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We study the robustness of the c μ -rule for the optimal allocation of a resource consisting of one unreliable server to parallel queues with two different classes of customers. The customers in queues can be served with respect to a FIFO retrial discipline, when the customers at the heads of queues repeatedly try to occupy the server at a random time. It is proved that for scheduling problems in the system without arrivals, the c μ -rule minimizes the total average cost. For the system with arrivals, it is difficult directly to prove the optimality of the same policy with explicit relations. We derived for an infinite-buffer model a static control policy that also prescribes the service for certain values of system parameters exclusively for the class-i customers if both of the queues are not empty, with the aim to minimize the average cost per unit of time. It is also shown that in a finite buffer case, the c μ -rule fails. [ABSTRACT FROM AUTHOR]

Optimal control problems are applied to a variety of dynamical systems with a random law of motion. In this paper we show that the random degradation processes defined on a discrete set of intermediate degradation states are also suitable for formulating and solving optimization problems and finding an appropriate optimal control policy. Two degradation models are considered in this paper: with random time to an instantaneous failure and with random time to a preventive maintenance. In both cases, a threshold-based control policy with two thresholds levels defining the signal state, after which an instantaneous failure or preventive maintenance can occur after a random time, and a maximum number of intermediate degradation states is applied. The optimal control problem is mainly solved in a steady-state regime. The main loss functional is formulated as the average cost per unit of time for a given cost structure. The Markov degradation models are used for numerical calculations of the optimal threshold policy and reliability function of the studied degrading units. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of optimal routing for the pair of two-server heterogeneous queues operating in parallel and subsequent optimal allocation of customers between the servers in each queue. Heterogeneity implies different servers in terms of speed of service. An open-loop control assumes the static resource allocation when a router has no information about the state of the system. We discuss here the algorithm to calculate the optimal routing policy based on specially constructed Markov-modulated Poisson processes. As an alternative static policy, we consider an optimal Bernoulli splitting which prescribes the optimal allocation probabilities. Then, we show that the optimal allocation policy between the servers within each queue is of threshold type with threshold levels depending on the queue length and phase of an arrival process. This dependence can be neglected by using a heuristic threshold policy. A number of illustrative examples show interesting properties of the systems operating under the introduced policies and their performance characteristics. [ABSTRACT FROM AUTHOR]

The paper deals with a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue. The main model has a non-preemptive service when the customer can not change the server during its service time. The optimal allocation problem is formulated as a Markov-decision one. We show numerically that the optimal policy which minimizes the long-run average number of customers in the system has a threshold structure. We derive the matrix expressions for performance measures of the system and compare the main model with alternative simplified queuing systems which are analysed for the arbitrary number of servers. We observe that the preemptive heterogeneous model operating under a threshold policy is a good approximation for the main model by calculating the mean number of customers in the system. Moreover, using the preemptive and non-preemptive queueing models with the faster server first policy the lower and upper bounds are calculated for this mean value. [ABSTRACT FROM AUTHOR]

This paper deals with heterogeneous queues where servers differ not only in service rates but also in operating costs. The classical optimisation problem in queueing systems with heterogeneous servers consists in the optimal allocation of customers between the servers with the aim to minimise the long-run average costs of the system per unit of time. As it is known, under some assumptions the optimal allocation policy for this system is of threshold type, i.e., the policy depends on the queue length and the state of faster servers. The optimal thresholds can be calculated using a Markov decision process by implementing the policy-iteration algorithm. This algorithm may have certain limitations on obtaining a result for the entire range of system parameter values. However, the available data sets for evaluated optimal threshold levels and values of system parameters can be used to provide estimations for optimal thresholds through artificial neural networks. The obtained results are accompanied by a simple heuristic solution. Numerical examples illustrate the quality of estimations. [ABSTRACT FROM AUTHOR]

The paper studies a controllable multi-server heterogeneous queueing system where servers operate at different service rates without preemption, i.e., the service times are uninterrupted. The optimal control policy allocates the customers between the servers in such a way that the mean number of customers in the system reaches its minimal value. The Markov decision model and the policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, is of threshold type defined as a sequence of threshold levels which specifies the queue lengths for the usage of any slower server. To avoid time-consuming calculations for systems with a large number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this solution with the real values. We develop also the simple lower and upper bound methods based on approximation by an equivalent heterogeneous queueing system with a preemption to measure the mean number of customers in the system operating under the optimal policy. Finally, the simulation technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of inter-arrival and service time distributions. [ABSTRACT FROM AUTHOR]

The paper provides reliability analysis of a cold double redundant renewable system assuming that both life-time and repair time distributions are arbitrary. The proposed approach is based on the theory of decomposable semi-regenerative processes. We derive the Laplace–Stieltjes transform of two main reliability measures like the distribution of the time between failures and the time to the first failure. The transforms are used to calculate corresponding mean times. It is further derived in closed form the time-dependent and time stationary state probabilities in terms of the Laplace transforms. Numerical results illustrate the effect of the type of distributions as well as their parameters on the derived reliability and probabilistic measures. [ABSTRACT FROM AUTHOR]