Abstract: We consider anM/G/1 retrial queue with infinite waiting space in which arriving customers who find the server busy join either (a) the retrial group with probabilityp in order to seek service again after a random amount of time, or (b) the infinite waiting space with probabilityq(=1−p) where they wait to be served. The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method. It is shown that our results are consistent with known results whenp=0 orp=1.

Abstract: Consider an open network of single-server stations, each with a first-in-first-out discipline. The network may be populated by various customer types, each with its own routing and service requirements. Routing may be either deterministic or stochastic, and the interarrival and service time distributions may be arbitrary. In this paper a general method for steady-state performance analysis is described and illustrated. This analytical method, called QNET, uses both first and second moment information, and it is motivated by heavy traffic theory. However, our numerical examples show that QNET compares favorably with W. Whitt's Queueing Network Analyzer (QNA) and with other approximation schemes, even under conditions of light or moderate loading. In the QNET method one first replaces the original queueing network by what we call an approximating Brownian system model, and then one computes the stationary distribution of the Brownian model. The second step amounts to solving a certain highly structured partial differential equation problem; a promising general approach to the numerical solution of that PDE problem is described by Harrison and Dai [8] in a companion paper. Thus far the numerical solution technique has been implemented only for two-station networks, and it is clear that the computational burden will grow rapidly as the number of stations increases. Thus we also describe and investigate a cruder approach to two-moment network analysis, called ΠNET, which is based on a product form approximation, or decomposition approximation, to the stationary distribution of the Brownian system model. In very broad terms, ΠNET is comparable to QNA in its level of sophistication, whereas QNET captures more subtle system interactions. In our numerical examples the performance of ΠNET and QNA is similar; the performance of QNET is generally better, sometimes much better.

Abstract: In this paper we compare several service disciplines commonly used in polling systems. We present a sample path comparison which allows us to evaluate the efficiency of the different policies based on thetotal amount of work found in the systemat any time. The analysis is carried out for a large variety of polling schemes under fairly general conditions and can be used to construct a hierarchy of the different service schemes.

Abstract: In this paper we study queueing networks which allow multiple changes at a given time. The model has a natural application to discrete-time queueing networks but describes also queueing networks in continuous time. It is shown that product-form results which are known to hold when there are single changes at a given instant remain valid when multiple changes are allowed.

Abstract: We analyze the time-dependent process in severalM/G/1 vacation models, and explicitly obtain the Laplace transform (with respect to an arbitrary point in time) of the joint distribution of server state, queue size, and elapsed time in that state. Exhaustive-serviceM/G/1 systems with multiple vacations, single vacations, an exceptional service time for the first customer in each busy period, and a combination ofN-policy and setup times are considered. The decomposition property in the steady-state joint distribution of the queue size and the remaining service time is demonstrated.

Abstract: Production systems, particularly those making use of a “pull” production control mechanism, are well-modeled as closed queueing networks. Average throughput is clearly one important performance measure for these systems. However, many control decisions require information concerning the variability of the output process as well as throughput. Because of this, the standard deviation of the number of outputs during a specified interval is a practical performance measure for production systems. In this paper, we consider the standard deviation of the number of outputs during a time interval from a closed queueing network consisting ofM single server exponential queues. Because computing this quantity exactly is extremely cumbersome, we introduce a simple approximation that makes use of (1) known results for the variance of the time a marked job takes to complete a round trip and (2) an approximate correction term for the covariance between successive round trips. We show through comparisons with simulation that our method is quite accurate under a variety of conditions.

Abstract: M/G/1 queues with server vacations have been studied extensively over the last two decades. Recent surveys by Boxma [3], Doshi [5] and Teghem [14] provide extensive summary of literature on this subject. More recently, Shanthikumar [11] has generalized some of the results toM/G/1 type queues in which the arrival pattern during the vacations may be different from that during the time the server is actually working. In particular, the queue length at the departure epoch is shown to decompose into two independent random variables, one of which is the queue length at the departure epoch (arrival epoch, steady state) in the correspondingM/G/1 queue without vacations. Such generalizations are important in the analysis of situations involving reneging, balking and finite buffer cyclic server queues. In this paper we consider models similar to the one in Shanthikumar [11] but use the work in the system as the starting point of our investigation. We analyze the busy and idle periods separately and get conditional distributions of work in the system, queue length and, in some cases, waiting time. We then remove the conditioning to get the steady state distributions. Besides deriving the new steady state results and conditional waiting time and queue length distributions, we demonstrate that the results of Boxma and Groenendijk [2] follow as special cases. We also provide an alternative approach to deriving Shanthikumar's [11] results for queue length at departure epochs.

