Waiting Time Problems Research Articles

On a generalization of a waiting time problem and some combinatorial identities

In this paper we give an extension of the results of the generalized waiting time problem given by El-Desouky and Hussen (1990). An urn contains m types of balls of unequal numbers, and balls are drawn with replacement until first duplication. In the case of finite memory of order k, let ni be the number of type i, i = 1, 2, …, m. The probability of success pi = ni/N, i = 1, 2, …, m, where ni is a positive integer and Let Ym,k be the number of drawings required until first duplication. We obtain some new expressions of the probability function, in terms of Stirling numbers, symmetric polynomials, and generalized harmonic numbers. Moreover, some special cases are investigated. Finally, some important new combinatorial identities are obtained.

The Study on the Effect of Background Music on Customer Waiting Time in Restaurant

Many scholars study shows that the background music is restaurant controllable environmental factor that can affect the customer waiting time, which has some management implications for solving customer wait time problem in the service sector. This article used experimental management method, randomly selected a large restaurant, and then ob- served each customer just coming in, recorded background music types (including no music, slow-paced and fast-paced) and customer waiting time. And we made comparative analysis of the differences between weekdays and weekends. Fi- nally, the experimental results and analysis are given: (1) Background music can really affect the customer's waiting time; (2) Slow-paced background music can extend the customer waiting time, meanwhile fast-paced background music will shorten the waiting time of customers; (3) On weekends average waiting time is longer than working day.

The waiting time problem in a model hominin population.

BackgroundFunctional information is normally communicated using specific, context-dependent strings of symbolic characters. This is true within the human realm (texts and computer programs), and also within the biological realm (nucleic acids and proteins). In biology, strings of nucleotides encode much of the information within living cells. How do such information-bearing nucleotide strings arise and become established?MethodsThis paper uses comprehensive numerical simulation to understand what types of nucleotide strings can realistically be established via the mutation/selection process, given a reasonable timeframe. The program Mendel’s Accountant realistically simulates the mutation/selection process, and was modified so that a starting string of nucleotides could be specified, and a corresponding target string of nucleotides could be specified. We simulated a classic pre-human hominin population of at least 10,000 individuals, with a generation time of 20 years, and with very strong selection (50 % selective elimination). Random point mutations were generated within the starting string. Whenever an instance of the target string arose, all individuals carrying the target string were assigned a specified reproductive advantage. When natural selection had successfully amplified an instance of the target string to the point of fixation, the experiment was halted, and the waiting time statistics were tabulated. Using this methodology we tested the effect of mutation rate, string length, fitness benefit, and population size on waiting time to fixation.ResultsBiologically realistic numerical simulations revealed that a population of this type required inordinately long waiting times to establish even the shortest nucleotide strings. To establish a string of two nucleotides required on average 84 million years. To establish a string of five nucleotides required on average 2 billion years. We found that waiting times were reduced by higher mutation rates, stronger fitness benefits, and larger population sizes. However, even using the most generous feasible parameters settings, the waiting time required to establish any specific nucleotide string within this type of population was consistently prohibitive.ConclusionWe show that the waiting time problem is a significant constraint on the macroevolution of the classic hominin population. Routine establishment of specific beneficial strings of two or more nucleotides becomes very problematic.

On sooner and later waiting time distributions associated with simple patterns in a sequence of bivariate trials

In this article, we study sooner/later waiting time problems for simple patterns in a sequence of bivariate trials. The double generating functions of the sooner/later waiting times for the simple patterns are expressed in terms of the double generating functions of the numbers of occurrences of the simple patterns. Effective computational tools are developed for the evaluation of the waiting time distributions along with some examples. The results presented here provide perspectives on the waiting time problems arising from bivariate trials and extend a framework for studying the exact distributions of patterns. Finally, some examples are given in order to illustrate how our theoretical results are employed for the investigation of the waiting time problems for simple patterns.

Coupon collector’s problems with statistical applications to rankings

Some new exact distributions on coupon collector’s waiting time problems are given based on a generalized Polya urn sampling. In particular, usual Polya urn sampling generates an exchangeable random sequence. In this case, an alternative derivation of the distribution is also obtained from de Finetti’s theorem. In coupon collector’s waiting time problems with \(m\) kinds of coupons, the observed order of \(m\) kinds of coupons corresponds to a permutation of \(m\) letters uniquely. Using the property of coupon collector’s problems, a statistical model on the permutation group of \(m\) letters is proposed for analyzing ranked data. In the model, as the parameters mean the proportion of the \(m\) kinds of coupons, the observed ranking can be intuitively understood. Some examples of statistical inference are also given.

