suggest exponential growth, logistic growth, or other functions. Queuing theory Another example of hyperbolic growth can be found in queueing theory: the average waiting time of randomly arriving customers grows hyperbolically as a function of the average load ratio of the server. The singularity in this case occurs when the average amount of work arriving to the server equals the server's processing capacity. If the processing needs exceed the server's capacity, then there is no well-defined average waiting time, as the queue can grow without bound. A practical implication of this particular example is that for highly loaded queuing systems the average waiting time can be extremely sensitive to the processing capacity. Enzyme kinetics A further practical example of hyperbolic growth can be found in enzyme kinetics. When the rate of reaction (termed velocity) between an enzyme and substrate is plotted against various concentrations of the substrate, a hyperbolic plot is obtained for many simpler systems. When this happens, the enzyme is said to follow Michaelis-Menten kinetics. Mathematical example The function exhibits hyperbolic growth with a singularity at time in the limit as the function goes to infinity. More generally, the function exhibits hyperbolic growth, where is a scale factor. Note that this algebraic function can be regarded as analytical solution for the function's differential: This means that with hyperbolic growth the absolute growth rate of the variable in the moment is proportional to the square of the value of in the moment . Respectively, the quadratic-hyperbolic function looks as follows: See also Exponential growth Logistic growth Mathematical singularity Notes References Alexander V. Markov, and Andrey V. Korotayev (2007). "Phanerozoic marine biodiversity follows a hyperbolic trend". Palaeoworld. Volume 16. Issue 4. Pages 311-318]. Kremer, Michael. 1993. "Population Growth and Technological Change: One Million B.C. to 1990," The Quarterly Journal of Economics 108(3): 681-716. Korotayev A., Malkov A., Khaltourina D. 2006. Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth. Moscow: URSS. . Rein Taagepera (1979) People, skills, and resources: An interaction model for world population growth. Technological Forecasting and Social Change 13, 13-30. Mathematical analysis Special functions Differential equations Population Curves Population ecology