Infinite-buffer Single Server Queue Research Articles

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

ABSTRACTWe 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.

A Note on the Waiting-Time Distribution in an Infinite-Buffer GI[X]/C-MSP/1 Queueing System

This paper deals with a batch arrival infinite-buffer single server queue. The interbatch arrival times are generally distributed and arrivals are occurring in batches of random size. The service process is correlated and its structure is presented through a continuous-time Markovian service process (C-MSP). We obtain the probability density function (p.d.f.) of actual waiting time for the first and an arbitrary customer of an arrival batch. The proposed analysis is based on the roots of the characteristic equations involved in the Laplace-Stieltjes transform (LST) of waiting times in the system for the first, an arbitrary, and the last customer of an arrival batch. The corresponding mean sojourn times in the system may be obtained using these probability density functions or the above LSTs. Numerical results for some variants of the interbatch arrival distribution (Pareto and phase-type) have been presented to show the influence of model parameters on the waiting-time distribution. Finally, a simple computational procedure (through solving a set of simultaneous linear equations) is proposed to obtain the “R” matrix of the corresponding GI/M/1-type Markov chain embedded at a prearrival epoch of a batch.

Stationary distributions of the R[X]/R/1 cross-correlated queue

ABSTRACTWe consider an infinite buffer single server queue wherein batch interarrival and service times are correlated having a bivariate mixture of rational (R) distributions, where R denotes the class of distributions with rational Laplace–Stieltjes transform (LST), i.e., ratio of a polynomial of degree at most n to a polynomial of degree n. The LST of actual waiting time distribution has been obtained using Wiener–Hopf factorization of the characteristic equation. The virtual waiting time, idle period (actual and virtual) distributions, as well as inter-departure time distribution between two successive customers have been presented. We derive an approximate stationary queue-length distribution at different time epochs using the Markovian assumption of the service time distribution. We also derive the exact steady-state queue-length distribution at an arbitrary epoch using distributional form of Little’s law. Finally, some numerical results have been presented in the form of tables and figures.

Stationary analysis of a BMAP/R/1 queue with R-type multiple working vacations

ABSTRACTWe consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple working vacation policy. The service time during a working vacation is generally distributed random variable which is independent of the service times during a normal busy period as well as the arrival process. Duration of service times during a normal busy period and duration of working vacation times follow the class of distributions whose Laplace-Stieltjes transforms are rational functions (R-type distributions). The service time during a normal busy period, working vacation time, and the service time during a working vacation are independent of each other as well as of the arrival process. If a working vacation terminates while service is going on for a customer at head of the queue in vacation mode then, the server switches to normal mode and the customer at head of the queue is entitled to receive a full service time in the normal busy period irrespective of the amount of service received by the customer at head of the queue during the previous working vacation period. We obtain system-length distributions at various epoch, such as post-departure, pre-arrival, arbitrary, and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain system-length distribution at post-departure epoch. Later, we use supplementary variable technique and simple algebraic manipulations to obtain system-length distribution at arbitrary epoch using the system-length distribution at post-departure epoch. Some important performance measures, such as mean system lengths and mean waiting time have been obtained. Finally, some numerical results have been presented in the form of tables and graphs to show the applicability of the results obtained in this article. The model has potential application in areas of computer and communication networks, such as ethernet passive optical network (EPON).

A simple analysis of system characteristics in the batch service queue with infinite-buffer and Markovian service process using the roots method:GI/C-MSP(a,b)/1/∞

We consider an infinite-buffer single-server queue with renewal input and Markovian service process where customers are served in batches according to a general bulk service rule. Queue-length distributions at epochs of pre-arrival, arbitrary and post-departure have been obtained along with some important performance measures such as mean queue lengths and mean waiting times in both the system as well as the queue. We also obtain the steady-state service batch size distributions as well as system-length distributions. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of queue-length distribution at a pre-arrival epoch. Also, we provide analytical and numerical comparison between the roots method used in this paper and the matrix geometric method in terms of computational complexities and required computation time to evaluate pre-arrival epoch probabilities for both the methods. Later, we have established heavy- and light-traffic approximations as well as an approximation for the tail probabilities at pre-arrival epoch based on one root of the characteristic equation. Numerical results for some cases have been presented to show the effect of model parameters on the performance measures.

Single server queues with a batch Markovian arrival process and bulk renewal or non-renewal service

We first consider an infinite-buffer single server queue where arrivals occur according to a batch Markovian arrival process(BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Next we obtain queue-length distributions at various other epochs such as, pre-arrival, arbitrary and pre-service using relations with post-departure epoch. Later we also obtain the system-length distributions at post-departure and arbitrary epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue-lengths and mean waiting times have been obtained. Total expected cost function per unit time is also derived to determine the locally optimal values of a and b. Secondly, we perform similar analysis for the corresponding infinite-buffer single server queue where arrivals occur according to a BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP).

Analysis and optimal control of a queue with infinite buffer under batch-size dependent versatile bulk-service rule

This article studies an infinite-buffer single server queue with versatile bulk service policy, where the service rates depend on the service batch sizes. Such queueing systems have potential applications in manufacturing, computer and telecommunication systems, semiconductor industry, group testing etc. Also from our present analysis, the results for several batch-service models can be derived. We use the matrix-geometric method to obtain joint distribution of the number of customers in the queue and the number of customers undergoing service, in steady-state. Based on these limiting probabilities, we present various performance measures, which are useful in determining optimal service policy and controlling congestion in the system. Some optimal policies has also been suggested based on a cost structure and the threshold values of the sensitive parameters.

Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations

We consider an infinite-buffer single server queue with batch Markovian arrival process (BMAP) and exhaustive service discipline under multiple adaptive vacation policy. That is, the server serves until system emptied and after that server takes a random maximum number H different vacations until either he finds at least one customer in queue or the server have exhaustively taken all the vacations. The maximum number H of vacations taken by the server is a discrete random variable. We obtain queue-length distributions at various epochs such as, service completion/vacation termination, pre-arrival, arbitrary, post-departure and pre-service. The proposed analysis is based on the use of matrix-analytic procedure to obtain queue-length distribution at a post-departure epoch. Later we use supplementary variable method and simple algebraic manipulations to obtain the queue-length distribution at other epochs using queue-length distribution at post-departure epoch. Some important performance measures, like mean queue lengths and mean waiting times have been obtained. Several other vacation queueing models can be obtained as a special case of our model, e.g., single-, multiple-vacation model and queues with exceptional first vacation time. Finally, the total expected cost function per unit time is considered to determine a locally optimal multiple adaptive vacation policy at a minimum cost.

Analyzing state-dependent arrival in [formula omitted] queues

We consider an infinite-buffer single-server queue with renewal input. The service to the queueing system is provided in batches of random size, according to a batch Markovian service process (BMSP). The queue length distribution of the number of customers in the system at pre-arrival and arbitrary epochs has been obtained along with some important performance measures, such as the mean number of customers in the system and the mean system sojourn time of a customer. Secondly, we study a similar queueing system with queue-length-dependent inter-arrival times and obtain the above-mentioned state probabilities and performance measures. These queueing models have potential applications in the areas of computer networks, telecommunication systems, manufacturing systems, etc.

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.