Batch Arrivals Research Articles

A batch arrival priority queue with recurrent repeated demands, admission control and hybrid failure recovery discipline

We consider an unreliable single server retrial queue with batch arrivals, accepting two types of customers, say ordinary and transit. In addition, there is a fixed number of recurrent customers in a retrial box. Ordinary and transit customers arrive in batches according to a compound Poisson process while in addition, each individual customer is subject to control admission policy upon arrival. Admitted ordinary and transit customers who find upon arrival the server unavailable, join a queue and the retrial box respectively. Upon a service completion, recurrent customers always return in retrial box while ordinary and transit leave the system. The server is subject to a hybrid failure recovery discipline. This discipline combines both the preemptive repeat and the preemptive resume disciplines according to the type of the interrupted customer. For such a system we investigate the stability conditions and the joint queue length distribution in steady state, while operating characteristics and numerical results are also presented. Reliability analysis is investigated while a stochastic decomposition result is also proved.

A generalised Little's law and its applications for a discrete-time G/D/1 queue with correlated arrivals

The discrete-time G/D/1 queues with serially correlated batch arrivals and unit service times have wide applications in modern telecommunication systems. Despite the rich literature in their performance analysis, no simple formula on the relation between system size and sojourn time is known. We show that for this specific type of queues, the Little's law can be extended to higher moments. The benefit of this generalised result is that once the moments of either performance measure are available, those of the other will be obtained simultaneously. This result is applied to a particular example of OO-G/D/1 system, where the mean, variance, and skewness of the sojourn delay are derived in closed-form. Numerical examples are given to examine how the correlation influences these performance measures.

To split or not to split: Selecting the right server with batch arrivals

We consider a dispatching system, where jobs, arriving in batches, are assigned to single-server FCFS queues. Batches can be split to different queues on per job basis. However, the holding costs are batch-specific and incurred until the last member of the batch completes the service. By using the first policy improvement step of the MDP framework, we are able to derive robust dispatching policies, which split arriving batches only when deemed advantageous. The approach is also demonstrated in numerical examples.

Traffic Offloading and Load Balancing to Enable Cloud Computing Connectivity

The distribution of computational tasks across a diverse set of geographically distributed heterogeneous resources is a critical issue in the realisation of true computational grids. Conventionally, workload allocation algorithms are divided into static and dynamic approaches. Whilst dynamic approaches frequently outperform static schemes, they usually require the collection and processing of detailed system information at frequent intervals - a task that can be both time consuming and unreliable in the real-world. This paper introduces a novel workload allocation algorithm for optimally distributing the workload produced by the arrival of batches of jobs. Results show that, for the arrival of batches of jobs, this workload allocation algorithm outperforms other commonly used algorithms in the static case. A hybrid scheduling approach (using this workload allocation algorithm), where information about the speed of computational resources is inferred from previously completed jobs, is then introduced and the efficiency of this approach demonstrated using a real world computational grid. These results are compared to the same workload allocation algorithm used in the static case and it can be seen that this hybrid approach comprehensively outperforms the static approach.

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.

A Feedback Retrial Queueing System with Two Types of Batch Arrivals

A retrial queueing system with two types of batch arrivals, called type I and type II customers, is considered. Type I customers and type II customers arrive in batches of variable sizes according to two different Poisson processes. Service time distributions are identical and independent and are different for both types of customers. If the arriving customers are blocked due to the server being busy, type I customers are queued in a priority queue of infinite capacity, whereas type II customers enter into a retrial group in order to seek service again after a random amount of time. A type I customer who has received service departs the system with a preassigned probability or returns to the priority queue for reservice with the complement probability. A type II call who has received service leaves the system with a preassigned probability or rejoins the retrial group with complement probability. For this model, the joint distribution of the number of customers in the priority queue and in the retrial group is obtained in a closed form. Some particular models and operating characteristics are obtained. A numerical study is also carried out.

