We discuss a rich family of directed series–parallel (SP) graphs grown by the simultaneous random series or parallel development of multiple edges. The family portrays a spectrum that spans a wide range of SP graphs: from simple models, where only as few as one edge is chosen for evolution at each discrete point in time, to complex hierarchical lattice networks grown by a take-all strategy, where all the edges in the existing network are developed.

The family of SP graphs we discuss is grown from an initial seed graph with τ0 edges under an arbitrary building sequence, $\{k_{n}\}_{n=1}^{\infty}$, of nonnegative integers (with $k_n \le \tau _0 + \sum\nolimits_{i = 1}^n {k_i} $, for arbitrary τ0 ≥ 1), that specifies the number of edges subjected to evolution at time n. We study the average north polar degree and show that we can go beyond averages to strong laws. We also find the exact average number of critical edges. The asymptotics of the critical edges are facilitated under the regularity condition that $k_n/\sum\nolimits_{i = 1}^n {k_i} $ converges to a constant (as n → ∞), a natural condition easily met by practical strategies, such as single-edge evolution and take-all choice, and much in between.

In this paper, we are going to study the strong limit theorem for the relative entropy density rates between two finite asymptotically circular Markov chains. Firstly, we prove some lammas on which the main result based. Then, we establish two strong limit theorem for non-homogeneous Markov chains. Finally, we obtain the main result of this paper. As corollaries, we get the strong limit theorem for the relative entropy density rates between two finite non-homogeneous Markov chains. We also prove that the relative entropy density rates between two finite non-homogeneous Markov chains are uniformly integrable under some conditions.

We consider a variation of the processor-sharing (PS) queue, inspired by freelance job websites where multiple freelancers compete for a single job. We develop fluid limit approximations for the overloaded PS-model with multiple (possibly infinitely many) service stages. Based on this approximation, we estimate what proportion of freelancers get the job they apply for. In addition, the PS model studied here is an instance of PS with routing and impatience, for which no Lyapunov function is known, and we suggest some partial solutions.

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ∞): = lim n→∞p(k, n) is decreasing in k.

This paper is devoted to the investigation of the G-network with multiple classes of positive and negative customers. The purpose of the investigation is to analyze such a network at a transient regime, to find the state probabilities of the network that depend on time. In the first part, a description of the functioning of G-networks with positive and negative customers is provided, when a negative customer when arriving to the system destroys a positive customer of its class. Streams of positive and negative customers arriving at each of the network systems are independent. Services of positive customers of all types occur in accordance with a random selection of them for service. For nonstationary probabilities of network states, a system of Kolmogorov's difference-differential equations (DDE) has been derived. A method for their finding is proposed. It is based on the use of a modified method of successive approximations, combined with the method of series. It is proved that successive approximations converge with time to a stationary probability distribution, the form of which is indicated in this paper, and the sequence of approximations converges to the unique solution of the DDE system. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations. A model example illustrating the determination of the time-dependent probabilities of network states using this technique has been calculated. The obtained results can be applied in modeling the behavior of computer viruses and attacks in information and telecommunication systems and networks, for example, as a model of the impact of some file viruses on server resources. variable.

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

Suppose there are n players, with player i having value vi > 0, and suppose that a game between i and j is won by i with probability vi/(vi + vj). In the winner plays random knockout tournament, we suppose that the players are lined up in a random order; the first two play, and in each subsequent game the winner of the last game plays the next in line. Whoever wins the game involving the last player in line, is the tournament winner. We give bounds on players’ tournament win probabilities and make some conjectures. We also discuss how simulation can be efficiently employed to estimate the win probabilities.

A cloud provider hosts virtual machines (VMs) of different types, with different resource requirements. There are bounds on the total amounts of each kind of resource that are available. Requests arrive in batches of different sizes. Under the ‘complete blocking’ policy, a request is accepted only if all the VMs in its batch can be accommodated. The ‘partial blocking’ policy would accept a request if there is room for at least one of the VMs in the batch. Blocked requests are lost, with an associated loss of revenue. The trade-offs between costs and benefits are evaluated by means of appropriate models, for which novel solutions based on fixed-point iterations are proposed. The applicability of those solutions is extended, by means of simplifications, to very large-scale systems. Numerical examples and comparisons with simulations are presented.

In this paper, we investigate the degree profile and Gini index of random caterpillar trees (RCTs). We consider RCTs which evolve in two different manners: uniform and nonuniform. The degrees of the vertices on the central path (i.e., the degree profile) of a uniform RCT follows a multinomial distribution. For nonuniform RCTs, we focus on those growing in the fashion of preferential attachment. We develop methods based on stochastic recurrences to compute the exact expectations and the dispersion matrix of the degree variables. A generalized Pólya urn model is exploited to determine the exact joint distribution of these degree variables. We apply the methods from combinatorics to prove that the asymptotic distribution is Dirichlet. In addition, we propose a new type of Gini index to quantitatively distinguish the evolutionary characteristics of the two classes of RCTs. We present the results via several numerical experiments.

