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We develop an approximation for a queue having an arrival process that is the superposition of independent renewal processes, i.e., ∑GI1/G/1. This model is useful, for example, in analyzing networks of queues where the arrival process to an individual queue is the superposition of departure processes from other queues. If component arrival processes are approximated by renewal processes, the ∑GI1/G/1 model applies. The approximation proposed is a hybrid that combines two basic methods described by Whitt. All these methods approximate the complex superposition process by a renewal process and yield a GI/G/1 queue that can be solved analytically or approximately. In the hybrid method, the moments of the intervals in the approximating renewal process are a convex combination of the moments determined by the basic methods. The weight in the convex combination is identified using the asymptotic properties of the basic methods together with simulation. When compared to simulation estimates, the error in hybrid approximations of the expected number in the queue is 3%; in contrast, the errors of the basic methods are 20–30%. The quality of the approximations suggests that the hybrid approach would be useful in approximating point processes in other contexts.

The quality of products or manufacturing processes is sometimes characterized by profiles or functions. A method is proposed to identify outlier profiles among a set of complex profiles which are difficult to model with explicit functions. It treats profiles as vectors in high-dimension space and applies a χ 2 control chart to identify outliers. This method is useful in Statistical Process Control (SPC) in two ways: (i) identifying outliers in SPC baseline data; and (ii) the on-line monitoring of profiles. The method does not require explicit expression of the function between the response and explanatory variables or fitting regression models. It is especially useful and sometimes the only option when profiles are very complex. Given a set of profiles (high-dimension vectors), the median of these vectors is derived. The variance among profiles is estimated by considering the pair-wise differences between profiles. A χ 2 statistic is derived to compare each profile to the center vector. A simulation experiment and manufacturing data are used to illustrate applications of the method. Comparing it with the existing non-linear regression method shows that it has a better performance: it misidentifies fewer non-outlier profiles as outliers than the non-linear regression method, and misidentifies similarly small fractions of outlier profiles as non-outliers.

We report on simulations of \Sigma i GI i/M/1 queues; the arrival process is the superposition (sum) of up to 1024 i.i.d. renewal processes and there is a single exponential server. As one might anticipate, the simulation estimate of the expected number of customers in a \Sigma i GI i/M/1 queueing system approaches the expected number in an M/M/1 queueing system as the number of arrival processes, n, increases. However, for a given n, the difference between the expected numbers in the M/M/1 and \Sigma i GI i/M/1 queueing systems dramatically increases as the traffic intensity increases from \rho = 0.5 to \rho = 0.9. This difference is approximated by a formula which is a function of the traffic intensity, the number of component arrival processes and the squared coefficient of variation of the component interarrival times.queues, approximations, superposition arrival processes

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