Tail probabilities of the delay in a batch-service queueing model with batch-size dependent service times and a timer mechanism

We deduce approximations for the tail probabilities of the customer delay in a discrete-time queueing model with batch arrivals and batch service. As in telecommunications systems transmission times are dependent on packet sizes, we consider a general dependency between the service time of a batch and the number of customers within it. The model also incorporates a timer mechanism to avoid excessive delays stemming from the requirement that a service can only be initiated when the number of present customers reaches or exceeds a service threshold. The service discipline is first-come, first-served (FCFS). We demonstrate in detail that our approximations are very useful for the purpose of assessing the order of magnitude of the tail probabilities of the customer delay, except in some special cases that we discuss extensively. We also illustrate that neglecting batch-size dependent service times or a timer mechanism can lead to a devastating assessment of the tail probabilities of the customer delay, which highlights the necessity to include these features in the model. The results from this paper can, for instance, be applied to assess the quality of service (QoS) of Voice over IP (VoIP) conversations, which is typically expressed in terms of the order of magnitude of the probability of packet loss due to excessive delays.

Balking and Re-service in a Vacation Queue with Batch Arrival and Two Types of Heterogeneous Service

In this paper we have considered a single server queue with customers or units arriving in groups or batches in a system providing two types of general heterogeneous service. The server provides type 1 or type 2 services on a first come first served basis. At the beginning of a service, a customer has the option to choose either type 1 or type 2 service. Also we added the concept of balking and re-service in this study. Balking is a kind of customer behavior. If a batch on arrival for service is reluctant or refuses to join the system for some reasons is said to balk. Once a service is completed the customer may leave the system or he has the option to demand re-service for the same service taken. Further as soon as service of any type gets completed, the server may take vacation or continue staying at the system even if the system is empty. For this model we have derived the steady state queue size distribution at random epoch and some particular cases have been developed and compared with known results.

Periodic Schedules for Bounded Timed Weighted Event Graphs

Timed event graphs (TEGs) and timed weighted event graphs (TWEGs), which have multiple arc cardinalities, have been widely used for automated production systems such as robotic work cells or embedded systems. TWEGs are useful for modeling batch flows of entities such as batch arrivals or processing of jobs. Periodic schedules, that combine an explicit description of starting times and an easy implementation, are particularly interesting, and have been proved to be optimal for ordinary timed event graphs (TEGs). In this paper, we present polynomial algorithms to check the existence of periodic schedules of bounded TWEGs and to compute their optimal throughput. These results can be considered as generalizations of those for ordinary timed event graphs. We then establish that periodic schedules are suboptimal for TWEGs and may not exist even for a live TWEG. The gap between optimal throughput and throughput of an optimal periodic schedule is experimentally investigated for a subclass of TWEGs, namely timed weighted circuits.

Minimizing makespan on parallel machines with batch arrivals

Most studies in the scheduling literature assume that jobs arrive at time zero, while some studies assume that jobs arrive individually at non-zero times. However, both assumptions may not be valid in practice because jobs usually arrive in batches. In this article, a scheduling model for an identical parallel machine problem with batch arrivals is formulated. Because of the NP-hardness of the problem, a heuristic based on a simplified version of lexicographical search is proposed. To verify the heuristic, two lower bounding schemes are developed, where one lower bound is tight, and the list scheduling heuristic is compared. Extensive computational experiments demonstrate that the proposed heuristic is quite efficient in obtaining near optimal solution with an average error of less than 1.58%. The percentage improvement (from the lower bound) of the heuristic solution on the solution by the list scheduling is as large as 31.68.

Combined Analysis of Transient Delay Characteristics and Delay Autocorrelation Function in the G e o X / G /1 Queue

We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state.