We use the random-product technique from [5] to study both the steady-state and time-dependent behavior of a Markovian reentrant-line model, which is a generalization of the preemptive reentrant-line model studied in the work of Adan and Weiss [2]. Our results/observations yield additional insight into why the stationary distribution of the reentrant-line model from [2] exhibits an almost-geometric product-form structure: indeed, our generalized reentrant-line model, when stable, admits a stationary distribution with a similar product-form representation as well. Not only that, the Laplace transforms of the transition functions of our reentrant-line model also have a product-form structure if it is further assumed that both Buffers 2 and 3 are empty at time zero.

The relation between extropy and variational distance is studied in this paper. We determine the distribution which attains the minimum or maximum extropy among these distributions within a given variation distance from any given probability distribution, obtain the tightest upper bound on the difference of extropies of any two probability distributions subject to the variational distance constraint, and establish an analytic formula for the confidence interval of an extropy. Such a study parallels to that of Ho and Yeung [3] concerning entropy. However, the proofs of the main results in this paper are different from those in Ho and Yeung [3]. In fact, our arguments can simplify several proofs in Ho and Yeung [3].

In dynamic networks, the quickest time is the least possible time required to transmit specified data from the source to the sink. When arcs in dynamic networks are independently subjected to failures, the quickest time is a random variable. Although previous studies have already explored the reliability of the quickest path, this work presents an algorithm that computes the probability distribution of the quickest time from the viewpoint of the quickest flow that contains all possible joint and disjoint paths. For moderate dynamic networks, the proposed algorithm can be easily modified to approximate the quickest time distribution with a trade-off between accuracy and running time. The performance and properties of the exact and modified algorithms are explored through computational experiments, which are conducted on 10 randomly generated networks. The exact algorithm is also compared with the exhaustive method which examines all state vectors.

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

The general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment.

We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of $O(\sqrt {(2^k/n} ))$ where 2k is the complexity of the system and n is the length of the second sample.

Stochastic ordering of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order of relations within each of these families.

The aim of the paper is to derive the distribution of the number of retrial of the tagged request and as a consequence to present the waiting time analysis of a finite-source M/M/1 retrial queueing system by using the method of asymptotic analysis under the condition of the unlimited growing number of sources. As a result of the investigation, it is shown that the asymptotic distribution of the number of retrials of the tagged customer in the orbit is geometric with given parameter, and the waiting time of the tagged customer has a generalized exponential distribution. For the considered retrial queuing system numerical and simulation software packages are also developed. With the help of several sample examples the accuracy and range of applicability of the asymptotic results in prelimit situation are illustrated showing the effectiveness of the proposed approximation.

The object of research is G-network with positive customers and signals of multiple classes. The present paper describes an analysis of this network at a non-stationary regime, also provided a description of method for finding non-stationary state probabilities.

At the beginning of the article, a description of the network with positive customers and signals is given. A signal when entering the system destroys a positive customer of its type or moves the customer of its type to another system. Streams of positive customers and signals arriving to each of the network systems are independent. Selection of positive customers of all classes for service – randomly. For non-stationary state probabilities of the network, the system of Kolmogorov difference-differential equations (DDE) has been derived. It is solved by a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time has been proved to the stationary distribution of probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the unique solution of the DDE system. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations.

The obtained results can be applied for modeling behavior of computer viruses and attack in computer systems and networks, for example, as model impact of some file viruses on server resources.

In this paper, we consider optimal components grouping in series–parallel and parallel–series systems composed of k subsystems. All components in each subsystem are drawn from a heterogeneous population consisting of m different subpopulations. Firstly, we show that when one allocation vector is majorized by another one, then the series–parallel (parallel–series) system corresponding to the first (second) vector is more reliable than that of the other. Secondly, we study the impact of changes in the number of subsystems on the system reliability. Finally, we study the influence of the selection probabilities of subpopulations on the system reliability.

In this paper, we extend the framework of Klein [15] [Journal of Banking & Finance 20: 1211–1229] to a general model under the double exponential jump model with stochastic volatility on the underlying asset and the assets of the counterparty. Firstly, we derive the closed-form characteristic functions for this dynamic. Using the Fourier-cosine expansion technique, we get numerical solutions for vulnerable European put options based on the characteristic functions. The inverse fast Fourier transform method provides a fast numerical algorithm for the twice-exercisable vulnerable Bermuda put options. By virtue of the modified Geske and Johnson method, we obtain an approximate pricing formula of vulnerable American put options. Numerical simulations are made for investigating the impact of stochastic volatility on vulnerable options.

This paper is devoted to the study of a clearing queueing system with a special discipline. As soon as the server receives N negative feedbacks from customers, all present customers are forced to leave the system and the server undergoes a maintenance procedure. After an exponential maintenance time, the system resumes its service immediately. Using the matrix analytic method, we derive the steady-state distributions, which are then used for the computation of other performance measures. Furthermore, using first step analysis, we obtain the Laplace–Stieltjes transform of the sojourn time of an arbitrary customer. We also study the busy period of the system and derive the generating function of the total number of lost customers in a busy period. Finally, we investigate a long-run rate of cost and explore the optimal N value that minimizes the total cost per unit time. We also present some numerical examples to illustrate the impact of several model parameters to the performance measures.