Abstract: This paper shows how a queueing network model helped to uncover the causes of delay in a health center appointment clinic. Patients, clerks, technicians, doctors and nurses agreed that the clerical registration area was the major bottleneck in the system. Our first reaction was to simulate the system with special attention on the complex registration procedure. Time constraints on data collection and program development led us to a queueing network model and QNA, a software tool for analyzing queueing networks developed by Whitt. The queueing analysis showed the registration area was not the bottleneck and we conjectured that delays were due to scheduling problems. A preliminary trial in the clinic of a modified appointment system showed promise with a 20 minute reduction in average time in the system (based on a small sample). Although there were significant differences between features of the real system and assumptions in the queueing network model, the queueing network model yielded insight into the operation of the appointment clinic.

Abstract: We study the effect of increasing the model parameters (e.g., arrival rates and traffic intensities) in the Erlang blocking model with heterogeneous user requirements. First-order (monotonicity) and second-order (concavity) qualitative results are obtained for the performance measures of interest (loss probabilities, throughput, channel occupancy, etc.) both in the transient and in the steady-state cases. Stochastic and likelihood-ratio orderings together with coupling techniques are used to indicate the effect of modifying the model parameters.

Abstract: A birth-death queueing system with two identical servers, first-come first-served discipline, and Poisson arrivals is considered Only one of the servers is active when the number of customers in the system does not exceed a prescribed threshold, whereas both are active above the threshold The problem of determining the equilibrium density of the waiting time is formulated A generating function is given for the Laplace transform of the density of the waiting time, and it is pointed out that it leads to an explicit expression for this quantity Explicit expressions are obtained for the first and second moments of the waiting and sojourn times, and they are compared with the corresponding quantities for a single-server system with the same state-dependent mean service rates

Abstract: We consider anM/G/1 retrial queue in which blocked customers may leave the system forever without service. Basic equations concerning the system in steady state are established in terms of generating functions. An indirect method (the method of moments) is applied to solve the basic equations and expressions for related factorial moments, steady-state probabilities and other system performance measures are derived in terms of server utilization. A numerical algorithm is then developed for the calculation of the server utilization and some numerical results are presented.

Abstract: This paper investigates a queueing system, which consists of Poisson input of customers, some of whom are lost to balking, and a single server working a shift of lengthL and providing a service whose duration can vary from customer to customer. If a service is in progress at the end of a shift, the server works overtime to complete the service. This process was motivated by the behavior of fishermen interviewed in the NY Great Lakes Creel Survey.

Abstract: We consider a queueing system with two stations served by a single server in a cyclic manner. We assume that at most one customer can be served at a station when the server arrives at the station. The system is subject to service interuption that arises from server breakdown. When a server breakdown occurs, the server must be repaired before service can resume. We obtain the approximate mean delay of customers in the system.

Abstract: This paper examines the properties of single server queueing systems with customers of several types, where the server rotates its effort among the customer classes and serves all the customers that have accumulated for each class before moving on to the next. The paper shows that expressions for the first two moments of the queue lengths, and for the mean waiting times, can be developed from two simple properties of the arrival and service processes. The properties, which include existing models as special cases, seem plausible descriptors of the complex arrival and service processes that arise in the transportation field.

Abstract: We consider the control of an infinite capacity shuttle which transports passengers between two terminals. The passengers arrive at each terminal according to a compound Poisson process and the travel time from one terminal to the other is a random variable following an arbitrary distribution. The following control limit policy is considered: dispatch the shuttle at terminali, at the instant that the total number of passengers waiting at terminali reaches or exceeds a predetermined control limitm i . The objective of this paper is to obtain the mean waiting time of an arbitrary passenger at each terminal for given control valuesm 1 andm 2. We also discuss a search procedure to obtain the optimal control values which minimize the total expected cost per unit time under a linear cost structure.

Abstract: We study a discrete-time single-server queue where batches of messages arrive. Each message consists of a geometrically distributed number of packets which do not arrive at the same instant and which require a time unit as service time. We consider the cases of constant spacing and geometrically distributed (random) spacing between consecutive packets of a message. For the probability generating function of the stationary distribution of the embedded Markov chain we derive in both cases a functional equation which involves a boundary function. The stationary mean number of packets in the system can be computed via this boundary function without solving the functional equation. In case of constant (random) spacing the boundary function can be determined by solving a finite-dimensional (an infinite-dimensional) system of linear equations numerically.