Unification of Dirichlet Methodology

In this article we provide a unified framework for solving Dirichlet related probability and waiting time problems. We consider a Pólya sampling scheme in which each time an object is selected, it is put back into the population along with c additional objects of the same type. By considering both fixed sample size and inverse sampling procedures, we unify the Dirichlet I, J, C, and D functions with their hypergeometric counterparts by extending these functions to Pólya sampling. We then use these functions to unify and extend the corresponding expected waiting time results.

Exact and asymptotic results for pattern waiting times

Waiting time problems for the occurrence of a pattern have attracted considerable research interest. Several results, including Poisson or Compound Poisson approximations as well as Normal approximations have appeared in the literature. In addition, a number of asymptotic results has been developed by making use of the finite Markov chain imbedding technique and the Perron–Frobenius eigenvalue. In the present paper we present a recursive scheme for the evaluation of the tail probabilities of the waiting time for the first and r-th occurrence of a pattern. A number of asymptotic results (along with their rates of convergence) that do not require the existence of the Perron–Frobenius eigenvalue are also offered. These results cover a quite wide class of pattern waiting time problems and, in most cases, perform better than the ones using the Perron–Frobenius eigenvalue.

Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials

We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional γ slack cells without quotas. For any given and , in our Model I we determine the waiting time until at least frequency cells and at least run cells reach their quotas. For any given τ ≤ α + β, in our Model II we determine the waiting time until τ cells reach their quotas. Computer algorithms are developed to calculate the distributions, expectations and standard deviations of the waiting time random variables of the two models. Numerical results demonstrate the efficiency of the algorithms.

Hospital waiting time: the forgotten premise of healthcare service delivery?

This is a national study which aims to determine the average waiting time in Malaysian public hospitals and to gauge the level of patient satisfaction with the waiting time. It also aims to identify factors perceived by healthcare providers which contribute to the waiting time problem. Self-administered questionnaires were the main method of data collection. Two sets of questionnaires were used. The first set solicited information from patients on their waiting time expereince. The second set elucidated information from hospital employees on the possible causes of lengthy waiting time. The questionnaires were administered in 21 public hospitals throughout all 13 states in Malaysia. A total of 13,000 responses were analysed for the patient survey and almost 3,000 were analysed for the employee survey. The findings indicate that on average, patients wait for more than two hours from registration to getting the prescription slip, while the contact time with medical personnel is only on average 15 minutes. Employee surveys on factors contributing to the lengthy waiting time indicate employee attitude and work process, heavy workload, management and supervision problems, and inadequate facilities to be among the contributory factors to the waiting time problem. Public healthcare in Malaysia is in a state of "excess demand", where demand for subsidised healthcare far outstrips supply, due to the large fee differential between public and private healthcare services. There is a need for hospital managers to reduce the boredom faced by patients while waiting, and to address the waiting time problem in a more scientific manner, as has been carried out in other countries through simulation and modelling techniques. Healthcare organisations are keen to address their waiting time problem. However, not much research has been carried out in this area. The study thus fills the lacuna in waiting time studies in healthcare organisations.

Generating Functions of Waiting Times and Numbers of Visits for Random Walks on Graphs

In this paper, we consider some cover time problems for random walks on graphs in a wide class of waiting time problems. By using generating functions, we present a unified approach for the study of distributions associated with waiting times. In addition, the distributions of the numbers of visits for the random walks on the graphs are also studied. We present the relationship between the distributions of the waiting times and the numbers of visits. We also show that these theoretical results can be easily carried out through some computer algebra systems and present some numerical results for cover times in order to demonstrate the usefulness of the results developed. Finally, the study of cover time problems through generating functions leads to more extensive development.

Studying the Evolution of Promoter Sequences: A Waiting Time Problem

To gain a better understanding of the evolutionary dynamics of regulatory DNA sequences, we address the following questions: (1) How long does it take until a given transcription factor (TF) binding site emerges at random in a promoter sequence? and (2) How does the composition of a TF binding site affect this waiting time? Using two different probabilistic models (an i.i.d. model and a neighbor dependent model), we can compute the expected waiting time for every k-mer, k ranging from 5 to 10, until it appears in a promoter of a species. Our findings indicate that new TF binding sites can be created on a short evolutionary time scale, i.e. in a time span below the speciation time of human and chimp. Furthermore, one can conclude that the composition of a TF binding site plays a crucial role concerning the waiting time until it appears and that the CpG methylation-deamination substitution process probably accelerates the creation of new TF binding sites. A screening of existing TF binding sites moreover reveals that k-mers predicted to have short waiting times occur more frequently than others. Supplementary Material is available at www.libertonline.com/cmb .

