed. Validationof the generated system model is one of the most critical considera-tions in simulations. This can be accomplished only if there is avail-able some numerical data from the actual system. The chi-squaredtest is referred to and its shortcoming for small sample size is pointedout; the variance test is recommended as an excellent substitute in thiscase. It is suggested that spectrum estimation be used as a means fordetermining independence of output samples; they fail to point outthat the periodogram so obtained will not converge to the true spec-trum unless aliasing is used (i.e., the variance of the estimate doesnot behave properly). Choice of sampling interval is discussed, butshould have been elaborated upon (it is here that they make thesignificant point that process activity is the critical driving influencein the simulation). Output variance reduction methods using the

powerful technique of antithetic random variables is mentioned.The authors close on the point that little work has been accomplishedin developing useful methods for deciding where in the parameterspace one should collect samples. This is a deep and difficult problem.

As the authors promised, they have provided in this paper a de-scription of the problem areas and their significance, with few solu-tions to these problems. This reviewer feels that they have succeededin this endeavor and would recommend the paper and many of itsreferences to anyone seriously interested in simulation methods.

LEONARD KLEINROCKDept. of Engrg.

University of CaliforniaLos Angeles, Calif.

B. HYBRID COMPUTATION

R69-6 Trajectory Optimization by a Direct Descent Process-L. E. Fogarty and R. M. Howe (Simulation, vol. 11, pp. 145-155,September 1968).

This paper develops and investigates a practical method by whichcertain aerospace trajectory optimization problems with terminal con-straints can be relatively easily solved on a hybrid computer. If agradient technique has been selected for the optimization process,then it has generally been the practice to solve the equations adjointto the equations of motion by utilizing a digital computer. In thepresent work no utilization is made of the adjoint variables. Thegradient is calculated by determining the response of the cost func-tion to approximately impulsive control perturbations, similar tothe method used by Wingrove and Raby. The variational problemswhich are considered are those of the Mayer type where the costfunction to be minimized is a function of the terminal state and ter-minal time. In this paper a technique is developed so that the gra-dient which is calculated is compatible with imposed terminal con-straints. The calculation of this modified gradient, which consists ofa linear combination of the cost gradient and the gradients of theterminal constraints, is relatively simple and can be rapidly per-formed on the hybrid computer. The circuitry required for handlingthe terminal constraints by this technique, as well as that requiredto simulate the other equations of the problem, is described in sub-stantial detail.

To illustrate the applicability of the method, two example prob-lems are presented. The first is the classical brachistochrone prob-

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IEEE TRANSACTIONS ON COMPUTERS, MARCH 1969

lem; the second consists of the determination of the optimum thrustdirection history leading to launch into circular orbit with minimumfuel or burning time. This latter example demonstrates the applica-bility of the method in solving optimum trajectory problems withmultiple (three) terminal constraints.

As the authors point out, a principal advantage of the presentmethod lies in its simplicity and in its ability to quickly generatesolutions with the aid of the hybrid computer. In this "impulse re-

sponse" method, moreover, the linearization of the equations about a

nominal set is not required, as would be the case where adjoint vari-ables are used. This characteristic can be advantageous when non-

analytical (tabular) functions are used in the analysis. This non-

linearization of the equations may also be a source of potential diffi-culty, however, in that the accuracy of hybrid results is generallylimited to three, or possibly four, significant figures. On the otherhand this type of accuracy would not be considered too serious ifapplied to linear variations about a nominal solution. Realizing thispotential source of difficulty, the authors present a method fordirectly computing the effects of small perturbations of the controlvariable on the state variables, rather than determining the small

effects of control variations directly through the differential equa-tions of motion. This approach, although not demonstrated inthe paper, appears to be a promising and logical improvement. Ap-parently the authors did not have to resort to this technique intheir study, since no accuracy difficulties were encountered in theirpresent investigation.

This paper shows that certain classes of trajectory optimizationproblems with terminal constraints can be rapidly solved by theimpulse response method with the aid of a hybrid computer. It isto be understood (as indicated by the authors) that the present in-vestigation is of an exploratory nature; nevertheless, the presentmethod appears to be a valuable tool for solving optimization prob-lems. It would be worthwhile to further investigate the applicabilityof the method in solving more complex problems of trajectory opti-mization. These would presumably include the consideration of a

larger ntumber of state and control variables to handle three dimen-sional trajectories and intermediate constraints.

F. T. SASAKIMartin Marietta Corp.

Denver, Colo.

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