In this thesis, we consider the equilibrium behaviour of a double-ended queueing system with dependent matching time in the context of taxi-passenger systems at airport terminal pickup. We extend the standard taxi-passenger model by considering random matching time between taxis and passengers in an airport terminal pickup setting. For two types of matching time distribution, we examine this model through analysis of equilibrium behaviour and optimal strategies. We demonstrate in detail how to derive the equilibrium joining strategies for passengers arriving at the terminal and the existence of a socially optimal strategies for partially observable and fully observable cases. Numerical experiments are used to examine the behaviour of social welfare and compare cases.