Working vacations queueing model with multiple types of server breakdowns A new hybrid-Trefftz triangular and quadrilateral plate elements. (English) Zbl 1185.90048
Summary: This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server’s status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix-geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters.
Summary: We develop 9 DOF triangular and 12 DOF quadrilateral plate bending elements based on the hybrid-Trefftz method. Among the two independent displacement fields, i.e. the internal and the boundary displacements, we use the Mindlin-Reissner’s thick plate solution with a particular solution under pressure load as the internal displacement field. For the boundary displacement field, we approximate transverse displacement \(w\) and rotations \(\beta_x\) and \(\beta_y\) by cubic and quadratic hierarchical shape functions, respectively. We derive the transverse shear strains from constitutive equations and equilibrium equations, respectively, and then, remove additional degrees of freedom of hierarchical shape functions using the relations between these two shear strains. Results of numerical tests reveal that the new triangular (\(T32\)-7) and quadrilateral (\(Q32\)-11) elements are robust, accurate and free of shear locking in the thin limit.
MSC:
| 90B22 | Queues and service in operations research |
| 90B25 | Reliability, availability, maintenance, inspection in operations research |
| 90C31 | Sensitivity, stability, parametric optimization |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74K20 | Plates |
Keywords:
state dependent queue; working vacation; multiple breakdown; matrix-geometric method; queue size thick plate; transverse shear locking; finite elementReferences:
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