The joint distribution in the stationary case of four random variables is studied in this paper. Two of these are discrete and arise as the numbers of customers left behind in the queue by two successive departing customers in the imbedded Markov chain analysis. The other two are continuous, being the time intervals between three successive departures. The marginal joint distribution of the latter two random variables is found; in particular, the autocorrelation of lag 1 for intervals in the departure process in a stationary system is evaluated. By extending the analysis to the study of three successive departure intervals the autocorrelation of lag 2 for the process is evaluated. Burke (1956) andFinch (1959) proved that with Poisson arrivals, and service times independent for a single-server queue in equilibrium, the departure intervals are independent if and only if service time is exponentially distributed. This paper gives a measure of the dependence in the important case when the service times are special Erlangian.
A problem of estimating waiting time in the statistical analysis of queues is investigated. The continuous time study of the M/M/1 queue made by Bailey is adapted to obtain the asymptotic variance of a direct estimate of waiting time as obtained under conditions of incomplete information. This is then compared with the asymptotic variance of the maximum likelihood estimate as obtained under conditions of complete information and based on the results of Clarke.
Summary
Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.
Surveys were returned from 429 two-year colleges regarding placement practices for calculus. The high school record was the most important factor indicated, followed by placement test results and SAT/M or ACT/M scores. Most schools rated their placement program as adequate. Placement was advisory in 145 schools and 70 schools used self-placement. There was no commonality with regard to cutoff scores or placement procedures.
INTRODUCTIONThe results of a nationwide survey indicate that two-year colleges use widely different placement practices for direct entrance into the calculus sequence. Some colleges used a single indicator such as a placement test while others had very elaborate linear regression models with several indicators. A number of colleges indicated that a majority of their students (70% to 99%) take precalculus first. This report concentrates only on the placement practices used for direct placement into calculus.
BACKGROUND
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.