In this article, some results on stochastic comparisons of the inspection paradox introduced by Ross [Probability in the Engineering and Informational Sciences 17: 47–51 (2003)] are established in the sense of the likelihood ratio order.

We investigate the tail distribution of the virtual waiting times in a LRD/GI/1 queue where the arrival process is long-range dependent (LRD) and the service times are independent and identically distributed (i.i.d.) random variables. We present two lower bounds on the stationary waiting time tail asymptotics, which illustrate the different dominating components that influence server performance under various conditions. In particular, we show that the tail distribution of the stationary waiting time is bounded below by that of the associated LRD/D/1 queues resulting from replacing all random service times by the mean. This shows the performance impact purely due to the long-range dependency of the arrival process. On the other hand, when the service times are subexponential, we show that the tail distribution of the stationary waiting time is bounded below by that of the corresponding D/GI/1 queue by replacing the dependent arrival process with its associated independent version. This shows the minimum performance impact due to the tail distribution of the service times. The above two lower bounds indicate that the performance of LRD/GI/1 queues will be dominated by the heavier tail of the corresponding LRD/D/1 and D/GI/1 queues. These features are further illustrated and quantified through examples and via numerous simulation experiments.

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.

In this note, we consider a simple immigration birth–death process with total catastrophes and we obtain the transient probabilities. Our approach involves a renewal argument. It is comparatively simpler and leads to more elegant expressions than other approaches that appeared in the literature recently.

The purpose of this article is to study several preservation properties of stochastic comparisons based on the mean inactivity time order under the reliability operations of convolution and mixture. Characterizations and relationships with the other well-known orders are given. Some examples of interest in reliability theory are also presented. Finally, testing in the increasing mean inactivity time class is discussed.

This article provides us with some simple criteria to compare Birnbaum reliability importance measure of components in a general binary coherent system. Such criteria are particularly useful in the absence of information concerning component reliabilities. We also find several simple (necessary and sufficient) conditions concerning system structure, under which such comparison is possible. Examples are given to illustrate our results.

The question of stability for the M/G/∞ queue with gated service is investigated using a Foster–Lyapunov drift criterion. The necessary and sufficient condition for positive recurrence is shown to be the finiteness of the first moment of the service time distribution, thus weakening the stability condition given in Browne et al. [3].

We obtain some techniques to study the shape of reliability functions (failure rate, mean residual life, etc.) by using the s-equilibrium distribution of a renewal process defined by Fagiuoli and Pellerey (Naval Res. Logist., 1993). We apply these techniques to study how to obtain distributions with bathtub shaped failure rate (BFR) from mixtures of two positive truncated normal distributions.

In this article, we introduce the notion of the Markov chain fields on the generalized Bethe trees or generalized Cayley trees, and some strong limit theorems on the frequencies of states and ordered couples of states, including the Shannon–McMillan theorem on Bethe tree TB,N and Cayley tree TC,N, are obtained. In the proof, a new technique in the study of the strong limit theorem in probability theory is applied.

We study the value of information sharing in a two-stage supply chain with a single manufacturer and a single retailer in an infinite time horizon, where the manufacturer has finite production capacity and the retailer faces independent demand. The manufacturer receives demand information even during periods of time in which the retailer does not order. Allowing for time-varying cost functions, our objective is to characterize the impact of information sharing on the manufacturer's cost and service level. We develop a new approach to characterize the induced Markov chains under cyclic order-up-to policy and provide a simple proof for the optimality of cyclic order-up-to policy for the manufacturer under the average cost criterion. Using extensive computational analysis, we quantify the impact of information sharing on the manufacturer's performance in an infinite time horizon under both i.i.d. demand and independent but nonstationary demand.

In this paper we develop a stochastic differential equation to describe the dynamic evolution of the congestion window size of a single TCP session over a network. The model takes into account recovery from packet losses with both fast recovery and time-outs, boundary behavior at zero and maximum window size, and slow-start after time-outs. We solve the differential equation to derive the distribution of the window size in steady state. We compare the model predictions with the output from the NS simulator.

We consider the problem of routing incoming airplanes to two runways of an airport. Due to air turbulence, the necessary separation time between two successive landing operations depends on the type of airplane. When viewed as a queuing problem, this means that we have dependent service times. The aim is to minimize the waiting times of aircrafts. We consider here a model in which arrivals form a stochastic process and the decision-maker does not know anything about future arrivals. We formulate this as a problem of stochastic dynamic programming and investigate the monotonicity of optimal routing strategies with respect to the workload of the runways, for example. We show that an optimal strategy is monotone (i.e., of switching type) only in a restricted case where decisions depend on the state of the runways only and not on the type of the arriving aircraft. Surprisingly, in the more realistic case where this type is also known to the decision-maker, monotonicity need not hold.

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.