Late Arrival System Research Articles

ANALYTICALLY CLOSED-FORM SOLUTIONS FOR THE DISTRIBUTION OF A NUMBER OF CUSTOMERS SERVED DURING A BUSY PERIOD FOR SPECIAL CASES OF THE GEO/G/1 QUEUE

AbstractThis paper presents the distribution of the number of customers served during a busy period for special cases of the Geo/G/1 queue when initiated with m customers. We analyze the system under the assumptions of a late arrival system with delayed access and early arrival system policies. It is not easy to invert the functional equation for the number of customers served during a busy period except for the simple case Geo/Geo/1 queue, as stated by several researchers. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We find the distribution of the number of customers served during a busy period for various service-time distributions such as geometric, deterministic, binomial, negative binomial, uniform, Delaporte, discrete phase-type and interrupted Bernoulli process. We compute the mean and variance of these distributions and also give numerical results. Due to the clarity of the expressions, the computations are very fast and robust. We also show that in the limiting case, the results tend to the analogous continuous-time counterparts.

Busy Period Analysis of the Geo/Geo/1/K Queue with a Single Vacation

Discrete-time Queueing models are frequently utilized to analyze the performance of computing and communication systems. The length of busy period is one of important performance measures for such systems. In this paper, we consider the busy period of the Geo/Geo/1/K queue with a single vacation. We derive the moments of the length of the busy (idle) period, the number of customers who arrive and enter the system during the busy (idle) period and the number of customers who arrive but are lost due to no vacancies in the system for both early arrival system (EAS) and late arrival system (LAS). In order to do this, recursive equations for the joint probability generating function of the busy period of the Geo/Geo/1/K queue starting with n, 1≤n≤K, customers, the number of customers who arrive and enter the system, and arrive but are lost during that busy period are constructed. Using the result of the busy period analysis, we also numerically study differences of various performance measures between EAS and LAS. This numerical study shows that the performance gap between EAS and LAS increases as the system capacity K decrease, and the arrival rate (probability) approaches the service rate (probability). This performance gap also decreases as the vacation rate (probability) decrease, but it does not shrink to zero.

Analysis of batch Bernoulli process subject to discrete-time renewal generated binomial catastrophes

With the evolution of internet, there has been increased interest in the use of computer-communication networks and digital communication systems. But unusual events, like virus attack, that often results in abrupt change in the state of the system poses a major threat to these systems. The occurrence of a virus may cause immediate removal of some elements (packets) from the system. Such type of situations can be well represented as discrete-time catastrophe model where interruptions due to various types of virus attacks are referred to as catastrophe. In this paper we study a discrete-time catastrophe model in which population grows according to batch Bernoulli process and catastrophes occur according to discrete-time renewal process. When a catastrophe occurs, an element (individual or packet) of the population survives with probability p or dies with probability $$1-p$$. The analysis of the model has been done using supplementary variable technique and steady-state probability generating function of the distribution of population size at post-catastrophe, arbitrary and pre-catastrophe epochs are obtained under late arrival system with delayed access. In order to make the model more useful for practitioners, a step-wise computing process for evaluation of distribution of population size at various epochs for commonly used inter-catastrophe time distributions viz. geometric, deterministic and arbitrary are given. We also present recursive formulae to compute factorial moments of the population size at various epochs. Further to study the effect of critical model parameters, numerical results and some graphs are included.

A Discrete-Time GIX/Geo/1 Queue with Multiple Working Vacations Under Late and Early Arrival System

This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.

The Queue Geo/G/1/N + 1 Revisited

This paper presents an alternative steady-state solution to the discrete-time Geo/G/1/N + 1 queueing system using roots. The analysis has been carried out for a late-arrival system using the imbedded Markov chain method, and the solutions for the early arrival system have been obtained from those of the late-arrival system. Using roots of the associated characteristic equation, the distributions of the numbers in the system at various epochs are determined. We find a unified approach for solving both finite- and infinite- buffer systems. We investigate the measures of effectiveness and provide numerical illustrations. We establish that, in the limiting case, the results thus obtained converge to the results of the continuous-time counterparts. The applications of discrete-time queues in modeling slotted digital computer and communication systems make the contributions of this paper relevant.

