In the Bitcoin, blockchain miners use computational resources to validate transactions and are rewarded with newly created Bitcoin and transaction fees appended by users for their transactions. The priority of a transaction is determined, among others, by the fee appended to it. Profit-maximising miners choose transactions that pay higher fees. As trading volume in Bitcoin increases, network congestion has become a key issue in the viability of Bitcoin as a medium of exchange. As congestion leads to large waiting times and soaring transaction fees, users with small-value transactions may no longer find it profitable to record transactions. We aim to examine the viability of Bitcoin blockchain when users pay no transaction fees so that there is no priority given to transactions. Users’ decisions to record a transaction through the Bitcoin blockchain are modelled through a queueing game in a processor-sharing system. The symmetric equilibrium threshold of this game sheds light on the limit of the queue length at which self-optimising users who seek to minimise transaction costs find it economically inviable to record a transaction through the Bitcoin blockchain. This analysis highlights the key role that transaction fees play in sustaining an increasing number of transactions in the Bitcoin blockchain.