Server farms, consisting of a collection of hosts and a front-end router that dispatches incoming jobs to hosts, are now commonplace. It is well known that when job service requirements (job sizes) are highly variable, then the Size-Interval task assignment policy is an excellent rule for assigning jobs to hosts, since it provides isolation for short jobs by directing short jobs to one host's queue and long jobs to another host's queue. What is not understood is how to classify a “short” job versus a “long” job. For a long time it was believed that the size cutoff separating “short” jobs from “long” ones should be chosen to balance the load at the hosts in the server farm. However, recent literature has provided empirical evidence that load balancing is not always optimal for minimizing mean response time. This article provides the first analytical criteria for when it is preferable to unbalance load between two hosts using Size-Interval task assignment and in which direction the load should be unbalanced. Some very simple sufficient criteria are provided under which we prove that the short job host should be underloaded, and likewise for the long job host. These criteria are then used to prove that the direction of load imbalance depends on moment index properties related to the job size distribution. For example, under the Bounded Pareto (BP) job size distribution with parameter α and a sufficiently high upper bound (the BP is well known to be a good model of empirical computer system workloads), we show that α determines the direction of load imbalance. For α<1, the short job host should be underloaded; for α=1, load should be balanced; and for α>1, the long job host should be underloaded. Many other job size distributions are considered as well. We end by showing that load unbalancing can have a dramatic impact on performance, reducing mean response time by an order of magnitude compared to load balancing in many common cases.

The Shannon entropy based on the probability density function is a key information measure with applications in different areas. Some alternative information measures have been proposed in the literature. Two relevant ones are the cumulative residual entropy (based on the survival function) and the cumulative past entropy (based on the distribution function). Recently, some extensions of these measures have been proposed. Here, we obtain some properties for the generalized cumulative past entropy. In particular, we prove that it determines the underlying distribution. We also study this measure in coherent systems and a closely related generalized past cumulative Kerridge inaccuracy measure.

In this paper, we present a pricing model for vulnerable options in discrete time. A Generalized Autoregressive Conditional Heteroscedasticity process is used to describe the variance of the underlying asset, which is correlated with the returns of the asset. As for counterparty default risk, we study it in a reduced form model and the proposed model allows for the correlation between the intensity of default and the variance of the underlying asset. In this framework, we derive a closed-form solution for vulnerable options and investigate quantitative impacts of counterparty default risk on option prices.

We consider a multiserver delay queue with finite additional waiting spaces and time-varying arrival rates, where the customers waiting in the buffer may abandon. These are features that arise naturally from the study of service systems such as call centers. Moreover, we assume rewards for successful service completions and cost rates for service resources. Finally, we consider service-level agreements that constrain both the fractions of callers who abandon and the ones who are blocked.

Applying the theory of Lagrangian mechanics to the fluid limit of a related Markovian service network model, we obtain near-profit-optimal staffing and provisioning schedules. The nature of this solution consists of three modes of operation. A key step in deriving this solution is combining the modified offered load approximation for loss systems with our fluid model. We use them to estimate effectively both our service-level agreement metrics and the profit for the original queuing model. Second-order profit improvements are achieved through a modified offered load version of the conventional square root safety rule.

This paper examines the availability of an inspected system where the rate of deterioration is governed by an exogenous random environment. We develop a qualitative result that exposes the relationship among remaining life, deterioration, and repairs. This result requires no specific distributional assumptions. We use this result in developing a simple bound on stationary availability. We investigate these relationships numerically for a Markovian system and provide insights useful to practitioners in selecting an inspection process that guarantees a target availability.

Let F be the common distribution function of the increments of a random walk {Sn, n ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.

The purpose of this work is to construct a bridge between two classical topics in applied probability: the finite-time ruin probability in insurance and the final outcome distribution in epidemics. The two risk problems are reformulated in terms of the joint right-tail and left-tail distributions of order statistics for a sample of uniforms. This allows us to show that the hidden algebraic structures are of polynomial type, namely Appell in insurance and Abel–Gontcharoff in epidemics. These polynomials are defined with random parameters, which makes their mathematical study interesting in itself.

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

We propose a geometric mixed fractional Brownian motion model for the stock price process with possible jumps superimposed by an independent Poisson process. Option price of the European call option is computed for such a model. Some special cases are studied in detail.

In this paper we analyze an asymmetric 2 × 2 buffered switch, fed by two independent Bernoulli input streams. We derive the joint equilibrium distribution of the numbers of messages waiting in the two output buffers. This joint distribution is presented explicitly, without the use of generating functions, in the form of a sum of two alternating series of product-form geometric distributions. The method used is the so-called compensation approach, developed by Adan, Wessels, and Zijm.

Recently, some authors have suggested usage models of Markov type as a technique of specifying the estimated operational use distribution of a given program. A main purpose of such models is the derivation of random test cases allowing unbiased estimates on the (un)reliability of the program in its intended environment. In this article, we show that by a shift of the transition probabilities of the Markov chain corresponding to such a model, prior information on the errorjproneness of single-program operations can be taken into account. An unbiased unreliability estimator with reduced variance is obtained. Furthermore, it is shown that minimization of the variance leads to a special stochastic optimization problem that can be demonstrated to be convex, such that efficient solution techniques apply. Some related questions are also treated in a more general, non-Markovian framework.

