Batch Arrival Queue Research Articles

Buffer overflow period in a batch-arrival queue with autocorrelated arrivals

Buffer overflow period in a batch-arrival queue with autocorrelated arrivals

Transient Solution of an Batch arrival Queue with Two Types of Service, Multiple Vacation, Random breakdown and Restricted Admissibility

In this paper we analyse a batch arrival queue with two types of service subject to random breakdowns having multiple server vacation. We assume that the server provides two types of service, type 1 with probability and type 2 with probability with the service times following general distribution and each arriving customer may choose either type of service. The server takes vacation each time the system becomes empty and the vacation period is assumed to be general. On returning from vacation if the server finds no customer waiting in the system, then the server again goes for vacation until he finds at least one customer in the system. The system may breakdown at random and repair time follows exponential distribution. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Average queue lenth and average system size are also computed.

Analysis of M X / G /1 feedback queuewith two-phase service, compulsory server vacation and random breakdowns

This paper investigates a batch arrival queueing system with two-phase heterogeneous service under compulsory vacation schedule. Upon arrival any batch that finds the server busy, on vacation or under repair joins the queue. Otherwise one customer from the arriving batch enters the server for service immediately while the rest join the queue. After completion of the second phase of service for each customer, the server goes for a vacation. Busy servers are interrupted by breakdowns and repair has to be done to resume service. If a customer is unsatisfied with his service he is allowed to join the tail of the queue as a feedback customer. We construct the mathematical model and derive the probability generating functions of transient state probabilities in terms of their Laplace transforms. The steady state distributions of the server state are deduced and the average number of customers in the system, the mean waiting time of a customer are obtained. Particular cases of interest of the model are discussed in order to verify the results. Numerical computations are provided to visualize the effect of parameters on system performance measures and also to validate our analytical results.

An M<SUP align="right">x</SUP>/G/1 retrial queue with two phase service under active server breakdowns, two types of repair and multiple vacations with N-policy

This paper analyses a single server batch arrival retrial queue with active breakdowns, two types of repair, a second optional service and multiple vacations with N-policy. The server may breakdown while providing first essential service. When the server fails, it is repaired immediately. On successful completion of first essential service, the customer may opt for second optional service with probability α. At a service completion epoch, without any request for re-service and the orbit size being zero, the server does a secondary job (vacation) repeatedly until the number of customers in the orbit reaches the threshold value N. At a secondary job completion epoch, if the orbit size in at least N, then the server remains in the system to render service for customers from main pool or from retrial group. The steady state queue size distribution of number of customers in the orbit and expected number of customers in the retrial group are obtained. Some particular cases and reliability measures are discussed. Also, some performance measures are obtained. Numerical illustrations are also provided.

AnMX/G/1 unreliable retrial queue with two phase service and persistence behaviour of customers in service

This paper describes an unreliable server batch arrival retrial queue with two types of repair and second optional service. The server provides preliminary first essential service (FES) to the primary arriving customers or customers from retrial group. On successful completion of FES, the customer may opt for second optional service (SOS) with probability α . The server is subject to active break downs. The customer under FES (or SOS) during the failure decides, with probability q , to join the orbit( impatientcustomer ) and, with complementary probability p , to remain in the server for repair in order to conclude his remaining service ( patientcustomer ). Both service and repair times are assumed to have general distribution. It is considered that the repair time of server during the presence of patient customer and the repair time of the server while the customer ( impatientcustomer ) joining the orbit due to failure, are different. For this queueing system, the orbit and system size distributions are obtained. Reliability of the proposed model is analysed. Some particular cases are also discussed. Other performance measures are also obtained. The effects of several parameters on the system are analysed numerically.

A Two Stage Batch Arrival Queue with Reneging during Vacation and Breakdown Periods

We study a two stage queuing model where the server provides two stages of service one by one in succession. We consider reneging to occur when the server is unavailable during the system breakdown or vacation periods. We concentrate on deriving the steady state solutions by using supplementary variable technique and calculate the mean queue length and mean waiting time. Further some special cases are also discussed and numerical examples are presented.

Discrete-Time Geo x /G/1 Queue with Geometrically Working Vacations and Vacation Interruption

This paper treats a discrete-time batch arrival queue with geometrically working vacations and vacation interruption. The main purpose of this paper is to present a performance analysis of this system. For this purpose, we first derive the probability generating function (PGF) of the stationary system length at the departure epochs based on the embedded Markov chain technique. Next, we present the stationary distributions of the system lengths as well as some performance measures at random epochs by using the supplementary variable method. Thirdly, still based on the supplementary variable method we give a distribution for the number of the customers at the beginning of a busy period and give a stochastic decomposition formula for the PGF of the stationary system length at the departure epochs. Additionally, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.

