A Mixed Exponential Time Series Model

There is considerable evidence that the distribution of the length of time that a patient occupies a bed in a hospital department is best described by a sum of two or three exponential terms, because of the presence of acute care, rehabilitation, and possibly long term care patients in the department. The patient flow models implied by these mixed exponential distributions are presented and fitting them to observed data when the admission rate fluctuates is discussed. Unlike single exponential distributions, mixed exponential distributions imply that the average length of stay of patients currently resident in the department is much longer than the average length of stay of a group of patients discharged over a period of time, so that the latter way of measuring will not correctly indicate what portion of the resources are being used by rehabilitation and long term care patients. Also, the expected additional length of stay increases dramatically with the time already spent in the department. Applications to predicting the effects of policy changes and to long term monitoring of hospital departments are presented. Two American hospitals are analyzed. The occupancy times in the government supported hospital follow a mixed exponential distribution similar to those found in the United Kingdom, but in the private hospital they fit a single exponential distribution, indicating markedly different management practices.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

We study the properties of a sequence of dependent random variables generated with the following scheme. A sequence of independent identically distributed random variables /spl alpha/ arrives at the input of a k-register; the random variables take values 0 and 1 with probabilities not equal to 1/2. The sequence /spl eta/ of random variables generated by the k-register for k/spl ges/2 is a stationary random sequence with dependent components. The sequence /spl eta/ is taken as the input to a memoryless binary symmetrical channel with input-independent noise, i.e. /spl eta/ is added coordinatewise modulo 2 to a sequence /spl beta/ of independent identically distributed random variables that also take the values 0 and 1 with probabilities not equal to 1/2 and are independent with the sequence /spl alpha/. In this paper we derive upper and lower estimates for the entropy of the stationary non-Markov random source identified with the channel output. The upper estimate is based on the well-known subadditivity property of the entropy of a finite-dimensional distribution. The main result is the proof of nontrivial lower estimate of the entropy for two particular k-registers: k=2 and k=3. If the probabilities of 0 and 1 in the sequences /spl alpha/ and /spl beta/ are close to 1/2, this estimate shows that the entropy of the source increases when k grows from 1 to 3. Pre-transformation of /spl alpha/ by the k-register, k>1, yields the increase of the entropy of the additive source /spl alpha/+/spl beta/ over the case k=1. This property of increase of the entropy is significant for constructing a strong random source from several weak ones.

SUMMARY The Laplace method for approximating integrals is applied to give a general approximation for the kth moment of a ratio of quadratic forms in random variables. The technique utilises the existence of a dominating peak at the boundary point on the range of integration. As closed form and tractable formulae do not exist in general, this simple approximation, which only entails basic algebraic operations, has evident practical appeal. We exploit the approximation to provide an approximate mean-bias function for the least squares estimator of the coefficient of the lag dependent variable in a first-order stochastic difference equation.

This paper is concerned on an M/M/1 queue with correlated interarrival and service times. In particular, we assume that the interarrival time and service time of a customer have a bivariate exponential distribution. By utilizing a Markov modulated fluid flow (MMFF) process associated with the age process of the customer in service, we obtain a number of queueing quantities in closed form. Using the solutions, the impact of the correlation coefficient of the interarrival and service times on a variety of queueing quantities is explored quantitatively and qualitatively. Specifically, we establish a monotonic relationship between the decay rates of queueing quantities and the correlation coefficient of the interarrival and service times. We also show that the decay rates are increasing in the correlation coefficient. In addition, queues with the maximum/minimum correlation coefficient are analyzed. Two examples are presented to demonstrate the importance of using the correlation coefficient in queueing performance/economic analysis.

General results on the applicability of the strong law of large numbers to a sequence of dependent random variables, as formulated in terms of estimates for the moments of sums of such variables, are applied to give new conditions of the applicability of this law to (in a wide sense) a stationary sequence of random variables.

