Abstract: Recently complex function techniques have been developed for the analysis of queueing systems which need for their modelling a two dimensional state space. A variety of computer- and communication networks gives rise to such two-dimensional queueing systems and their analysis is needed for the performance evaluation of these aggregates. The present study reviews these developments

Abstract: 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.

Abstract: This paper considers the maintenance of an unreliable M/G/1 queue-like job shop, integrating the maintenance process and the resulting queue operating characteristics. The system may breakdown, leading to unscheduled maintenance. Otherwise, preventive maintenance is done whenn jobs have been processed — whichever comes first. Using arguments from renewal theory, basic results regarding the queue-maintenance policy are established. Both an analytical and numerical example are studied in detail.

Abstract: This paper shows that certain basic descriptions of the time-dependent behavior of the M/M/1 queue have very simple representations as mixtures of exponentials. In particular, this is true for the busy-period density, the probability that the server is busy starting at zero, the expected queue length starting at zero and the autocorrelation function of the stationary queue-length process. In each case the mixing density is a minor modification of a beta density. The last two representations also apply to regulated or reflected Brownian motion (RBM) by virtue of the heavy-traffic limit. Connections are also established to the classical spectral representations of Ledermann and Reuter (1954) and Karlin and McGregor (1958) and the associated trigonometric integral representations of Ledermann and Reuter, Vaulot (1954), Morse (1955), Riordan (1961) and Takacs (1962). Overall, this paper aims to provide a more unified view of the M/M/1 transient behavior and show how several different approaches are related.

Abstract: In a service operation where worker requirements have to be determined for short scheduling time periods with nonstationary customer demand, the assumptions necessary for applying steady-state solutions to elementary queueing models are usually violated. This paper describes a simulation study of the behavior of such a service operation. The results are compared with the steady-state solutions to a queueing model where individual scheduling time periods are assumed to be independent. It is found that if the system utilization is below a derived maximum value (based on a service level criterion), then the steady-state solutions are robust enough to explain the behavior of the system and can be used to schedule worker requirements.

Abstract: Given a finite number of empty ./M/1 queues, let customers arrive according to an arbitrary arrival process and be served at each queue exactly once, in some fixed order. The process of departing customers from the network has the same law, whatever the order in which the queues are visited. This remarkable result, due to R. Weber [4], is given a simple probabilistic proof.

Abstract: We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run “state” behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both “actual” and “virtual” delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural “explanations” for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Benes's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.

Abstract: The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot

Abstract: This paper considers a class of two discrete-time queues with infinite buffers that compete for a single server. Tasks requiring a deterministic amount of service time, arrive randomly to the queues and have to be served by the server. One of the queues has priority over the other in the sense that it always attempts to get the server, while the other queue attempts only randomly according to a rule that depends on how long the task at the head of the queue has been waiting in that position. The class considered is characterized by the fact that if both queues compete and attempt to get the server simultaneously, then they both fail and the server remains idle for a deterministic amount of time. For this class we derive the steady-state joint generating function of the state probabilities. The queueing system considered exhibits interesting behavior, as we demonstrate by an example.

Abstract: In this paper we consider a single server queueing system with Poisson input, general service and a waiting room that allows only a maximum of ‘b’ customers to wait at any time. A minimum of ‘a’ customers are required to start a service and the server goes for a vacation whenever he finds less than ‘a’ customers in the waiting room after a service. If the server returns from a vacation to find less than ‘a’ customers waiting, he begins another vacation immediately. Using the theory of regenerative processes we derive expressions for the time dependent system size probabilities at arbitrary epochs.

Abstract: An infinite server queue is considered where customers have a choice of individual service or batch service. Transient results have been obtained for the first two moments of the system size distribution. Waiting time distribution is important in system evaluation and steady state results are obtained.

