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Random routing protocols in sensor networks forward packets to randomly selected neighbors. These packets are agents carrying information about events, or queries seeking such information. We derive the probability of a packet visiting a given node in a given step as well as the rendezvous probability of agents and queries within a specific number of hops at a given node(s) in a 2-D grid-based sensor network. The utility of the model is demonstrated by determining the protocol parameters to optimize performance of rumor routing protocol under different constraints, e.g., to evaluate the number of queries and agents to maximize the probability of rendezvous for a given amount of energy. Monte Carlo simulations are used to validate the model. The closed form exact solution presented, unlike existing models relying on asymptotic behavior, is applicable to small and medium-scale networks as well. An upper bound is provided for the case where the packet is not sent back to its immediate forwarding node. Simulation results indicate that the model is a good approximation even for sparse arrays with 75 % of the nodes. The model can be used to set parameters and optimize performance of several classes of random routing protocols.