General Bulk Service Rule Research Articles

Analytic computation for stationary distributions of D-BMAP/D -MSP(a,b)/1 queueing system

ABSTRACT This paper analyses single server correlated batch arrival and correlated batch service queue with innumerable waiting space in discrete-time. The arrival and service processes are governed by discrete batch Markovian arrival process and discrete Markovian service process with general bulk service rule, respectively. Observing two consecutive random epochs, we construct general structure transition probability matrix and it is then reblocked to the desired M/G/1 structure. The UL-type RG-factorization approach combined with spectral expansion method is adopted to unravel the queue length distribution at random epoch. The relationships are undertaken to obtain the queue length distributions at pre-arrival, outside observer's, intermediate and post-departure epochs. Our most challenging effort is to derive the probability mass functions of waiting time and service batch size for arbitrary customer in an arriving batch. We address the warehouse management process for finished products from dealer to retailer as a potential practical use of our batch queue with correlation in both arrival and service, and provide the effect of our queueing behaviour in this application based on our analytical and computational investigation. Based on the implementation of parametrized numeric experiments with numerous categories, we impart our computational results to validate the analytical findings reported in this investigation.

Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation

In this paper, a bulk arrival and two-phase bulk service with active Bernoulli feedback, vacation, and breakdown is considered. The server provides service in two phases as mandatory according to the general bulk service rule, with minimum bulk size ′a′ and maximum bulk size ′b′. In the first essential service (FES) completion epoch, if the server fails, with probability ′δ′, then the renewal of the service station is considered. On the other hand, if there is no server failure, with a probability ′1−δ′, then the server switches to a second essential service (SES) in succession. A customer who requires further service as feedback is given priority, and they join the head of the queue with probability β. On the contrary, a customer who does not require feedback leaves the system with a probability ′1−β′. If the queue length is less than ′a′ after SES, the server may leave for a single vacation with probability ′1−β′. When the server finds an inadequate number of customers in the queue after vacation completion, the server becomes dormant. After vacation completion, the server requires some time to start service, which is attained by including setup time. The setup time is initiated only when the queue length is at least ′a′. Even after setup time completion, the service process begins only with a queue length ‘N’ (N > b). The novelty of this paper is that it introduces an essential two-phase bulk service, immediate Bernoulli feedback for customers, and renewal service time of the first essential service for the bulk arrival and bulk service queueing model. We aim to develop a model that investigates the probability-generating function of the queue size at any time. Additionally, we analyzed various performance characteristics using numerical examples to demonstrate the model’s effectiveness. An optimum cost analysis was also carried out to minimize the total average cost with appropriate practical applications in existing data transmission and data processing in LTE-A networks using the DRX mechanism.

Performance study of the M/M(a,b)/1/MWV queuing system with types of breakdowns

Abstract This paper analyses the M/M(a,b)/1/MWV queuing model with various types of breakdowns. During peak times, the system has breakdown due to server unavailability. The model considers system with two types of breakdowns. The system may breakdown in two different ways during busy and working vacation stages. Customer arrives to the system with parameter λv which follows Poisson distribution and server provides service in regular busy period with parameter µrb and under the multiple working vacations it provides service with parameter µwv with exponential distribution. In this model batches of customers are served under General Bulk Service Rule. The system may breakdown at any time. The breakdown that occurs during working vacation is denoted as βv 1 and the breakdown during busy state is denoted as βv 2. The steady-state equation, the performance of measures for the system and particular cases of described model are derived. Finally, a real life example demonstrated the validity of the model in a proper way.

Analysis of batch size-dependent bulk service queue with multiple working vacation

ABSTRACT The present article analyses an infinite buffer batch size-dependent bulk service queue with multiple working vacation (MWV). The customer's arrival pattern follows the Poisson process, and they get the service in batches according to the general bulk service (GBS) rule in regular period (RP) as well as in working vacation period (WVP). After one service in RP, if the queue size is greater or equal to the lower threshold of the GBS rule, then the server performs the service in RP, otherwise, it starts the working vacation (WV) following exponential vacation time distribution. During the WVP, the server serves the customers in batches (if any) at a lower service rate than the usual service rate. The service time in the RP as well as in WVP is generally distributed. At an arbitrary epoch, the joint probabilities of the queue size and batch size with the server in RP as well as in WVP have been obtained using the supplementary variable technique (SVT) and the bivariate generating function method. Finally, some numerical observations are presented to enhance the applicability of the analytical results.