Abstract: In response to the Editorial Introduction (Queueing Systems I (1986), 1-4) the author received several suggestions for additional books that deserved mention. Rather than publishing a supplementary note to that article, the author thought it might perhaps be worthwhile to compile a more complete bibliography of books and important papers on queueing theory. The following is such a bibliography. In section 1 we list important books that serve as valuable sources for the work done during the pioneering phase of queueing theory. This list includes the two survey papers by Kendall. Several other papers should perhaps have been included, but the author decided against it. Many of the books listed here are now out of print, and others are out of date. Even so this list provides a historical perspective. Section 2 contains books on elementary queueing theory with emphasis on applications, and in section 3 are listed advanced books and monographs on a wide variety of special topics. Books on computer performance and data communications systems are listed separately in sections 4 and 5, as these two areas have become increasingly important. Problems in teletraffic have motivated much of the earlier research in queueing theory; nevertheless this is an independent topic by itself, so it is covered by section 6, along with other related topics such as road traffic and scheduling. Numerical tables and simulation are included in section 7. In the final section 8 we list books of general interest whose scope goes beyond queueing theory, but are considered to be very important sources on the subject. Many textbooks on probability, statistics and operations research include some material on queueing systems, but they do not merit a place in a bibliography of this type. A few books in languages other than English are included, but this list is far from complete in this respect. The author is grateful to the associate editors for bringing to his attention several important works of reference.

Abstract: This paper deals with a queueing system with finite capacity in which the server passes from the active state to the inactive state each time a service terminates withv customers left in the system. During the active (inactive) phases, the arrival process is Poisson with parameter λ (λ0). Denoting byu n the duration of thenth inactive phase and byx n the number of customers present at the end of thenth inactive phase, we assume that the bivariate random vectors {(v n ,x n ),n ⩾ 1} are i.i.d. withx n ⩾v+l a.s. The stationary queue length distributions immediately after a departure and at an arbitrary instant are related to the corresponding distributions in the classical model.

Abstract: An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.

Abstract: The Sokolov procedure is described and used to obtain an explicit and easily applied approximation for the waiting time distribution in the FIFO GI/G/1 queue.

Abstract: It is proven that the steady-state probability of the PH/PH/1 queue is a linear combination of product forms. The method of linear combination of product forms is introduced, and simple formulae are obtained.

Abstract: For a double channel Markovian queue with finite waiting space the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities are obtained in a simple closed form which can be easily used to find the important parameters of the system.

Abstract: The information processing industry has shifted from the conventional mathematical computation to information management. Interactive use of a system with large data bases is now widespread, and the data base has become one of the key factors central to an overall system design. This, in turn, has prompted the development of new peripheral devices.

Abstract: Simulation is a widely used methodology for queueing systems. Its superficial simplicity hides a number of pitfalls which are not all as well known as they should be. In particular simulation experiments need careful design and analysis as well as good presentations of the results. Even the elements of simulation such as the generation of arrival and service times have a chequered history with major problems lying undiscovered for 20 years. On the other hand, good simulation practice can offer much more than is commonly realized.

Abstract: The present remarks concern my paper On Davenport’s bound for the degree of f3− g2 and Riemann’s Existence Theorem, published in this journal, 71 (2) (1995), 107–137. I recently discovered the paper Polynomial identities and Hauptmoduln, Quart. J. Math. (2) 32 (1981), 349–370, by W. W. Stothers, which actually covers part of my results, using a method of the same nature. In fact my Proposition 2, p. 120, which proves the existence of cases of equality in Davenport’s bound, is completely contained in Stothers’s results, who actually finds a remarkable exact formula for the number of solutions (Thm. 4.6, p. 362). He also briefly discusses rationality (Thm. 2.4, p. 355), with a method different from mine. I would also like to avail myself of this opportunity to remark that the genus zero case of the result now well known as the abc-theorem for function fields appears, already in 1981, as Theorem 1.1 of the paper by Stothers.

Abstract: Consider aG/M/s/r queue, where the sequence{An}n=−∞∞ of nonnegative interarrival times is stationary and ergodic, and the service timesSnare i.i.d. exponentially distributed. (SinceAn=0 is possible for somen, batch arrivals are included.) In caser < ∞, a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr=∞ (withρ=ES0/EA0