This book has its origin in the need for developing and analyzing mathematical models for phenomena that evolve in time and influence each another, and aims at a better understanding of the structure and asymptotic behavior of stochastic processes. This monograph has double scope. First, to present tools for dealing with dependent structures directed toward obtaining normal approximations. Second, to apply the normal approximations presented in the book to various examples. The main tools consist of inequalities for dependent sequences of random variables, leading to limit theorems, including the functional central limit theorem (CLT) and functional moderate deviation principle (MDP). The results will point out large classes of dependent random variables which satisfy invariance principles, making possible the statistical study of data coming from stochastic processes both with short and long memory. Over the course of the book different types of dependence structures are considered, ranging from the traditional mixing structures to martingale-like structures and to weakly negatively dependent structures, which link the notion of mixing to the notions of association and negative dependence. Several applications have been carefully selected to exhibit the importance of the theoretical results. They include random walks in random scenery and determinantal processes. In addition, due to their importance in analyzing new data in economics, linear processes with dependent innovations will also be considered and analyzed.

Let {Xn: n ≧ 1} be a sequence of dependent random variables and let {wnk: 1 ≦ k ≦ n, n ≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.

This study mainly concerned with the K-capacity queueing system with recurrent input and two heterogeneous servers. Interarrival times are independent and have an arbitrary distribution. There are two servers and server k has an exponential distribution with parameter \(\mu _k\). Arriving customers choose server from the empty servers with equal probability. At an arrival time the customer joins the queue when both servers are busy. In addition an arrival leaves without having service when the system capacity is achieved. The defined system is represented by semi-Markov process and embedded Markov chain is obtained. Steady-state probabilities are found and loss probability is calculated by analyzing stream of overflows. Moreover loss probabilities are computed numerically for the queueing systems where the interarrival times are assumed as exponential, Erlang and deterministic distribution.

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

This paper considers a single server queue in which the service time is exponentially distributed and the service station may breakdown according to a Poisson process with the rates γ and γ' in busy period and idle period respectively. Repair will be performed immediately following a breakdown. The repair time is assumed to have an exponential distribution. Let g(t) and G(t) be the probability density function and the cumulative distribution function of the interarrival time respectively. When t tends to infinity, the rate of g(t)/[1 – G(t)] will tend to a constant. A set of equations will be derived for the probabilities of the queue length and the states of the arrival, repair and service processes when the queue is in a stationary state. By solving these equations, numerical results for the stationary queue length distribution can be obtained.

Stochastic weather generators are statistical models widely used to produce climate time series with similar statistical properties to observed data. They are also used as downscaling tools to generate climate change scenarios for impact studies. Precipitation is one of the main variables simulated by weather generators and is also a key variable for impact studies, especially for hydrology. Precipitation is usually simulated by multiple precipitation models which have been proposed for simulating site-specific precipitation. However, these models’ performance in simulating watershed-averaged extreme precipitation, especially in representing hydrological extremes, has not been well-investigated. Accordingly, this study compares the performance of six probability distributions (exponential, gamma, skewed normal, mixed exponential, hybrid exponential/Pareto, and Weibull distributions) and a polynomial curve-fitting method in generating precipitation for simulating hydrological extremes over three basins using a set of extreme indices. The results show that except for the exponential distribution (EXP), all of the methods produce the distribution of observed precipitation at the daily, monthly and annual scales reasonably well for all three river basins. Although the three-parameter hybrid exponential/Pareto distribution (EXPP) overestimates precipitation extremes, other three-parameter models produce extremes accurately. The three-parameter mixed exponential (MEXP) distribution outperforms other models for simulating precipitation extremes. However, with respect to representing hydrological extremes, the MEXP distribution is not the best model. When simulating extreme streamflows with synthetic weather data, the EXP distribution shows the worst performance, while the curve fitting method (PN) performs the best. The inferiority of the EXPP distribution in generating extreme precipitation does not propagate to extreme flow simulations. Meanwhile, the performance of WEB is moderate in terms of representing hydrological extremes. Overall, finding the model that best reproduces precipitation for simulating hydrological extremes is not as clear-cut, since the performance of each model is extreme-indices dependent. Taking all of the indices into account, the MEXP and the PN appear to be superior in representing extreme precipitation and hydrological extremes.

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

In this paper mathematical methods are given for finding the distributions of the virtual waiting time and the occupation time for single-server queues in which the interarrival times and the service times are discrete random variables taking on positive or non-negative integral values. First we shall consider deterministic queues in which the interarrival times and the service times are numerical (non-random) quantities and deduce a fundamental relation for such queues. Then we shall consider random queues with recurrent input and general service times and study the time dependent behavior of such queues.