DIFFICULTIES WITH TESTING FOR CAUSATION*

This paper examines the use of econometric tests for causality that are based on the work of Granger and Sims. The main conclusion is that it is not possible to infer the direction of causation from observed correlations. There are two reasons why the causality tests cannot reveal the direction of causation from observed correlations. First, the null hypothesis tested is necessary but not sufficient to imply causality. Second, any specification error renders the causality test results uninterpretable.     

APPROXIMATIONS FOR THE GI/G/m QUEUE

Queueing models can usefully represent production systems experiencing congestion due to irregular flows, but exact analyses of these queueing models can be difficult. Thus it is natural to seek relatively simple approximations that are suitably accurate for engineering purposes. Here approximations for a basic queueing model are developed and evaluated. The model is the GI/G/m queue, which has m identical servers in parallel, unlimited waiting room, and the first-come first-served queue discipline, with service and interarrival times coming from independent sequences of independent and identically distributed random variables with general distributions. The approximations depend on the general interarrival-time and service-time distributions only through their first two moments. The main focus is on the expected waiting time and the probability of having to wait before beginning service, but approximations are also developed for other congestion measures, including the entire distributions of waiting time, queue-length and number in system. These relatively simple approximations are useful supplements to algorithms for computing the exact values that have been developed in recent years. The simple approximations can serve as starting points for developing approximations for more complicated systems for which exact solutions are not yet available. These approximations are especially useful for incorporating GI/G/m models in larger models, such as queueing networks, wherein the approximations can be components of rapid modeling tools.