Analysis of bulk arrival queueing system with batch size dependent service and working vacation

A Mx/G/1 Queueing system with two service modes under (a, c, b) policy is considered. A single server Poisson queue with state dependent service is considered for analysis. The service is rendered with different service patterns (different distributions) as singly or in bulk according to the queue length. If the queue length reaches the threshold value ‘a’ then the server does single service and if the queue length reaches the value ‘c’(c > a), then the server selects batch service with variable batch size with a maximum batch size of ‘b’ (b > c). The server starts the service only if number of customers in the queue is at least ‘a’. The Server switches over from single service to batch service or vice-versa only at service initiation epochs depending on the queue length. The probability generating function of the queue size at an arbitrary time epoch is obtained using supplementary variables technique. Some important performance measures are also obtained. Numerical example is presented in order to illustrate the effectiveness of the model.

In this paper, we discuss about the steady state behaviour of M/G/1 retrial queueing system with two phases of services and immediate feedbacks under working vacation policy where the regular busy server is affected due to the arrival of negative customers. Upon arrival if the customer finds the server busy, breakdown or on working vacation it enters an orbit; otherwise the customer enters into the service area immediately. After service completion, the customer is allowed to make finite number of immediate feedback. The feedback service also consists of two phases. At the service completion epoch of a positive customer, if the orbit is empty the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. Using the supplementary variable technique, we found out the steady state probability generating function for the system and in orbit. System performance measures and reliability measures are discussed. Finally, some numerical examples are presented to validate the analyticalresults.

Analysis of an unreliable retrial G-queue with working vacations and vacation interruption under Bernoulli schedule

This paper deals with the analysis of two classes of batch arrivals, one is high priority and other is low priority(retrial) customers with non-preemptive priority service, working vacations, negative arrival, emergency vacation for an unreliable server, which consists of a break-down by negative arrival and delay time to start repair. When there are no customers in the queue and in the orbit at the time of service completion for a positive customers, the server goes for a working vacation. The server works at a lower service rate during working vacation period. If there are no customers in the system at the end of vacation, the server becomes idle and ready for serving the new arrivals with probability p(single working vacation) or it remains on vacation with probability $$q=(1{-}p)$$ (multiple working vacations). Here we assume that customers arrive according to compound Poisson process in which high priority customers are assigned to class one and class two customers are of a low priority type. The priority customers that find the server busy are queued and then served in accordance with FCFS discipline. The arriving low priority customers on finding the server busy are queued in the orbit in accordance with FCFS constant retrial policy. The arrival of negative customer remove the customer being in service and makes the server breakdown and the server fail for a short interval of time. When the server fails it will send to repair. But the repair time do not start immediately, there is a delay time to start the repair process. We consider an emergency vacation which means that during the service time the server suddenly go for a vacation and the interrupted customer waits to getting the remaining service. The retrial time, service time, emergency vacation time, delay time and repair time are all follows general(arbitrary) distribution, working vacation time and negative arrival follows exponential distribution. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results are obtained explicitly. Also some performance measures of our model such as the system state probabilities, reliability indices, the average number of customer in the high priority queue and the low priority in the orbit and the average waiting time are derived. Numerical results are computed and graphical representations are presented.

We study a queueing system of an unreliable server providing services to customers with two different service rates. When the system is empty of customers, the server takes a vacation and provides a low service rate. Once the vacation is completed, the server is resumed to serve customers, if any, at a higher service rate, or takes another vacation if the server found an empty system. The server is subject to random breakdowns when serving customers and if failed, it enters a repair stage. Under different system information levels, we investigate customers’ equilibrium strategies and follow-up system measures. Numerical comparisons among different information levels reveal that when the traffic load is large enough, the service provider (SP) prefers to disclose the server’s status and queue length information to customers to maximize the system throughput. While when the traffic load is small enough, there exist two cases with regard to the failure rate: (i) If the failure rate is smaller than a certain threshold, the SP tends to conceal the system information. (ii) If the failure rate is greater than the threshold, revealing the information is helpful to maximize the system throughput.

The objective of this paper is to analyze an $$M^{[X]}/G(a,b)/1$$ queueing model with admission control, multiple vacation, restricted re-service, setup and closedown times. At the moment of service completion, the leaving batch of customers may request for re-service with probability $$\eta $$ and it is not compulsory to accept the request. The server accepts the request with probability $$\zeta $$ and restricts it with probability $$1-\zeta $$ . All the batches of customers are not allowed all the time into the system. On completion of a service or re-service, if the queue length is $$\omega $$ , where $$\omega <a$$ , then the server resumes closedown work. After that, the server leaves for a vacation of random length. When the server returns from a vacation if the queue length is still less than a, he leaves for another vacation and so on until he finds at least a customers in the queue. If the queue length is at least a, then the server commences setup work before starting the service. Using supplementary variable technique, the steady-state probability generating function of the system size at an arbitrary time is obtained. The performance measures and cost model are also derived. Numerical illustrations are presented to visualize the effect of system parameters.

