Random search algorithms for redundancy allocation problem of a queuing system with maintenance considerations

Highlights

• In this paper a Redundancy Allocation problem (RAP) for a queueing system is modeled.

• Optimal number of repairmen and redundant servers are found by which queueing and maintenance costs are minimized.

• Four random search algorithms are developed to deal with different structures of solution space.

• Two algorithms for unimodal objective functions and two algorithms for multimodal objective functions are developed.

Abstract

The Redundancy Allocation Problem (RAP) is becoming all the time more important in system reliability design. Therefore, RAP has been studied vastly for different systems and under various assumptions. One of the important systems that RAP can be applied to them is queuing systems. In many applications, reliability of queuing systems need to be improved with regard to some queuing and reliability cost constraints. Despite the importance of queuing systems in real world situations, RAP of these systems have not gained attention in the literature. Therefore, RAP of a queuing system with maintenance considerations and including queueing costs in the modeling of RAP is considered in this paper. To find optimal solutions of the problem, four random search algorithms are developed and compared based on the problem structures. According to the experiments, for unimodal solution spaces algorithms 1 and 2 and for multimodal solution spaces algorithms 3 and 4 are recommended.

Introduction

In every service system, there is a number of servers to provide a set of services to the incoming customers. Normally, customers wait in a queue till receiving their services in such a service system. Consequently, a service system can be modeled/analyzed as a queuing system. In addition, customer satisfaction is a central point of attention for the owner of the system. One of the factors that shows the attention of system's owner to customers is the reliability of the system to provide the service over the time. Reliability of a service system can be measured by the probability of remaining active in providing the service to the customers without stop. In other words, if the servers fail to provide the service for a while, the system stops temporarily/permanently its service. Like most of the systems in modern and industrial environments, failures in queuing systems can also cause great damages and costs. In this regard, standby and maintenance techniques are extensively adjusted to improve reliability and availability of the systems, i.e., in case of system failure and before replacing failed components of a system, engineers usually allocate some hot or cold standbys to the system. Evidently, as the number of standbys augments, the cost of the system and need for maintenance crew proportionally grows. In this context, achieving an optimal number of standby servers and repairmen by which total costs of the queuing system is minimized could be interesting. This problem can be seen as configuring queuing systems in terms of redundancy allocation with cost and maintenance considerations.

In most of the queuing literature, it is assumed that the server is operational on an everlasting basis. However, these assumptions are apparently idealistic. Such a type of queuing system is identified as queue with service stoppages or queue with unreliable server. Maintenance and redundancy allocation strategies such as considering standbys can be applied to these systems for improving the reliability. Many researchers have developed maintenance strategies and models for queuing systems. For example, Kumar et al. [1] analyzed the performance of a single queue server with maintenance considerations and close down period. Taleb and Aissani [2] studied an unreliable retrial queue with persistent and impatient customers. They considered corrective and preventive maintenances for this system and derived some performance measures and analyzed the time to the first failure of the server. Unreliable queuing systems can be designed based on reliability targets and cost considerations. Redundancy Allocation Problem (RAP) is a well-known problem in addressing this problem.

Redundancy Allocation Problem (RAP) is an important problem in the area of reliability optimization, which has been vastly investigated in the literature. In RAP, system configuration is determined by either maximizing system reliability under budget constraints or by minimizing system cost under constraints on the lower bound of system reliability. It is evident that when RAP is considered in the context of queuing systems, the difficulty of the resulting problem becomes more. This problem has not been deeply studied, especially when maintenance exists. Due to the importance of queuing systems for modeling a service system and capability of RAP in enhancing reliability of such a system, we are going to formulate and solve an RAP for a queuing system.

