We consider a production–inventory control model with two reflecting boundaries, representing the finite storage capacity and the finite maximum backlog. Demands arrive at the inventory according to a Poisson process, their i.i.d. sizes having a common phase-type distribution. The inventory is filled by a production process, which alternates between two prespecified production rates ρ1 and ρ2: as long as the content level is positive, ρ1 is applied while the production follows ρ2 during time intervals of backlog (i.e., negative content). We derive in closed form the various cost functionals of this model for the discounted case as well as under the long-run-average criterion. The analysis is based on a martingale of the Kella–Whitt type and results for fluid flow models due to Ahn and Ramaswami.

In this paper, we carry out stochastic comparisons of the largest order statistics arising from multiple-outlier gamma models with different both shape and scale parameters in the sense of various stochastic orderings including the likelihood ratio order, star order and dispersive order. It is proved, among others, that the weak majorization order between the scale parameter vectors along with the majorization order between the shape parameter vectors imply the likelihood ratio order between the largest order statistics. A quite general sufficient condition for the star order is presented. The new results established here strengthen and generalize some of the results known in the literature. Numerical examples and applications are also provided to explicate the theoretical results.

Random k-set pool designs are known to be efficient but suffer the presence of clones having identical test sets (hence indistinguishable). We propose a random pool design with distinct k-sets. We give an algorithm to generate the design and compute the expected numbers of unresolved negatives and unresolved positives. We show a relation between the two numbers which holds also for other random designs; hence, it can be used to obtain their expected numbers of unresolved positives which are not known. Among these other random designs, the random size-k design is a new design.

We investigate the evolution of an urn of balls of two colors, where one chooses a pair of balls and observes rules of ball addition according to the outcome. A nonsquare ball addition matrix of the form $\left( \matrix{a & b \cr c & d \cr e & f}\right)$ corresponds to such a scheme, in contrast to pólya urn models that possess a square ball addition matrix. We look into the case of constant row sum (the so-called balanced urns) and identify a linear case therein. Two cases arise in linear urns: the nondegenerate and the degenerate. Via martingales, in the nondegenerate case one gets an asymptotic normal distribution for the number of balls of any color. In the degenerate case, a simpler probability structure underlies the process. We mention in passing a heuristic for the average-case analysis for the general case of constant row sum.

This paper deals with two related approximations that were recently proposed for the loss probability in finite-buffer queues. The purpose of the paper is twofold: first, to provide better insight and more theoretical support for both approximations, and second, to show by an experimental study how well both approximations perform. An interesting empirical finding is that in many cases of practical interest the two approximations provide upper and lower bounds on the exact value of the loss probability.

In this paper, we study the asymptotic equipartition property (AEP) for a nonhomogeneous Markov information source. We first give a limit theorem for the averages of the functions of two variables of this information source by using the convergence theorem for the martingale difference sequence. As corollaries, we get several limit theorems and a limit theorem of the relative entropy density, which hold for any nonhomogeneous Markov information source. Then, we get a class of strong laws of large numbers for nonhomogeneous Markov information sources. Finally, we prove the AEP for a class of nonhomogeneous Markov information sources.

