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We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e−x. This leads to two distinct cases, heavy tailed and moderately heavy tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little’s law to establish tail asymptotics for steady-state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

We investigate the question of when sampling a stochastic process X={X(t) : t⩾0} at the times of an independent point process ψ leads to the same empirical distribution as the time-average limiting distribution of X. Two main cases are considered. The first is when X is asymptotically stationary and ergodic, and ψ satisfies a mixing condition. In this case, the pathwise limiting distributions in function space are shown to be the same. The second main case is when X is only assumed to have a constant finite time average and ψ is assumed a positive recurrent renewal processes with a spread-out cycle length distribution. In this latter case, the averages are shown to be the same when some further conditions are placed on X and ψ.

The inventory equation, Z(t) = X(t) + L(t), where X = X(t):t ≥ 0 is a given netput process and L(t):t ≥ 0 is the corresponding lost potential process, is explored in the general case when X is a negative drift stochastic process that has asymptotically stationary increments. Our results show that if (as s → ∞) X s ≜ X(s + t) − X(s):t ≥ 0 converges in some sense to a process X∗ with stationary increments and negative drift, then, regardless of initial conditions, θ s Z ≜ Z(s + t):t ≜ 0 converges in the same sense to a stationary version Z∗ . We use coupling and shift-coupling methods and cover the cases of convergence in total variation and in total variation in mean, as well as strong convergence in mean. Our approach simplifies and extends the analysis of Borovkov (1976). We remark upon an application in regenerative process theory.

The stability of open Jackson networks is established where service times are i.i.d. general distribution, exogeneous interarrival times are i.i.d. general distribution, and the routing is Markovian. The service time distributions are only required to have finite first moment. The system is modeled (at arrival epochs) as a general state space Markov chain. Explicit regeneration points are found (even in the case when the system never empties) and the chain is shown to be Harris ergodic if standard rate conditions are enforced, that is, if at each node, the long run average amount of work per unit time that arrives exogenously destined for that node is strictly less than one. In addition, we prove that if the system is modeled in continuous time then convergence to a steady-state occurs in total variation if the interarrival time distribution is spread-out. Extensions of the results to multi-server nodes, non-Markovian routing and Markov modulated arrivals are given.

We present, in discrete time, general-state-space dualities between content and insurance risk processes that generalize the stationary recursive duality of Asmussen and Sigman (1996, Probab. Eng. Inf. Sci. 10, 1–20) and the Markovian duality of Siegmund (1976, Ann. Probab. 4, 914–924) (both of which are one dimensional). The main idea is to allow a risk process to be set-valued, and to define ruin as the first time that the risk process becomes the whole space. The risk process can also become infinitely rich which means that it eventually takes on the empty set as its value. In the Markovian case, we utilize stochastic geometry tools to construct a Markov transition kernel on the space of closed sets. Our results connect with strong stationary duality of Diaconis and Fill (1990, Ann. Probab. 18, 1483–1522). As a motivating example, in multidimensional Euclidean space our approach yields a dual risk process for Kiefer–Wolfowitz workload in the classic G/G/c queue, and we include a simulation study of this dual to obtain estimates for the ruin probabilities.

Let X={X(t):t⩾0} be a positive recurrent synchronous process (PRS), that is, a process for which there exists an increasing sequence of random times τ={τ(k)} such that for each k the distribution of θ τ(k) X = def {X(t+τ(k)):t⩾0} is the same and the cycle lengths T n = def τ(n+1)−τ(n) have finite first moment. Such processes (in general) do not converge to steady-state weakly (or in total variation) even when regularity conditions are placed on the cycles (such as non-lattice, spread-out, or mixing). Nonetheless, in the present paper we first show that the distributions of {θsX:s > 0} are tight in the function space D (0,∞) . Then we investigate conditions under which the Cesaro averaged functionals μ i = def ( 1 t )∫ t 0 E(ƒ(θ s X)) d s converge uniformly (over a class of functions) to π(ƒ) , where π is the stationary distribution of X. We show that μ t (ƒ)→π(ƒ) uniformly over ƒ satisfying ‖ƒ‖ ∞ ⩽1 (total variation convergence). We also show that to obtain uniform convergence over all ƒ satisfying |ƒ|⩽g (gϵL + 1 (π) fixed) requires placing further conditions on the PRS. This is in sharp contrast to both classical regenerative processes and discrete time Harris recurrent Markov chains (where renewal theory can be applied) where such uniform convergence holds without any further conditions. For continuous time positive Harris recurrent Markov processes (where renewal theory cannot be applied) we show that these further conditions are in fact automatically satisfied. In this context, applications to queueing models are given.

We present a framework for representing a queue at arrival epochs as a Harris recurrent Markov chain (HRMC). The input to the queue is a marked point process governed by a HRMC and the queue dynamics are formulated by a general recursion. Such inputs include the cases of i.i.d, regenerative, Markov modulated, Markov renewal and the output from some queues as well. Since a HRMC is regenerative, the queue inherits the regenerative structure. As examples, we consider split & match, tandem, G/G/c and more general skip forward networks. In the case of i.i.d. input, we show the existence of regeneration points for a Jackson type open network having general service and interarrivai time distributions.