Analytically simple and computationally efficient solution to GI[formula omitted]/Geom/1 queues involving heavy-tailed distributions

This paper provides an analytically simple and computationally efficient solution to the discrete-time queueing model GIX /Geom/1. Using the imbedded Markov chain method, the analysis has been carried out for the late-arrival system and the results for the early-arrival system have been derived from those of the late-arrival system. The distributions of the numbers in the system at various epochs are all found in terms of roots of the underlying characteristic equation. Numerical results are also discussed.

GEOM/GEOM[a]/1/ queue with late arrival system with delayed access and delayed multiple working vacations

This paper considers a discrete-time bulk-service queue with infinite buffer space and delay multiple working vacations. Considering a late arrival system with delayed access (LAS-AD), it is assumed that the inter-arrival times, service times, vacation times are all geometrically distributed. The server does not take a vacation immediately at service complete epoch but keeps idle period. According to a bulk-service rule, at least one customer is needed to start a service with a maximum serving capacity 'a'. Using probability analysis method and displacement operator method, the queue length and the probability generating function of waiting time at pre-arrival epochs are obtained. Furthermore, the outside observer?s observation epoch queue length distributions are given. Finally, computational examples with numerical results in the form of graphs and tables are discussed.

Analysis of discrete-time multi-server queue with balking

This paper analyses a discrete-time multi-server queue with balking behavior. The inter-arrival and service times are assumed to be independent and geometrically distributed. If all the servers are busy, an arriving customer either enters with probability b or balks with probability 1 − b. Otherwise, the service of the arriving customer commences immediately. The system is analysed under the assumptions of an early arrival system and a late arrival system with delayed access. Closed-form expressions for the steady-state probabilities at arbitrary epochs and an outside observer's observation epoch probabilities are obtained. Various performance measures along with some numerical results in the form of a table and graphs are presented. Some special cases of the model have also been discussed. We have shown that the obtained results converge to the results of the continuous-time counterparts in the limiting case.

Discrete-time GI/G/1 retrial queues with time-controlled vacation policies

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation times using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death (LDQBD) process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies: deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system (EAS) and the late arrival system with delayed access (LAS-DA). Significant difference between EAS and LAS-DA is illustrated by some numerical examples.

Performance analysis of a discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

This paper presents the analysis of a discrete-time Geo/G/1 queue with randomized vacations. Using the probability decomposition theory and renewal process, two variants on this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both the cases, recursive solution for queue length distributions at arbitrary, just before a potential arrival, pre-arrival, immediately after potential departure, and outside observer’s observation epochs are obtained. Further, various performance measures such as potential blocking probability, turned-on period, turned-off period, vacation period, expected length of the turned-on circle period, average queue length and sojourn time, etc. have been presented. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

Analysis of discrete-time queues with batch renewal input and multiple vacations

This paper analyzes a discrete-time multiple vacations finite-buffer queueing system with batch renewal input in which inter-arrival time of batches are arbitrarily distributed. Service and vacation times are mutually independent and geometrically distributed. The server takes vacations when the system does not have any waiting jobs at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the imbedded Markov chain techniques, the authors obtain the queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs for partial-batch rejection policy. The blocking probability of the first-, an arbitraryand the last-job in a batch have been discussed. The analysis of actual waiting-time distributions measured in slots of the first-, an arbitrary- and the last-job in an accepted batch, and other performance measures along with some numerical results have also been investigated.

The recursive solution of queue length for Geo/G/1 queue with N-policy

This paper considers a discrete-time queue with N-policy and LAS-DA (late arrival system with delayed access) discipline. By using renewal process theory and probability decomposition techniques, the authors derive the recursive expressions of the queue-length distributions at epochs n −, n +, and n. Furthermore, the authors obtain the stochastic decomposition of the queue length and the relations between the equilibrium distributions of the queue length at different epochs (n −, n +, n and departure epoch D n ).

Analysis of Finite-Buffer Discrete-Time Batch-Service Queue with Multiple Vacations

This paper investigates a discrete-time single-server finite-buffer batch-service queue with multiple vacations wherein arrivals occur according to a discrete-time renewal process. Service and vacation times are mutually independent and geometrically distributed. The service is performed in batches and the server takes vacations when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. The system is analyzed under the assumptions of late arrival system with delayed access and early arrival system. Using the supplementary variable and the embedded Markov chain techniques, we obtain the queue-length distributions at pre-arrival, arbitrary and outside observer's observation epochs. Some performance measures, waiting-time distribution in the system and cost analysis have also been discussed. Finally, a variety of numerical results as tables and graphs showing the effect of model parameters on key performance measures are presented.

