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We study the sojourn time in a queueing system with a single exponentialserver, serving a Poisson stream of customers in order of arrival. Service isprovided at low or high rate, which can be adapted at exponential inspectiontimes. When the number of customers in the system is above a given threshold,the service rate is upgraded to the high rate, and otherwise, it is downgradedto the low rate. The state dependent changes in the service rate make theanalysis of the sojourn time a challenging problem, since the sojourn time nowalso depends on future arrivals. We determine the Laplace transform of thestationary sojourn time and describe a procedure to compute all moments aswell. First we analyze the special case of continuous inspection, where theservice rate immediately changes once the threshold is crossed. Then we extendthe analysis to random inspection times. This extension requires thedevelopment of a new methodological tool, that is "matrix generatingfunctions". The power of this tool is that it can also be used to analyzegeneralizations to phase-type services and inspection times.

Sojourn time in a single server queue with threshold service rate control