Level-dependent quasi-birth-and-death processes: Application to cost analysis of multi-server systems

Analysing costs is crucial for optimising the operational efficiency and resource allocation in systems evolving under uncertainty. In this paper, we study the distribution of costs associated with the evolution of level-dependent quasi-birth-and-death (LD-QBD) processes, which are useful in modelling many multi-server systems. We derive analytical expressions for the Laplace-Stieltjes transforms (LSTs) of the distribution of total costs accumulated during the times the LD-QBD processes spend in a specified set of levels. We present algorithms for the numerical evaluation of these LSTs. We also give memory efficient versions of the algorithms and discuss their algorithmic complexity. To assess the robustness of the distribution of costs with respect to model parameters, we develop algorithms for the sensitivity analysis of the corresponding LSTs.
To illustrate the application potential of our results, we construct LD-QBD example models for a finite capacity multi-server queueing systems with admissions policies including redirection, preemptive transfer, and guard-channel threshold. The analysis is based on a large dataset obtained from a tertiary referral hospital in Australia. We compute the long-run performance measures, the distribution of time until some number of beds become available following congestion, and the distribution of the associated costs. We present valuable insights into how the system behaves under the various policies. We also perform the sensitivity analysis of the distribution of costs with respect to model parameters.