Analysis of Flexible Batch Service Queueing System to Constrict Waiting Time of Customers

When the required number of customers is available in the general bulk service (GBS) queueing system, the server begins service. Otherwise, the server will remain inactive until the number of consumers in the queue reaches that minimum required number. Customers that have already come must wait throughout this time, regardless of their arrival time. In some circumstances, like specimens awaiting testing in a clinical laboratory or perishable commodities awaiting delivery, it is necessary to finish services before the expiration date. It might only be achievable if consumers’ waiting times are kept under control. As a result, the flexible general bulk service (FGBS) rule is developed in this article to provide flexibility in batching. The effectiveness of FGBS implementation has been demonstrated using two examples: a clinical laboratory and a distribution center. To justify the suggested model, a simulation study and numerical illustration are provided.

Complete analysis of a discrete-time batch service queue with batch-size-dependent service time under correlated arrival process: D-MAP/Gn(a,b)/1

Discrete-time queueing models find a large number of applications as they are used in modeling queueing systems arising in digital platforms like telecommunication systems and computer networks. In this paper, we analyze an infinite-buffer queueing model with discrete Markovian arrival process. The units on arrival are served in batches by a single server according to the general bulk-service rule, and the service time follows general distribution with service rate depending on the size of the batch being served. We mathematically formulate the model using the supplementary variable technique and obtain the vector generating function at the departure epoch. The generating function is in turn used to extract the joint distribution of queue and server content in terms of the roots of the characteristic equation. Further, we develop the relationship between the distribution at the departure epoch and the distribution at arbitrary, pre-arrival and outside observer’s epochs, where the first is used to obtain the latter ones. We evaluate some essential performance measures of the system and also discuss the computing process extensively which is demonstrated by some numerical examples.

Analysis of a k-Stage Bulk Service Queuing System with Accessible Batches for Service

In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process (MAP). An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity cj,2≤j≤k), customer arrivals occur according to distinct Poisson processes with rates λj,2≤j≤k. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate μjm, which depends on the service stage j,1≤j≤k, and the size of the batch m,a≤m≤b. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained.

Computing stationary distributions of the D-MAP/D-MSP$$^{(a,b) }/1$$ queueing system

The present study investigates a single-server batch service queueing model with arrival process as discrete-time Markovian arrival process and service process as discrete-time Markovian service process. The server serves the customers in batches followed by the general bulk-service rule. We first determine the random epoch probabilities using the matrix-geometric method, where the rate matrix $${\mathbf{R}}$$ is determined by an efficient approach based on the eigenvalues and the corresponding eigenvectors of the associated characteristic equation. Next we obtain the explicit closed-form expressions for the pre-arrival, intermediate, outside observer’s and post-departure epoch probabilities by developing the relations among them in equilibrium state. Further, we provide an analytically simple approach to carry out the waiting-time distribution in the queue measured in slots as well as the distribution of the size of a service batch of an arriving customer. We also demonstrate a cost function to evaluate the optimum value of the minimum service batch size and the corresponding expected cost of the system. Finally, an adequate variety of numerical experiments are performed for validation purpose of our analytical results and they are conferred in the form of tables and graphs.

Analysis of a Queueing Model with Batch Markovian Arrival Process and General Distribution for Group Clearance.

In this paper we consider a single server queueing model with under general bulk service rule with infinite upper bound on the batch size which we call group clearance. The arrivals occur according to a batch Markovian point process and the services are generally distributed. The customers arriving after the service initiation cannot enter the ongoing service. The service time is independent on the batch size. First, we employ the classical embedded Markov renewal process approach to study the model. Secondly, under the assumption that the services are of phase type, we study the model as a continuous-time Markov chain whose generator has a very special structure. Using matrix-analytic methods we study the model in steady-state and discuss some special cases of the model as well as representative numerical examples covering a wide range of service time distributions such as constant, uniform, Weibull, and phase type.

Queue-Length, Waiting-Time and Service Batch Size Analysis for the Discrete-Time GI/D-MSP$^{\text {(a,b)}}/1/\infty $ Queueing System

This paper analyzes an infinite-buffer single-server bulk-service queueing system in which customers arrive according to a discrete-time renewal process. The customers are served under the discrete-time Markovian service process according to the general bulk-service rule. The matrix-geometric method is used to obtain the queue-length distribution at prearrival epoch. The queue-length distributions at other various time epochs are also derived based on prearrival epoch probabilities. A simple approach has been developed to compute the waiting-time distribution of an arriving customer. We also carried out closed-form analytical expression for the service batch size distribution of an arriving customer. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions and model parameters to understand the system behaviour.

