We develop and evaluate time-series models of call volume to the emergency medical service of a major Canadian city. Our objective is to offer simple and effective models that could be used for realistic simulation of the system and for forecasting daily and hourly call volumes. Notable features of the analyzed time series are: a positive trend, daily, weekly, and yearly seasonal cycles, special-day effects, and positive autocorrelation. We estimate models of daily volumes via two approaches: (1) autoregressive models of data obtained after eliminating trend, seasonality, and special-day effects; and (2) doubly-seasonal ARIMA models with special-day effects. We compare the estimated models in terms of goodness-of-fit and forecasting accuracy. We also consider two possibilities for the hourly model: (3) a multinomial distribution for the vector of number of calls in each hour conditional on the total volume of calls during the day and (4) fitting a time series to the data at the hourly level. For our data, (1) and (3) are superior.

A popular approach for modeling dependence in a finite-dimensional random vector X with given univariate marginals is via a normal copula that fits the rank or linear correlations for the bivariate marginals of X. In this approach, known as the NORTA method, the normal distribution function is applied to each coordinate of a vector Z of correlated standard normals to produce a vector U of correlated uniform random variables over (0 1); then X is obtained by applying the inverse of the target marginal distribution function for each coordinate of U. The fitting requires finding the appropriate correlation between any two given coordinates of Z that would yield the target rank or linear correlation r between the corresponding coordinates of X. This root-finding problem is easy to solve when the marginals are continuous but not when they are discrete. In this paper, we provide a detailed analysis of this root-finding problem for the case of discrete marginals. We prove key properties of r and of its derivative as a function of . It turns out that the derivative is easier to evaluate than the function itself. Based on that, we propose and compare alternative methods for finding or approximating the appropriate . The case of discrete distributions with unbounded support is covered as well. In our numerical experiments, a derivative-supported method is faster and more accurate than a state-of-theart, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of ) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.

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We propose and examine a probabilistic model for the multivariate distribution of the number of calls in each period of the day (e.g., 15 or 30 minutes) in a call center, where the marginal distribution of the number of calls in any given period is arbitrary, and the dependence between the periods is modeled via a normal copula. Conditional on the number of calls in a period, their arrival times are independent and uniformly distributed over the period. This type of model has the advantage of being simple and reasonably flexible, and can match the correlations between the arrival counts in different periods much better than previously proposed models. For the situation where the number of periods is large, so the number of correlations to estimate can be excessive, we propose simple parametric forms for the correlations, defined as functions of the time lag between the periods. We test our proposed models on three data sets taken from real-life call centers and compare their goodness of fit to the best previously-proposed methods that we know. In the three cases, the new models provide a much better match of the correlations and of the coefficients of variation of the arrival counts in individual periods.