The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States

We consider the stationary distribution of the M/GI/1 type queue when background states are countable. We are interested in its tail behavior. To this end, we derive a Markov renewal equation for characterizing the stationary distribution using a Markov additive process that describes the number of customers in system when the system is not empty. Variants of this Markov renewal equation are also derived. It is shown that the transition kernels of these renewal equations can be expressed by the ladder height and the associated background state of a dual Markov additive process. Usually, matrix analysis is extensively used for studying the M/G/1 type queue. However, this may not be convenient when the background states are countable. We here rely on stochastic arguments, which not only make computations possible but also reveal new features. Those results are applied to study the tail decay rates of the stationary distributions. This includes refinements of the existence results with extensions.     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.     

Single server retrial queues with two way communication

The main aim of this paper is to study the steady state behavior of an M/G/1-type retrial queue in which there are two flows of arrivals namely ingoing calls made by regular customers and outgoing calls made by the server when it is idle. We carry out an extensive stationary analysis of the system, including stability condition, embedded Markov chain, steady state joint distribution of the server state and the number of customers in the orbit (i.e., the retrial group) and calculation of the first moments. We also obtain light-tailed asymptotic results for the number of customers in the orbit. We further formulate a more complicate but realistic model where the arrivals and the service time distributions are modeled in terms of the Markovian arrival process (MAP) and the phase (PH) type distribution.     

The Discrete-Time GI/Geo/1 Queue with Multiple Vacations

We study a discrete-time GI/Geo/1 queue with server vacations. In this queueing system, the server takes vacations when the system does not have any waiting customers at a service completion instant or a vacation completion instant. This type of discrete-time queueing model has potential applications in computer or telecommunication network systems. Using matrix-geometric method, we obtain the explicit expressions for 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.     

Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue.     

The impact of production time variability on make-to-stock queue performance

In this paper, we exploit the distributional Little’s law to obtain the steady-state distribution of the number of customers in a GI/G/1 make-to-stock queueing system. Non-exponential service times in make-to-stock queue modeling are usually avoided or at best, considered in approximations due to difficulties in developing an exact method. By providing a numerical solution of the GI/G/1 make-to-stock queue, we observed the impact of production time variability on optimal inventory control policies. The numerical results prove the degree of errors in the results if an exponential service time distribution were assumed instead of the actual distribution.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times

In this paper we study a single-server queue where the inter-arrival times and the service times depend on a common discrete time Markov chain. This model generalizes the well-known MAP/G/1 queue by allowing dependencies between inter-arrival and service times. The waiting time process is directly analyzed by solving Lindley's equation by transform methods. The Laplace–Stieltjes transforms (LST) of the steady-state waiting time and queue length distribution are both derived, and used to obtain recursive equations for the calculation of the moments. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times. An erratum to this article is available at .     

Analysis of the MAP/G/1/N queue with multiple vacations

This paper deals with a batch service queue and multiple vacations. The system consists of a single server and a waiting room of finite capacity. Arrival of customers follows a Markovian arrival process (MAP). The server is unavailable for occasional intervals of time called vacations, and when it is available, customers are served in batches of maximum size ‘b’ with a minimum threshold value ‘a’. We obtain the queue length distributions at various epochs along with some key performance measures. Finally, some numerical results have been presented.     

On the Stationary Distribution of the GI X /M Y /1 Queueing System

For the GI ^X /M/1 queue, it has been recently proved that there exist geometric distributions that are stochastic lower and upper bounds for the stationary distribution of the embedded Markov chain at arrival epochs. In this note we observe that this is also true for the GI ^X /M ^Y /1 queue. Moreover, we prove that the stationary distribution of its embedded Markov chain is asymptotically geometric. It is noteworthy that the asymptotic geometric parameter is the same as the geometric parameter of the upper bound. This fact justifies previous numerical findings about the quality of the bounds.     

