A hooking network is built by stringing together components randomly chosen from a set of building blocks (graphs with hooks). The vertices are endowed with “affinities” which dictate the attachment mechanism. We study the distance from the master hook to a node in the network chosen according to its affinity after many steps of growth. Such a distance is commonly called the depth of the chosen node. We present an exact average result and a rather general central limit theorem for the depth. The affinity model covers a wide range of attachment mechanisms, such as uniform attachment and preferential attachment, among others. Naturally, the limiting normal distribution is parametrized by the structure of the building blocks and their probabilities. We also take the point of view of a visitor uninformed about the affinity mechanism by which the network is built. To explore the network, such a visitor chooses the nodes uniformly at random. We show that the distance distribution under such a uniform choice is similar to the one under random choice according to affinities.

In this paper, we study the estimation of a scale parameter from a sample of lifetimes of coherent systems with a fixed structure. We assume that the components are independent and identically distributed having a common distribution which belongs to a scale parameter family. Some results are obtained as well for dependent (exchangeable) components. To this end, we will use the representations for the distribution function of a coherent system based on signatures. We prove that the efficiency of the estimators depends on the structure of the system and on the scale parameter family. In the dependence case, it also depends on the baseline copula function.

In this paper, we discuss a generalization of the classical compound Poisson model with claim sizes following a compound distribution. As applications, we consider models involving zero-truncated geometric, zero-truncated negative-binomial and zero-truncated binomial batch-claim arrivals. We also provide some ruin-related quantities under the resulting risk models. Finally, through numerical examples, we visualize the behavior of these quantities.

Finite-state-and-action Markov branching decision chains are studied with bounded endogenous expected population sizes and interest-rate-dependent one-period rewards that are analytic in the interest rate at zero. The existence of a stationary strong-maximum-present-value policy is established. Miller and Veinott's [1969] strong policy-improvement method is generalized to find in finite time a stationary n-present-value optimal policy and, when the one-period rewards are rational in the interest rate, a stationary strong-maximum-present-value policy. This extends previous studies of Blackwell [1962], Miller and Veinott [1969], Veinott [1974], and Rothblum [1974, 1975], in which the one-period rewards are independent of the interest rate, and Denardo [1971] in which semi-Markov decision chains with small interest rates are studied. The problem of finding a stationary n-present-value optimal policy is also formulated as a staircase linear program in which the objective function and right-hand sides, but not the constraint matrix, depend on the interest rate, and solutions for all small enough positive interest rates are sought. The optimal solutions of the primal and dual are polynomials in the reciprocal of the interest rate. A constructive rule is given for finding a stationary n-present-value optimal policy from an optimal solution of the asymptotic linear program. This generalizes the linear programming approaches for finding maximum-reward-rate and maximum-present-value policies for Markov decision chains studied by Manne [1960], d'Epenoux [1960, 1963], Balinski [1961], Derman [1962], Denardo and Fox [1968], Denardo [1970], Derman and Veinott [1972], Veinott [1973], and Hordijk and Kallenberg [1979, 1984].

In this paper we consider one-machine scheduling problems with or without a perfect machine and random processing times and derive among other results elimination criteria for different classes of cost functions.

We study the problem of scheduling multiclass parallel server queues where the cost for each job is a general class-dependent function of the job's completion time and a second random variable. We characterize the structure of the optimal policy under fairly general conditions on the random variables.

We consider N unreliable machines that are maintained by M repairmen. The time until failure of machine i, and its repair by repairman j, are exponentially distributed random variable with parameters λi, and μj, respectively. All failure times and repair times are independent. Machine i earns ci, per unit time, while it is working. We wish to maximize the expected total discounted reward. It is shown that when the following conditions are satisfied — c1 ≥ … ≥ cN and c1/λ1≥…≥cN/λN, the policy that assigns the fastest repairman to the machine with the lowest index is optimal. Moreover, it is shown that when the second condition is replaced by λ≤ … ≤λN, then this policy maximizes, stochastically, the number of the most reliable machines at every time t.

We discuss the asymptotic behavior of time-inhomogeneous Metropolis chains for solving constrained sampling and optimization problems. In addition to the usual inverse temperature schedule (βn)n∈[hollow N]*, the type of Markov processes under consideration is controlled by a divergent sequence (θn)n∈[hollow N]* of parameters acting as Lagrange multipliers. The associated transition probability matrices (Pβn,θn)n∈[hollow N]* are defined by Pβ,θ = q(x, y)exp(−β(Wθ(y) − Wθ(x))+) for all pairs (x, y) of distinct elements of a finite set Ω, where q is an irreducible and reversible Markov kernel and the energy function Wθ is of the form Wθ = U + θV for some functions U,V : Ω → [hollow R]. Our approach, which is based on a comparison of the distribution of the chain at time n with the invariant measure of Pβn,θn, requires the computation of an upper bound for the second largest eigenvalue in absolute value of Pβn,θn. We extend the geometric bounds derived by Ingrassia and we give new sufficient conditions on the control sequences for the algorithm to simulate a Gibbs distribution with energy U on the constrained set [Ω with tilde above] = {x ∈ Ω : V(x) = minz∈ΩV(z)} and to minimize U over [Ω with tilde above].

