In this paper, we develop a novel game theoretic model of the interactions between an EDoS attacker and the defender based on a signaling game that is a dynamic game of incomplete information. We then derive the best defense strategies for the network defender to respond to the EDoS attacks. That is, we compute the perfect Bayesian Nash Equilibrium (PBE) of the proposed game model such as the pooling PBE, separating PBE and mixed strategy PBE. In the pooling equilibrium, each type of the attacker takes the same action and the attacker's type is not revealed to the defender, whereas in the separating equilibrium, each type of the attacker uses different actions and hence the attacker's type is completely revealed to the defender. On the other hand, in the mixed strategy PBE, both the attacker and the defender randomize their strategies to optimize their payoffs. Numerical illustration is also presented to show the efficacy of the proposed model.

One of the interesting problems on the stochastic behavior of random recurrent events in a random time interval is to obtain the conditions under which the reliability properties of a random time T are inherited by N(T), where {N(t):t≥0} is a stochastic process. Most of the studies on the topic has been done under the assumption that the random time T and the stochastic process {N(t):t≥0} are stochastically independent. However, in practice, there can be different cases when appropriate dependence structure is more appropriate. In this paper, we study the preservation of a renewal process stopped at a random time when they are “stochastically dependent.” We discuss the stochastic ordering properties and the preservation of reliability classes for the random counting variables N(T) when the corresponding counting process is a renewal process. Furthermore, we study the preservation of NBUE (NWUE) reliability class when the counting process is a homogeneous Poisson process.

In this article, we derive closed-form expressions of the large deviations local rate function in terms of the arrival and service parameters for the zero path in exponential queueing networks corresponding to a coupled-processors system.

We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman's infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank. However, the majority of the ranks have a weighted path length with average behavior similar to that of the weighted path to the maximal node.

A retrial queue with classical retrial policy, where each blocked customer in the orbit retries for service, and general retrial times is modeled by a piecewise deterministic Markov process (PDMP). From the extended generator of the PDMP of the retrial queue, we derive the associated martingales. These results are used to derive the conditional expected number of customers in the orbit in the transient regime.

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.

Relying only on the classical Bahadur–Rao approximation for large deviations (LDs) of univariate sample means, we derive strong LD approximations for probabilities involving two sets of sample means. The main result concerns the exact asymptotics (as n→∞) of

$$ {\open P}\left({\max_{i\in\{1,\ldots,d_x\}}\bar X_{i,n} \les \min_{i\in\{1,\ldots,d_y\}}\bar Y_{i,n}}\right),$$

${\bar X}_{i,n}{\rm s}$

${\bar Y}_{i,n}{\rm s}$

${\open E}X \gt {\open E}Y$

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.

In this article, we investigate single-server finite-capacity queues where the partial acceptance rule is applied. In particular, we focus on the monotonicity of the amount of lost (processed) work in the queues with respect to the directionally convex order of work or interarrival processes. We first compare the queues that differ only in their work processes and show that if the work processes are directionally convex ordered, so is the amount of work lost (or processed) in the systems. Next, we compare the queues that differ only in their interarrival processes and show that if the interarrival processes are directionally convex ordered, so is the amount of work lost (or processed) in the systems. Using these results, we establish the formula that gives the upper bound of work-loss probability based only on the marginal distributions of work and interarrival processes. Numerical experiments using the data of actual-LAN (local area network) traffic show that the derived formula gives tight bounds sufficient for practical use.

We analyze a continuous review inventory model with the marginal carrying cost of a unit of inventory given by an increasing function of its shelf age and the marginal delay cost of a backlogged demand unit by an increasing function of its delay duration. We show that, under a minor restriction, an (r, q)-policy is optimal when the demand process is a renewal process, and a state dependent (r, q)-policy is optimal when the demand is a Markov-modulated renewal process. We also derive various monotonicity properties for the optimal policy parameters r* and r* + q*.

We consider the problem of estimation of the standard deviation σ and the process capability indices Cp and Cpm for a normally distributed process. The problem is addressed with the objective of obtaining optimal estimators under the classic criterion of minimum variance unbiased estimation, as well as under the Pitman measure of closeness.

We consider a simple model of sequential decisions made by a fusion agent that receives binary-passive reports from distributed sensors. The main result is an explicit formula for the probability of making a decision before a fixed budget is exhausted. These results depend on the relationship between a special ruin problem for a “lazy random walk” and a traditional biased walk.

We consider a two-echelon production inventory system with a manufacturer having limited production capacity and a distribution center (DC). There is a positive transportation time between the manufacturer and the DC. Customers gain a value by receiving the product and incur a waiting cost when facing a delay. We assume that customers' waiting cost depends on their degree of impatience with respect to delay, which is captured by a convex waiting cost function. Customers are strategic with respect to joining the system and either place an order or balk from the system upon their arrival depending on their expected waiting time. We study the Stackelberg equilibrium assuming that the DC acts as a Stackelberg leader and customers are the followers. We first obtain the total expected revenue and then provide a heuristic to derive the optimal base-stock levels in the warehouse and the DC as well as the optimal price of the product.

We study call admission rates in a linear communication network with each call identified by an arrival time, duration, bandwidth requirement, and origin-destination pair. Network links all have the same bandwidth capacity, and a call can be admitted only if there is sufficient bandwidth available on every link along the call's path. Calls not admitted are held in a queue, in contrast to the protocol of loss networks. We determine maximum admission rates (capacities) under greedy call allocation rules such as First Fit and Best Fit for several baseline models and prove that the natural necessary condition for stability is sufficient. We establish the close connections between our new problems and the classic problems of bin packing and interval packing. In view of these connections, it is surprising to find that Best Fit allocation policies are inferior to First Fit policies in the models studied.

In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

This paper proposes a logistic approximation to the upper tail of the t distribution and its inverse that is more convenient and more accurate than those of Shah and Allen [3], and Koehler [1] and Lin [2], when measured in terms of the number of keystrokes and absolute relative errors, respectively.

We give conditions on the optimality of an index policy for multiarmed bandits when arms expire independently. We also give a new simple proof of the optimality of the Gittins index policy for the classic multiarmed bandit problem.

Consider a population having size X(t) at time t that undergoes a sequence of periods of linear birth, death, and immigration. This paper provides a general method for calculating the absolute probabilities pi, (t) = P(X(t) = i) at any stage in this sequence. In general, the different periods of birth, death, and immigration will have different parameters, although some special cases are also considered where certain parameters are common to all periods. Applications of this model can be found in fiber optics when considering a general cascade of erbium-doped fiber amplifiers. In this application, population size represents photon numbers, transmission through fibers causes attenuation (death), and amplification can be described by a linear birth, death, and immigration process.

In this article, we consider an insurance risk model where the claim and premium processes follow some time series models. We first consider the model proposed in Gerber [2,3]; then a model with dependent structure between premium and claim processes modeled by using Granger's causal model is considered. By using some martingale arguments, Lundberg-type upper bounds for the ruin probabilities under both models are obtained. Some special cases are discussed.

We investigate the sensitivity of the leaky-bucket filter, which is known as a traffic-shaping mechanism of source traffic entering an ATM network. Adopting the gradual input source model and using the infinitesimal perturbation analysis, we derive the strongly consistent estimators for the gradients of mean cell delay and a smoothness index of output process with respect to the design parameters of the system. Simulation results show the validity of our estimators.