Transient Analysis of Batch Service Queue with Second Optional Service and Reneging

This paper investigates a single server batch service queueing system with second optional service and reneging. The transient state probabilities of the queueing model are computed by using fourth order Runge Kutta method. Cost analysis is performed to determine the optimal values of the service rates of First Essential Service (FES) and Second Optional Service (SOS) simultaneously at the minimum total expected cost per unit time. Some important performance measures and numerical illustrations are provided in order to show the managerial intuitions of the model.


1.Introduction
In queueing theory, a service process is among the important characteristics of a queueing system. Recently, most of the service systems adopted the provision of more than one service. For instance, in transport industry passengers get transport service from buses/airline and other services like food, soft drinks, etc. However, majority of them may prefer only main service while few of them may demand an optional service. Complete details of these situations have been presented by [1][2][3][4][5] and considered an / /1 queue with second optional service which assumed to follow exponential distribution while first essential service is a general distribution. [6][7][8] considered both SOS and FES having a general service time distribution. Balking and reneging are two impatient acts where a customer tends to leave without receiving the service due to longer waiting. Queuing problems with balking and reneging was studied by [2]. A single server queueing model with impatience has been presented by [9,10], where the customers lose patience if the wait is more than the threshold they fixed. Later on, [11,12] studied a finite buffer multiple working vacation queues with balking, reneging and vacation interruption under N-Policy and obtained the solution for the steady state probabilities using recursive. The study of the transient solution of the performance measures of the queue system has been analysed by [14-18] etc. A time dependent solution of a single server queueing system with a Poisson input process has been considered by [19,20]. The transient behaviour of the infinite capacity / /1 modelwith batch arrivals and server vacations has been discussed by [21]. The transient behaviour analysis of an / /1/ queue with working breakdowns and server vacations has been studied by [22]. From the above literature study, most of the heterogeneous server queueing systems are analysed in steady state. However, the steady state measures do not explain the real picture of the system behaviour, since they ignore the transient and the initial behaviour of a system [18]. In most of the practical applications, the state of the system experiences a change and such changes can be comprehended by the transient behaviour. Therefore, this article aims to study a / [ ] /1/ queue with SOS and reneging in transient state. The transient probabilities of the queueing model are obtained using fourth order Runge-Kutta method.

Model Description
We consider an / [ ] /1/ queueing model with SOS and reneging customers in which the customers arrive according to a Poisson process with mean rate . The service times distribution of both FES and SOS are exponential with mean rates µ 1 and µ 2 , respectively and the services are given in batches of size not more than such that if the server finds the customers less or equal to in the waiting queue, the server takes all of them in the batch for service, but if the server finds the customers more than waiting in the queue, then the he takes a batch of size while others remain waiting in the queue. The FES is required by all arriving customers and after completing FES, they may opt SOS with probability or may leave the system with probability (1 − ). The capacity of the system is finite ( ). A customer upon joining the queueing system and waiting in queue for some time, may get impatient and renege from the queue if his waiting time is beyond the expected time limit. Reneging times follow an exponential distribution with rate .

Formulation of Mathematical Model
The / [ ] /1/ queue with SOS and reneging can be modeled by a two-dimensional continuous time Markov process {( ( ), ( )); ≥ 0}, where, ( ) is the number of customers in the queue, and ( ) is the server state with The state space of the Markov process is given by The transient probabilities are defined as Here, , ( ) is the transient probability that there are customers in the queue at time and the server is providing FES (SOS) service, and ( ) is the probability that the queue is empty at time and the server is idle. Using Markov theory, thedifferential-difference equations of the model are as follows: The above is an implicit set of differential equations whose analytical solution is a difficult task. However, we shall use the numerical technique to derive the approximate solutions assuming some initial conditions.

Transient Solution of the Model
In this section, we apply the fourth order Runge-Kutta method (R-K-4) to obtain the numerical solutions to the above equations. We assume that the system is empty at the starting point when the server is idle and the probability of the state of the initial system at time = 0 are given as follows: (0) = 1, , (0) = 0, = 1, 2, 0 ≤ ≤ The set of differential difference equations (1) to (8) can be expressed as: The formula used for Runge-Kutta method of fourth order is given by: where; ∆ is the step size, and 0 ≤ ≤ − 1

Performance Measures
In this section, we present the performance measures for anticipating the system behaviour.
The various performance measures are as given below.

