# Towards Practical Applications in Modeling Blockchain System
## 2 Related Works
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Research** & **Presented Solution** & **Tool/Technology** & **Parameters to be Considered and Characteristics to be Modeled** \\ \hline \multirow{4}{*}{[10]} & Modeling the process using multiple queues based on four phases (waiting for inclusion in the block; waiting for confirmation; waiting for service; servicing) & A state transition graph; Mass service theory; Markov processes & Simulation of block generation; State transition probabilities; Access delays \\ \hline \multirow{4}{*}{[11]} & The mining organization model is defined by & & \\  & M/M/n/L. The queue capacity is set as TxB, the queue policy is first-come, first-served, and the & & Average number of transactions per block; \\  & discard rule is block after service, which means that only transactions of block size TxB remain in the & & Total mining capacity; \\  & dynamic memory of the mining nodes, while other transactions, even if they are processed, are in the & & Number of transactions per second \\  & memory pool & & \\ \hline \multirow{4}{*}{[12]} & The M/M/1 model is used to model the blockchain memory pool, and the mining pool is modeled by the M/M/n model. There can only be one block in a mining pool at any one time. However, within a mining pool, processes can be divided into multiple tasks or threads for parallel processing by multiple mining nodes in the network. & & Average number of transactions per block; \\  & The mining process is modeled using a queuing system, analyzing transaction confirmation times. The authors consider the M/G/1 model with batch service, in which a newly arrived transaction cannot reach the service facility even if the number of transactions in the service facility does not reach the maximum batch size. In this model, the dwell time of a transaction corresponds to its confirmation time. & & Average block generation time; Average number of transactions in the system \\ \hline \multirow{4}{*}{[13]} & The system is considered using the M/G/1 model as an example. Data arrivals to nodes are modeled as a heterogeneous Poisson process, where the distribution of arrival rate to nodes is derived from an analytical model of the data delivery protocol. & & Block and transaction allocation time probabilities; Node response time; The likelihood of a branching chain; Duration of the period of the inconsistency of the register \\ \hline \multirow{4}{*}{[15]} & Stochastic network models are proposed to capture the evolution and dynamics of the blockchain. A combination of analytical calculations and simulation experiments is used to investigate both steady-state and transient performance characteristics & Stochastic models & The effect of a block’s propagation delay; Hashing power of nodes \\ \hline \multirow{4}{*}{[16]} & Game theories are proposed for modeling to address common problems in the blockchain network, such as security, problems related to mining management, and issues related to the economics of the blockchain. & Game theory & Economic aspects \\ \hline \multirow{4}{*}{[17]} & The system is examined using the M/G/\(\infty\) model as an example. Equivalence between two specific service disciplines is used to obtain the stationary distribution of the model. & Mass service theory & Distribution of busy periods; service delays \\ \hline \multirow{4}{*}{[18]} & The theory of mass service in blockchain systems is developed, and system performance is evaluated. & & Average number of transactions per queue, \\  & For this purpose, a Markov queuing system of packet service is developed with two different stages, which are suitable for explicitly expressing the process of mining in a pool of miners and constructing a new blockchain. & Mass service theory; Markov processes & Average number of transactions per block; \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analytical modeling solutions for blockchain systems.