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Although queueing models in the mathematical field of queueing theory are mainly studied in the steady-state regime and practical applications are interested mostly in queues in transient situations subject to burst arrivals and congestion, their use is still justified as a first step towards a more complex and thorough analysis. In these practical applications, parameters such as the traffic intensity, which is the ratio between the arrival rate and service rate, are unknown and need to be estimated statistically. In this study, a Markovian arrival and deterministic single-server queueing system, known in Kendall notation as an M ∕ D ∕ 1 queueing model, is considered. This is one of the simplest queueing models with deterministic service time and may be seen as an approximation of a variety of applications in the performance evaluation of production management, telecommunications networks, and other areas. The main goal of this manuscript is to propose a methodology to determine the sample size for an M ∕ D ∕ 1 queueing system under the Bayesian setup by observing the number of customer arrivals during the service time of a customer. To verify the efficiency and efficacy of the proposed approach, an extensive set of numerical results is presented and discussed.

A certain type of queueing system that is quite common in manufacturing systems occurs when the time between the arrivals of items approximately follows an exponential distribution with rate λ , the services are mechanized and their times may be considered approximately constant ( b ). In Kendall notation, such a queueing system is well known as an M ∕ D ∕ 1 queue; despite being one of the simplest queueing models, it has wide applicability to numerous practical situations as a first approximation by a steady-state model before a deeper analysis can be performed by means of more sophisticated transient-regime stochastic models that consider, for example, burst arrival, block arrivals, congestion, and so on. In queues, one very important parameter that must estimated is the traffic intensity, defined for an M ∕ D ∕ 1 queue as ρ = λ b . This article aims to investigate statistical methods to estimate ρ , namely, the maximum likelihood and Bayes estimators, by considering the number of customers present in the system at successive departure epochs, which is a very natural way to collect data. An extensive set of computational results from Monte Carlo simulations is shown to establish the efficiency and effectiveness of the proposed approaches, which will possibly enhance practical applications.

An M / M / C queueing system operating in a Markovian environment is studied. This paper focuses on the stationary behavior and presents the theoretical framework. For a special case, analytical results are derived that are analogous to the classical solutions for the simple M / M / C queue. The elaborate analysis of a specific case is given to illustrate the basic idea of the framework. A technical proof with respect to the existence of d − 1 roots is displayed to sustain the corresponding theory.

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the non-preemptive case) can be represented by the duration of the busy period of a special Markovian fluid model. By making use of the recent results on the busy period analysis of Markovian fluid models it is possible to calculate several queueing performance measures in an efficient way including the sojourn time distribution (both in the time domain and in the Laplace transform domain), the moments of the sojourn time, the generating function of the queue length, the queue length moments and the queue length probabilities.

This article discusses the steady state analysis of the M / G / 2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served ‘(FCFS)’ queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate μ or by server-II with a general service time distribution B ( t ) . Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.

Closed, finite queueing networks are applicable to many different manufacturing and service system settings. The incorporation of material handling and transportation networks in finite buffer closed queueing networks is studied. A novel queue decomposition approach using state dependent queues to capture the buffer of finite M/M/1/K queues is shown to be a viable approach for modelling these systems. Each M / M / 1 / K queue is replaced with a coupled state dependent queue plus an M/M/1 queue. An extended mean value analysis (MVA) algorithm is employed to demonstrate the integration of the state dependent queues for the buffers in the approach. Under certain restrictions concerning the network population, finite queueing networks with the state dependent queues acting as buffers should have a product form distribution. This paper focuses on M/M/1/K queues and their transformation while future papers will treat the multi-server case. Several different closed series (i.e. cyclic), merge, and split topological systems of finite queues are analyzed and presented.

By studying the spectral properties of the underlying operator corresponding to the M/G/1 queueing model with optional second service we obtain that the time-dependent solution of the model strongly converges to its steady-state solution. We also show that the time-dependent queueing size at the departure point converges to the corresponding steady-state queueing size at the departure point.

An efficient numerical evaluation technique is presented for families of finite Markov chains indexed by a parameter e, that possess a monotonicity property for the chain at e = 0. The monotonicity allows for retrieving the series expansion of the steady state solution in e = 0 in O( NM ) operations, where N is the size of the state space and M is the number of terms in the series expansion. We then apply the method to a layered queueing network with four queues, three queues in tandem and one queue for the servers operating this tandem queue.

