COST OPTIMIZATION OF SINGLE SERVER RETRIAL QUEUEING MODEL WITH BERNOULLI SCHEDULE WORKING VACATION, VACATION INTERRUPTION AND BALKING

In present paper an M/M/1 retrial queueing model with working vacation interruption using Bernoulli schedule and Balking is analyzed under classical retrial policy. When sever is occupied in regular busy period customer enters in system with probability b and during working vacation of server, customer joins the orbit with probability v. Whenever no customers are in the system after vacation completion then either server returns to normal free state with probability q or goes for multiple working vacation with probability 1-q. Server provides the service at lower rate during working vacation than normal busy period. In this paper steady state probabilities have been obtained using probability generating function technique. Some important performance measures of this model are also evaluated numerically and some results are shown graphically using MATLAB software.


INTRODUCTION
Retrial queues have wide applications in field of communication networking, computer systems, call centres and telephone switching systems etc. In retrial queues arriving customers, on finding the busy server, instead of joining the queue in front of server, join the virtual room called orbit and 2509 SINGLE SERVER RETRIAL QUEUEING MODEL retry for their request after some random period of time. Otherwise, arrival will get service immediately, if server is free at the time of arrival. Retrial queues have been studied extensively in literature. Yang,Templeton [23] and Falin [8] analyzed the retrial queues. Gomez-Corral [9] studied stochastic analysis of retrial queues with general retrial times. Artalejo and Corral [3] did the pioneer work on retrial queueing system.
The impatient customer behaviour is a challenge in modeling queueing system. The pioneer work had been done by Haight [7] to study queueing model with balking. Multi server queue with fixed probability of balking is presented by Al-Seedy et al. [19]. Choudhury and Medhi [1] analyzed M/M/C queueing model with balking and reneging. Ammar et al. [22] studied busy period of single sever queueing model with balking and reneging. Yue et al. [6] investigate M/M/1/N system with dynamic balking probability. Vijaya Laxmi et al. [16] analyzed finite buffer queueing model with working vacation and impatient behaviour of customers. Kumar and Sharma [17] present multiserver finite capacity queueing model with impatient behaviour of customers. Kumar and Sharma [18] also extended their work to the case of infinite capacity. Some researchers studied queueing model with dynamic probability of balking.
A special class of vacation queueing models where system don't stop working completely rather provides service at relatively slow rate is working vacation. The vacation models with Bernoulli schedule where server goes on vacation or remains in system with probability 1-q or q respectively provides a control on congestion of system. Many researchers worked on working vacation queueing models. In this respect an important work on GI/G/1 queueing model with Bernoulli schedule vacation was done by Keilson and Servi [10]. Servi and Finn [12,13] did pioneer work on working vacation queueing model. Takagi [5] also studied single server queueing model with Bernoulli vacation. Later on a general working vacation queueing model was analyzed by Banik et al. [2].
Arivudainambi et al. [4] analyzed single server retrial queue with working vacation. This paper analyses M/M/1 retrial queue with working vacation and vacation interruption with different levels of customer impatience at the time of arrival in busy state of server in normal as well in working vacation period. Bernoulli schedule of working vacation at vacation completion instant provides an option that server may go on multiple working vacation with probability 1-q or may return to normal state with probability q, when there is no customer in the system i.e. a generalization of single and multiple working vacation. The concept of vacation interruption is also important for effective utilization of server where vacation is interrupted on service completion 2510 POONAM GUPTA, NAVEEN KUMAR instant if customers are present in the system. Li and Tian [11] studied queuing model with vacation interruption. Later on Zhang and Hou [15] analyzed M/G/1 queueing model with working vacation and vacation interruption. A pioneer work on M/G/1 retrial queueing model with Bernoulli schedule working vacation and vacation interruption has been given by Gao et al. [20,21]. GI/M/1 queues with vacation interruption under controlled Bernoulli schedule was analyzed by Tao [14].
In this paper, we have considered classical retrial queue with Bernoulli Schedule of working vacation interruption along with impatient customer behaviour in both normal and vacation state using the method of probability generating functions. The various sections of the paper are described as below: The model description of the paper is given in section 2. The steady state equations are described in Section 3. Section 4 describes some system performance measures and normalization condition.
Section 5 illustrates the graphical results of the model and finally conclusion is given in section 6.

MODEL DESCRIPTION
In present paper, M/M/1 queueing model with impatient behaviour of customers, Bernoulli working vacation and vacation interruption under classical retrial policy is considered. Description of model of present paper is given as follows: (1) Customer arrives in the system with arrival rate which follows Poisson distribution. On arrival customer decides to join the system or balk depending on the state of server. If server is not busy then arriving customer gets service immediately otherwise if the server is busy in normal working state then arrival either joins the orbit with probability b or balk with probability 1-b. On the other hand if server is busy in working vacation state then arriving customer joins the orbit with probability v or balk the system with probability 1-v (2) Customers get service using first come first served (FCFS) basis. In normal busy state service time is assumed to follow exponential distribution with mean 1/µ.
(3) Customers in the orbit retry for service with retrial rate ξ, which follows Poisson distribution. If retrial customers find free server then request is accepted immediately otherwise retrial customers waits in the orbit for his turn. 2511 SINGLE SERVER RETRIAL QUEUEING MODEL (4) When system is empty then server goes for working vacation. Vacation period is exponentially distributed with parameter φ. In working vacation state customer gets service with lower service rate θ (<µ) which is exponentially distributed. On completion of vacation if server finds customer in the system then server goes to normal busy period otherwise if there is no customer in the system then either server goes to regular free sate in normal service period with probability q or continues vacation with probability 1-q, thereby giving rise to a generalization of single or multiple working vacations.
Let N(t) is number of customers in the orbit (a free pool) at a given time t and H(t) denotes the state of server at a given time t. The possible values of the server states H(t) can be: , the server is free in normal service state 1, the server is busy in normal service state 2, the server is free in working vacation state 3, the server is busy in working vacation state In present model, we consider Bernoulli Schedule working vacation and vacation interruption policy. So, the states {(n, 2), n ≥ 1} doesn't exists.

SYSTEM PERFORMANCE MEASURES
From equation (20), we have From equation (11) 0 From equation (19) we get Using L-Hospital rule On differentiating equation (19) and using L-Hospitals rule twice we obtain Expected orbit length is given by The probability of server being in busy state is: The probability of server being in free state is: The probability of server being in regular service state is The probability of server being in working vacation state is

GRAPHICAL RESULTS
In this section, we illustrate the effect of various parameters like ξ, θ, μ on expected orbit length E[ ] along with effect of ξ on probabilities of server being in busy and working vacation state.
Further we have optimized the cost with respect to θ using parabolic method.
In below graphs, we have fixed the parameters λ = 0.9, μ=1.8, ξ=0.9, θ=0.3, φ=0.4, b=0.7, v=0.5 unless they are used as a variables in the graph.   Figure 2 shows that E[ ] varies inversely with θ, as expected. Since as θ increases the mean service time in vacation decreases resulting in decrease in expected queue length. This decrease is not so obvious for large values of θ because we have considered that as soon as service of a customer is completed in working vacation period server resumes the normal working state, if any customer is waiting in the system.      The new value obtained here replaces one of the three points to improve the current 3 point pattern.

a) SENSITIVITY ANALYSIS
This process is used iteratively till optimum value is obtained up to desire degree of accuracy. Table 1 shows that optimum value ( ) = 131.446 corresponding to θ = 0.2676 with permissible error of 10 −3 , which agrees with results of Figure 6.

CONFLICT OF INTERESTS
The authors declare that there is no conflict of interests.