This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = a x. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

We consider compound geometric approximation for a nonnegative, integer-valued random variable W. The bound we give is straightforward but relies on having a lower bound on the failure rate of W. Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth–death processes and Poisson processes.

We consider an M/G/1 queue in which an arriving customer does not enter the system whenever its virtual waiting time, i.e. the amount of work seen upon arrival, is larger than a certain random patience time. We determine the busy period distribution for various choices of the patience time distribution. The main cases under consideration are exponential patience and a discrete patience distribution.

In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

We study the expected time to ruin in a risk process in which dividends are paid when the surplus is above the barrier. We consider the case in which the dividend rate is smaller than the premium rate. We obtain results for the classical compound Poisson risk process with phase-type claim size. When the ruin probability is 1, we derive the expected time to ruin and the expected dividends paid. When the ruin probability is less than 1, these quantities are derived conditioning on the event that ruin occurs.

We consider an infinite capacity buffer where the incoming fluid traffic belongs to K different types modulated by K independent Markovian on-off processes. The kth input process is described by three parameters: (λk, μk, rk), where 1/λk is the mean off time, 1/μk is the mean on time, and rk is the constant peak rate during the on time. The buffer empties the fluid at rate c according to a first come first served (FCFS) discipline. The output process of type k fluid is neither Markovian, nor on-off. We approximate it by an on-off process by defining the process to be off if no fluid of type k is leaving the buffer, and on otherwise. We compute the mean on time τkon and mean off time τkoff. We approximate the peak output rate by a constant rko so as to conserve the fluid. We approximate the output process by the three parameters (λko, μko, rko), where λko = 1/τkoff, and μko = 1/τkon. In this paper we derive methods of computing τkon, τkoff and rko for k = 1, 2,…, K. They are based on the results for the computation of expected reward in a fluid queueing system during a first passage time. We illustrate the methodology by a numerical example. In the last section we conduct a similar output analysis for a standard M/G/1 queue with K types of customers arriving according to independent Poisson processes and requiring independent generally distributed service times, and following a FCFS service discipline. For this case we are able to get analytical results.

We consider an M/G/1 queue with the special feature that the speed of the server alternates between two constant values sL and sH > sL. The high-speed periods are exponentially distributed, and the low-speed periods have a general distribution. Our main results are: (i) for the case that the distribution of the low-speed periods has a rational Laplace–Stieltjes transform, we obtain the joint distribution of the buffer content and the state of the server speed; (ii) for the case that the distribution of the low-speed periods and/or the service request distribution is regularly varying at infinity, we obtain explicit asymptotics for the tail of the buffer content distribution. The two cases in which the offered traffic load is smaller or larger than the low service speed are shown to result in completely different asymptotics.

A recent study on the GI/G/1 queue derives the Maclaurin series for the moments of the waiting time and the delay to respect to some parameters. By the same approach, we obtain an identity on the moments of the transient delay of the M/G/1 queue. This identity allows us to understand the transient behavior of the process better. We apply the identity with other established results to study convergence rate and stochastic concavity of the transient delay process, and to derive bounds and approximations of the moments. Our approximation and bound both have simple closed forms and are asymptotically exact as either the traffic intensity goes to zero or the process approaches stationarity. Performance of the approximation of several M/G/1 queues is illustrated by numerical experiments. It is interesting to note that our results can also help to gain variance reduction in simulation.

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.

This paper considers a single-server queue with Poisson arrivals and multiple customer feedbacks. If the first service attempt of a newly arriving customer is not successful, he returns to the end of the queue for another service attempt, with a different service time distribution. He keeps trying in this manner (as an ‘old' customer) until his service is successful. The server operates according to the ‘gated vacation' strategy; when it returns from a vacation to find K (new and old) customers, it renders a single service attempt to each of them and takes another vacation, etc. We study the joint queue length process of new and old customers, as well as the waiting time distribution of customers. Some extensions are also discussed.

A disaster occurs in a queue when a negative arrival causes all the work (and therefore customers) to leave the system instantaneously. Recent papers have addressed several issues pertaining to queueing networks with negative arrivals under the i.i.d. exponential service times assumption. Here we relax this assumption and derive a Pollaczek–Khintchine-like formula for M/G/1 queues with disasters by making use of the preemptive LIFO discipline. As a byproduct, the stationary distribution of the remaining service time process is obtained for queues operating under this discipline. Finally, as an application, we obtain the Laplace transform of the stationary remaining service time of the customer in service for unstable preemptive LIFO M/G/1 queues.

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.

Exponential bounds are derived for the tail probabilities of various compound distributions, generalizing the classical Lundberg inequality of insurance risk theory. Failure rate properties of the compounding distribution including log-convexity and log-concavity are considered in some detail. Mixed Poisson compounding distributions are also considered. A ruin theoretic generalization of the Lundberg inequality is obtained in the case where the number of claims process is a mixed Poisson process. An application to the M/G/1 queue length distribution is given.

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

This paper is concerned with the determination of the optimal release rate for a continuous storage system. The input to the store is assumed to be a compound Poisson process (with non-negative jumps), and its content is released either at a constant rate or at a linear rate. Rewards are collected at an output-dependent rate and are continuously discounted at a constant rate. The problem consists in finding the optimal release rate which maximizes the infinite-horizon expected discounted return. This problem is solved by deriving an explicit solution for the expected discounted return, from which the optimal release rate can be found.

This paper considers the two-stage cyclic queueing model consisting of one general (G) and one exponential (M) server. The strong connection between the present model and the M/G/1 model (with finite waiting room) is exploited to yield the joint distribution of the successive response times of a customer at the G queue and the M queue. This result reveals a surprising phenomenon: in general there is a difference between the joint distribution of the two successive response times at (first) the G queue and (then) the M queue, and the joint distribution of the two successive response times at (first) the M queue and (then) the G queue.

Another associated result is an expression for the cycle-time distribution. Special consideration is given to the case that the number of customers in the system tends to ∞, while the mean service times tend to 0 at an inversely proportional rate.