A simulation model for dielectric relaxation based on defect diffusion model and waiting time problems

We propose a simulation model for dielectric relaxation based on defect diffusion. The defect diffusion model (DDM) has been used to interpret dielectric relaxation and other relaxation phenomena. The essential feature of the model is a cooperative interaction between the relaxing dipole and its nearest-neighbors, containing defects, and the relaxation can only occur when a defect encounters a dipole. In our model we have taken the motion of defect as a stochastic process characterized by successive waiting time problem rather than classical random walk motion. We present computer simulation result for a simple dipolar model system and the dipole correlation function obtained from this new model under various physical conditions appears to be in the form of a stretched exponential function.

Rates of convergence in certain limit theorem for extreme values

Let X n 1 * , … , X n n * be independent random variables with the common negative binomial distribution with parameters r > 0 and 1 / n , where r is not necessarily an integer. We determine the limiting distribution of the random variable M n * = max { X n 1 * , … , X n n * } as n → ∞ , corresponding normalizing constants and the rate of convergence. For an integer r the connection with certain waiting time problems is indicated.

A waiting time problem arising from the study of multi-stage carcinogenesis

We consider the population genetics problem: how long does it take before some member of the population has $m$ specified mutations? The case $m=2$ is relevant to onset of cancer due to the inactivation of both copies of a tumor suppressor gene. Models for larger $m$ are needed for colon cancer and other diseases where a sequence of mutations leads to cells with uncontrolled growth.

Football fever: goal distributions and non-Gaussian statistics

Analyzing football score data with statistical techniques, we investigate how the not purely random, but highly co-operative nature of the game is reflected in averaged properties such as the probability distributions of scored goals for the home and away teams. As it turns out, especially the tails of the distributions are not well described by the Poissonian or binomial model resulting from the assumption of uncorrelated random events. Instead, a good effective description of the data is provided by less basic distributions such as the negative binomial one or the probability densities of extreme value statistics. To understand this behavior from a microscopical point of view, however, no waiting time problem or extremal process need be invoked. Instead, modifying the Bernoulli random process underlying the Poissonian model to include a simple component of self-affirmation seems to describe the data surprisingly well and allows to understand the observed deviation from Gaussian statistics. The phenomenological distributions used before can be understood as special cases within this framework. We analyzed historical football score data from many leagues in Europe as well as from international tournaments, including data from all past tournaments of the ``FIFA World Cup'' series, and found the proposed models to be applicable rather universally. In particular, here we analyse the results of the German women's premier football league and consider the two separate German men's premier leagues in the East and West during the cold war times and the unified league after 1990 to see how scoring in football and the component of self-affirmation depend on cultural and political circumstances.

Methods for Studying Generalized Birthday and Coupon Collection Problems

In this paper, we consider generalizations of two classical probability problems: the birthday problem and the coupon collector's problem. These problems are discussed in terms of urn models and captured through generating functions. Some methods for the study of the problems are presented. Furthermore, we also formulate the generalized birthday and coupon collector's problems as the waiting time problems. In each case, numerical examples are given in order to illustrate the feasibility of our methods.

On waiting time distributions associated with compound patterns in a sequence of multi-state trials

In this article, waiting time distributions of compound patterns are considered in terms of the generating function of the numbers of occurrences of the compound patterns. Formulae for the evaluation of the generating functions of waiting time are given, which are very effective computational tools. We provide several viewpoints on waiting time problems associated with compound patterns and develop a general workable framework for the study of the corresponding distributions. The general theory is employed for the investigation of some examples in order to illustrate how the distributions of waiting time can be derived through our theoretical results.

A Survey of the Coupon Collector’s Problem with Random Sample Sizes

This paper surveys the coupon collector’s waiting time problem with random sample sizes and equally likely balls. Consider an urn containing m red balls. For each draw, a random number of balls are removed from the urn. The group of removed balls is painted white and returned to the urn. Several approaches to addressing this problem are discussed, including a Markov chain approach to compute the distribution and expected value of the number of draws required for the urn to contain j white balls given that it currently contains i white balls. As a special case, E[N], the expected number of draws until all the balls are white given that all are currently red is also obtained.

On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.