Various solution methods are proposed in the literature for solving RAP. Generally, we can classify these solution methods to:

• exact approaches, such as branch and bound [3,4], and dynamic programming [[5], [6], [7]],

• special-purpose heuristics [8,9] and metaheuristics like simulated annealing [10], genetic algorithms [[11], [12], [13]], tabu search [14,15], ant colony optimization [16], variable neighborhood search [17], particle swarm optimization [18], artificial bee colony [17] and hybrid algorithms [19]. For many of the complicated problem formulations, it is not possible to show the convexity of the solution space and/ or the convexity (concavity) of the objective function. In addition, even if the problem is recognized as a convex optimization problem (COP), a set of equations corresponding to Karush–Kuhn–Tucker or Fritz–John optimality conditions should be solved in order to find an algebraic closed form solution (ACFS). But, even many COPs cannot be solved parametrically to find an ACFS due to the inherent difficulty of solving such set of optimality equations. Furthermore, since RAP is an NP-hard combinatorial optimization problem [20], it is expected to observe an extensive growth in the needed computational efforts in the worst case, when the size of the system grows. Hence, most of the previous studies lie in the non-exact general-purpose approaches. Also, developing an efficient solution method depends on the problem structures. In addition, developing efficient algorithms according to the structure of the problem (called special-purpose algorithms) can make considerable reduction in computational efforts for finding good quality feasible solutions, within certain distance from optimal solution. Among others, special-purpose random search algorithms can find such good quality solutions in reasonable CPU times in the absence of convexity guarantee, especially when the objective function is multi-modal [21]. For instance, Karimi–Nasab and Sabri–Laghaie [22] developed three random search algorithms to find optimal rates of main and rework processes in a batch production system with the possibility of imperfect production. Jeżowski et al. [21] addressed a random search optimization method for highly multi-modal nonlinear problems with continuous variables. Nagaraju et al. [23] developed an adaptive EM algorithm for Non-homogeneous Poisson process (NHPP) software reliability growth models (SRGM) which is a combination of EM algorithm with unconstrained search of the model parameter space.

Therefore, in this paper four random search algorithms are developed to solve RAP in a queuing system. In other words, an unconstrained redundancy allocation for a queuing system with just one type of component under hot and cold standby policies is considered. The aim of the studied problem is to find the optimal number of redundant servers and repairmen by minimizing the total costs. Among the developed algorithms, the first one deterministically selects the best neighbor solution in each step. The second one randomly moves among the best neighbor solutions. In the third and fourth algorithms random movements to all neighbor solutions can take place according to a probability that is updated based on the cost of each solution. Performance analysis of the algorithms shows that algorithms 1 and 2 for unimodal solution spaces and algorithms 3 and 4 for multimodal solution spaces are better. Although algorithms 1 and 2 are faster than algorithms 3 and 4, but they are not always able to find optimal solution in multimodal spaces.

Queuing systems with standby redundancy of unreliable servers have particularly wide applications. Typical examples of such systems are assembly lines, multi-channel data-transmission lines, duplex or triplex computer systems, etc. For example, Garg and Sharma [24] studied the Reliability-Redundancy Allocation Problem (RRAP) of a pharmaceutical plant. However, they didn't take into account the queuing costs of the system in the analysis. The approach of the current paper can be useful in a more comprehensive analysis of the RAP and RRAP where queuing costs are also considered in finding the optimal allocation. In another work, Savsar and Aldaihani [25] modeled a Flexible Manufacturing Cell (FMC) with two machines served with one robot. They analyzed performance measures of the FMC under different operational conditions. However, they did not consider queuing cost of the FMC. The approach of the current paper can be used for reliability analysis and redundancy allocation of the mentioned FMC.

The main contributions of this paper can be summarized as:

• Formulating a Redundancy Allocation Problem for a queuing system with maintenance considerations,

• Considering queuing costs for a Redundancy Allocation Problem,

• Developing four random search algorithms based on the structure of the formulated RAP.

The rest of this paper is organized as follows. In Section 2 the problem assumptions are explained and the problem is analyzed within the context of a queuing system. Cost analysis of the problem is discussed in Section 3. Solution methods are described in Section 4. Numerical examples are reported in Section 5. Finally, a concise conclusion is given in Section 6.