In this paper we present an exact steady-state analysis of a discrete-time Geo/G/1 queueing system with working vacations, where the server can keep on working, but at a slower speed during the vacation period. The transition probability matrix describing this queuing model can be seen as an M/G/1-type matrix form. This allows us to derive the probability generating function (PGF) of the stationary queue length at the departure epochs by the M/G/1-type matrix analytic approach. To understand the stationary queue length better, by applying the stochastic decomposition theory of the standard M/G/1 queue with general vacations, another equivalent expression for the PGF is derived. We also show the different cases of the customer waiting to obtain the PGF of the waiting time, and the normal busy period and busy cycle analysis is provided. Finally, we discuss various performance measures and numerical results, and an application to network scheduling in the wavelength division-multiplexed (WDM) system illustrates the benefit of this model in real problems.

We consider a non-cooperative queueing environment where a finite number of customers independently choose when to arrive at a queueing system that opens at a given point in time and serves customers on a last-come first-serve preemptive-resume (LCFS-PR) basis. Each customer has a service time requirement which is identically and independently distributed according to some general probability distribution, and they want to complete service as early as possible while minimizing the time spent in the queue. In this setting, we establish the existence of an arrival time strategy that constitutes a symmetric (mixed) Nash equilibrium, and show that there is at most one symmetric equilibrium. We provide a numerical method to compute this equilibrium and demonstrate by a numerical example that the social efficiency can be lower than the efficiency induced by a similar queueing system that serves customers on a first-come first-serve (FCFS) basis.

Finite buffer, single-server queueing systems and networks are difficult to analyze since the length of time a customer spends in the system does not follow the Markovian property. A two-moment approximation schema is developed for the probability distribution of M/G/1/K systems and extended to the analysis of M/G/1/K queueing networks. The general purpose of this paper is to develop a flexible and practical transform-free approach for computing the probability distribution and performance measures of the system as well as identify the underlying properties of these systems. It is shown that for most performance measures, a sigmoid or S-shaped curve with an inflection point at @r=1 appears as K->~. This has direct implications for the analysis and optimization of such systems. The performance modelling of the M/G/1/K queueing networks of general topologies along with extensive numerical results accompany the paper along with the linear concave performance measures for these systems.

We consider an infinite-buffer single-server queue with renewal input. The service to the queueing system is provided in batches of random size, according to a batch Markovian service process (BMSP). The queue length distribution of the number of customers in the system at pre-arrival and arbitrary epochs has been obtained along with some important performance measures, such as the mean number of customers in the system and the mean system sojourn time of a customer. Secondly, we study a similar queueing system with queue-length-dependent inter-arrival times and obtain the above-mentioned state probabilities and performance measures. These queueing models have potential applications in the areas of computer networks, telecommunication systems, manufacturing systems, etc.

This paper presents an approximating model for a 2-class weighted fair queueing (or random polling) model. The approximating system can be analyzed analytically to obtain mean performance measures such as expected delay. We show through a formal argument that the approximation works well when the overall utilization of the system @r is small. Based on simulation experiments, we develop a modified version of the approximation that is accurate for a wide range of @r. Finally, we extend the approximation to more complex queueing scenarios, such as the low-latency-queueing discipline and systems with more than 2 classes.

We consider an M/M/c queue with c=2, in which the number of working servers can be adjusted one at a time at arrival epochs or at service completion epochs depending on the number of customers in the system. Analytic closed-form solutions of the infinite capacity M/M/2 queueing system operating under the triadic (0,Q,N,M) policy are derived. The total expected cost function per unit time is developed, to obtain the optimal operating (0,Q,N,M) policy and the optimal service rate, at minimum cost. Some illustrative examples are provided and the genetic algorithm is employed to search for the optimal management policy of the multi-server queueing system.

This paper studies the transientsolution of the Markovian queue, namely M/Ek/1model. Anew approach is introduced for finding exact solution for this queueing system. The transient probabilities are expressed recurrently.