This paper deals with the analysis of an M/G/1 retrial queue with starting failure and working vacation which arise in many industries. The single server provides service in both a normal busy period and in a vacation period, which turn as working vacations. We assume that, during the normal busy period, the server faces unreliability due to starting failure. In working vacation period the server will provide service lower rate rather than completely stopping the service. After the vacation completion instant, the server will come back to the normal working and starts serving the customers. The necessary and suffcient condition for the system to be stable have been derived. Using supplementary variable method, we obtain the stationary probability distribution and some performance measures. To validate the proposed model some numerical examples are illustrated. Further, we carry out some special cases for the proposed model.

In this paper, we analyze finite buffer M / G / 1 queue where service is offered in groups/batches according to ‘general bulk service’ rule by a single server. The service time distribution is considered to be generally distributed and allowed to change dynamically depending on the batch size under service and queue length just before the service initiation of the batch under consideration. Using the embedded Markov chain technique and supplementary variable technique, we obtain the joint distribution of the queue length and batch size at various epoch. At the end, we present several numerical results in the form of self-explanatory table and graphs to bring out some interesting features of the model.

In this paper, we study a renewal input working vacations queue with state dependent services and Bernoulli-schedule vacations. The model is analyzed with single and multiple working vacations. The server goes for exponential working vacation whenever the queue is empty and the vacation rate is state dependent. At the instant of a service completion, the vacation is interrupted and the server resumes a regular busy period with probability 1 − q (if there are customers in the queue), or continues the vacation with probability q (0 ≤ q ≤ 1). We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, using some numerical results, we present the parameter effect on the various performance measures.

This research article investigates the stationary N-policy with a set-up time of a single server bulk arrival queueing model rendering two-heterogeneous optional repeated service (THORS) with the factor of unreliability. The server studied here is allowed to take a single vacation after each busy period following Bernoulli Schedule (BS). An extensive analysis of the proposed model is carried out using the Supplementary Variable Technique (SVT) under some suitable transformations. The joint distribution of the server’s state and queue length are obtained under both elapsed and remaining times along with their double transforms. Probability generating function (PGF) of the queue size distribution together with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides the busy period distribution and various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, some numerical experiments are carried out in support of the theory.

A single server retrial queueing model with multiple working vacations and starting failure is discussed in this paper. The server goes for multiple working vacations when orbit becomes empty at any service completion instant. We obtain the steady-state probability generating function (PGF) for this model’s system size and orbit size, mean orbit size, and mean system size by using the supplementary variable technique. Necessary system performance measures and some exceptional cases are discussed.KeywordsRetrial queueWorking vacationStarting failure

This paper considers an infinite buffer single server batch service queue with single exponential working vacation policy. The inter-arrival times are generally independent and identically distributed random variables and the service times are exponential. The server accesses new arrivals even after service has started on any batch of initial number a. This operation continues till the service time of the ongoing batch is completed or the maximum accessible limit d of the batch being served is attained whichever occurs first. The supplementary variable technique and the recursive method are used to develop the steady-state queue length distributions at pre-arrival and arbitrary epochs. Some performance measures and numerical results are discussed.

ABSTRACTThis paper analyses an M/M/c queueing system with variant working vacations. The service times during regular busy periods, working vacation periods and vacation times are exponentially distributed and are mutually independent. During working vacation, customers may renege due to impatience and the impatient timer follows an exponential distribution. We derive the probability generating function of the steady-state probabilities and obtain some performance measures. The cost function associated with the model is also constructed and its optimization is investigated using quadratic fit search method. We have also presented the numerical illustrations of the effect of some parameters on the performance measures using graphs.

In this paper, we model a discrete-time renewal input queue with change over times and Bernoulli schedule vacation interruption under batch service (a, c, b) policy. The service, working vacation and change over times are geometrically distributed. At the instants of a service completion, the vacation is interrupted and the server is resumed to a regular busy period with probability 1 − q if there are c or more customers in the system, or continues the vacation with probability q. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at various epochs. Some performance measures and a cost model have been presented and an optimum service rate has been obtained using geneticalgorithm. Numerical results showing the effect of the parameters of the modelon the key performance measures are presented.

In this paper, a bulk service queueing system with variant threshold policy for secondary jobs is considered. On completion of service, if the queue length is less than ‘ a ’, the server performs a secondary job of type one, repeatedly, until the queue length reaches ‘ a ’. On returning from a secondary job of type one, if the queue length is at least ‘ a ’, the server performs another secondary job of type two, repeatedly, until the queue length reaches the threshold value ‘ N ’ ( N ≥ b > a ), then the server serves a batch of ‘b’ customers. Various performance measures and a cost model are presented.