In a rare life-threatening disease setting the number of patients in the trial is a high proportion of all patients with the condition (if not all of them). Further, this number is usually not enough to guarantee the required statistical power to detect a treatment effect of a meaningful size. In such a context, the idea of prioritizing patient benefit over hypothesis testing as the goal of the trial can lead to a trial design that produces useful information to guide treatment, even if it does not do so with the standard levels of statistical confidence. The idealized model to consider such an optimal design of a clinical trial is known as a classic multi-armed bandit problem with a finite patient horizon and a patient benefit objective function. Such a design maximizes patient benefit by balancing the learning and earning goals as data accumulates and given the patient horizon. On the other hand, optimally solving such a model has a very high computational cost (many times prohibitive) and more importantly, a cumbersome implementation, even for populations as small as a hundred patients. Several computationally feasible heuristic rules to address this problem have been proposed over the last 40 years in the literature. In this paper, we study a novel heuristic approach to solve it based on the reformulation of the problem as a Restless bandit problem and the derivation of its corresponding Whittle Index (WI) rule. Such rule was recently proposed in the context of a clinical trial in Villar, Bowden, and Wason [16]. We perform extensive computational studies to compare through both exact value calculations and simulated values the performance of this rule, other index rules and simpler heuristics previously proposed in the literature. Our results suggest that for the two and three-armed case and a patient horizon less or equal than a hundred patients, all index rules are a priori practically identical in terms of the expected proportion of success attained when all arms start with a uniform prior. However, we find that a posteriori, for specific values of the parameters of interest, the index policies outperform the simpler rules in every instance and specially so in the case of many arms and a larger, though still relatively small, total number of patients with the diseases. The very good performance of bandit rules in terms of patient benefit (i.e., expected number of successes and mean number of patients allocated to the best arm, if it exists) makes them very appealing in context of the challenge posed by drug development and treatment for rare life-threatening diseases.

In this paper, we establish a fluid limit for a two-sided Markov order book model. The main result states that in a certain asymptotic regime, a pair of measure-valued processes representing the “sell-side shape” and “buy-side shape” of an order book converges to a pair of deterministic measure-valued processes in a certain sense. We also test the fluid approximation on data. The empirical results suggest that the approximation is reasonably good for liquidly traded stocks in certain time periods.

Consider the GI/GI/1 queue with the Last-Come First-Served Preemptive-Resume service discipline. We give intuitive explanations for (1) the geometric nature of the stationary queue length distribution and (2) the mutual independence of the residual service requirements of the customers in the queue, both considered at arbitrary time points. These distributions have previously been established in the literature by either first considering the system at arrival instants or using balance equations. Our direct arguments provide further understanding of properties 1 and 2.

In this paper, we consider multi-state coherent systems that can be regarded as a series/parallel/recurrent connection of multi-state modules with binary/multi-state components. The multi-state (survival) signatures of such systems are presented in terms of multi-state (survival) signatures of related modules based on the structures. For a recurrent structure, the multi-state survival signature of the structure is also needed. The results established here are finally illustrated with a number of examples.

Formulas are derived for moments of the number of refused customers in a busy period for the M/GI/1/n and the GI/M/1/n queueing systems. As an interesting special case for the M/GI/1/n system, we note that the mean number is 1 when the mean interarrival time equals the mean service time. This provides a more direct argument for a result given in Abramov [1].

This paper deals with design of an alternative secure Blockchain network framework to prevent damages from an attacker. The alliance concept from the strategic management perspectives is applied on the top of a general stochastic game framework. This new enhanced hybrid theoretical model is designed to find the best strategies toward preparation for preventing a network malfunction from an attacker through strategic alliances with other genuine nodes and it is developed based on the combination of a strategic management framework and a conventional stochastic model based on the Blockchain Governance Game. Analytically, tractable results for decision-making parameters are fully obtained to predict of the moment for operations and also to provide the optimal number of allegiance nodes to protect a Blockchain network. This research helps those whom are considering initial coin offering or launching new Blockchain-based services by enhancing security features through strategic alliances in a decentralized network.

We consider a generic continuous-time system in which events of random magnitudes occur stochastically and study the system's extreme-value statistics. An event is described by a pair (t,x) of coordinates, where t is the time at which the event took place and x is the magnitude of the event. The stochastic occurrence of the events is assumed to be governed by a Poisson point process.

We study various issues regarding the system's extreme-value statistics, including (i) the distribution of the largest-magnitude event, the distribution of the nth “runner-up” event, and the multidimensional distribution of the “top n” extreme events, (ii) the internal hierarchy of the extreme-value events—how large are their magnitudes when measured relative to each other, and (iii) the occurrence of record times and record values. Furthermore, we unveil a hidden Poissonian structure underlying the system's sequence of order statistics (the largest-magnitude event, the second largest event, etc.). This structure provides us with a markedly simple simulation algorithm for the entire sequence of order statistics.

