We study an M/G/1 queue with second optional service. Poisson arrivals with mean arrival rate l (>0) all demand the first ‘essential’ service, whereas only some of them demand the second ‘optional’ service. The service times of the first essential service are assumed to follow a general (arbitrary) distribution with distribution function B(v) and that of the second optional service are exponential with mean service time 1/m_2 (m_2>0). The time-dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been derived explicitly. Also the mean queue length and the mean waiting time have been found explicitly. The well-known Pollaczec–Khinchine formula and some other known results including M/D/1, M/E_{k}/1} and M/M/1 have been derived as particular cases.

An M/G/1 queue with second optional service

Blockchain has been envisioned to be a disruptive technology with potential for applications in various industries. As more and more different blockchain platforms have emerged, it is essential to assess their performance in different use cases and scenarios. In this paper, we conduct a systematic survey on the blockchain performance evaluation by categorizing all reviewed solutions into two general categories, namely, empirical analysis and analytical modelling. In the empirical analysis, we comparatively review the current empirical blockchain evaluation methodologies, including benchmarking, monitoring, experimental analysis and simulation. In analytical modelling, we investigate the stochastic models applied to performance evaluation of mainstream blockchain consensus algorithms. Through contrasting, comparison and grouping different methods together, we extract important criteria that can be used for selecting the most suitable evaluation technique for optimizing the performance of blockchain systems based on their identified bottlenecks. Finally, we conclude the survey by presenting a list of possible directions for future research.

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Performance Evaluation of Blockchain Systems: A Systematic Survey

An M/G/1 with second optional service and unreliable server is studied in this paper. We assume that customers arrive to the system according to a Poisson process with rate @l. All demand the first ''essential'' service, whereas only some of them demand the second ''optional'' service. The service times of the first essential service are i.i.d. random variables, and that of the second optional service are i.i.d. exponential random variables. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a service-phase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest.

An M/G/1 queue with second optional service and server breakdowns

Bitcoin is a virtual currency based on a transaction-ledger database called blockchain. The blockchain is maintained and updated by mining process in which a number of nodes called miners compete for finding answers of very difficult puzzle-like problem. Transactions issued by users are grouped into a block, and the block is added to the blockchain when an algorithmic puzzle specialized for the block is solved. A recent study reveals that newly arriving transactions are not included in the block being under mining. In this paper, we model the mining process with a queueing system with batch service, analyzing the transaction-confirmation time. We consider an \(M/\text{ G }^B/1\) with batch service, in which a newly arriving transaction cannot enter the service facility even when the number of transactions in the service facility does not reach the maximum batch size, i.e., the block-size limit. In this model, the sojourn time of a transaction corresponds to its confirmation time. We consider the joint distribution of the number of transactions in system and the elapsed service time, deriving the mean transaction-confirmation time. In numerical examples, we show how the block-size limit affects the transaction-confirmation time.

M.L. Chaudhry

Papers

The queuing system M/GB/l and its ramifications

On the service system MX/G/∞

On bounds for bulk arrival queues