INFINITE-SERVER QUEUES WITH BATCH ARRIVALS AND DEPENDENT SERVICE TIMES

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

Performance Analysis of Probabilistic Multipath Transmission of Video Streaming Traffic over Multi-Radio Wireless Devices

Popular smart wireless devices become equipped with multiple radio interfaces. Multihoming support can be enabled to allow for multiple simultaneous associations with heterogeneous networks. In this study, we focus on video streaming traffic and propose analytical approaches to evaluate the packet-level and call-level performance of a multipath transmission scheme, which sends video traffic bursts over multiple available channels in a probabilistic manner. A probability generation function (PGF) and z-transform method is applied to derive the PGF of packet delay and any arbitrary moment in general. Particularly, we can obtain the average delay, delay jitter, and delay outage probability. The essential characteristics of video traffic are taken into account, such as deterministic burst intervals, highly dynamic burst length, and batch arrivals of transmission packets. The video substream traffic resulting from the probabilistic flow splitting is characterized by means of zero-inflated models. Further, the call-level performance, in terms of flow blocking probability and system throughput, is evaluated with a three-dimensional Markov process and compared with that of an always-best access selection. The numerical and simulations results demonstrate the effectiveness of our analysis framework and the performance gain of multipath transmission.

A catastrophic-cum-restorative queuing system with correlated batch arrivals and general service time distribution

In this paper, a stochastic queuing model for a catastrophic-cum-restorative queuing system with correlated batch arrivals and general service time distribution has been developed. The transient analysis of the queuing model has been performed. We The Laplace Transform of the probability generating function of system size has been obtained. Finally, some particular cases of the model have been derived and discussed .

Transient and Stationary Losses in a Finite‐Buffer Queue with Batch Arrivals

We present an analysis of the number of losses, caused by the buffer overflows, in a finite‐buffer queue with batch arrivals and autocorrelated interarrival times. Using the batch Markovian arrival process, the formulas for the average number of losses in a finite time interval and the stationary loss ratio are shown. In addition, several numerical examples are presented, including illustrations of the dependence of the number of losses on the average batch size, buffer size, system load, autocorrelation structure, and time.

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.

Mathematical Models of Multiserver Queuing System for Dynamic Performance Evaluation in Port

We discuss dynamic system performance evaluation in the river port utilizing queuing models with batch arrivals. The general models of the system are developed. This system is modelled by MX/M/n/m queue with finite waiting areas and identical and independent cargo‐handling capacities. The models are considered with whole and part batch acceptance (or whole and part batch rejections) and the interarrival and service times are exponentially distributed. Results related to the batch blocking probability and the blocking probability of an arbitrary vessel in nonstationary and stationary states have been obtained. Numerical results and computational experiments are reported to evaluate the efficiency of the models for the real system.

A Complete and Simple Solution to a Discrete-Time Finite-Capacity BMAP/D/c Queue

We consider a discrete-time multi-server finite-capacity queueing system with correlated batch arrivals and deterministic service times (of single slot), which has a variety of potential applications in slotted digital telecommunication systems and other related areas. For this queueing system, we present, based on Markov chain analysis, not only the steady-state distributions but also the transient distributions of the system length and of the system waiting time in a simple and unified manner. From these distributions, important performance measures of practical interest can be easily obtained. Numerical examples concerning the superposition of certain video traffics are presented at the end.

Limit Analysis of Oscillating Batch Arrival M[x]/G/1 Systems with Finite Capacity: EMC Approach

Limit Analysis of Oscillating Batch Arrival M[x]/G/1 Systems with Finite Capacity: EMC Approach

A queueing system with batch arrival of customers in sessions

This paper describes and analyzes a single-server queueing model with a finite buffer and session arrivals. Generation of the sessions is described by the Markov Arrival Process (MAP). Arrival of the groups of the requests within any admitted session is described by the Terminating Batch Markov Arrival Process (TBMAP). Service time of the request has Phase (PH) type distribution. The number of the sessions that can be simultaneously admitted to the system is under control.Analysis of the joint distribution of the number of sessions and requests in the system is implemented by means of the matrix technique. Analysis of the sojourn time of an arbitrary and admitted session is performed by means of the extension of the method of catastrophes. Effect of control on the main performance measures of the system is numerically demonstrated.