Analysis of a MX/G(a,b)/1 queueing system with vacation interruption

In this paper, a batch arrival general bulk service queueing system with interrupted vacation (secondary job) is considered. At a service completion epoch, if the server finds at least ‘a ’ customers waiting for service say ξ , he serves a batch of min (ξ , b) customers, where b ≥ a . On the other hand, if the queue length is at the most ‘a -1’, the server leaves for a secondary job (vacation) of random length. It is assumed that the secondary job is interrupted abruptly and the server resumes for primary service, if the queue size reaches ‘a ’, during the secondary job period. On completion of the secondary job, the server remains in the system (dormant period) until the queue length reaches ‘a ’. For the proposed model, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained. Various performance measures are derived. A cost model for the queueing system is also developed. To optimize the cost, a numerical illustration is provided.

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M[x]/G/1 queueing system with an unreliable server and a delayed repair, in which the server operates a randomized vacation policy with multiple vacations. Whenever the system is empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be immediately activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 −p. Whenever one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. The server may also meet an unpredictable breakdown and the repair may be delayed. For such a system the authors derive the distributions of some important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle period and the busy period. The authors perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.

A batch arrival queue under randomised multi-vacation policy with unreliable server and repair

This article examines an M[x]/G/1 queueing system with an unreliable server and a repair, in which the server operates a randomised vacation policy with multiple available vacations. Upon the system being found to be empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1 − p. When one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. It is possible that an unpredictable breakdown may occur in the server, in which case a repair time is requested. For such a system, we derive the distributions of several important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle and busy periods. We perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximisation model is constructed to show the benefits of such a queueing system.

On an M<SUP align="right">[X]</SUP>/G/1 queuing system with random breakdowns, server vacations, delay times and a standby server

In this work, we study a batch arrival queue with a Bernoulli schedule vacation where after every service completion the server may take a vacation of random length with probability p or may stay in the system providing service with probability 1 - p . Moreover, we assume that the server may breakdown randomly, and the repair process does not start immediately after a breakdown; consequently, there is a delay time waiting for repairs to start. The main new contribution in this work is that we assume that the system is equipped with a standby server to serve the costumers during the vacation period of the main server. The service times, vacation times, repair times and delay times are assumed to follow different general arbitrary distributions while the service times of the standby server follows an exponential distribution.

A Batch Arrival Queue System with Coxian-2 Server Vacations and Admissibility Restricted

The M/G/1 classic queueing system was extended by many authors in last two decades. The systems with server’s vacation are important models that extend the M/G/1 queueing system. Also another condition such as admissibility restricted may occur in systems. From this motivation, in this system I consider a single server queue with batch arrival Poisson input. There is a restricted admissibility of arriving batches in which not all batches are allowed to join the sys-tem at all times. At each service completion epoch, the server may apt to take a vacation with probability θ or else with probability 1 ? θ may continue to be available in the system for the next service. The vacation period of the server has two heterogenous phases. Phase one is compulsory, and phase two follows the phase one vacation in such a way that the server may take phase two with probability p or may return back to the system with probability 1 ? p. The vacation times are assumed to be general. All stochastic processes involved in this system (service and vacation times) are inde-pendent of each other. We derive the PGF’s of the system and by using them the informance measures are obtained. Some numerical approaches are examined the validity of results.

The queue length in an M/G/1 batch arrival retrial queue

An M/G/1 retrial queue with batch arrivals is studied. The queue length K μ is decomposed into the sum of two independent random variables. One corresponds to the queue length K ? of a standard M/G/1 batch arrival queue, and another is compound-Poisson distributed. In the case of the distribution of the batch size being light-tailed, the tail asymptotics of K μ are investigated through the relation between K ? and its service times.

An M[X]/G/1 Queue with Bernoulli Schedule General Vacation Times, General Extended Vacations, Random Breakdowns, General Delay Times for Repairs to Start and General Repair Times

This paper studies a batch arrival queueing system in which the server may face occasional random breakdowns. The repair process does not start immediately after a breakdown and there is a delay time waiting for repairs to start. Further, after every service completion the server has the option to leave for a vacation with probability p or continue serving with probability . The main new assumption in this paper is that the server has the option to go on extended vacation after the original vacation completion with probability r or rejoins the system to provide service directly after the original vacation with probability . The service times, vacation times, extended vacation times, delay times and repair times are all assumed to follow general arbitrary distributions, while only the breakdown times are exponentially distributed .

