We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter $\alpha$ where $0\lt\alpha \leq 1$. Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time $t$ excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time $t$ pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time $t$ pdfs as $t\rightarrow \infty$ do not exist. To reasonably observe the large $t$ pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large $t$ pdfs is given by an $L_{1}$ metric taking values in $(0,1)$, where “near $0$” means “close” and “near $1$” means “far”.

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$, the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$, the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$, in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$.

This paper concentrates on the fundamental concepts of entropy, information and divergence to the case where the distribution function and the respective survival function play the central role in their definition. The main aim is to provide an overview of these three categories of measures of information and their cumulative and survival counterparts. It also aims to introduce and discuss Csiszár's type cumulative and survival divergences and the analogous Fisher's type information on the basis of cumulative and survival functions.

Innovations in medicine provide us longer and healthier life, leading lower mortality. Sooner rather than later, much greater longevity would be possible for us due to artificial intelligence advances in health care. Similarly, Advanced Driver Assistance Systems (ADAS) in highly automated vehicles may reduce or even eventually eliminate accidents by perceiving dangerous situations, which would minimize the number of accidents and lead to fewer loss claims for insurance companies. To model the survivor function capturing greater longevity as well as the number of claims reflecting less accidents in the long run, in this paper, we study a Cox process whose intensity process is piecewise-constant and decreasing. We derive its ultimate distributional properties, such as the Laplace transform of intensity integral process, the probability generating function of point process, their associated moments and cumulants, and the probability of no more claims for a given time point. In general, this simple model may be applicable in many other areas for modeling the evolution of gradually disappearing events, such as corporate defaults, dividend payments, trade arrivals, employment of a certain job type (e.g., typists) in the labor market, and release of particles. In particular, we discuss some potential applications to insurance.

This paper is concerned with developing low variance simulation estimators of probabilities related to the sum of Bernoulli random variables. It shows how to utilize an identity used in the Chen-Stein approach to bounding Poisson approximations to obtain low variance estimators. Applications and numerical examples in such areas as pattern occurrences, generalized coupon collecting, system reliability, and multivariate normals are presented. We also consider the problem of estimating the probability that a positive linear combination of Bernoulli random variables is greater than some specified value, and present a simulation estimator that is always less than the Markov inequality bound on that probability.

In this note, we consider dynamic assortment optimization with incomplete information under the capacitated multinomial logit choice model. Recently, it has been shown that the regret (the cumulative expected revenue loss caused by offering suboptimal assortments) that any decision policy endures is bounded from below by a constant times $\sqrt {NT}$, where $N$ denotes the number of products and $T$ denotes the time horizon. This result is shown under the assumption that the product revenues are constant, and thus leaves the question open whether a lower regret rate can be achieved for nonconstant revenue parameters. In this note, we show that this is not the case: we show that, for any vector of product revenues there is a positive constant such that the regret of any policy is bounded from below by this constant times $\sqrt {N T}$. Our result implies that policies that achieve ${{\mathcal {O}}}(\sqrt {NT})$ regret are asymptotically optimal for all product revenue parameters.

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.

This paper considers optimal admission and routing control in multi-class service systems in which customers can either receive quality regular service which is subject to congestion or can receive congestion-free but less desirable service at an alternative service station, which we call the self-service station. We formulate the problem within the Markov decision process framework and focus on characterizing the structure of dynamic optimal policies which maximize the expected long-run rewards. For this, value function and sample path arguments are used. The congestion sensitivity of customers is modeled with class-independent holding costs at the regular service station. The results show how the admission rewards of customer classes affect their priorities at the regular and self-service stations. We explore that the priority for regular service may not only depend on regular service admission rewards of classes but also on the difference between regular and self-service admission rewards. We show that optimal policies have monotonicity properties, regarding the optimal decisions of individual customer classes such that they divide the state space into three connected regions per class.

