We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

For several pairs (P, Q) of classical distributions on ℕ0, we show that their stochastic ordering P ≤stQ can be characterized by their extreme tail ordering equivalent to P({k*})/Q({k*}) ≥ 1 ≥ limk→k*P({k})/Q({k}), with k* and k* denoting the minimum and the supremum of the support of P + Q, and with the limit to be read as P({k*})/Q({k*}) for finite k*. This includes in particular all pairs where P and Q are both binomial (bn1,p1 ≤stbn2,p2 if and only if n1 ≤ n2 and (1 - p1)n1 ≥ (1 - p2)n2, or p1 = 0), both negative binomial (b−r1,p1 ≤stb−r2,p2 if and only if p1 ≥ p2 and p1r1 ≥ p2r2), or both hypergeometric with the same sample size parameter. The binomial case is contained in a known result about Bernoulli convolutions, the other two cases appear to be new. The emphasis of this paper is on providing a variety of different methods of proofs: (i) half monotone likelihood ratios, (ii) explicit coupling, (iii) Markov chain comparison, (iv) analytic calculation, and (v) comparison of Lévy measures. We give four proofs in the binomial case (methods (i)-(iv)) and three in the negative binomial case (methods (i), (iv), and (v)). The statement for hypergeometric distributions is proved via method (i).

Staffing and equipment levels in the UK are impacting heavily on waiting times for radiotherapy. This paper presents the results of financial and organisational interventions on the radiotherapy waiting list at the Derbyshire Royal Infirmary. Efficacy of these interventions was assessed using audits before and after their application. The interventions included working extended days and weekends, employment of agency staff and “out-of-hours” machine servicing and maintenance. A Waiting List and Data Quality radiotherapist was appointed to streamline the radiotherapy referral system. A waiting list database application was developed to book patients in the correct order of priority and within target waiting times. The resulting increase in radiotherapy workload was achieved over a 3-month period at an approximate cost of £30,000, albeit with several implications. Staff felt that the pace of work was unsustainable, although sickness rates did not increase. After the interventions were removed, and waiting times increased, staff morale dropped. Extended working hours impacted on the reliability of the machines, causing more unscheduled interruptions than normal. This indicated the need for additional maintenance or servicing sessions.

This paper shows that full compliance with RCR guidelines for waiting times is achievable, given sufficient financial and organisational assistance.

Purpose: Radiation techniques employed for breast cancer must be efficient as well as effective in order to minimise waiting lists. Protocol restrictions, or the technical application of treatment, may influence planning and treatment times as radiographers follow departmental policies. A national survey of UK radiotherapy centres was undertaken to establish trends in waiting times for breast cancer irradiation; and to investigate relationships of waiting times with the deployment of equipment and personnel and technical procedures adopted.

Method: A questionnaire was posted to the Head of Radiotherapy Services and a Clinical Oncologist with an interest in breast cancer in the radiotherapy centres in the UK. Survey questions investigated a number of issues, including the number of breast patients planned per week; protocols chosen; average treatment and planning times; levels of personnel and equipment; and the population served.

Results: A total of 53 centres were contacted, of which 51 centres responded to some aspect of the survey (96%). Average waiting times for treatment fluctuated from 1 to 7 weeks and maximum waiting times of 11 weeks were reported. Variation in clinical practice was found, including procedure times and the number of radiographers employed per linear accelerator. A multiple regression analysis indicated that a combination of equipment levels, simulation times, and the number of breast contours taken best predicted the average waiting time for breast treatment.

Conclusion: Waiting times reported were influenced by a combination of levels of equipment available and protocols adopted.

Let ε be a (single or composite) pattern defined over a sequence of Bernoulli trials. This article presents a unified approach for the study of the joint distribution of the number Sn of successes (and Fn of failures) and the number Xn of occurrences of ε in a fixed number of trials as well as the joint distribution of the waiting time Tr till the rth occurrence of the pattern and the number STr of successes (and FTr of failures) observed at that time. General formulae are developed for the joint probability mass functions and generating functions of (Xn,Sn), (Tr,STr) (and (Xn,Sn,Fn),(Tr,STr,FTr)) when Xn belongs to the family of Markov chain imbeddable variables of binomial type. Specializing to certain success runs, scans and pattern problems several well-known results are delivered as special cases of the general theory along with some new results that have not appeared in the statistical literature before.

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is waiting at the polled queue. Naive intuition would lead one to believe that the SD version should perform better than the SI version. We characterize the difference in the expected waiting times of these two versions, and we uncover some surprising facts. In particular, we show that, regardless of the server utilization or the service-time distribution, the SD version performs (i) the same as, (ii) worse than, or (iii) better than its SI counterpart if the switchover and setup times are, respectively, (i) both constants, (ii) variable (i.e. non-deterministic) and constant, or (iii) constant and variable. Only (iii) is consistent with naive intuition.

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

The statistical properties of a population of immigrant pairs of individuals subject to loss through emigration are calculated. Exact analytical results are obtained which exhibit characteristic even–odd effects. The population is monitored externally by counting the number of emigrants leaving in a fixed time interval. The integrated statistics for this process are evaluated and it is shown that under certain conditions only even numbers of individuals will be observed.

It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.

We consider a new embedded Markov chain for the PH/G/1 queue by recording the queue length, the phase of the arrival process and the number of services completed during the current busy period at the successive departure epochs. Algorithmically tractable matrix formulas are obtained which permit the analysis of the fluctuations of the queue length and waiting times during a typical busy cycle. These are useful in the computation of certain profile curves arising in the statistical analysis of queues. In addition, informative expressions for the mean waiting times in the stable GI/G/1 queue and a simple new algorithm to evaluate the waiting-time distributions for the stationary PH/PH/1 queue are obtained.

Waiting-time distributions for M/G/1 systems with priority dependent on class, order of arrival, service length, etc., are difficult to obtain. For single-server multipurpose processors the difficulties are compounded. A certain ergodic post-arrival depletion time is shown to be a true maximum for all delay times of interest. Explicit numerical evaluation of the distribution of this time is available. A heavy-traffic distribution for this time is shown to provide a simple and useful engineering tool with good results and insensitivity to service-time distribution even at modest traffic intensity levels. The relationship to the diffusion approximation for heavy traffic is described.

We investigate conditions under which the transition probabilities of various Markovian storage processes approach a stationary limiting distribution π at an exponential rate. The models considered include the waiting time of the M/G/1 queue, and models for dams with additive input and state-dependent release rule satisfying a ‘negative mean drift' condition. A typical result is that this exponential ergodicity holds provided the input process is ‘exponentially bounded'; for example, in the case of a compound Poisson input, a sufficient condition is an exponential bound on the tail of the input size distribution. The results are proved by comparing the discrete-time skeletons of the Markov process with the behaviour of a random walk, and then showing that the continuous process inherits the exponential ergodicity of any of its skeletons.

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.