Heavy-usage asymptotic expansions for the waiting time in closed processor-sharing systems with multiple classes

Abstract

We present new results based on novel techniques for the problem of characterizing the waiting-time distribution in a class of closed queueing networks in heavy usage, which in practical terms means that the processor is utilized more than about 80 per cent. This paper extends recent work by Mitra and Morrison [10] on the same system in normal usage. The closed system has a CPU operating under the processor-sharing (‘time-slicing’) discipline and a bank of terminals. The presence of multiple job-classes allows distinctions in the user’s behavior in the terminal and in the service requirements. This work is primarily applicable to the case of large numbers of terminals. We give an effective method for calculating, for the equilibrium waiting time, the first and second moments and the leading term in the asymptotic approximation to the distribution. Our results are in the form of asymptotic expansions in inverse powers of , where N is a large parameter. The expansion coefficients depend on the classical parabolic cylinder functions.