This paper is concerned with how coupling can be used to enhance the efficiency of a certain class of terminating simulations, in Markov process settings in which the stationary distribution is known. We are able to theoretically establish that our coupling-based estimator is often more efficient than the naive estimator. In addition, we discuss extensions of our methodology to Markov process settings in which conventional coupling fails and show (for Doeblin chains) that knowledge of the stationary distribution is sometimes unnecessary.

It is proved that the total average waiting time is a convex function of the routing densities for regular routing to parallel queues.

We consider shot-noise and max-shot-noise processes driven by spatial stationary Cox (doubly stochastic Poisson) processes. We derive their upper and lower bounds in terms of the increasing convex order, which is known as the order relation to compare the variability of random variables. Furthermore, under some regularity assumption of the random intensity fields of Cox processes, we show the monotonicity result which implies that more variable shot patterns lead to more variable shot noises. These are direct applications of the results obtained for so-called Ross-type conjectures in queuing theory.

This note introduces bivariate dual generalized Marshall–Olkin distribution and builds a comparison result on the copula of this distribution. Several applications in survival analysis and actuarial science are presented as well.

Diameter measurements on successive metal hubs from a machining operation are modeled using a random walk with observation error and linear drift corresponding to tool wear. After producing and measuring a hub, the depth of the cutting tool on the lathe can be adjusted in integer multiples of 0.0001 inches. How should the tool be adjusted?

An optimal discrete adjustment strategy is derived assuming that the lathe automatically corrects for deterministic tool wear. The objective is to minimize expected run costs proportional to the sum of squared diameter deviations from a target plus fixed charges for manual tool adjustments. The optimal strategy makes no manual adjustment if an estimate of the process mean is near target. Otherwise, an adjustment is made to return the estimated mean as near to target as possible within the adjustment resolution.

The region where no adjustments are made widens near the end of the production run where adjustments have only short-term impact. The region converges as the number of remaining periods increases. Plots of expected run costs show that the extra cost of discreteness is small at high resolution but is substantial when the adjustment grid is coarse.

In this paper, we consider a generalized Pólya urn model with multiple drawings and time-dependent reinforcements. Suppose an urn initially contains w white and r red balls. At the nth action, m balls are drawn at random from the urn, say k white and m−k red balls, and then replaced in the urn along with cnk white and cn(m − k) red balls, where {cn} is a given sequence of positive integers. Repeat the above procedure ad infinitum. Let Xn be the proportion of the white balls in the urn after the nth action. We first show that Xn converges almost surely to a random variable X. Next, we give a necessary and sufficient condition for X to have a Bernoulli distribution with parameter w/(w + r). Finally, we prove that X is absolutely continuous if {cn} is bounded.

In this paper, we study the credit default swap (CDS) pricing with counterparty risk in a reduced form model. The default jump intensities of the reference firm and counterparty are both assumed to follow the mean-reverting CIR processes with independent jumps respectively and a common jump. The approximate closed-form solutions of the joint survival probability density and the probability density of the first default can be obtained by using the PDE method. Then with the expressions of the probability densities, we can get the formula for the CDS price with counterparty risk in a reduced form model with a common jump. In the numerical analysis part, we find that the default of the reference asset has a greater impact on the CDS price than that of the default of counterparty after introducing the common jump process.

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results.

In this paper, we consider some dividend problems in the perturbed compound Poisson model under a constant barrier dividend strategy. We approximate the expected present value of dividend payments before ruin and the expected discounted penalty function based on the COS method, and construct some nonparametric estimators by using a random sample on claim number and individual claim sizes. Under a large sample size setting, we perform an error analysis of the estimators. We also provide some simulation results to verify the effectiveness of this estimation method when the sample size is finite.

In this paper, we investigate the pricing of vulnerable European options in a market where the underlying stocks are not perfectly liquid. A liquidity discount factor is used to model the effect of liquidity risk in the market, and the default risk of the option issuer is incorporated into the model using a reduced-form model, where the default intensity process is correlated with the liquidity risk. We obtain a semiclosed-form pricing formula of vulnerable options through the inverse Fourier transform. Finally, we illustrate the effects of default risk and liquidity risk on option prices numerically.

The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed.

For the GI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.