We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies limt→∞varD(t) / t = λ(1–2/π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

We show that the waiting time in queue and the sojourn time of every customer in the G/G/1 and G/D/c queue are jointly convex in mean interarrival time and mean service time, and also jointly convex in mean interarrival time and service rate. Counterexamples show that this need not be the case, for the GI/GI/c queue or for the D/GI/c queue, for c ≧ 2. Also, we show that the average number of customers in the M/D/c queue is jointly convex in arrival and service rates.

These results are surprising in light of the negative result for the GI/GI/2 queue (Weber (1983)).

It is shown that a set of random variables with increasing and convex (concave) survival functions are stochastically increasing and convex (concave) in the sample path sense. This stochastic convexity (concavity) result is then used to establish convexity (concavity) results for (i) a single-server queueing system with a time-out control policy, (ii) residual life, (iii) stationary renewal excess life and (iv) M/G/1 queues. These results are new and could not be derived without the direct or indirect aid of the above stochastic convexity (concavity) result. Furthermore, we illustrate that the above stochastic convexity (concavity) result can be applied to obtain new bounds for queueing systems. Specifically, let be the waiting time of the nth customer in a GI/G/1 queue with inter-arrival time survival function and service time survival function . Using the above convexity result it is shown that if and for some such that then for all increasing convex functions φ, whenever the expectations exist. A similar result for and is also obtained. Other examples are also included.

Recent work by Athreya and Ney and by Nummelin on the limit theory for Markov chains shows that the close connection with regeneration theory holds also for chains on a general state space. Here this is used to study extremal behaviour of stationary (or asymptotically stationary) Markov chains. Many of the results center on the ‘clustering’ of extremes of adjacent values of the chains. In addition one criterion for convergence of extremes of general stationary sequences is derived. The results are applied to waiting times in the GI/G/1 queue and to autoregressive processes.

We prove strong approximations for partial sums indexed by a renewal process. The obtained results are optimal. The established probability inequalities are also used to get bounds for the rate of convergence of some limit theorems in queueing theory.

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

Let Sn = X1 + · · · + Xn be a random walk with negative drift μ < 0, let F(x) = P(Xk ≦ x), v(u) =inf{n : Sn > u} and assume that for some γ > 0 is a proper distribution with finite mean Various limit theorems for functionals of X1,· · ·, Xv(u) are derived subject to conditioning upon {v(u)< ∞} with u large, showing similar behaviour as if the Xi were i.i.d. with distribution For example, the deviation of the empirical distribution function from properly normalised, is shown to have a limit in D, and an approximation for by means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and the GI/G/1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for the GI/G/1 queue. For example uniformly in N, with WN the waiting time of the Nth customer.

The problem of controlling input to a stochastic input-output system by accepting or rejecting arriving customers is analyzed as a semi-Markov decision process. Included as special cases are a GI/G/1 model and models with compound input and/or output processes, as well as several previously studied queueing-control models. We establish monotonicity of socially and individually optimal acceptance policies and the more restrictive nature of the former, with random rewards for acceptance and both customer-oriented and system-oriented non-linear waiting costs. Distinctive features of our analysis are (i) that it allows dependent interarrival times and (ii) that the monotonicity proofs do not rely on the standard concavity-preservation arguments.

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.

Various authors have derived the necessary and sufficient conditions for optimality in semi-Markov decision processes in which the state remains constant between jumps. In this paper similar results are presented for a generalized semi-Markov decision process in which the state varies between jumps according to a Markov process with continuous sample paths. These results are specialized to a general storage model and an application to the service rate control in a GI/G/1 queue is indicated.

A new upper bound on the stationary waiting-time distribution of a GI/G/1 queue is derived following Kingman's martingale approach. This bound is generally stronger than Kingman's upper bound and is sometimes stronger than an upper bound derived by Ross.

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.

Consider a stable GI/G/1 queue with non-lattice interarrival time distribution. Let G and H be the limiting actual and virtual waiting time distributions respectively. Two separate statements of the relationship between G and H are found in a classical theorem of Takàcs and a more recent (and previously unpublished) theorem of Hooke. A simplified proof of Takàcs's theorem, based on a sample path relationship between the virtual and actual waiting time processes, has recently been advanced. This paper gives a similar proof of Hooke's theorem, based on the same sample path relationship, and demonstrates the utility of the result in analyzing the special case of Poisson input. In particular, by combining the Takàcs and Hooke results one can obtain the Pollaczek–Khintchine formula without any reference to the imbedded Markov chain.

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.

Let {Xk: k ≧ 1} be a sequence of independent, identically distributed random variables with EX1 = μ < 0. Form the random walk {Sn: n ≧ 0} by setting S0 = 0, Sn = X1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X1) that Sn, conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

The limiting behavior of the number of runs up to time n of a regenerative phenomenon on the positive integers is related to the one of the number of regenerations. Applications are considered in fluctuation theory, and for the GI/G/1 queue.

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.