Asymptotics in the MAP/G/1 Queue with Critical Load

When the offered load ρ is 1, we investigate the asymptotic behavior of the stationary measure for the MAP/G/1 queue and the asymptotic behavior of the loss probability for the finite buffer MAP/G/1/K + 1 queue. Unlike Baiocchi [Stochastic Models 10(1994):867–893], we assume neither the time reversibility of the MAP nor the exponential moment condition for the service time distribution. Our result generalizes the result of Baiocchi for the critical case ρ = 1 and solves the problem conjectured by Kim et al. [Operations Research Letters 36(2008):127–132].     

A Regularly Preemptive Model and its Approximations

A regularly preemptive model D,MAP/D ₁,D ₂/1 is studied. Priority customers have constant inter-arrival times and constant service times. On the other hand, ordinary customers' arrivals follow a Markovian Arrival Process (MAP) with constant service times. Although this model can be formulated by using the piecewise Markov process, there remain some difficult problems on numerical calculations. In order to solve these problems, a novel approximation model MAP/MR/1 with Markov renewal services is proposed. These two queueing processes become different due to the existence of idle periods. Thus, a MAP/MR/1 queue with a general boundary condition is introduced. It is a model with the exceptional first service in each busy period. In particular, two special models are studied: one is a warm-up queue and the other is a cool-down queue. It can be proved that the waiting time of ordinary customers for the regular preemption model is stochastically smaller than the waiting time of the former model. On the other hand, it is stochastically larger than the waiting time of the latter model.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.     

A Discrete-Time Geo /G/1 Retrial Queue with General Retrial Times

We consider a discrete-time Geo/G/1 retrial queue in which the retrial time has a general distribution and the server, after each service completion, begins a process of search in order to find the following customer to be served. We study the Markov chain underlying the considered queueing system and its ergodicity condition. We find the generating function of the number of customers in the orbit and in the system. We derive the stochastic decomposition law and as an application we give bounds for the proximity between the steady-state distributions for our queueing system and its corresponding standard system. Also, we develop recursive formulae for calculating the steady-state distribution of the orbit and system sizes. Besides, we prove that the M/G/1 retrial queue with general retrial times can be approximated by our corresponding discrete-time system. Finally, we give numerical examples to illustrate the effect of the parameters on several performance characteristics.     

A discrete-time Geo[X]/G/1 retrial queue with control of admission

This paper analyses a discrete-time Geo/G/1 retrial queue with batch arrivals in which individual arriving customers have a control of admission. We study the underlying Markov chain at the epochs immediately after the slot boundaries making emphasis on the computation of its steady-state distribution. To this end we employ numerical inversion and maximum entropy techniques. We also establish a stochastic decomposition property and prove that the continuous-time M/G/1 retrial queue with batch arrivals and control of admission can be approximated by our discrete-time system. The outcomes agree with known results for special cases.     

A Recursive Algorithm for State Dependent GI/M/1/N Queue with Bernoulli-Schedule Vacation

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1???q (if there are customers in the queue), or continues the vacation with probability q (0?≤?q?≤?1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.     

A G/M/1 retrial queue with constant retrial rate

In this paper, we are concerned with the analytical treatment of an GI/M/1 retrial queue with constant retrial rate. Constant retrial rate is typical for some real world systems where the intensity of individual retrials is inversely proportional to the number of customers in the orbit or only one customer from the orbit is allowed to make the retrials. In our model, a customer who finds the server busy joins the queue in the orbit in accordance with the FCFS (first-come-first-out) discipline and only the oldest customer in the queue is allowed to make the repeated attempts to reach the server. A distinguishing feature of the considered system is an arbitrary distribution of inter-arrival times, while the overwhelming majority of the papers is devoted to the retrial systems with the stationary Poisson arrival process. We carry out an extensive analytical analysis of the queue in steady state using the well-known matrix analytic technique. The ergodicity condition and simple expressions for the stationary distributions of the system states at pre-arrival, post-arrival and arbitrary times are derived. The important and difficult problem of finding the stationary distribution of the sojourn time is solved in terms of the Laplace–Stieltjes transform. Little’s formula is proved. Numerical illustrations are presented.