Arrival Count Research Articles

The Challenge of Predicting Demand for Emergency Department Services

The objective was to develop methodology for predicting demand for emergency department (ED) services by characterizing ED arrivals. One year of ED arrival data from an academic ED were merged with local climate data. ED arrival patterns were described; Poisson regression was selected to represent the count of hourly ED arrivals as a function of temporal, climatic, and patient factors. The authors evaluated the appropriateness of prediction models by whether the data met key Poisson assumptions, including variance proportional to the mean, positive skewness, and absence of autocorrelation among hours. Model accuracy was assessed by comparing predicted and observed histograms of arrival counts and by how frequently the observed hourly count fell within the 50 and 90% prediction intervals. Hourly ED arrivals were obtained for 8,760 study hours. Separate models were fit for high- versus low-acuity patients because of significant arrival pattern differences. The variance was approximately equal to the mean in the high- and low-acuity models. There was no residual autocorrelation (r = 0) present after controlling for temporal, climatic, and patient factors that influenced the arrival rate. The observed hourly count fell within the 50 and 90% prediction intervals 50 and 90% of the time, respectively. The observed histogram of arrival counts was nearly identical to the histogram predicted by a Poisson process. At this facility, demand for ED services was well approximated by a Poisson regression model. The expected arrival rate is characterized by a small number of factors and does not depend on recent numbers of arrivals.

Semi-Markov modulated Poisson process: probabilistic and statistical analysis

We consider a Poisson process that is modulated in such a way that the arrival rate at any time depends on the state of a semi-Markov process. This presents an interesting generalization of Poisson processes with important implications in real life applications. Our analysis concentrates on the transient as well as the long term behaviour of the arrival count and the arrival time processes. We discuss probabilistic as well as statistical issues related to various quantities of interest.

Modeling Daily Arrivals to a Telephone Call Center

We develop stochastic models of time-dependent arrivals, with focus on the application to call centers. Our models reproduce three essential features of call center arrivals observed in recent empirical studies: a variance larger than the mean for the number of arrivals in any given time interval, a time-varying arrival intensity over the course of a day, and nonzero correlation between the arrival counts in different periods within the same day. For each of the new models, we characterize the joint distribution of the vector of arrival counts, with particular focus on characterizing how the new models are more flexible than standard or previously proposed models. We report empirical results from a study on arrival data from a real-life call center, including the essential features of the arrival process, the goodness of fit of the estimated models, and the sensitivity of various simulated performance measures of the call center to the choice of arrival process model.

Stochastic bound on delay for guaranteed rate nodes

We find that the probabilistic bound on delay in the article by Goyal et al. (see Springer Multimedia Syst., vol.5, p.157-63, 1997) is incorrect. The problem originates from: 1) the difference between stationary and Palm probabilities and 2) treating the arrival counting process over some random time intervals as if the time interval would be fixed. The error is propagated to some later work, e.g., recently by Bensaou et al.(see IEEE/ACM. Trans. Networking, vol.9, p.591-603, Oct. 2001), which derived some probabilistic delay bounds based on Goyal et al. in 1997. We give fixes to the above problem.

Comparisons of multi-server queues with finite waiting rooms

In this paper we consider s-server queues with capacity c, 1 ≤ s ≤ c ≤ ∞, the first-come first-served queue discipline and very general arrival and service processes. We show that the admission epochs and departure epochs decrease, so that the throughput increases, when any of the following changes occur: (1) the number of servers s increases, (2) the capacity c increases, (3) the external arrival counting process increases or (4) the service times decrease, provided that the service times are assigned in order of service initiation and that a subsequence ordering is used to compare arrival counting processes. The subsequence ordering for the arrival processes is very important for obtaining positive results with finite waiting rooms. The subsequence ordering holds between a superposition point process and its component point processes. The subsequence ordering can often be applied via its stochastic generalization, the stochastic subsequence ordering, which is implied by a failure rate ordering.

A new view of the heavy-traffic limit theorem for infinite-server queues

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

A new view of the heavy-traffic limit theorem for infinite-server queues

This paper presents a new approach for obtaining heavy-traffic limits for infinite-server queues and open networks of infinite-server queues. The key observation is that infinite-server queues having deterministic service times can easily be analyzed in terms of the arrival counting process. A variant of the same idea applies when the service times take values in a finite set, so this is the key assumption. In addition to new proofs of established results, the paper contains several new results, including limits for the work-in-system process, limits for steady-state distributions, limits for open networks with general customer routes, and rates of convergence. The relatively tractable Gaussian limits are promising approximations for many-server queues and open networks of such queues, possibly with finite waiting rooms.

An LIL Version of L = λW

This paper establishes a law-of-the-iterated-logarithm (LIL) version of the fundamental queueing formula L = λW: Under regularity conditions, the continuous-time arrival counting process and queue-length process jointly obey an LIL when the discrete-time sequence of interarrival times and waiting times jointly obey an LIL, and the limit sets are related. The standard relation L = λW appears as a corollary. LILs for inverse processes and random sums are also established, which are of general probabilistic interest because the usual independence, identical-distribution and moment assumptions are not made. Moreover, an LIL for regenerative processes is established, which can be used to obtain the other LILs.

Inequalities between Arrival Averages and Time Averages in Stochastic Processes Arising from Queueing Theory

A common scenario in stochastic models, especially in queueing theory, is that an arrival counting process both observes and interacts with another continuous time stochastic process. In the case of Poisson arrivals, Wolff (Wolff, R. W. 1982. Poisson arrivals see time averages. Opns. Res. 30 223–231.) recently proved that the proportion of arrivals finding the process in some state is equal to the proportion of time it spends there under a lack of anticipation assumption. Inspired by Wolff's approach, in this paper we study the related interesting question of when do we have inequalities between these proportions. We establish two-sided inequalities under the following three assumptions: (i) the interarrival time distributions are of type NBUE or NWUE, (ii) the process being observed have monotone sample paths between arrival epochs, and (iii) the state of the process does not depend on future jumps of the arrival process. These assumptions are typically true in all standard queueing models and hence our results have wide implications. Stochastic inequalities between limiting distributions of interest, when they exist, also follow easily from our main result.