This paper proposes a new approach for the time-dependent analysis of stochastic and non-stationary queueing systems. The analysis of a series of stationary queueing models leads to a new approximation of time-dependent performance measures. Based on a stationary backlog-carryover (SBC) approximation of the time-dependent expected utilization, different approximations of the time-dependent expected queue length and the number of customers in the system are discussed. Limiting results are given for the case of constant rates. The accuracy of the SBC approach is shown for non-stationary M(t)/M(t)/c(t) queueing systems with time-dependent and piecewise constant arrival rates. In numerical experiments we demonstrate the reliability of this approach and compare it with the (lagged) stationary independent period by period (SIPP) approach. In addition, the approximation is applied to temporarily overloaded systems that cannot be analyzed by the variants of the SIPP approach.

In this paper, we study the discrete time GI^X/G^Y/1 queueing system. First, some general results are obtained for the stability condition, stationary distributions of the queue lengths and waiting times. Then we show that, for some practical situations, the queueing system of interest has the properties of both the M/G/1 and GI/M/1 non-skip-free Markov chains described by [H.R. Gail, S.L. Hantler, B.A. Taylor, Non-skip-free M/G/1 and GI/M/1 types of Markov chains, Advances in Applied Probability, 29 (3) (1997) 733-758]. For such cases, the queueing system can be analyzed as a QBD process, a M/G/1 type Markov chain, or a GI/M/1 type Markov chain with finite blocks after re-blocking. Computational procedures are developed for computing a number of performance measures for the queueing system of interest. In addition, we study a GI/M/1 type Markov chain associated with the age process of the customers in service.

This paper deals with the optimal control of a finite capacity G/M/1 queueing system combined the F-policy and an exponential startup time before start allowing customers in the system. The F-policy queueing problem investigates the most common issue of controlling arrival to a queueing system. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distribution of the number of customers in the system. We illustrate a recursive method by presenting three simple examples for exponential, 3-stage Erlang, and deterministic interarrival time distributions, respectively. A cost model is developed to determine the optimal management F-policy at minimum cost. We use an efficient Maple computer program to determine the optimal operating F-policy and some system performance measures. Sensitivity analysis is also studied.

In this paper we consider the problem of designing a GI/M/c queueing system. Given arrival and service data, our objective is to choose the optimal number of servers so as to minimize an expected cost function which depends on quantities, such as the number of customers in the queue. A semiparametric approach based on Erlang mixture distributions is used to model the general interarrival time distribution. Given the sample data, Bayesian Markov Chain Monte Carlo methods are used to estimate the system parameters and the predictive distributions of the usual performance measures. We can then use these estimates to minimize the steady-state expected total cost rate as a function of the control parameter c. We provide a numerical example based on real data obtained from a bank in Madrid.

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.

The independence of processes in queueing systems is generally assumed when developing queueing models. However, real systems often involve several process dependencies, and failure to take these into consideration can lead to serious underestimation of the performance measures. We consider herein a single server queueing system with a Markov renewal process (MRP) for its arrival process and a general service time distribution, and derive the distribution function and correlation coefficient of the departure process. Since the departure process also often corresponds to an arrival process in downstream queues, the results obtained here can be used to derive a better approximation of the performance measures of a non-product form general queueing network.

In this paper, we study an open and nested tandem queueing network, where the population constraint within each subnetwork is controlled by a semaphore queue. The total number of customers that may be present in the subnetwork can not exceed a given value. Each node has a constant service time and the arrival process to the queueing network follows an arbitrary distribution. A major characteristic of this queueing network is that the low layer flow is halted by the state of the high layer. We develop a simple and equivalent queueing network that has the same performance characteristics as the original queueing network. Using this model, the waiting time on the queueing network can be easily derived. It is interesting to see how the simplification process can be applied to multi-layered queueing network.

In this paper a class of finite length Markovian queueing models is studied that, in general, does not exhibit a product form solution. Good approximations can be derived for the marginal queue size distributions in this case, and hence measures such as the average response time can be calculated. However, because no product form exists, expressions for the joint queue size distribution are much more costly to derive, hence many performance measures of interest cannot be easily computed. An approximation for the joint queue size distributions is explored here, which improves on a naive product form assumption by considering various boundary cases. The approximations are explored numerically by example.

Optimal flow control problems of multiple-server (M/M/n) queueing systems are studied. Due to enhanced flexibility of the decision making, intuitively, we expect that grouping together separated systems into one system provides improved performance over the previously separated systems. This paper presents a result counterintuitive against such an expectation. We consider a non-cooperative optimal flow control scheme M/M/n queueing systems where each of multiple players strives to optimize unilaterally its own power where the power of a player is the quotient of the throughput divided by the mean response time for the player. We report a counter-intuitive case where the power of every user degrades after grouping together K(>1) separated M/M/N systems into a single M/M/(K × N) system. Some numerical results are presented.

