On the Busy Cycle Maxima in a Heterogeneous Fork-Join Queue

This work investigates the distribution of the maximum response time during a busy cycle in a Fork-Join queueing system with two parallel servers having heterogeneous deterministic service times and Poisson arrivals. We formulate a renewal-type Fredholm integral equation of the second kind that explicitly characterizes the cumulative distribution function (CDF) of the maximum response time. An infinite series representation is derived from this integral equation, expressing the probability distribution in terms of recursively defined integrals, each corresponding to a fixed number of customer arrivals during the busy cycle. A computationally efficient numerical scheme, using the infinite series, is developed to accurately approximate this distribution. We validate the accuracy and efficiency of our numerical method through comprehensive simulations, demonstrating strong agreement between analytical and empirical results. Our analysis provides clear insights into the behavior of heterogeneous Fork-Join queue dynamics, facilitating more precise design and performance evaluation of parallel processing and service systems.