This paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

Tail asymptotics of the solution R to a fixed-point problem of the type R=DQ+∑1NRm are derived under heavy-tailed conditions allowing both dependence between Q and N and the tails to be of the same order of magnitude. Similar results are derived for a K-class version with applications to multi-type branching processes and busy periods in multi-class queues.

This paper is devoted to the study of a clearing queueing system with a special discipline. As soon as the server receives N negative feedbacks from customers, all present customers are forced to leave the system and the server undergoes a maintenance procedure. After an exponential maintenance time, the system resumes its service immediately. Using the matrix analytic method, we derive the steady-state distributions, which are then used for the computation of other performance measures. Furthermore, using first step analysis, we obtain the Laplace–Stieltjes transform of the sojourn time of an arbitrary customer. We also study the busy period of the system and derive the generating function of the total number of lost customers in a busy period. Finally, we investigate a long-run rate of cost and explore the optimal N value that minimizes the total cost per unit time. We also present some numerical examples to illustrate the impact of several model parameters to the performance measures.

We study the exact asymptotics for the distribution of the first time, τx, a Lévy process Xt crosses a fixed negative level -x. We prove that ℙ{τx >t} ~V(x) ℙ{Xt≥0}/t as t→∞ for a certain function V(x). Using known results for the large deviations of random walks, we obtain asymptotics for ℙ{τx>t} explicitly in both light- and heavy-tailed cases.

A semi-numerical method is derived to compute the Laplace transform of the equilibrium busy period probability density function in a fluid queue with constant output rate when the buffer is nonempty. The input process is controlled by a continuous-time semi-Markov chain (CTSMC) with n states such that in each state the input rate is constant. The holding time in states with net positive output rate - so-called emptying states - is assumed to be an exponentially distributed random variable, whereas in states with net positive input rate - so-called filling states - it may have an arbitrary probability distribution. The result is demonstrated by applying it to various systems, including fluid queues with two on-off input sources. The latter exercise in part shows consistency with prior results but also solves the problem in the case where there are two emptying states. Numerical results are presented for selected examples which expose discontinuities in the busy period distribution when the number of emptying states changes, e.g. as a result of increasing the fluid arrival rate in one or more states of the controlling CTSMC.

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 the problem of first-crossing from above for a partial sums process m+St, t ≥ 1, with the diagonal line when the random variables Xt, t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the Xts belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

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.

We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the MX/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).

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.

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.

It is shown that the time to ruin and the recovery time in a risk process have the same distribution as the busy period in a certain queueing system. Similarly, the deficit at the time of ruin is distributed as the idle period in a single-server queueing system. These duality results are exploited to derive upper bounds for the expected time to ruin and the expected recovery time as defined by Egídio dos Reis (2000). When the claim size is generally distributed, Lorden's inequality is applied to derive the bounds. When the claim-size distribution is of phase type, tighter upper bounds are derived.

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

We exploit the structural properties of the BMAP/D/k system to carry out its algorithmic analysis. Specifically, we use these properties to develop algorithms for studying the distributions of waiting times in discrete time and the busy period. One of the structural properties used results from considering the system as having customers assigned in a cyclic order—which does not change the waiting-time distribution—and then studying only one arbitrary server. The busy period is defined as the busy period of an arbitrary single server based on this cyclic assignment of customers to servers. Finally, we study the marginal distribution of the joint queue length and phase of customer arrival. The structural property used for studying the queue length is based on the observation of the system every interval that is the length of one customer service time.

We present a recursive method of computation for the probability that at most k customers were served during the busy period of an M/G/1 retrial queue.

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

The present paper provides simple inequalities for the number of lost customers during a busy period of a GI/M/1/n queueing system.

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.

For truncated birth-and-death processes with two absorbing or two reflecting boundaries, necessary and sufficient conditions on the transition rates are given such that the transition probabilities satisfy a suitable spatial symmetry relation. This allows one to obtain simple expressions for first-passage-time densities and for certain avoiding transition probabilities. An application to an M/M/1 queueing system with two finite sequential queueing rooms of equal sizes is finally provided.

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI/G/1 queueing process