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.

Transient Analysis of $$M^{[X_{1}]},M^{[X_{2}]}/G_{1},G_{2}/1$$ M [ X 1 ] , M [ X 2 ] / G 1 , G 2 / 1 Retrial Queueing System with Priority Services, Working Vacations and Vacation Interruption, Emergency Vacation, Negative Arrival and Delayed Repair

Bulk arrival retrial queueing model with M stages of service and Bernoulli vacation is discussed in this chapter. Customers arrive in batches according to Poisson process. Server provides M stages of heterogenous service in succession to each customer. After receiving service from one stage, the customer may either move to next stage or depart the system. If the customer leaves the system, the server takes Bernoulli vacation. The vacation time, service time and retrial time are assumed to be generally distributed. The stability condition of the system is derived. Using generating function technique, the steady state distributions of the server state and the number of customers in the orbit are obtained along with important performance measures. Stochastic decomposition law is verified. The effects of various parameters on the system performance are analysed numerically. The model is further analysed to construct the membership function of the mean orbit size with fuzzy parameters.

Bulk arrival multi-stage retrial queue with vacation

This paper deals with the analysis of batch arrival retrial queue with two classes non-preemptive priorityunits, negative arrival, balking as well as reneging, feedback, emergency and Bernoulli vacation for anunreliable server. Here we assume that customers arrive according to compound Poisson process inwhich priority customers are assigned to class one and class two customers are of a low-priority type.If the server is free at the time of any batch arrivals, the customers of this batch begins to be servedimmediately. The low-priority customer may join the orbit with feedback if the service is not satisfied(or) may leave the system if the service is satisfied. The priority customers that find the server busyare queued and then served in accordance with FCFS discipline. The priority customers may renegethe queue if the server is not avilable in the system and there is no optional for feedback service tothe priority customers. The arriving low-priority customers on finding the server busy then they arequeued in the orbit in accordance with FCFS retrial policy without balking (or) may balk the orbit.While the server is serving to the customers, it faces two types of break-down there are breakdowns bythe arrival of negative customer and break-down at any instant of service and server will be down for ashort interval of time. Further concept of the delay time of repair is also introduced for breakdowns. Weconsider two different kinds of vacations, one is an emergency and the other one is Bernoulli vacation, theemergency vacation means at the time of the server serving the customer suddenly go for a vacation andthe interrupted customer waits to get the remaining service and after the completion of each service, theserver either goes for a vacation or may continue to serve for the next customer; if any . The retrial time,service time, vacation time, delay time and repair time are all follows general(arbitrary) distribution.Finally, we obtain some important performance measures of this model. Keywords: Batch arrival, priority queue, retrial queue, negative arrival, emergency and Bernoulli vacation, unreliableserver, breakdown and repair. AMSC : 60K25; 60K30; 90B22

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Transient Analysis of M[X1];M[X2]=G1;G2=1 Queueing Model with Retrial Priority Service, Negative Arrival, Two kinds of Vacations, Breakdown, Delayed Repair, Balking, Reneging and Feedback

Bulk arrival retrial G-queue with two phases of service and feedback is considered. The two phases include the first phase of essential service and the second phase of multistage of services having multi options. In addition to this the breakdowns due to starting failure and due to arrival of negative customers are included. If the server is free one of the arriving batch of positive customers receives the essential service, otherwise the batch joins the orbit. After completing essential service the customer moves to second phase, leaves the system or joins the orbit as feedback customer. In second phase the customer has to move the stages in succession. The customer has provision to leave the system or join the orbit after service completion at any stage of second phase. The generating functions of the queue size distribution and the steady state probabilities of the server state are analysed explicitly. Numerical results are presented to explore the effect of several parameters on the system measures.

Multi stage and multi optional retrial G-queue with feedback and starting failure

In present paper we have introduced the concept of differentiated vacations in a retrial queueing model with state dependent arrival rates of customers. The arrival rate of customers is different in various states of the server. The vacation types are differentiated by means of their durations as well as the previous state of the server. In type I vacation, server goes just after providing service to at least one customer whereas in type II, it comes after remaining free for some time. In steady state, we have obtained the system size probabilities and other system performance measures. Finally, sensitivity and cost analysis of the proposed model is also performed. The probability generating function technique, parabolic method and MATLAB is used for the purpose.

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Analysis of classical retrial queue with differentiated vacation and state dependent arrival rate.

This paper deals with a single server retrial queueing model in which customers arrive according to a Markovian arrival process. An arriving customer on finding a free server enters into service immediately; otherwise the customer enters into an orbit of infinite size. An orbiting customer competes for service by sending out signals at random times until a free server is captured. The server operates under two vacation mechanisms. During service the server may go for vacation to attend an emergency call and after completing the vacation, the server continues the service for the same customer. Upon completion of a service, with a certain probability the server takes multiple vacations depending on the orbit size. Assuming the service, retrial and vacation times follow a general distribution, the joint distribution of the state of the server and the number of customer in the orbit is derived in the steady state. The explicit expressions of some performances measures are given. In addition numerical results are presented.

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Single Server Retrial Queuing System with Two Different Vacation Policies

Abstract — An M/G/1 retrial queueing system with k stages of heterogeneous services and feedback is considered. Primary customers get into the system according to Poisson process. If the server is free, an arriving customer receives first stage service immediately otherwise, he enters a retrial orbit. After the completion of the first stage, the customer may proceed to second stage with probability θ1, or feedback to the retrial group with probability p 1 or depart the system with probability 1 – θ1 – p1 = q1. In general, after the completion of the i, (i = 1, 2, ...., k−1) stage, the customer may opt (i+1) stage with probability θ1, or feedback to the retrial group with probability p i or depart the system with probability 1 – θ1 – pi = qi. The customer in final stage will feedback to the retrial group with probability pk or depart the system with probability 1 – pk = qk. It is assumed that the service times and the retrial times are arbitrarily distributed. The condition under which the steady state exists is investigated. Performance measures are obtained. A stochastic decomposition is presented.

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Multi-Stage Retrial Queueing System with Bernoulli Feedback

A single server batch arrival retrial queue with se rver vacation under Bernoulli schedule is considere d. Arrivals are controlled according to the state of the server. The necessary and sufficient condition for the system to be stab le is derived. Explicit formulae for the stationary distributions and performance measur es of the system in steady state are obtained. Nume rical examples are presented to illustrate the influence of the parameters on se veral performance characteristics.

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Anna Bagyam

Papers

Single Server Retrial Queuing System with Two Different Vacation Policies

Multi-Stage Retrial Queueing System with Bernoulli Feedback

Batch arrival retrial queueing system with state dependent admission and bernoulli vacation