Showing papers in "Queueing Systems in 1990"
TL;DR: A survey of the main results and methods of the theory of retrial queues, concentrating on Markovian single and multi-channel systems, as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffer.
Abstract: We present a survey of the main results and methods of the theory of retrial queues, concentrating on Markovian single and multi-channel systems. For the single channel case we consider the main model as well as models with batch arrivals, multiclasses, customer impatience, double connection, control devices, two-way communication and buffer. The stochastic processes arising from these models are considered in the stationary as well as the nonstationary regime. For multi-channel queues we survey numerical investigations of stationary distributions, limit theorems for high and low retrial intensities and heavy and light traffic behaviour.
485 citations
TL;DR: A simplifying approximation is made, based on understanding of the physical behavior of the system, which yields an infinitesimal generator with a modified matrix-geometric equilibrium probability vector that can be very efficiently computed even for high congestion levels.
Abstract: We consider a queueing model in which customers arrive in a Poisson stream to be served by one ofc servers. Each arriving customer enters a pool of active customers and starts generating requests for service at exponentially distributed time intervals at rate σ until he finds a free server and begins service. An analytical solution of this model is difficult and does not lend itself to numerical implementation. In this paper, we make a simplifying approximation, based on understanding of the physical behavior of the system, which yields an infinitesimal generator with a modified matrix-geometric equilibrium probability vector. That vector can be very efficiently computed even for high congestion levels. Illustrative numerical examples demonstrate the effectiveness of the approximation as well as the effect of the retrial rate on the system behavior for various levels of congestion. This study shows how numerical results for analytically intractable systems can be obtained by combining intuition with efficient algorithmic methods.
181 citations
TL;DR: A single server retrial queue where the server is subject to breakdowns and repairs, and the limiting behavior of the system is studied by using the tools of Markov regenerative processes.
Abstract: In this paper we consider a single server retrial queue where the server is subject to breakdowns and repairs. New customers arrive at the service station according to a Poisson process and demand i.i.d. service times. If the server is idle, the incoming customer starts getting served immediately. If the server is busy, the incoming customer conducts a retrial after an exponential amount of time. The retrial customers behave independently of each other. The server stays up for an exponential time and then fails. Repair times have a general distribution. The failure/repair behavior when the server is idle is different from when it is busy. Two different models are considered. In model I, the failed server cannot be occupied and the customer whose service is interrupted has to either leave the system or rejoin the retrial group. In model II, the customer whose service is interrupted by a failure stays at the server and restarts the service when repair is completed. Model II can be handled as a special case of model I. For model I, we derive the stability condition and study the limiting behavior of the system by using the tools of Markov regenerative processes.
144 citations
TL;DR: A product form equilibrium distribution is derived for a class of queueing networks in either discrete or continuous time, in which multiple customers arrive simultaneously and batches of customers complete service simultaneously.
Abstract: A product form equilibrium distribution is derived for a class of queueing networks in either discrete or continuous time, in which multiple customers arrive simultaneously and batches of customers complete service simultaneously.
125 citations
TL;DR: A cruder approach to two-moment network analysis, called ΠNET, is described and investigated, which is based on a product form approximation, or decomposition approximation, to the stationary distribution of the Brownian system model.
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.
112 citations
TL;DR: The joint generating function of the numbers of customers in the two groups is derived by using the supplementary variable method and it is shown that the results are consistent with known results whenp=0 orp=1.
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.
110 citations
TL;DR: A sample path comparison is presented which allows us to evaluate the efficiency of the different policies based on the total amount of work found in the system at any time.
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.
106 citations
TL;DR: This paper analyzes a model of a queueing system in which customers can only call in to request service: if the server is free, the customer enters service immediately, but if the service system is occupied, the unsatisfied customer must break contact and reinitiate his request later.
Abstract: We analyze a model of a queueing system in which customers can only call in to request service: if the server is free, the customer enters service immediately, but if the service system is occupied, the unsatisfied customer must break contact and reinitiate his request later. Such a customer is said to be in “orbit”. In this paper we consider three models characterized by the discipline governing the order of re-request of service from orbit. First, all customers in orbit can reapply, but are discouraged and reduce their rate of demand as more customers join the orbit. Secondly, the FCFS discipline operates for the unsatisfied customers in orbit. Finally, the LCFS discipline governs the customers in orbit and the server takes an exponentially distributed vacation after each service is completed. We calculate several characteristics quantities of such systems, assuming a general service-time distribution and different exponential distributions for the times between arrivals of first and repeat requests.
92 citations
TL;DR: 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: 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.
79 citations
TL;DR: A modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times, which satisfies a modified version of the classical Lindley recursion.
Abstract: We consider a modification of the standardG/G/1 queue with unlimited waiting space and the first-in first-out discipline in which the service times and interarrival times depend linearly and randomly on the waiting times. In this model the waiting times satisfy a modified version of the classical Lindley recursion. We determine when the waiting-time distributions converge to a proper limit and we develop approximations for this steady-state limit, primarily by applying previous results of Vervaat [21] and Brandt [4] for the unrestricted recursionYn+1=CnYn+Xn. Particularly appealing for applications is a normal approximation for the stationary waiting time distribution in the case when the queue only rarely becomes empty. We also consider the problem of scheduling successive interarrival times at arrival epochs, with the objective of achieving nearly maximal throughput with nearly bounded waiting times, while making the interarrival time sequence relatively smooth. We identify policies depending linearly and deterministically upon the work in the system which meet these objectives reasonably well; with these policies the waiting times are approximately contained in a specified interval a specified fraction of time.
