Multi-server machine repair model with standbys and synchronous multiple vacation

Abstract

This paper investigates a machine repair problem with homogeneous machines and standbys available, in which multiple technicians are responsible for supervising these machines and operate a (R, V, K) synchronous vacation policy. With such a policy, if any V idle technicians exist in the system, these V (V < R) technicians would take a synchronous vacation. Upon returning from vacation, they would take another vacation if there is no broken machine waiting in the queue. This pattern continues until at least one failed machine arrives. It is assumed that the number of teams/groups on vacation is less than or equal to K (0 ≦ KV < R). The matrix analytical method is employed to obtain a steady-state probability and the closed-form expression of the system performance measures. Efficient approaches are performed to deal with the optimization problem of the discrete/continuous variables while maintaining the system availability at a specified acceptable level.

Introduction

In many industrial processes, production machines are unreliable and may have a breakdown. When a machine fails, it is sent to a maintenance facility and repaired by a group of technicians (servers). In order to achieve the production quota and reduce the loss of production capacity, the plant usually keeps standby machines that could substitute for a failed machine. In this paper, a machine repair problem, which includes M identical machines, S standby machines, and R technicians with synchronous multiple vacation policy is investigated. There are numerous researches on the machine repair problem or the multi-server queueing system with various vacation policies.

This paper first conducts a literature review on non-vacation servers (i.e. servers do not perform secondary tasks during their idle period). Ke and Wang (1999) analyzed machine repair problems with constant balking probability, negative exponential distributed reneging, and unreliable servers. A subsequent study, by Ke and Wang, 1999, Wang and Ke, 2003, revisited this model with reneging behavior. The system steady-state availability, MTTF, and some system performance measures were presented. Wang, Ke, and Ke (2007) investigated the profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures. They employed the direct search method and the steepest descent method in order to determine global maximum values to satisfy system constraints. A comprehensive and exhaustive discussion of machine repair problems was given by Haque and Armstrong (2007). Ke and Lin (2008) modeled manufacturing systems using two queueing systems with different repair rates and different numbers of technicians. As for vacation servers, Gupta (1997) first investigated a machine interference problem with warm spares and server vacations, including multiple vacations, single vacation, and hybrid multiple/single vacation schemes. A transform free, closed form expression of the probability distribution for the number of operating machines and performance measures was developed in Gupta’s work. Ke (2006) generalized Gupta’s work to unreliable-server cases. Numerical investigation and sensitivity analysis of the reliability and availability measures of a repair system were investigated by Ke and Lin (2005), in which the servers were imperfect, and applied a multiple vacation policy. Ke and Wang (2007) dealt with machine repair problems with a single/multiple vacation policy and two type spares, and a cost analysis for both vacation models was developed. Recently, Wang, Chen, and Yang (2009) studied the M/M/1 machine repair problem with a working vacation policy, where the server may work with different repair rates rather than completely terminate during a vacation period. In their work, the optimal number of machines and two different repair rates were determined using a direct search method and Newton’s method.

The second category of studies is in regard to a queueing system with synchronous vacation policy. Zhang and Tian, 2003a, Zhang and Tian, 2003b first introduced the multi-server queueing system with single/multiple synchronous vacations, in which some idle servers would take a vacation of random duration when finished serving customers, with no customers waiting. Moreover, Tian and Zhang (2003) investigated a GI/M/c queueing system with phase-type vacations where all servers take multiple vacations together until the system is not empty. Tian and Zhang (2006) considered a multi-server queueing system with a (d, N) vacation policy, in which d idle servers may take multiple vacations together until the number is equal to or more than a predetermined threshold N. They also conducted a computational study of the optimal value of controllable variable d, while N was presented under non-controllable parameter. Yue et al. (2006) studied a finite capacity queueing system with balking, reneging, and single synchronous vacations policy. They also obtained a matrix-form solution for the steady-state probability vector and some performance measures. A multi-server queueing model, with Markovian arrivals and synchronous phase type vacations, was investigated by Srinivas (2007), who performed several cases of MAP processes and numerical examples, including the tables of optimum values of system parameters, the corresponding system performances, and total expected costs were presented. Recently, Srinivas (2009) presented a steady-state analysis of the MAP/M/c queueing system with the phase type vacation and provided some interesting numerical results. However, existing research works regarding synchronous vacation do not include machine repair problems, and mainly focused on the infinite capacity queueing system.

In the photolithography process (see Uzsoy et al., 1992, Uzsoy et al., 1994), each job is processed by stepper machines, which are unreliable and are subject to unpredictable breakage. When a machine fails it is immediately sent to the maintenance department and repaired by technicians, as the stepper machines are critical resources in the photolithography process, thus, maintaining the machines operational performance in the system is very important. In the repair facility, an arriving broken stepper machine undergoes a random process. The service/repair time of each failed machine is by provided a technician, and could be regarded as a random variable. For the convenience of labor management, the technicians usually are divided into teams/groups of fixed size. Whenever there is an idle team, they would take a multiple vacation and leave the repair facility at random periods. The primary goal of leaving the repair facility is to improve the utility of the work force (support for other departments), or increase the abilities of personnel by joining a training course. A broken machine must wait for repair service in a queue when there is no available technician/server in the system. Therefore, the plant always maintains some standby machines to substitute the failed machines. As mentioned above, to allocate labor and maintain the operations of product machines are important for the engineers and management. However, regarding production or manufacturing systems, there are no studies on partial server multiple vacation (synchronous vacation) policies for the machine repair problems or finite arrival resources.

The objectives of this paper are as follows: (1) provide a matrix-analytical computational algorithm to develop the steady-state probability vectors; (2) derive the steady-state availability, the mean time to failure (MTTF), and other system performance measures; (3) construct a cost model to determine the optimal number of technicians (servers), the optimal vacation policy, the optimal service rate, and the optimal vacation rate; (4) conduct numerical study on the effect of parameters on the system characteristics.