Abstract:
This paper examines the non-stationary performance metrics of a three-phase queuing system with a Poisson input flow, exponentially distributed service time across all phases, and a constraint on the total size of the shared buffer. A system of Kolmogorov differential equations is written using specially introduced functions that account for the system's operating principles. A probability translation matrix method is used to solve the system of equations. Expressions are derived for finding the loss probability and system performance. A system with a buffer size of three is considered as an example. The duration of the transient mode is analyzed as a function of the ratios of service rates in individual service phases. It is concluded that the service rates in the first and second phases have the greatest impact on the duration of the transient mode. The dependences of the maximum values of the non-stationary loss probability and the corresponding stationary probabilities for various service rates are analyzed. The analysis of the system performance metrics is conducted for parameters corresponding to modern optical networks. The obtained conclusions are of interest for the design of high-performance computing systems.
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