Cost Analysis
In this section, we develop the total expected cost function per unit time based on the steady state system performance measure when → ∞. Our objective is to determine the optimal value of service rates of FES and SOS i.e., (µ 1 and µ 2 ), respectively, so that the total cost function is minimized. Let us define the following cost elements: ≡ Cost per unit time per customer present in the queue, 1 ≡ Cost per unit time when the server is serving in FES, 2 ≡ Cost per unit time when the server is serving in SOS, ≡ Cost per unit time per lost customer due to reneging.
The above expected cost function is highly complex and non-linear in terms of (µ 1 , µ 2 )and the derivatives of the cost function are not easily available. Therefore, we solve the above optimization problem using Quadratic Fit Search Method (QFSM), since this method is highly suitable for the non-differentiable objective functions.

Numerical Investigation
In this section we demonstrate the applicability of the Runge-Kutta method in transient state models using MATLAB software. The numerical computations are presented in the form of tables and graphs. The values of the system parameters are assumed to be = 2, µ 1 = 3, µ 2 = 2.5, = 0.1, = 0.36, = 10, = 5. Table 1, the transient probabilities obtained by Runge-Kutta method tend to the corresponding steady state probabilities for large values of . This confirms correctness, accuracy and efficiency of the model.   Table 3 provides the sensitivity analysis for optimum serviceµ 1 * ,µ 2 * , optimum cost * and some performance measures. We observe that for fixed batch size the optimum service rates, cost, , . , , and are increasing as the arrival rate is increasing, which is in accordance with the practical situations. The trend is reversed when we fix and increase batch size value .   Figure 3 shows the effect of arrival rate on the expected queue size with respect to time. We observe that the expected number of customers in the queue gradually increases as increases. In figure 4 one can observe that the expected queue size increases progressively from zero initially up to a certain value where it attains the steady state. In addition, we observed that the expected queue size in SOS is lower than in FES. Figures 5 and 6 demonstrates the impact of α and r, respectively on the expected number of customers in the queue versus time. We can see that as increase the expected queue size decrease. Furthermore, we observed that initially the customers were patience with a variation of reneging rate, but as time goes on, some customers' loss patience and decide to leave the queue. In figure 6 we observe that as increases with time the queue size in both case SOS and FES increase. In addition, when increase the expected queue size in SOS increase, this is due to the fact that more customers demand SOS.

Practical Application of the model
The model has numerous applications such as in hospital management systems, bank services, computer and communication networks, production system, etc. For example, in the hospital management systems the model can be applied to the situation where the outpatients request the appointment in the clinic system. The clinic officer monitors the length of booking windows for appointment of the outpatients, but since in most cases there is limited number of physicians/doctors,it leads to the unbalanced ratio between the number of outpatients and that of the physicians/doctors. This situation leads to an increase in the length of the booking window and brings the long waiting for appointments. Therefore, in order to shorten outpatients waiting time, we limit the length of the booking window and we assume the system has a limited slots capacity during a limited length of booking window. The system capacity can be divided into equal amount of slots i.e. ( = ). The slots are termed as a maximum size of a batch of the outpatients appointment and is equal to the number of sessions/periods. In practice, the new requests for appointments come individually and the service can be provided from single patients up to the maximum batch size b of the patients. And after the first service the patients may opt for another service provided by a clinic. On the other hand, some patients cancel the appointment and those patients are termed as reneging patients.

Conclusion
In this paper, we present / [ ] /1/ queue with second optional service and reneging in transient state. The fourth order Runge Kutta method is used to determine the transient solution of the queueing model and obtained the transient probabilities of the model. We developed the cost analysis to determine the optimal service rates of FES and SOS.
Furthermore, we presented some performance measures along with the numerical results to show the effect of various system parameters on the transient behaviour of the model. [15]Hofkens, T, Spacy, K and Blondia, C.