Pre-arrival Epochs Research Articles

Analysis of stationary discrete-time GI/D-MSP/1 queue with finite and infinite buffers

This paper considers a single-server queueing model with finite and infinite buffers in which customers arrive according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process (D-MSP). This service process is similar to the discrete-time Markovian arrival process (D-MAP), where arrivals are replaced with service completions. Using the imbedded Markov chain technique and the matrix-geometric method, we obtain the system-length distribution at a prearrival epoch. We also provide the steady-state system-length distribution at an arbitrary epoch by using the supplementary variable technique and the classical argument based on renewal-theory. The analysis of actual-waiting-time (in the queue) distribution (measured in slots) has also been investigated. Further, we derive the coefficient of correlation of the lagged interdeparture intervals. Moreover, computational experiences with a variety of numerical results in the form of tables and graphs are discussed.

On the batch arrival batch service queue with finite buffer under server’s vacation: MX/GY/1/N queue

This paper considers a finite-buffer batch arrival and batch service queue with single and multiple vacations. The steady-state distributions of the number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Finally, various performance measures such as average queue length, average waiting time, probability that the server is busy, blocking probabilities, etc. are discussed along with some numerical results. The effect of certain model parameters on the key performance measures have also been investigated. The model has potential application in several areas including manufacturing, internet web-server and telecommunication systems.

GI/M/1 Queues with Server Vacations and Dependent Interarrival and Service Times

We derive the equilibrium distribution at pre-arrival and arbitrary epochs and the waiting time distribution in a GI/M/1 queueing system with dependence between the service time of each customer and the subsequent interarrival times. In addition, the server takes exponentially distributed vacations when there are no customers left to serve in the queue.

Analyzing discrete-time D- BMAP/ G/1/ N queue with single and multiple vacations

This paper analyzes a single-server finite-buffer vacation (single and multiple) queue wherein the input process follows a discrete-time batch Markovian arrival process (D-BMAP). The service and vacation times are generally distributed and their durations are integral multiples of a slot duration. We obtain the state probabilities at service completion, vacation termination, arbitrary, and prearrival epochs. The loss probabilities of the first-, an arbitrary- and the last-customer in a batch, and other performance measures along with numerical aspects have been discussed. The analysis of actual waiting time of these customers in an accepted batch is also carried out.

Recursive method to calculate reliability of (M, N) systems with service facility and general vacation for repairman

A fast and memory efficient recursive method is proposed to calculate the steady state probabilities of M-out-of-N:G reliability model with vacation for repairman. It is assumed that the repair time is exponential, failure time has general distribution and vacation times are independently and identically distributed random variables with general distribution. Using the embedded Markov chain techniques steady state probabilities are computed at pre-arrival epochs and at random epochs. The distribution of waiting time of a unit before the service station (excluding the one with the service facility) has been derived.

Discrete-time [formula omitted] queues with single and multiple vacations

In this paper, we consider a discrete-time single-server finite-buffer batch arrival queue in which customers are served in batches according to a general bulk-service rule, that is, at least ‘ a ’ customers are needed to start a service with a maximum serving capacity of ‘ b ’ customers. The server takes a (single and multiple) vacation as soon as the queue length falls below ‘ a ’ at the completion of service. The interarrival times of batches are assumed to be independent and geometrically distributed. The service times of the batches and vacation times of the server are generally distributed and their durations are integral multiples of slot duration. We obtain queue length distributions at service completion, vacation termination, arbitrary, and prearrival epochs. Finally, we discuss various performance measures and numerical results.

Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations

This paper treats a discrete-time single-server finite-buffer exhaustive (single- and multiple-) vacation queueing system with discrete-time Markovian arrival process (D-MAP). The service and vacation times are generally distributed random variables and their durations are integral multiples of a slot duration. We obtain the queue-length distributions at departure, service completion, vacation termination, arbitrary and prearrival epochs. Several performance measures such as probability of blocking, average queue-length and the fraction of time the server is busy have been discussed. Finally, the analysis of actual waiting time under the first-come-first-served discipline is also carried out.

Computing queue length distributions in MAP/ G/1/ N queue under single and multiple vacation

This paper studies a single server queue with finite waiting room in which the server takes vacation(s) whenever the system becomes empty and we consider both single and multiple vacation(s). Whereas the input process is a Markovian Arrival Process (MAP), the service and vacation times are arbitrarily distributed. The distributions of number of customers in the queue at service completion, vacation termination, departure, arbitrary and pre-arrival epochs have been obtained. Computational procedure has been given when the service- and vacation-time distributions are of phase type (PH-distribution).

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).

On a finite-buffer bulk-service queue with disasters

We deal with a finite-buffer bulk-service queue with disasters. The arrival streams of units and disasters are Markovian arrival processes (MAPs). We study the stationary distribution of the embedded Markov chain at post-departure epochs. The block structure allows us to derive a general approach amenable to numerical calculation following results of the theory for censored Markov chains and level-dependent quasi-birth-and-death processes. We give tractable analytical formulas for the departure process and the stationary distributions of the system state at arbitrary and pre-arrival epochs. The effect of the disaster stream on certain probabilistic descriptors is graphically illustrated.

