Asymptotic expansions of moments of the waiting time in closed and open processor-sharing systems with multiple job classes

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Abstract
We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of queueing networks. We give effective methods for computing, for each of possibly several job-classes, the second moment of the equilibrium waiting time for closed systems as well as for open systems. Both open and closed systems have a CPU operating under the processor-sharing (‘time-slicing') discipline in which service-time requirements may depend on job-class. The closed system also includes a bank of terminals grouped according to job-classes, with the class structure allowing distinctions in the user's behavior in the terminal. In the contrasting open system, the job streams submitted to the CPU are Poisson with rate parameters dependent on job-classes.Our results are exact for the open system and, for the closed system, in the form of an asymptotic series in inverse powers of a parameter N. In fact, the result for open networks is simply the first term in the asymptotic series. For larger closed systems, the parameter N is larger and thus fewer terms of the series need be computed to achieve a desired degree of accuracy. The complexity of the calculations for the asymptotic expansions is polynomial in number of classes and, importantly, independent of the class populations. Only the results on the single-class systems, closed and open, were previously known.