The single server semi-markov queue

A general model for the single server semi-Markov queue is studied. Its solution is reduced to a matrix factorization problem. Given this factorization, results are obtained for the distributions of actual and virtual waiting times, queue lengths both at arrival epochs and in continuous time, the number of customers during a busy period, its length and the length of a busy cycle. Two examples are discussed for which explicit factorizations have been obtained

How to Staff when Customers Arrive in Batches

In settings as diverse as autonomous vehicles, cloud computing, and pandemic quarantines, requests for service can arrive in near or true simultaneity with one another. This creates batches of arrivals to the underlying queueing system. In this paper, we study the staffing problem for the batch arrival queue. We show that batches place a significant stress on services, and thus require a high amount of resources and preparation. In fact, we find that there is no economy of scale as the number of customers in each batch increases, creating a stark contrast with the square root safety staffing rules enjoyed by systems with solitary arrivals of customers. Furthermore, when customers arrive both quickly and in batches, an economy of scale can exist, but it is weaker than what is typically expected. Methodologically, these staffing results follow from novel large batch and hybrid large-batch-and-large-rate limits of the general multi-server queueing model. In the pure large batch limit, we establish the first formal connection between multi-server queues and storage processes, another family of stochastic processes. By consequence, we show that the limit of the batch scaled queue length process is not asymptotically normal, and that, in fact, the fluid and diffusion-type limits coincide. This is what drives our staffing analysis of the batch arrival queue, and what implies that the (safety) staffing of this system must be directly proportional to the batch size just to achieve a non-degenerate probability of customers waiting

Optimal inspection and maintenance for stochastically deteriorating systems

This thesis concerns the optimisation of maintenance and inspection for stochastically deteriorating systems. The motivation for this thesis is the problem of determining condition based maintenance policies, for systems whose degradation may be modelled by a continuous time stochastic process. Our emphasis is mainly on using the information gained from inspecting the degradation to determine efficient maintenance and inspection policies. The system we shall consider is one in which the degradation is modelled by a Levy process, and in which failure is defined to occur when the degradation reaches a critical level. It is assumed that the system may be inspected or repaired at any time, and that the costs of inspections and repairs may depend on the level of system degradation. Initially we look at determining optimal inspection policies for systems whose degradation may be directly and perfectly observed, before extending this analysis to the case where the degradation is unobservable, and a related covariate process is used to determine maintenance decisions. In both cases it is assumed the replacement policy is fixed and known in advance. Finally we consider the case of joint optimisation of maintenance and inspection, for cases in which the maintenance action has either deterministic or random effect on the degradation level. In all of these cases we use the properties of the Levy process degradation model to form a recursive relationship which allows us to determine integral and functional equations for the maintenance cost of the system. Solutions to these determine optimal periodic and non-periodic inspection and maintenance policies. Throughout the thesis we use the gamma process degradation model as an example. For this model we determine optimal perfect inspection policies for the cases when inspections are periodic and non-periodic. As a special case of a covariate process we consider the optimal imperfect periodic inspection policy. Finally we obtain jointly optimal deterministic-maintenance and periodic-inspection policies