A difference equation approach for analysing a batch service queue with the batch renewal arrival process

We consider an infinite buffer single server queue wherein customers arrive according to the batch renewal arrival process and are served in batches following the random serving capacity rule. The service-batch times follow exponential distribution. This model has been studied in the past using the embedded Markov chain technique and probability generating function. In this paper we provide an alternative yet simple methodology to carry out the whole analysis which is based on the supplementary variable technique and the theory of difference equations. The procedure used here is simple in the sense that it does not require the complicated task of constructing the transition probability matrix. We obtain explicit expressions of the steady-state system-content distribution at pre-arrival and arbitrary epochs in terms of roots of the associated characteristic equation. We also present few numerical results in order to illustrate the computational procedure.     

On the distributions of infinite server queues with batch arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

     

Optimal control for parallel queues with a single batch server

We consider a queueing system with multiple Poisson arrival queues and a single batch server that has infinite capacity and a fixed service time. The problem is to allocate the server at each moment to minimize the long-run average waiting cost. We propose a Cost-Arrival Weighted (CAW) policy for this problem based on the structure of the optimal policy of a corresponding fluid model. We show that this simple policy enjoys a superior performance by numerical experiments.     

Batch arrival queues with vacations and server setup

We study a single server queueing system whose arrival stream is compound Poisson and service times are generally distributed. Three types of idle period are considered: threshold, multiple vacations, and single vacation. For each model, we assume after the idle period, the server needs a random amount of setup time before serving. We obtain the steady-state distributions of system size and waiting time and expected values of the cycle for each model. We also show that the distributions of system size and waiting time of each model are decomposed into two parts, whose interpretations are provided. As for the threshold model, we propose a method to find the optimal value of threshold to minimize the total expected operating cost.     

The infinite-buffer single server queue with a variant of multiple vacation policy and batch Markovian arrival process

We consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process (BMAP). The server serves until system emptied and after that server takes a vacation. The server will take a maximum number H of vacations until either he finds at least one customer in the queue or the server has exhaustively taken all the vacations. We obtain queue length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue lengths and mean waiting times, etc. have been obtained. Several other vacation queueing models like, single and multiple vacation model, queues with exceptional first vacation time, etc. can be considered as special cases of our model.     

On the loss process in a batch arrival queue

In this paper, we investigate the loss process in a finite-buffer queue with batch arrivals and total rejection discipline. In such a model, if the buffer has insufficient capacity to accept all the customers included in an arriving batch, the whole batch is blocked and lost. This scheme is especially useful in performance evaluation of buffering processes in IP (internet protocol) networks. The main result of this paper is a closed-form formula for the joint distribution of the length of the first lost series of batches and the time of the first loss. Moreover, the limiting distribution (as the buffer size grows to infinity) is shown.     

Departures inGR^{X{n}} /G_n /\infty

Departure processes in infinite server queues with non-Poisson arrivals have not received much attention in the past. In this paper, we try to fill this gap by considering the continuous time departure process in a general infinite server system with a Markov renewal batch arrival process ofM different types of customers. By a conditional approach, analytical results are obtained for the generating functions and binomial moments of the departure process. Special cases are discussed, showing that while Poisson arrival processes generate Poisson departures, departure processes are much more complicated with non-Poisson arrivals.This research has been supported in part by the Natural Science and Engineering Research Council of Canada (Grant A5639).     

Product form in networks of queues with batch arrivals and batch services

A product form equilibrium distribution is derived for a class of queueing networks in either discrete or continuous time, in which multiple customers arrive simultaneously and batches of customers complete service simultaneously.     

