A networked system often uses a shared communication network to transmit the measurements to a remotely located estimation center. Due to the limited bandwidth of the channel, a delay may appear while receiving the measurements. This delay can be arbitrary step random, and packets are sometimes dropped during transmission as it exceeds a certain permissible number. In this paper, such measurements are modeled with the Poisson distribution, which allows the user to determine the maximum delay the system might suffer. When the measurement delay exceeds the permissible number, the packet dropout happens. Based on the proposed model, we solve the problem by assuming that the prior and posterior densities of states are Gaussian and derive the expression of the estimated state and the error covariance. Later, relaxing the Gaussian assumption for densities, we propose a solution with the help of the sequential Monte Carlo (SMC) approach. The proposed SMC method divides the set of particles into several groups, where each group supports the possibility that the received measurement is delayed by a certain number of steps. The strength of an individual group is determined by the probability of a measurement being delayed with the same number of steps that the group represents. This approach estimates the states and also assesses the amount of delay from the received measurements. Finally, the developed estimators are implemented on two nonlinear estimation problems, and the simulation results are compared. The proposed SMC approach shows better results compared to the designed Gaussian delay filters and existing particle filters with delay.