Recent measurement studies of teletraffic data in modern telecommunication networks have shown that self-similar processes may provide better models of teletraffic than Poisson processes. If this is not taken into account, it can lead to inaccurate conclusions about performance of telecommunication networks. We show how arrival processes with self-similar input influences the run-length of a distributed steady-state simulation of queueing systems in telecommunication networks. For this purpose, the simulation run-length of SSM/M/1/∞ queueing systems in the method based on the batch means, conducted for estimating steady-state mean waiting times is compared with the results obtained from simulations of M/M/1/∞ queueing systems when a single processor and multiple processors are used. We also investigate speedup conducted stochastic simulation of SSM/M/1/∞ queueing systems on multiple processors under a scenario of distributed stochastic simulation known as MRIP (Multiple Replications In Parallel) in a local area network (LAN) environment on Solaris operating system. We show that, assuming self-similar inter-event processes (i.e., SSM/M/1/∞ queueing systems), many more observations are required to obtain the final simulation results with a required precision, as the value of the Hurst parameter H increases, than when assuming Poisson models, exhibiting short-range dependence (i.e., M/M/1/∞ queueing systems) on a single processor and multiple processors. Our results show that the time for collecting many numbers of observations under the MRIP scenario is clearly reduced as traffic intensity and the value of the Hurst parameter increase, and as the engaged processor increases one to four. In particular, the value of H influences much more the speedup than traffic intensity and the engaged processor.

We study a Markovian queueing system with multi-task servers. Each server can perform two types of job-serving the queue as primary jobs and taking vacations as secondary jobs. In such a system with c>1 servers, if at a service completion instant the server finds no customer waiting in line and the number of servers already on vacations is less than d(⩽c), he or she will take a vacation. At any time, the number of servers attending the queue or staying idle is at least c−d. By changing the parameter d, the queueing manager can better allocate the servers’ time to performing the primary jobs and the secondary jobs. Using the matrix analytic method, we provide a new computational algorithm for the stationary distributions of the queue length and waiting time. The conditional stochastic decomposition properties have been established for such a system.

We consider a queueing system consisting of multiple identical servers and a common queue. The service time follows an exponential distribution and the batch arrival process is governed by a semi-Markov process (SMP). Our analysis is based on the theory of piecewise Markov process. We derive the distributions of the queue size and the waiting time. When the sojourn time of SMP follows an exponential distribution all the unknown constants contained in the generating function of queue size can be determined through the zeros of the denominator for this generating function. A numerical example is shown.

In this paper, we study the regular M/G/1 queue with a combination of FCFS and STF queueing disciplines. The advantage of this combined discipline is the reduction of mean waiting time. Customers are divided into two groups: an FCFS group and an STF group. Customers are first put into FCFS group and are then moved into STF group. An embedded Markov chain is formulated at the times when the STF group is empty. Results are compared with regular M/G/1 queues of FCFS. Scope and purpose First-come-first-served (FCFS) queueing discipline probably is the most commonly used rule in queueing systems. To reduce the mean waiting time, shortest-time-first (STF) rule may be considered, but the disadvantage associated with STF is the long waiting time for customers with long service times. In this paper, we attempt to balance these two queueing disciplines, hoping to achieve better system performance. The impact of the combined queueing disciplines is investigated and compared with FCFS.

We study an M/M/c queue with server vacations. In this queueing system, d (≤ c) of c servers take synchronous vacations when these d servers become idle at a service completion instant. This type of queueing model captures the major characteristics of a stochastic service system which processes both random arriving jobs and constantly available jobs. In this paper, the multi-server vacation queueing system has been analyzed as a quasi-birth and death process. Using matrix geometric method, we obtain the stationary distributions of queue length and waiting time and demonstrate the conditional stochastic decomposition property of the queue length and waiting time in this system.