Geometric equilibrium distributions for queues with interactive batch departures

Gelenbe et al. [1, 2] consider single server Jackson networks of queues which contain both positive and negative customers. A negative customer arriving to a nonempty queue causes the number of customers in that queue to decrease by one, and has no effect on an empty queue, whereas a positive customer arriving at a queue will always increase the queue length by one. Gelenbe et al. show that a geometric product form equilibrium distribution prevails for this network. Applications for these types of networks can be found in systems incorporating resource allocations and in the modelling of decision making algorithms, neural networks and communications protocols.In this paper we extend the results of [1, 2] by allowing customer arrivals to the network, or the transfer between queues of a single positive customer in the network to trigger the creation of a batch of negative customers at the destination queue. This causes the length of the queue to decrease by the size of the created batch or the size of the queue, whichever is the smallest. The probability of creating a batch of negative customers of a particular size due to the transfer of a positive customer can depend on both the source and destination queue.We give a criterion for the validity of a geometric product form equilibrium distribution for these extended networks. When such a distribution holds it satisfies partial balance equations which are enforced by the boundaries of the state space. Furthermore it will be shown that these partial balance equations relate to traffic equations for the throughputs of the individual queues.     

Multiclass batch arrival retrial queues analyzed as branching processes with immigration

TheM/G/1 batch arrival retrial queue is studied by means of branching processes with immigration. We shall investigate this queue when traffic intensity is less than one, tends to one or is greater than one.     

TheGr X n/G n/∞ system: System size

An important property of most infinite server systems is that customers are independent of each other once they enter the system. Though this non-interacting property (NIP) has been instrumental in facilitating excellent results for infinite server systems in the past, the utility of this property has not been fully exploited or even fully recognized. This paper exploits theNIP by investigating a general infinite server system with batch arrivals following a Markov renewal input process. The batch sizes and service times depend on the customer types which are regulated by the Markov renewal process. By conditional approaches, analytical results are obtained for the generating functions and binomial moments of both the continuous time system size and pre-arrival system size. These results extend the previous results on infinite server queues significantly.     

Activity periods of an infinite server queue and performance of certain heavy tailed fluid queues

A fluid queue with ON periods arriving according to a Poisson process and having a long-tailed distribution has long range dependence. As a result, its performance deteriorates. The extent of this performance deterioration depends on a quantity determined by the average values of the system parameters. In the case when the the performance deterioration is the most extreme, we quantify it by studying the time until the amount of work in the system causes an overflow of a large buffer. This turns out to be strongly related to the tail behavior of the increase in the buffer content during a busy period of the M/G/∞ queue feeding the buffer. A large deviation approach provides a powerful method of studying such tail behavior. This revised version was published online in June 2006 with corrections to the Cover Date.     

The randomized vacation policy for a batch arrival queue

This paper examines an M^[x]/G/1${\text{M}}^{[\text{x}]}/\text{G}/1$ queueing system with a randomized vacation policy and at most J 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$1-p$. This pattern continues until the number of vacations taken reaches J. If the system is empty by the end of the J  th vacation, the server is dormant idly in the system. If there is one or more customers arrive at server idle state, the server immediately starts his services for the arrivals. For such a system, we derive the distributions of important characteristics, such as system size distribution at a random epoch and at a departure epoch, system size distribution at busy period initiation epoch, idle period and busy period, etc. Finally, a cost model is developed to determine the joint suitable parameters (p^∗,J^∗)$(p^{\ast }\text{,}{J}^{\ast })$ at a minimum cost, and some numerical examples are presented for illustrative purpose.     

Two queues with alternating service and server breakdown

We consider a queueing system with two stations served by a single server in a cyclic manner. We assume that at most one customer can be served at a station when the server arrives at the station. The system is subject to service interuption that arises from server breakdown. When a server breakdown occurs, the server must be repaired before service can resume. We obtain the approximate mean delay of customers in the system.     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

Batch arrival queues under vacation policies with server breakdowns and startup/closedown times

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under vacation policies with startup/closedown times, where the vacation time, the startup time, and the closedown time are generally distributed. When all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time. After shutdown, the server operates one of (1) multiple vacation policy and (2) single vacation policy. When the server reactivates since shutdown, he needs a startup time before providing the service. If a customer arrives during a closedown time, the service is immediately started without a startup time. The server may break down according to a Poisson process while working and his repair time has a general distribution. We analyze the system characteristics for the vacation models.