Several devices of microelectromechanical systems (MEMS) are analyzed in the presented work, using a novel numerical meshless method called the random differential quadrature (RDQ) method. The differential quadrature (DQ) is an effective derivative discretization technique but it requires all the field nodes to be arranged in a collinear manner with a pre-defined pattern. This limitation of the DQ method is overcome in the RDQ method using the interpolation function by the fixed reproducing kernel particle method (fixed RKPM). The RDQ method extends the applicability of the DQ method over a regular as well as an irregular domain discretized by uniform or randomly distributed field nodes. Due to the strong-form nature, RDQ method captures well the local high gradients. These features of the RDQ method enable it to efficiently solve the MEMS problems with different boundary conditions. In the presented work, several MEMS devices that are governed by the nonlinear electrostatic force are analyzed using the RDQ method, and their results are compared with the other simulation results presented in the existing literature. It is seen that the RDQ method effectively and accurately solves the MEMS devices problems.