The controlled Mx/G/1-type queueing model with two modes of operation is considered. The modes are characterized by different service time distributions and input rates. The switchover times are imposed in the model. The embedded stationary queue-length distribution and the explicit dependence of operation criteria on switchover levels are derived.

This paper discusses an optimal dynamic policy for a queueing system with M servers, no waiting room, and two types of customers. Customer types differ with respect to the reward that is paid on commencement of service, but service times are exponentially distributed with the same mean for both types of customers. The arrival stream of one customer type is generated by a Poisson process, and the other customer type arrives according to the overflow process of an M/M/m/m queue. The objective is to determine a policy for admitting customers to maximize the expected long-run average reward.

By posing the problem in the framework of Markov decision processes and exploiting properties of submodular functions, the optimal policy is shown to be a “generalized trunk reservation policy”; in other words, the optimal policy accepts higher-paying customers whenever possible and accepts lower-paying customers only if fewer than c1 servers are busy, where i is the number of busy servers in the overflow queue. Computational issues are also discussed. More specifically, approximations of the overflow process by an interrupted Poisson process and a Poisson process are investigated.

We present a new estimator for the standard deviation parameter in the standard quality control model. We argue that it has a smaller mean square error than the usual estimator when the data are normal and that, unlike the usual estimator, it remains consistent even when the data are not normal.

We consider the problem of routing calls dynamically in a multiskill call center. Calls from different skill classes are offered to the call center according to a Poisson process. The agents in the center are grouped according to their heterogeneous skill sets that determine the classes of calls they can serve. Each agent group serves calls with independent exponentially distributed service times. We consider two scenarios. The first scenario deals with a call center with no buffers in the system, so that every arriving call either has to be routed immediately or has to be blocked and is lost. The objective in the system is to minimize the average number of blocked calls. The second scenario deals with call centers consisting of only agents that have one skill and fully cross-trained agents, where calls are pooled in common queues. The objective in this system is to minimize the average number of calls in the system. We obtain nearly optimal dynamic routing policies that are scalable with the problem instance and can be computed online. The algorithm is based on one-step policy improvement using the relative value functions of simpler queuing systems. Numerical experiments demonstrate the good performance of the routing policies. Finally, we discuss how the algorithm can be used to handle more general cases with the techniques described in this article.

Falls are a major health concern in older adults. Falls lead to mortality, immobility and high costs to social and health care services. Early detection and classification of falls is imperative for timely and appropriate medical aid response. Traditional machine learning models have been explored for fall classification. While newly developed deep learning techniques have the ability to potentially extract high-level features from raw sensor data providing high accuracy and robustness to variations in sensor position, orientation and diversity of work environments that may skew traditional classification models. However, frequently used deep learning models like Convolutional Neural Networks (CNN) are computationally intensive. To the best of our knowledge, we present the first instance of a Hybrid Multichannel Random Neural Network (HMCRNN) architecture for fall detection and classification. The proposed architecture provides the highest accuracy of 92.23% with dropout regularization, compared to other deep learning implementations. The performance of the proposed technique is approximately comparable to a CNN yet requires only half the computation cost of the CNN-based implementation. Furthermore, the proposed HMCRNN architecture provides 34.12% improvement in accuracy on average than a Multilayer Perceptron.

The purpose of this paper is to study definitions and characterizations of orders based on reliability measures related with the doubly truncated random variable X[x, y] = (X|x ≤ X ≤ y). The relationship between these orderings and various existing orderings of life distributions are discussed. Moreover, we give two new characterizations of the likelihood ratio order based on double truncation. These new orders complete a general diagram between orders defined from truncation.

Components which have bathtub shapes are reasonable candidates for burn-in in the sense that if these components are exposed to standard or elevated operating conditions for a short period of time, they tend to improve. In this paper we study the principle which states that burn-in should occur at or before the point at which a bathtub-shaped failure function starts increasing. To this end we develop three sign change results which characterize bathtub-shaped failure rates. These results are used to show that if a failure rate function is bathtub-shaped, then various other related functions have bathtub or upside-down bathtub shape. Finally, we give a framework for determining when the above principle holds.

We consider the virtual queuing time (vqt, also known as work-in-system, or virtual-delay) process in an M/G/s queue with impatient customers. We focus on the vqt-based balking model and relate it to reneging behavior of impatient customers in terms of the steady-state distribution of the vqt process. We construct a single-server system, analyze its operating characteristics, and use them to approximate the multiserver system. We give both analytical results and numerical examples. We conduct simulation to assess the accuracy of the approximation.

This paper presents an exact derivation of the statistical distribution of the signal to interference-plus-noise ratio (SINR) for optimum linear combining in wireless systems with multiple cochannel interferes, Rayleigh fading, and additive white Gaussian noise. The distribution of the SINR is shown to be remarkably simple and leads to bounds on the bit error rate and outage probabilities which are tighter, simpler, and more robust than any previous results. The simplicity of the SINR distribution permits extremely fast computation of outage probabilities for any number of interference channels and diversity levels. Hence for wireless systems it enables performance studies to be performed over a much wider range of conditions, such as shadow fading, specific channel allocation methods, etc. Previously such studies were extremely limited due to the intensive computational requirements of simulating these systems.