Stationary queue and server content distribution of a batch-size-dependent service queue with batch Markovian arrival process:

Queueing systems with batch Markovian arrival process (BMAP) have paramount applications in the domain of wireless communication. The BMAP has been used to model the superposition of video sources and to approximate the super-position of data, voice and video traffic. This article analyzes an infinite-buffer generally distributed batch-service queue with BMAP, general bulk service (a, b) rule and batch-size-dependent service time. In this proposed analysis, we mainly focus on deriving the bivariate vector generating function of queue and server content distribution together at departure epoch using supplementary variable technique. The mathematical procedure for the complete extraction of distribution at departure epoch has been discussed and using those extracted probabilities, we achieve the queue and server content distribution at arbitrary epoch. Finally, numerical illustrations have been carried out in order to make a deep insight to the readers which contains deterministic as well as phase-type service time distributions.

Extended analysis and computationally efficient results for the queueing system

We consider an infinite waiting space queueing system in which customers arrive according to a renewal process. The server serves the customers in batches under general bulk-service rule. The successive service times of batches are mutually independent and have a common exponential distribution. To obtain the queue-length distributions at prearrival and random epochs, we have used the displacement operator which assists to solve simultaneous non-homogeneous difference equations. An analytically simple and computationally efficient procedures have developed to compute the waiting-time distribution of an arrival customer. Our formula to determine the waiting-time distribution is also useful even when multiple poles occur in the Laplace-Stieltjes transform of the interarrival time distribution. We have determined closed-form analytical expression for the distribution of size of a service batch of an arrived customer. We also have derived the results of some particular well-known queueing models from our model. Some numerical results are provided in the form of tables for a variety of interarrival-time distributions to demonstrate the variation of performance measures of the system.

Analysis of an <i>M</i><SUP align="right">[<i>X</i>]</SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/1 queueing model with multiple vacation controllable arrival during multiple vacation and two phases of repair with delay

The objective of this paper is to analyse an M[X]/G(a, b)/1 queueing model with multiple vacation, controllable arrival during multiple vacation and two phases of repair with delay. Whenever the queue size is less than 'a', the server resumes multiple vacation and continues this until at least 'a' customers are waiting in the queue. After finishing a batch of service, if the server is breakdown with probability γ, the server will be sent for repair after a short interval of time called delay time. After this delay, two consecutive phases of repair (first phase repair, second phase repair) are considered. After the second phase of repair or when there is no breakdown with probability 1-γ, the server starts a vacation if the queue length is less than 'a'. Otherwise, the server starts a service under the general bulk service rule. Using supplementary variable technique, the probability generating function of the queue size at an arbitrary time is obtained for the steady-state case. Also some performance measures and cost model are derived. Numerical illustrations are presented to visualise the effect of various system parameters.

Numerical Study on Transient Behaviour of Finite Bulk Arrival or Service Queues with Multiple Working Vacations

The transient behaviour of single server bulk queues with finite-buffer capacity is discussed numerically under the assumption that the server may take multiple working vacations after completing a busy period. It is considered that the inter-arrival and the service times are exponentially distributed and independent of each other. The buffer space is limited, therefore partial and full batch rejection policies are studied for the bulk arrival queueing model. However, for the batch service, the general bulk service rule is considered. Using probabilistic arguments and relating the state of the systems at two consecutive time epochs, differential equations are obtained to model such phenomena. Further, these equations are solved numerically by Runge-Kutta method and the time dependent numerical solutions are compared with the exact stationary solutions. The blocking probability and the mean waiting time of the first, last and an arbitrary customer are also reviewed mathematically and computed numerically.

On Finite Buffer Bulk Arrival Bulk Service Queue with Queue Length and Batch Size Dependent Service

This article deals with the mathematical modeling and steady state analysis of bulk arrival bulk service queue with finite buffer space under queue length and serving batch size dependent service policy. The service time distribution follows any general distribution and allowed to change only at service initiation instant, depending on queue length and serving batch size. A single server is providing service following general bulk service rule. We employed the supplementary variable technique and the embedded Markov chain technique to compute the joint probabilities of queue length and serving batch size at various epoch, viz., arbitrary epoch, departure epoch and pre-arrival epoch, in steady state. This paper ends with a numerical optimal control analysis to demonstrate the effect of the threshold limits for general bulk service rule on the total system cost of the system.