Motivated by a desire to understand a GI/∞ node in a closed queueing network environment, we study in this paper a finite population multiserver model, where the number of processes is equal to the population in the network. We study its steady-state behavior under various scenarios, e.g., the first three moments of the arrival and service distributions. We show that when the arrivals are bursty, an increase in service time variation can actually decrease congestion. Furthermore, such a decrease is also observed for an increase in the skewness in the arrivals. Additionally, several insensitivity properties in limiting behaviors are observed. In particular, the finite population GI/GI///s-loop is insensitive to the service time distribution when the arrival distribution either has an infinite variance (a form of heavy tail), or is highly skewed. In the latter case, the congestion approaches that of the all Markovian system M/M///s-loop. We conclude that the influence of the third moment can be very significant, indicating that the first two moments do not suffice to characterize the performance measures under consideration.

The queueing delay and packet loss performance of priority-based differentiated services (DiffServ) networks are analyzed through the nonpreemptive M/M/m/K finite queueing bulk arrival model. The analytical results reveal the queueing time relation of different priority classes of data traffic due to the differentiated services priority queueing and scheduling control network operations.

We consider a single-server, two-phase queueing system with N-policy. Customers arrive at the system according to a Poisson process and receive batch service in the first phase followed by individual services in the second phase. If the system becomes empty at the moment of the completion of the second-phase services, it is turned off. After an idle period, when the queue length reaches N (threshold), the server is turned on and begins to serve customers. We obtain the system size distribution and show that the system size decomposes into three random variables. The system sojourn time is provided. Analysis for the gated batch service model is also provided. Finally we derive a condition under which the optimal operating policy is achieved.

We consider a two node multiclass queueing network given by two machines each with two classes. There are two streams of jobs. One stream originates in machine 1, which feeds it for further processing to machine 2, and the other stream moves in the opposite direction. We describe a policy for this system which is stable and which keeps both machines busy at all times. We obtain explicit expressions for its steady-state behavior under M/M/. and under M/G/. assumptions. We also describe a similar M machine system.

This paper studies a single removable server in a G/M/1 queueing system with finite capacity operating under the N policy. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. Numerical results for various system performance measures are presented for four different interarrival time distributions such as exponential, 2-stage hyperexponential, 4-stage Erlang, and deterministic.

By using C0-semigroup theory of linear operators, the existence of a unique non-negative time-dependent solution of M/G/1 queueing model with single vacations has been proved.

We introduce P λ M -service policy for an M / G /1 queueing system. The stationary distribution of the workload under this policy is explicitly obtained through a decomposition technique, renewal reward theorem, and level crossing argument.

An r-quorum queueing system under N-policy discipline is considered. The service discipline consists in providing no service until the queue length reaches N, then service begins and continues until the queue length drops below r. We use Neuts' matrix analytic methodology to study this hysteretic queueing model. Numerical implementation is carried out and sample results are presented along with graphs.

We consider a storage model that can be either interpreted as a certain queueing model with dependence between a service request and the subsequent interarrival time, or as a fluid production/inventory model with a two-state random environment. We establish a direct link between the workload distributions of the queueing model and the production/inventory model, and we present a detailed analysis of the workload and waiting time process of the queueing system.

G-networks (or queueing networks with negative customers, signals, triggers, etc.) are characterized by the following feature: in contrast with the normal positive customers, negative customers arriving to a non-empty queue remove an amount of work from the queue. In its simplest version, a negative customer deletes an ordinary positive customer according to some strategy. Extensions of the model result when a negative customer removes a random batch of customers, all the work from the queue or indeed a random amount of work that does not necessarily correspond to an integer number of positive customers. Since Gelenbe (E. Gelenbe, Neural Computation 1 (1989) 502–510; E. Gelenbe, Journal of Applied Probability 28 (1991) 656–663) introduced the notion of negative customers, there has been an increasing interest not only in queueing networks but also in the single server node case. Significant progress in the analysis of this versatile class of networks has enriched queueing theory as well as contributed to the development of real applications in fields such as computers, communications and manufacturing. This paper presents a survey of the main results and methods of the theory of G-networks.

Classical, analytical models of queueing systems are used only under the assumptions that mean service times of resPective channels differ between themselves insignificantly. In practice, such a situation hardly ever happens. In this case applications of the classical models cause certain errors. A new class of analytical models with individual characteristics of each service channel is presented. The developed models are described on the example of M / M / m /FIFO/ m + N queueing system. System of equation for two and three channels with individual service is introduced and solved in the paper. The obtained analytical results are generalised onto m service channels.