We investigate fast simulation techniques for estimating the unreliability in large Markovian models of highly reliable systems for which analytical/numerical techniques are difficult to apply. We first show mathematically that for “small” time horizons, the relative simulation error, when using the importance sampling techniques of failure biasing and forcing, remains bounded as component failure rates tend to zero. This is in contrast to naive simulation where the relative error tends to infinity. For “large” time horizons where these techniques are not efficient, we use the approach of first bounding the unreliability in terms of regenerative-cycle-based measures and then estimating the regenerative-cycle-based measures using importance sampling; the latter can be done very efficiently. We first use bounds developed in the literature for the asymptotic distribution of the time to hitting a rare set in regenerative systems. However, these bounds are “close” to the unreliability only for a certain range of time horizons. We develop new bounds that make use of the special structure of the systems that we consider and are “close” to the unreliability for a much wider range of time horizons. These techniques extend to non-Markovian, highly reliable systems as long as the regenerative structure is preserved.

Viterbi training (VT) provides a fast but inconsistent estimator of hidden Markov models (HMM). The inconsistency is alleviated with a little extra computation when we enable VT to asymptotically fix the true values of the parameters. This relies on infinite Viterbi alignments and associated with them limiting probability distributions. First in a sequel, this article is a proof of concept; it focuses on mixture models, an important but special case of HMM where the limiting distributions can be calculated exactly. A simulated Gaussian mixture shows that our central algorithm (VA1) can significantly improve the accuracy of VT with little extra cost. Next in the sequel, we present elsewhere a theory of the adjusted VT for the general HMMs, where the limiting distributions are more challenging to find. Here, we also present another, more advanced correction to VT and verify its fast convergence and high accuracy; its computational feasibility requires additional investigation.

When starting from 0, a standard M/M/k queueing process has a second-order stochastic monotonicity property of a strong kind: its increments are stochastically decreasing (the SDI property). A first attempt to generalize this to the Jackson queueing network fails. This gives us reason to reexamine the underlying theory for stochastic monotonicity of Markov processes starting from a zero-point, in order to find a condition on a function of a Jackson network process to have the SDI property. It turns out that the total number of customers at time t has the desired property, if the network is idle at time O. We use couplings in our analysis; they are also of value in the comparison of two networks with different parameters.

This note extends a recently proposed approximation for the loss probability to the batch-arrival Finite-buffer queues with complete rejection. Also, a condition is stated under which the approximation is exact.

We consider a K-out-of-N system maintained by R repairmen, where the lifetime of the ith component is an exponentially distributed random variable with parameter μi. Repairmen are distinguishable, and the time it takes the rth repairman to repair a failed component is an exponentially distributed random variable with parameter λr. Repaired components are as good as new and preemptions are allowed. We show that the policy that assigns the faster repairmen to the most reliable components is optimal with respect to several optimality criteria.

The approach we take in establishing stochastic optimality with respect to the number of functioning components is of wide applicability to different classes of stochastic optimization problems.

Achieving high throughput in input-queued switches has been found to be difficult, especially when traffic is nonuniform in the sense that different inputs have very different cell generation rates. We show that for general arrival processes, 100% throughput can be achieved with a simple algorithm that is very easy to implement.

We consider a switch in which in each time slot, at most one cell may be transmitted from each input, and at most one cell may be received at each output. Cells that are destined for output j arrive at input i according to a stationary and ergodic process, and arrivals are queued at the input. The problem is to decide which inputs are to transmit to which outputs in each time slot in order to maximize throughput. Necessary conditions for stability are that the total arrival rate to each input must be less than 1, and the total arrival rate destined to each output must be less than 1. We propose a simple scheduling algorithm and show that with this algorithm the necessary conditions for stability are also sufficient.

In this note, we consider a simple immigration birth–death process with total catastrophes and we obtain the transient probabilities. Our approach involves a renewal argument. It is comparatively simpler and leads to more elegant expressions than other approaches that appeared in the literature recently.