Analysis of Departure Process in Batch Arrival Queue with Multiple Vacations and Exhaustive Service

A batch arrival single-server queueing system of the M X /G/1 type is considered. At the end of each busy period the server begins a multiple vacation period when the service process is stopped. A new method of the study of departure process in such a system in transient state is proposed. Using the formula of total probability, we direct the analysis to that in the system without vacations. General results are obtained using the renewal-theory approach. An explicit representation for the probability generating function of Laplace transform of departure process is derived. The formula is written down by means of transforms of distributions describing the arrival and service processes, vacation times, and components of certain factorization identity of Wiener-Hopf type connected with these distributions.

Performance analysis for power saving class type III of IEEE 802.16 in WiMAX

In this paper, we present an effective method to analyze the performance of the power saving class type III initiated by the base station with unsolicited MOB_SLP-RSP or DL sleep control extended subheader in a self-similar traffic. We build a batch arrival queueing model, in which the batch size is described by a random variable following a Pareto( c, α) distribution in order to capture the self-similar property in multimedia WiMAX based on the IEEE 802.16. By using the discrete-time embedded Markov chain, we can examine and analyze the power saving class type III systems for the following items: the probability generating functions of the number of data frames and batches when the busy period begins, the queueing length, the waiting time and finally the busy cycle. With the first and higher derivatives of the probability generating functions, we can give the averages and the standard deviations for the system performance using the diffusion approximation for the operation process of the system. Next, we give the formulas for the system performance such as the energy saving ratio, the switching ratio, the system utility and the response time of data frames. Moreover, we construct a cost function with the aim of determining the optimal sleep window length that minimizes the cost function. Numerical results are provided with analysis and simulation to show the average performance measures, standard deviations and a cost function with different degrees of self-similarity and different sleep window lengths.

Dynamic Classified Buffer Control for QoS-Aware Packet Scheduling in IEEE 802.16/WiMAX Networks

WiMAX is an advanced broadband wireless access technology that supports multimedia transmission. Here we propose a batch-arrival queueing model for evaluating WiMAX QoS-aware packet scheduling systems and obtaining packet loss rate (PLR), queueing delay (QD), and bandwidth utilization (BU) performance measures. These are used to solve a utilization optimization problem associated with QoS constraints. Specifically, a given traffic statistic and maximum PLR and QD constraints were applied to optimize BU for a classified packet-scheduling buffer. It is the writer's hope that operators can apply the model to dynamically control classified buffer size and to enhance profit tied to QoS constraints.

Some Results for the Actual Waiting Time in Batch Arrival Queueing Systems

The article deals with the queueing system of GI/G/1 type with individual service and batch arrival of customers. The method of integral equations on a half-axis [0, ∞) is applied to obtain new results for the actual waiting time w n of nth arriving customer. The transient and steady state as n tends to infinity are considered. Some simplifications and numerical results for M/M/1, M/E 2/1, and M 2/M/1 queueing systems are derived as well.

Steady-state analysis of a discrete-time batch arrival queue with working vacations

This paper analyzes a discrete-time batch arrival queue with working vacations. In a Geo X / G / 1 system, the server works at a lower speed during the vacation period which becomes a lower speed operation period. This model is more appropriate for the communication systems with the transmit units arrived in batches. We formulate the system as an embedded Markov chain at the departure epoch and by the M/G/1-type matrix analytic approach, we derive the probability generating function (PGF) of the stationary queue length. Then, we obtain the distribution for the number of the customers at the busy period initiation epoch, and use the stochastic decomposition technique to present another equivalent PGF of the queue length. We also develop a variety of stationary performance measures for this system. Some special models and numerical results are presented. Finally, a real-world example in an Ethernet Passive Optical Network (EPON) is provided.

Batch Arrival Vacation Queue with Second Optional Service and Random System Breakdowns

Abstract A single server queue with a second optional service, Bernoulli schedule server vacations, and random system breakdowns was analyzed. It is assumed that customers arrive to the system in batches of variable size, but served one by one. As soon as the first service of a customer is completed, then with probability k he may opt for the second service. After a customer is served, the server may decide to leave the system with probability p, or to continue serving customers with probability 1 - p. If the system breaks down, it enters a repair process immediately. The repair time and the vacation time both are assumed to have general distributions. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue. Some special cases of interest are presented and a numerical illustration is provided.