Abstract: Recently developed methods of qualitative analysis for regenerative processes arising in queueing are presented. These methods are essentially qualitative and use notions such as coupling, probability metrics, etc. They are developed for studying various properties of regenerative models, including convergence rate to a stationary regime, continuity of their characteristics with respect to some parameters and first-occurrence time of an event such as queue overflowing. In spite of their qualitative nature they lead to good quantitative estimates of underlying properties with computer methods available to calculate them.

Abstract: The customer response times in the egalitarian processor sharing queue are shown to be associated random variables under renewal inputs and general independent service times assumptions.

Abstract: The tandem behavior of a telecommunication system with finite buffers and repeated calls is modeled by the performance of a finite capacityG/M/1 queueing system with general interarrival time distribution, exponentially distributed service time, the first-come-first-served queueing discipline and retrials. In this system a fraction of the units which on arrival at a node of the system find it busy, may retry to be processed, by merging with the incoming arrival units in that node, after a fixed delay time. The performance of this system in steady state is modeled by a queueing network and is approximated by a recursive algorithm based on the isolation method. The approximation outcomes are compared against those from a simulation study. Our numerical results indicate that in steady state the non-renewal superposition arrival process, the non-renewal overflow process, and the non-renewal departure process of the above system can be approximated with compatible renewal processes.

Abstract: We consider a single server loss system in which arrivals occur according to a doubly stochastic Poisson process with a stationary ergodic intensity functionλt. The service times are independent, exponentially distributed r.v.'s with meanμ−1, and are independent of arrivals. We obtain monotonicity results for loss probabilities under time scaling as well as under amplitude scaling ofλt. Moreover, using these results we obtain both lower and upper bounds for the loss probability.

Abstract: This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state. Both unsaturated (ρ 1) conditions are considered. Numerical evidence is presented to indicate that the quality of the approximations is usefully good, especially when ease of computation is an issue. Further, the methodology is adapted to assess expected waiting time when inference must be made from a random sample of service times, and the decision is made to do so nonparametrically, i.e., without fitting a specific function. The results appear reasonable and potentially useful, and are not burdensome to obtain. The methodology investigated can also be applied to the variety of queueing models that are close siblings ofM/G/1: priority and breakdowns and “vacations” being examples. Of course other approximating and inferential options remain to be investigated.

Abstract: This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled in continuous time as Markov processes. The theory is presented for theM/E k /φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output, departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works, as well as providing some new results. Several extensions to the model are discussed.

Abstract: In this paper we present an effective load balancing algorithm for a multi-processor architecture designed for the real time switching of telephone calls. By modifying an algorithm developed for an abstract queueing model, which is of independent interest by itself, we propose a hybrid load balancing algorithm and study its performance in a simulation test-bed. This case study demonstrates how simple abstractions and theoretically intractable but intuitively appealing ideas can be combined to effectively solve a real problem.

Abstract: A class of service stage Petri net models whose equilibrium probabilities satisfy local balance equations is presented. Examples of their applicability include bus oriented multiprocessor interconnections, the classical dining philosophers problem and the alternating bit protocol. The natural topological space for embedding the state transition lattice for this class of SSPN is shown to be a multidimensional toroidal manifold.

Abstract: A multi-server Markovian queueing system is considered such that an idle server will take the entire batch of waiting customers into service as soon as their number is as large as some control limit. Some new results are derived. These include the distribution of the time interval between two consecutive commencements of service (including itsrth moment) and the actual service batch size distribution. In addition, the average customer waiting time in the queue is derived by a simple combinatorial approach.

Abstract: This paper, motivated by the need to predict performance of production systems with random arrivals, setup times and revisitation, presents an imbedded Markov chain analysis of the underlyingM/G/1 queue with two customer classes, changeover times and instantaneous Bernoulli feedback. It is assumed that jobs are scheduled according to the exhaustive alternative priority queue discipline. Expressions for the mean waiting time and the nonsaturation condition are derived under two different priority assignments to the repeat customers. Sojourn times under these priority assignments are shown to possess a convex ordering. Results of the study are also applicable to data communication networks that operate under cyclic switching mechanisms.

Abstract: Explicit formulas for the individual call loss probabilities are derived which arise when a finite collection of independent general stationary traffic streams with exponentially distributed service times are offered simultaneously to a single server. The formulas show a modified insensitivity property of the given model.

Abstract: Under weak conditions the average virtual waiting time converges exponentially fast to its limit. For this reason this quantity has been suggested as a measure of performance for queueing systems.