## 2 Preliminaries and Literature Review
Proposed Variable Bulk Arrival and Variable Bulk Service (VBAVBS) Model with \(\lambda_{F}\) for Real-Time Chain  
In the proposed real-time chain model, an embedded Markovian single-server exponential queueing system (i.e., \(M^{1,n}/M^{1,n}/1\)) is considered without loss of generality, and the server (the server is the equivalent of the group of miners to select the transactions to be posted) serves the entire batch of customers (the customers are the equivalent of the transactions to be posted in the block) in the queue (a queue is the equivalent of a block to be mined and posted) all at once, at the same time. Whenever the server completes a service (a service is the equivalent of a process of posting a block), it then purges the queue (i.e., the equivalence of posting a block) and then serves the influx of new incoming customers. Note that it is assumed that the service takes place within a certain amount of time, yet no transaction is assumed to arrive in the meantime. However, note that it is not unlikely to have new customer arrive if a significant amount of service time is assumed, from a practical point of consideration. It is assumed that the service time is exponential at \(\frac{1}{\mu}\) when the server is serving the entire queue of any size between 0 and \(n\), inclusively and equivalently, posting and purging the entire queue. Without loss of generality, it is assumed that customers arrive at an exponential rate of \(\lambda\) successfully within the deadline yet, at the rate of \(\lambda_{F}\), the customers are assumed to fail to arrive within a specific deadline to meet the real-time requirement. The underlying queueing process is assumed to take place with variable-sized slots and the status of the queue is determined by the number of slots of any size in the current block. Based on the assumptions above, the proposed VBAVBS model with \(\lambda_{F}\) also employs an embedded Markovian queueing model like the VBASBS (Variable Bulk Arrival And Static Bulk Service) in [19], and it defines the states as expressed in terms of the number of slots assigned to a block, and it traces the normalized number of slots allocated for the transactions in steady state rather than the number of transactions whose size varies in the number of slots. * \(P_{0}\): The state in which no transaction (i.e., no slot) has arrived in the queue as of yet for the posting in the block, currently [19]. * \(P_{n}\): The state in which there are \(n\) number of slots (i.e., the capacity of the queue, equivalently, the maximum number of slots set and voted by the miners or voters) arrived in the queue for the posting in the block, currently [19]. * \(P_{i}\): The state in which there are \(i\) number of slots (where \(0<i<n\)) arrived in the queue for the posting in the block, currently [19]. The random variables employed to express the state transition rates are specified as follows. * \(\lambda\): The rate for a slot of a transaction to arrive successfully within the real-time deadline requirement, and the rate for a transaction to arrive is determined by the number of slots allocated for the transaction in a prorated manner such that a transaction with a size of \(j\) number of slots arrives at the rate of \(j\lambda\), without loss of generality and practicality as well. * \(\lambda_{F}\): The rate for a slot of a transaction to arrive unsuccessfully past the real-time deadline requirement, and at the rate, the state will self-loop without making any state transition. This rate is the unique one to distinguish VBAVBS from VBASBS in which there was no real-time deadline requirement taken into consideration. * \(\mu\): The rate for the slots of the transactions in the entire queue to be posted and purged. Notice that this is a single and unique state transition 1, 2 and 3. The balance equations for VBAVBS with \(\lambda_{F}\) are shown in the following Equations.