We suggest four new measures of importance for repairable multistate systems based on the classical Birnbaum measure. Periodic component life cycles and general semi-Markov processes are considered. Similar to the Birnbaum measure, the proposed measures are generic in the sense that they only depend on the probabilistic properties of the components and the system structure. The multistate system model encodes physical properties of the components and the system directly into the structure function. As a result, calculating importance is easy, especially in the asymptotic case. Moreover, the proposed measures are composite measures, combining importance for all component states into a unified quantity. This simplifies ranking of the components with respect to importance. The proposed measures can be characterized with respect to two features: forward-looking versus backward-looking and with respect to how criticality is measured. Forward-looking importance measures focus on the next component states, while backward-looking importance measures focus on the previous component states. Two approaches to measuring criticality are considered: probability of criticality versus expected impact. Examples show that the different importance measures may result in unequal rankings.

Reliability theory and survival analysis, the residual entropy is known as a measure suitable to describe the dynamic information content in stochastic systems conditional on survival. Aiming to analyze the variability of such information content, in this paper we introduce the variance of the residual lifetimes, “residual varentropy” in short. After a theoretical investigation of some properties of the residual varentropy, we illustrate certain applications related to the proportional hazards model and the first-passage times of an Ornstein–Uhlenbeck jump-diffusion process.

In this paper, we consider a d-node GI|GI|1 tandem queue with i.i.d. inter-arrival process and service processes that are independent of each other. Our main interest is to estimate the probability to reach a high level N in a busy cycle of the system using simulation. As crude simulation does not give a sufficient precision in reasonable time, we use importance sampling. We introduce a method to find a state-independent change of measure and we show that this is equivalent to a change of measure that was earlier, but implicitly, described by Parekh and Walrand [8]. We also show that this change of measure is the only exponential state-independent change of measure that may result in an asymptotically efficient estimator. Lastly, we provide necessary conditions for this state-independent change of measure to give an asymptotically efficient estimator.

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.

We focus on Erol Gelenbe's scientific and technical contributions to probability models in the computer and information sciences, but limit our survey to the last fifteen years. We start with a brief overview of his work as a single author, as well as his work in collaboration with over 200 co-authors. We discuss some of his recent and innovative work regarding a new probability model that represents Intermittent Energy Sources for Computing and Communications, introducing Energy Packet Networks which are a probabilistic representation of the flow, storage and consumption of electrical energy at the microscopic level (in electronic chips), and at the macroscopic level (e.g. in buildings or data centers) and for its routing and dynamic usage by consuming units (such as computer elements, chips or machines). We next discuss his work on designing computer and communication systems that parsimoniously use energy in order to achieve a satisfactory level of quality of service (QoS). Trade-offs between system QoS and energy consumption are also considered. Then we turn to Prof. Gelenbe's pioneering work on Autonomic Communications and the design and implementation of CPN, the Cognitive Packet Network, and we also briefly review his spiking random neural network that was used in CPN. This is followed by a brief review of work that he conducted since 1999 on human evacuation from dangerous or catastrophic environments, and the design of technology driven Emergency Management Systems. His research since the late 2000s on Gene Regulatory Networks is then covered together with its application to the detecting possible disease from microarray data. Finally, we briefly discuss some novel analytical models that he developed in this period with publications appearing in journals of physics and applied mathematics.

Autonomic computing is emerging as a significant new approach to the design of computer services. Its goal is the development of services that are able to manage themselves with minimal direct human intervention, and, in particular, are able to sense their environment and to tune themselves to meet end-user needs. However, the impact on performance of the interaction between multiple uncoordinated self-optimizing services is not yet well understood. We present some recent results on a non-cooperative load-balancing game which help to better understand the result of this interaction. In this game, users generate jobs of different services, and the jobs have to be processed on one of the servers of a computing platform. Each service has its own dispatcher which probabilistically routes jobs to servers so as to minimize the mean processing cost of its own jobs. We first investigate the impact of heterogeneity in the amount of incoming traffic routed by dispatchers and present a result stating that, for a fixed amount of total incoming traffic, the worst-case overall performance occurs when each dispatcher routes the same amount of traffic. Using this result we then study the so-called Price of Anarchy (PoA), an oft-used worst-case measure of the inefficiency of non-cooperative decentralized architectures. We give explicit bounds on the PoA for cost functions representing the mean delay of jobs when the service discipline is PS or SRPT. These bounds indicate that significant performance degradations can result from the selfish behavior of self-optimizing services. In practice, though, the worst-case scenario may occur rarely, if at all. Some recent results suggest that for the game under consideration the PoA is an overly pessimistic measure that does not reflect the performance obtained in most instances of the problem.