The recursive solution for Geom/G/1(E, SV) queue with feedback and single server vacation

Using recursive method, this paper studies the queue size properties at any epoch n + in Geom/G/1(E, SV) queueing model with feedback under LASDA (late arrival system with delayed access) setup. Some new results about the recursive expressions of queue size distribution at different epoch (n +, n, n −) are obtained. Furthermore the important relations between stationary queue size distribution at different epochs are discovered. The results are different from the relations given in M/G/1 queueing system. The model discussed in this paper can be widely applied in many kinds of communications and computer network.

Transient Queue Size Distribution Solution of Geom / G / 1 Queue with Feedback-A Recursive Method

This paper considers the Geom / G / 1 queueing model with feedback according to a late arrival system with delayed access (LASDA). Using recursive method, this paper studies the transient property of the queue size from the initial state N(0+) = i. Some new results about the recursive expression of the transient queue size distribution at any epoch n+ and the recursive formulae of the equilibrium distribution are obtained. Furthermore, the recursive formulae of the equilibrium queue size distribution at epoch n−, and n are obtained, too. The important relations between stationary queue size distributions at different epochs are discovered (being different from the relations given in M / G / 1 queueing system). The model discussed in this paper can be widely applied in all kinds of communications and computer network.

Analysis of finite capacity discrete-time GI/Geo/1 queueing system with multiple vacations

This paper investigates a discrete-time single-server finite-buffer queueing system with multiple vacations in which arrivals occur according to a discrete-time renewal process. Service and vacation times are mutually independent and geometrically distributed. We obtain steady-state system length distributions at prearrival, arbitrary and outside observer's observation epochs under the late arrival system with delayed access and early arrival system. The analysis of actual waiting-time for both the systems has also been carried out. The model has potential application in high-speed computer network, digital communication systems and other related areas.

Transient solution for queue-length distribution of Geometry/G/1 queueing model

In this paper, the Geometry/G/1 queueing model with inter-arrival times generated by a geometric(parameter p) distribution according to a late arrival system with delayed access and service times independently distributed with distribution {gj}, j ≥ 1 is studied. By a simple method (techniques of probability decomposition, renewal process theory) that is different from the techniques used by Hunter (1983), the transient property of the queue with initial state i(i ≥ 0) is discussed. The recursion expression for u-transform of transient queue-length distribution at any time point n+ is obtained, and the recursion expression of the limiting queue length distribution is also obtained.

Analyzing discrete-time bulk-serviceGeo/Geob/mqueue

This paper analyzes a discrete-time multi-server queue in which service capacity of each server is a minimum of one and a maximum of b customers. The interarrival- and service-times are assumed to be independent and geometrically distributed. The queue is analyzed under the assumptions of early arrival system and late arrival system with delayed access. Besides, obtaining state probabilities at arbitrary and outside observer's observation epochs, some performance measures and waiting-time distribution in the queue have also been discussed. Finally, it is shown that in limiting case the results obtained in this paper tend to the continuous-time counterpart.

Computing queue length and waiting timedistributions in finite-buffer discrete-time multiserver queues with late and early arrivals

This paper presents modelling and analysis of discrete-time multiserver finite-buffer queue with general interarrival and geometric service time. Using the supplementary variable technique, and considering the remaining interarrival time as a supplementary variable, two variants of this model, namely the late arrival system with delayed access (LAS-DA) and early arrival system (EAS), have been examined. For both the cases, steady-state system length distributions at arbitrary, prearrival, and outside observer's observation epochs have been obtained. Further, the waiting time distribution in the queue is also discussed. Various performance measures such as probability of loss, average number of busy servers and average waiting time in the queue etc. have been presented. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.

Complete analytic and computational analyses of the discrete-time bulk-arrival infinite-server system: GIX/ Geom/∞

In this paper, we consider the discrete-time queueing system with bulk arrivals, generally distributed interbatch times, geometrically distributed service times, and infinite number of servers: the GI X / Geom/∞ queue. Under the assumptions of early arrival system (EAS) and late arrival system (LAS), we derive the system size distributions at two different epochs—prearrival and random. In addition, simple relations among the binomial moments of the steady-state system-size distributions of the two systems are derived. Special cases, such as the GI/ Geom/∞ or GI/ D/∞ queues, are also presented. Some numerical results are included at the end.