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Let {X(t), V(t), t ≥ 0} be a telegraph process, with V(0+) = 1. The distribution of X(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levels a and −a are studied for M/G and for G/M processes.

Zacks (Failure distribution associated with general renewal damage processes. In: Nikulin M, Commenges D, Haber C (eds) Probability statistics and modelling in public health. Springer, Berlin, pp 465–475, 2006 ) studied the reliability function, the hazard function and the distribution of the failure time when a system is subject to a cumulative, compound renewal damage process. The failure occurs when the damage process crosses a threshold β . In the present paper these results are generalized to the model where the system is replaced after failures. Two cases are considered: instant replacement and random positive replacement time. The distribution of the age of the current renewal cycle, as well as its excess life, and the availability function are studied. We derive also the distribution of total time in (0, t ) at which the system has been operational.

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

We study a one-dimensional telegraph process (M t ) t≥0 describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution of M t and its first two moments are derived. We characterize the level hitting times of M t in terms of integro-differential equations which can be solved in special cases.

Distributions of the first-exit times from a region with concave upper boundary are discussed for ordinary and compound Poisson processes. Explicit formulae are developed for the case of ordinary Poisson processes. Recursive formulae are given for the compound Poisson case, where the jumps are positive, having discrete or continuous distributions with finite means. Applications to sequential point estimation and insurance are illustrated. [PUBLICATION ABSTRACT]

The management of patients with midgut neuroendocrine tumors (MNET) is rapidly evolving. Current preoperative detection rates of primary tumor sites are higher than ever and progression-free survival in patients with already advanced disease is expanding due to the implementation of novel efficacious treatment strategies. This survival benefit may potentially translate into a need for a multidisciplinary approach to an even more heterogenous variety of clinical conditions, among these, carcinoid syndrome (CS) and carcinoid heart disease (CHD). The latter often triggers substantial morbidity and mortality, hence a systematic screening, an accurate diagnosis, as well as effective interventions are critically important. The rarity of the disease has result in a relative lack of statistically powerful evidence, which in turn may have rendered significant variability between practices. In this regard, despite recent guidelines, the optimal follow-up of patients with CHD remain debatable to some authors, perhaps due to the preponderance of certain schools throughout the manuscript. Herein, we present a concise and practical guidance document on clinical screening and echocardiographic surveillance of patients with CHD based on a comprehensive review of the literature, and complemented by our experience at the Center for Carcinoid and Neuroendocrine Tumors at The Mount Sinai Hospital.

The paper reviews recent results of D. Perry, W. Stadje and S. Zacks, on functionals of stopping times and the associated compound Poisson process with lower and upper linear boundaries. In particular, formulae of these functionals are explicitly developed for the total expected discounted cost of discarded service in an M/G/1 queue with restricted accessibility; for the expected total discounted waiting cost in an M/G/1 restricted queue; for the shortage, holding and clearing costs in an inventory system with continuous input; for the risk in sequential estimation and for the transform of the busy period when the upper boundary is random.

Prior investigation has shown that intravenous β-blockers decrease T-wave alternans (TWA) positivity in patients undergoing electrophysiology study (EPS). The present study examined whether oral β-blocker use within 24 hours of TWA influences yield and predictive value of TWA and EPS. We prospectively evaluated 387 patients (312 [81%] men, mean age 67 ± 11 years) with coronary artery disease, left ventricular ejection fraction ≤40%, and nonsustained ventricular tachycardia who underwent EPS and were followed for a mean of 2.8 ± 1.4 years. T-Wave alternans was performed using an atrial pacing protocol and interpreted using standard criteria. β-Blocker status was determined based on oral β-blocker use in the 24 hours preceding the test: β-blocker (−) (n = 62), β-blocker (+) (n = 325). Follow-up for ventricular tachycardia, ventricular fibrillation, and death was obtained from chart review, device interrogation, and the Social Security Death Index. Estimated sensitivity and specificity of TWA and EPS stratified by β-blocker use were calculated based on event-free 2-year survival. There was no difference in EPS (31 [50%] inducible off β-blockers vs 166 [51%] on β-blockers [ P = .89]) or TWA (26 [42%] positive, 17 [27%] indeterminate off β-blockers vs 136 [42%] positive, 81 [25%] indeterminate on β-blockers [ P = .89]). β-Blocker use within 24 hours of testing did not affect the predictive value of TWA or EPS for overall or 2-year event-free survival. Oral β-blocker therapy appears to have no effect on yield or predictive value of EPS or TWA in patients with coronary artery disease, diminished left ventricular function, and a history of nonsustained ventricular tachycardia.

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.

We consider the first-exit time of a compound Poisson process from a region that is bounded from below by an increasing straight line, while its upper boundary has positive jumps of i.i.d. sizes at Poisson times and increases linearly between jumps. An integral equation for the corresponding Laplace-Stieltjes transforms is derived and solved. The case of exponential jumps is treated separately. The problem has applications in queueing and risk theory. [PUBLICATION ABSTRACT]