Optimal control of an N - policy two - phase M X / M / 1 queueing system with server startup subject to the server breakdowns and delayed repair

In most queueing systems the server may be subjected to lengthy and unpredictable breakdowns while serving a customer. For instance, in manufacturing systems the machine may breakdown due to malfunction or job related problems; in computer systems, the machine may be subject to scheduled backups and predicted failures. In these systems, server breakdown results in a period of server unavailability time until it is fixed. Therefore, it is necessary to see how the breakdowns affect the level of performance of the system. Regarding the queueing systems with two phases of service Krishna and Lee [5] and Doshi[4] studied the distributed systems where all customers waiting in the queue receive batch service in the first phase of service followed by individual service in the second phase. Selvam and Sivasankaran [7] introduced the two phase queueing system with server vacations. Kim and Chae [6] investigated the two phase system with Npolicy.The server startups correspond to the preparatory work of the server before starting the service. In some actual situations, the server often requires a startup time before starting each service period. Baker[1]first proposed the N-policy M/M/1 queueing system with exponential startup time. Several authors have investigated queueing models with server breakdowns and vacations in different frameworks in recent past. Wang [13] for the first time proposed the Markovian queueing system under the N-policy with server breakdowns. Vasanta Kumar and Chandan [8] presented the optimal operating policy for the two-phase M x /Ek/1 queueing system under N-policy. Vasanta Kumar et, al. [9,10]. presented the optimal control of M x / M/1 and M/Ek/1 gated queueing systems with server startup and break downs. Vasanta Kumar et al.[11,12] presented the optimal operating policy for a two-phase M x /M/1 and M x /Ek/1 queueing systems under N-policy with server breakdowns and without gating. Choudhury and Tadj [2] considered an M/G/1 model with an additional phase of optional service with the assumption that the server is subject to breakdowns and delayed repair. Later Choudury et al. [3] investigated such a type of model, where the concept of N-policy is also introduced along with a delayed repair for batch arrival queueing system. In the two-phase M x /M/1 queueing system without gating analyzed by Vasanta Kumar et al.[11] breakdowns are considered in second phase of service, without taking into consideration the concept of delayed repair. Abstract: This paper investigates the economic behaviour of the two-phase M x /M/1 queueing system with Npolicy, server startup time and unreliable server, consisting of breakdown and delay periods. The service is in two phases of which the first phase is batch service provider to all customers waiting in the queue and the second phase is individual provider to each customer in the batch. The server is turned off and takes a vacation whenever the system is empty and turned on when the total number of customers in the system reaches the threshold N (N≥1), and starts preparatory work before providing the batch service. While the server is working with any phase of service, it may breakdown at any instant and the service channel will fail for a short interval of time. There may be delay in repair due to non-availability of the repairing facility. The startup times, batch service times, individual service times, breakdown and delay periods are assumed to follow an exponential distribution. The closed form of expressions for the performance measures of interest is obtained. The mean queue waiting time by heuristic interpretation and the reliability indices of this model are also derived. The total expected cost function per unit time is developed to determine the optimal threshold of N at a minimum cost. The sensitivity analysis is presented through numerical illustration.