The Impact of Self-Generation of Priorities on Multi-Server Queues with Finite Capacity

This paper deals with multi-server queues with a finite buffer of size N in which units waiting for service generate into priority at a constant rate, independently of other units in the buffer. At the epoch of a unit's priority generation, the unit is immediately taken for service if there is one unit in service that did not generate into priority while waiting; otherwise such a unit leaves the system in search of immediate service elsewhere. The arrival stream of units is a Markovian arrival process (MAP) and service requirements are of phase (PH) type. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a finite quasi-birth-and-death (QBD) process. We give formulas well suited for numerical computation for a variety of performance measures, including the blocking probability, the departure process, and the stationary distributions of the system state at pre-arrival epochs, at post-departure epochs and at epochs at which arriving units are lost. Illustrative numerical examples show the effect of several parameters on certain probabilistic descriptors of the queue for various levels of congestion.

Analysis of the discrete-time bulk-service queue Geo/G Y/1/ N+ B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G Y/1/ N+ B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.

The Analysis of a General Input Queue with N Policy and Exponential Vacations

This paper studies a single removable server in a G/M/1 queueing system with finite capacity where the server applies an N policy and takes multiple vacations when the system is empty. 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 and deterministic interarrival time distributions. We establish the distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting time and the busy period.

The operating characteristic analysis on a general input queue with N policy and a startup time

This paper studies a single removable server in a finite capacity G/M/1/K queueing system with combined N policy and a exponential startup before each service period. 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. The distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting time (in the queue) and the busy period are established.

The Relations among the queue Size Distributions at Departure, Arbitrary and Pre-arrival Epochs in the MAP/G/1 Queue With Finite/infinite Buffer — an Alternative Approach

In this paper we consider a single server finite buffer queue wherein arrivals are governed by Markovian arrival process (MAP) and the service times are arbitrarily distributed. Using the supplementary variable method, the relations among the limiting queue size distributions at post-departure, arbitrary and pre-arrival epochs are obtained. Furthermore, similar results have also been derived for the infinite buffer queue. These relations are consistent with the known results and the method of derivation is simple and elegant.

Analysis of the MAP/G a , b /1/N Queue

The Markovian arrival process (MAP) is used to represent the bursty and correlated traffic arising in modern telecommunication network. In this paper, we consider a single server finite capacity queue with general bulk service rule in which arrivals are governed by MAP and service times are arbitrarily distributed. The distributions of the number of customers in the queue at arbitrary, post-departure and pre-arrival epochs have been obtained using the supplementary variable and the embedded Markov chain techniques. Computational procedure has been given when the service time distribution is of phase type.

Performance modelling of the M/G/1 machine repairman problem with cold-, warm- and hot-standbys

This paper presents the modelling and analysis of the M/G/1 machine repairman problem consisting of K working machines, Y spare machines and single repairman. Spare machines are considered to be either cold-standby, warm-standby or hot-standby. Using supplementary variable technique, an efficient recursive algorithm has been developed to obtain the distributions of the number of down machines at arbitrary, departure and pre-arrival epochs. The relations among the distributions at various epochs and some performance measures are also discussed. In addition, the waiting time analysis has been carried out. Finally, some numerical results have been presented.

On the analysis of single server finite queue with state dependent arrival and service processes:M(n)/G(n)/1/K

This paper discusses the analysis of a single server finite queue with Poisson arrival and arbitrary service time distribution, wherein the arrival rates are state dependent which are all distinct or all equal, service times are conditioned on the system length at the moment of service initiation. The analytic analysis of the queue is carried out and the final results have been presented in the form of recursive equations which can be easily programmed on any PC to obtain the distributions of number of customers in the system at arbitrary, departure and pre-arrival epochs. It is shown that the method works for all service time distributions including the non-phase type and also for low and high values of the model parameters. Some performance measures, and relations among the state probabilities at arbitrary, departure and pre-arrival epochs are also discussed. Furthermore, it is shown that results for a number of queueing models can be obtained from this model as special cases. To demonstrate the effectiveness of our method some numerical examples have been presented.

On the queueing system

This paper considers an infinite server queue in discrete time in which arrivals are in batches of variable size X and service is provided in batches of fixed size R. We obtain analytical results for the number of busy servers and waiting customers at pre-arrival and arbitrary epochs. For the number of busy servers we obtain a recursive relation for the partial binomial moments and explicit expressions for the marginal binomial moments both in transient and steady states. Special cases are also considered

Queue-length and waiting-time distributions of discrete-time GI^{X}/Geom/1 queueing systems with early and late arrivals

In this paper, we give a unified approach to solving discrete-time GI^{X}/Geom/1 queues with batch arrivals. The analysis has been carried out for early- and late-arrival systems using the supplementary variable technique. The distributions of numbers in systems at prearrival epochs have been expressed in terms of roots of associated characteristic equations. Furthermore, distributions at arbitrary as well as outside observer’s observation epochs have been obtained using the relation derived in this paper. We also present delay analyses for both the systems. Numerical results are presented for various interarrival-time and batch-size distributions.