49 citations
IBM1
TL;DR: The time-dependent process in several M/G/1 vacation models is analyzed, and the Laplace transform of the joint distribution of server state, queue size, and elapsed time in that state is explicitly obtained.
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.
TL;DR: In this article, an iterative numerical technique for the evaluation of queue length distributions is applied to multi-queue systems with one server and cyclic service discipline with Bernoulli schedules.
Abstract: An iterative numerical technique for the evaluation of queue length distributions is applied to multi-queue systems with one server and cyclic service discipline with Bernoulli schedules. The technique is based on power-series expansions of the state probabilities as functions of the load of the system. The convergence of the series is accelerated by applying a modified form of the epsilon algorithm. Attention is paid to economic use of memory space. The technique is based on power-series expansions of the state probabilities as functions of one parameter (the traffic intensity) of the systems. The coefficients of these power series can be recursively calculated for a large class of multi-queue models. The coefficients of the power-series expansions of the moments of the queue length distributions follow directly from those of the state probabillities. In most instances a bilinear transformation ensures convergence of the power series over the whole range of traffic intensities for which the system is stable. We have introduced in Blanc (2,3) extrapolations of the coefficients of the power series in order to accelerate the convergence of the series. One of these extrapolations will be combined with the epsilon algorithm (cf. Brezinski (6), Wynn (13)) in the present paper. The advantages of the present technique are that quantities are calculated iteratively, that it is relatively easy to compute additional terms of the power series in order to increase accuracy, that algorithms for accelerating the convergence of sequences can be applied, and that, once the coefficients of the
TL;DR: It is deduced that the stationary virtual and actual waiting times in aGI/PH/1 queue are of phase type and the equivalence of the methods of Neuts and those of Sengupta is established.
Abstract: We consider the single server queuesN/G/1 andGI/N/1 respectively in which the arrival process or the service process is a Neuts Process, and derive the matrix-exponential forms of the solution of relevant nonlinear matrix equations for such queues. We thereby generalize the matrix-exponential results of Sengupta forGI/PH/1 and of Neuts forMMPP/G/1 to substantially more general models. Our derivation of the results also establishes the equivalence of the methods of Neuts and those of Sengupta. A detailed analysis of the queueGI/N/1 is given, and it is noted that not only the stationary distribution at arrivals but also at an arbitrary time is matrix-geometric. Matrix-exponential steady state distributions are established for the waiting times in the queueGI/N/1. From this, by appealing to the duality theorem of Ramaswami, it is deduced that the stationary virtual and actual waiting times in aGI/PH/1 queue are of phase type.
TL;DR: An open queueing network having multiple classes, priorities, and general service time distributions is considered, and it is conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion.
Abstract: In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.
TL;DR: This paper considers models similar to the one in Shanthikumar but uses the work in the system as the starting point of the investigation and derives new steady state results and conditional waiting time and queue length distributions, and demonstrates that the results of Boxma and Groenendijk follow as special cases.
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.
TL;DR: This paper introduces a simple approximation that makes use of known results for the variance of the time a marked job takes to complete a round trip and an approximate correction term for the covariance between successive round trips for standard deviation of the number of outputs.
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.
TL;DR: First-order and second-order qualitative results are obtained for the performance measures of interest both in the transient and in the steady-state cases of the Erlang blocking model.
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.
TL;DR: This paper shows how a queueing network model helped to uncover the causes of delay in a health center appointment clinic and yielded insight into the operation of the appointment clinic.
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.
TL;DR: A birth-death queueing system with two identical servers, first-come first-served discipline, and Poisson arrivals is considered, and explicit expressions are obtained for the first and second moments of the waiting and sojourn times.
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
TL;DR: AnM/G/1 retrial queue in which blocked customers may leave the system forever without service is considered, and a numerical algorithm is developed for the calculation of the server utilization.
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.
TL;DR: 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.
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.
TL;DR: This work considers a queueing system with two stations served by a single server in a cyclic manner, and obtains the approximate mean delay of customers in the system.
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.
TL;DR: 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.
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.
TL;DR: A search procedure is discussed to obtain the mean waiting time of an arbitrary passenger at each terminal for given control valuesm1 andm2 to minimize the total expected cost per unit time under a linear cost structure.
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.
TL;DR: A discrete-time single-server queue where batches of messages arrive is studied, considers the cases of constant spacing and geometrically distributed (random) spacing between consecutive packets of a message and derives a functional equation which involves a boundary function.
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.
TL;DR: Recently developed methods of qualitative analysis for regenerative processes arising in queueing are presented, which lead to good quantitative estimates of underlying properties with computer methods available to calculate them.
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.
TL;DR: 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 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.
TL;DR: Numerical results indicate that in steady state the non-renewal superposition arrival process, the Non-Renewal overflow process, and theNon-renewsal departure process of the above system can be approximated with compatible renewal processes.
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.
TL;DR: Monotonicity results for loss probabilities under time scaling as well as under amplitude scaling ofλt.v.'s are obtained and both lower and upper bounds for the loss probability are obtained.
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.
TL;DR: This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state and can be applied to the variety of queueing models that are close siblings of M/g/1.
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.