Analysis of an M[X]/G(a, b)/1 queueing model with multiple vacation controllable arrival during multiple vacation and two phases of repair with delay

The objective of this paper is to analyse an M[X]/G(a, b)/1 queueing model with multiple vacation, controllable arrival during multiple vacation and two phases of repair with delay. Whenever the queue size is less than 'a', the server resumes multiple vacation and continues this until at least 'a' customers are waiting in the queue. After finishing a batch of service, if the server is breakdown with probability γ, the server will be sent for repair after a short interval of time called delay time. After this delay, two consecutive phases of repair (first phase repair, second phase repair) are considered. After the second phase of repair or when there is no breakdown with probability 1-γ, the server starts a vacation if the queue length is less than 'a'. Otherwise, the server starts a service under the general bulk service rule. Using supplementary variable technique, the probability generating function of the queue size at an arbitrary time is obtained for the steady-state case. Also some performance measures and cost model are derived. Numerical illustrations are presented to visualise the effect of various system parameters.

An <i>M</i><SUP align="right">[<i>X</i>]</SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/1 queueing system with server breakdown and repair, stand-by server and single vacation

In this paper, we discuss a non-Markovian batch arrival general bulk service single server queueing system with server breakdown and repair, single vacation and stand-by server. There is a stand-by server which is employed during the period for which the regular server remains under repair. Both the servers serve the customer under general bulk service rule. Suppose at the instant main servers service completion if the queue length is less than 'a' then the main server goes for a vacation. Suppose at the instant of repair completion or vacation completion if the system size is less than 'a' then the main server stays in the system and waits for the next batch of arriving customers. The probability generating function of queue size at an arbitrary time and some performance measures of the system are derived. An extensive numerical result for a particular case of the model is illustrated.

An <i>M</i><SUP align="right">[<i>X</i>]</SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/1 queueing system with server breakdown and repair, stand-by server and single vacation

In this paper, we discuss a non-Markovian batch arrival general bulk service single server queueing system with server breakdown and repair, single vacation and stand-by server. There is a stand-by server which is employed during the period for which the regular server remains under repair. Both the servers serve the customer under general bulk service rule. Suppose at the instant main servers service completion if the queue length is less than 'a' then the main server goes for a vacation. Suppose at the instant of repair completion or vacation completion if the system size is less than 'a' then the main server stays in the system and waits for the next batch of arriving customers. The probability generating function of queue size at an arbitrary time and some performance measures of the system are derived. An extensive numerical result for a particular case of the model is illustrated.

Performance characteristics of a batch service queueing system with functioning server failure and multiple vacations

This paper examines bulk arrival and batch service queueing system with functioning server failure and multiple vacations. Customers are arriving into the system in bulk according to Poisson process with rate λ. Arriving customers are served in batches with minimum of ‘a’ and maximum of ‘b’ number of customers according to general bulk service rule. In the service completion epoch if the queue length is less than ‘a’ then the server leaves for vacation (secondary job) of random length. After a vacation completion, if the queue length is still less than ‘a’ then the server leaves for another vacation. The server keeps on going vacation until the queue length reaches the value ‘a’. The server is not stable at all the times. Sometimes it may fails during functioning of customers. Though the server fails service process will not be interrupted.It will be continued for the current batch of customers with lower service rate than the regular service rate. The server will be repaired after the service completion with lower service rate. The probability generating function of the queue size at an arbitrary time epoch will be obtained for the modelled queueing system by using supplementary variable technique. Moreover various performance characteristics will also be derived with suitable numerical illustrations.

State dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations and threshold

The objective of this paper is to analyse state dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations, threshold and constant retrial policy. Primary customers are arriving into the system in bulk with different arrival rates λa and λb. If arriving customers find the server is busy then the entire batch will join to orbit. Customer from orbit request service one by one with constant retrial rate γ. On the other hand if an arrival of customers finds the server is idle then customers will be served in batches according to general bulk service rule. After service completion, customers may request service again with probability δ as feedback or leave from the system with probability 1 − δ. In the service completion epoch, if the orbit size is zero then the server leaves for multiple vacations. The server continues the vacation until the orbit size reaches the value ‘N’ (N > b). At the vacation completion, if the orbit size is ‘N’ then the server becomes ready to provide service for customers from the main pool or from the orbit. For the designed queueing model, probability generating function of the queue size at an arbitrary time will be obtained by using supplementary variable technique. Various performance measures will be derived with suitable numerical illustrations.