We study G-networks with positive and negative customers and signals. We consider two types of signals: they can make a subnetwork of queues operational or down. As signals are sent by queues after a customer service completion, one can model the availability of a sub-network of queues controlled by another network of queues. We prove that under classical assumptions for G-networks and assumptions on the rerouting probabilities when a subnetwork is not operational, the steady-state distribution, if it exists, has a product form steady state distribution. Some examples are given.

Mobile networks are universally used for personal communications, but also increasingly used in the Internet of Things and machine-to-machine applications in order to access and control critical services. However, they are particularly vulnerable to signaling storms, triggered by malfunctioning applications, malware or malicious behavior, which can cause disruption in the access to the infrastructure. Such storms differ from conventional denial of service attacks, since they overload the control plane rather than the data plane, rendering traditional detection techniques ineffective. Thus, in this paper we describe the manner in which storms happen and their causes, and propose a detection framework that utilizes traffic measurements and key performance indicators to identify in real-time misbehaving mobile devices. The detection algorithm is based on the random neural network which is a probabilistic computational model with efficient learning algorithms. Simulation results are provided to illustrate the effectiveness of the proposed scheme.

Motivated by applications in areas such as cloud computing and information technology services, we consider GI/GI/1 queueing systems under workloads (arrival and service processes) that vary according to one discrete time scale and under controls (server capacity) that vary according to another discrete time scale. We take a stochastic optimal control approach and formulate the corresponding optimal dynamic control problem as a stochastic dynamic program. Under general assumptions for the queueing system, we derive structural properties for the optimal dynamic control policy, establishing that the optimal policy can be obtained through a sequence of convex programs. We also derive fluid and diffusion approximations for the problem and propose analytical and computational approaches in these settings. Computational experiments demonstrate the benefits of our theoretical results over standard heuristics.

The purpose of this paper is to analyze the so-called back-off technique of the IEEE 802.11 protocol in broadcast mode with waiting queues. In contrast to existing models, packets arriving when a station (or node) is in back-off state are not discarded, but are stored in a buffer of infinite capacity. As in previous studies, the key point of our analysis hinges on the assumption that the time on the channel is viewed as a random succession of transmission slots (whose duration corresponds to the length of a packet) and mini-slots during which the back-off of the station is decremented. These events occur independently, with given probabilities. The state of a node is represented by a two-dimensional Markov chain in discrete-time, formed by the back-off counter and the number of packets at the station. Two models are proposed both of which are shown to cope reasonably well with the physical principles of the protocol. The stability (ergodicity) conditions are obtained and interpreted in terms of maximum throughput. Several approximations related to these models are also discussed.

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.

These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

Previous studies on emergency management of large-scale urban networks have commonly concentrated on system development to off-load intensive computations to remote cloud servers or improving communication quality during a disaster and ignored the effect of energy consumption of vehicles, which can play a vital role in large-scale evacuation owing to the disruptions in energy supply. Hence, in this paper we propose a cloud-enabled navigation system to direct vehicles to safe areas in the aftermath of a disaster in an energy and time efficient fashion. A G-network model is employed to mimic the behaviors and interactions between individual vehicles and the navigation system, and analyze the effect of re-routing decisions toward the vehicles. A gradient descent optimization algorithm is used to gradually reduce the evacuation time and fuel consumption of vehicles by optimizing the probabilistic choices of linked road segments at each intersection. The re-routing decisions arrive at the intersections periodically and will expire after a short period. When a vehicle reaches an intersection, if the latest re-routing decision has not expired, the vehicle will follow this advice, otherwise, the vehicle will stick to the shortest path to its destination. The experimental results indicate that the proposed algorithm can reduce the evacuation time and the overall fuel utilization especially when the number of evacuated vehicles is large.