# Transaction of Fee and Confirmation Time in Cryptocurrencies

### Vishu Teja Kunda

**1 Introduction** The is the summary of the tasks undertaken by Vishnu Teja Kunde, for being a CNI MTech fellow for the year 2021-2022.

**2 Research contribution** In well-known cryptocurrencies like Bitcoin, there is a network of nodes that arrive at a distributed consensus on transactions between users. The network nodes are called miners and the collection of broadcasted transactions to the network is called mempool. Miners select transactions from the mempool, verify it, and aggregate them for creation of a potential block, that can be added to the blockchain. The block generation is referred to as mining. The inclusion of a transaction is incentivised by offering higher transaction fee to the miner. The current work, titled “Transaction Fee and Confirmation Time in Cryptocurrencies” is about modelling and analysis of a cryptocurrency transaction waiting time and its relation to the fee offered to the miner mining the transaction. We analyse an K queue priority model for K values of fee that can be chosen by an incoming transaction and compute the mean waiting time for probabilistically paying one of the K fee values. We develop tandem queue model to incorporate the inclusion of transactions that wait sufficiently longer as in Bitcoin. Next, we look at the problem of an incoming transaction choosing to either pay a non-zero fee, F > 0 and do not pay any fee to the miner. The ones paying fee F are queued into mempool MF and those not paying are queued into M0. The decision of the incoming transaction depends on the state of the system (NF , N0) which is the number of transactions waiting in the queue paying fee F and 0 respectively. We model two service disciplines that capture higher quality of service (less delay) offered for transactions paying the fee. The first one is probabilistic service where a transaction from MF is picked with higher probability pF . In this case, the optimal decision policy is like join-theshortest-queue (JSQ) policy. The second service discipline is priority service with non-pre-emption. In this case, we design a worst-case policy and a probabilistically best-case policy. The design of the latter involves a recursive relation that converges to some stationary policy that is probabilistically best-case optimal. This policy gives guarantees on optimality in some states. We derive some properties of this policy. The worst-case policy also gives guarantees on optimality in certain states. We observe that the worst-case policy is represented by a linear switching curve in first quadrant of the 2D plane. We comment on the dependence of this decision policy on various system parameters through numerical studies. We observe that a method to derive an optimal policy would work in a more general setting of batch services, through numerical studies.

**3 General contribution** I am involved in the design of tutorial notes for mathematics related to blockchain technology and cryptography. The notes contain the selected mathematics of finite fields and the techniques used for the cryptographic key-generation and sharing. I am planning to include simplified proofs from well-known papers related to blockchain technology to get a flavour of the analytical research in this area.