This paper studies the classification problem of Markov processes of M/G/1 type with a tree structure. It is shown that the classification of positive recurrence, null recurrence, and transience of the Markov processes of interest is determined completely by the Perron-Frobenius eigenvalue of a nonnegative matrix. The results are used to find classification criteria for a number of discrete time or continuous time queueing systems with multiple types of customers.

This paper studies a multi-server queueing system with multiple types of customers and last-come-first-served (LCFS) non-preemptive service discipline. First, a quasi-birth-and-death (QBD) Markov process with a tree structure is defined and some classical results of QBD Markov processes are generalized. Second, the MMAP[K]/PH[K]/N/LCFS non-preemptive queue is introduced. Using results of the QBD Markov process with a tree structure, explicit formulas are derived and an eAcient algorithm is developed for computing the stationary distribution of queue strings. Numerical examples are presented to show the impact of the correlation and the pattern of the arrival process on the queueing process of each type of customer. ” 2000 Elsevier Science B.V. All rights reserved.

In this note, we study a Markovian queueing system with accessible batches for service, but units depart individually. This system generalizes the M/M/C, bulk service, and accessible batch queueing systems. We compute the system size probabilities in transient and steady states, waiting time distribution, busy period distribution, expected queue length, etc. Little's formula is shown to hold for the present set up. We investigate the optimal values of the maximum batch size for service and capacity of the system.

A BMAP/G/1 retrial queueing system is considered. A sufficient condition for existence of a steady-state distribution is obtained. A matrix differential equation for the vector generating function of the stationary queue distribution is derived. An approximate algorithm for calculating the stationary state probabilities is proposed.

In this paper, we present a @?GI"i/D/1/~ queue with heterogeneous input/output slot times. This queueing model can be regarded as an extension of the ordinary GI/D/1/~ model. For this @?GI"i/D/1/~ queue, we assume that several input streams arrive at the system according to different slot times. In other words, there are different slot times for different input/output processes in the queueing model. The queueing model can therefore be used for an ATM multiplexer with heterogeneous input/output link capacities. Several cases of the queueing model are discussed to reflect different relationships among the input/output link capacities of an ATM multiplexer. In the queueing analysis, two approaches: the Markov model and the probability generating function technique, are adopted to develop the queue length distributions observed at different epochs. This model is particularly useful in the performance analysis of ATM multiplexers with heterogeneous input/output link capacities.

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems , 1986, 1 , 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc. , 1994, 36 , 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M x /M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M x /M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models. This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M x /M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.

We consider a MAP/M/2/K queueing model in which messages should leave the system in the order in which they entered into the system. In the case of infinite resequencing buffer, the steady-state probability vector is shown to be of matrix-geometric type. The total sojourn time of an admitted message into the system is shown to be of phase type. Efficient algorithmic procedures for computing various performance measures are given, and some interesting numerical examples are discussed.

Arrival Instant Theorem is one of the most important theoretical results established in the studies of classical product-form queueing networks. Closed queueing networks with blocking have been shown to have product-form equilibrium distributions in certain cases, e.g. with repetitive service blocking mechanism. In this paper, theoretical results on acceptance instant distribution which is defined as queue length distribution of such networks seen by a customer at the instant of its acceptance to a service node are established and their extensions discussed.

This article concerns the production/inventory system with periodic-review policy. The system is modelled as an M/M/c-type of queue and is analysed by using the embedded Markov chain technique. The probability generating functions for the queue length are derived, which are used to determine the mean queue length, mean job waiting times, and higher moments of these quantities. Analytic results are obtained for M/M/1 queues and M/M/∞ queues with periodic-review. For M/M/c queues of 1 < c < ∞, analytic solutions do not exist. Thus, numerical methods have to be applied. An optimization model is imposed on the queueing model that balances the inventory (waiting) cost and the cost of reviewing the system. The cost is minimized by determining the length of the review period. Both constant and state-dependent lengths of the review period are considered and compared. The optimal review periods are given for various numerical examples.

We consider an open tandem queueing network with population constraint and constant service times. The total number of customers that may be present in the network can not exceed a given value K . Customers arriving at the queueing network when there are more than K customers are forced to wait in an external queue. The arrival process to the queueing network is assumed to be arbitrary. We show that this queueing network can be transformed into a simple network involving only two nodes. Using this simple network, we obtain an upper and lower bound on the mean waiting time. These bounds can be easily calculated. Validations against simulation data establish the tightness of these bounds.