A state-dependent GI/G/1 queue

We consider a state-dependent GI/G/1 queueing system characterized by the unfinished work U(t) in the system at time t. We introduce state-dependence by allowing (i) the arrival process to depend on the instantaneous value of U(t), (ii) the service rate, that is, the rate at which U(t) decreases in the absence of arrivals, to depend on U(t), and (iii) the customer's service requirement to depend on U(t*) where t* denotes the instant in which that customer entered the system. We consider the limit of short inter-arrival times and small service requests and compute asymptotic approximations to the stationary density of the unfinished work, including the stationary probability of finding the system empty, using the WKB method and the method of matched asymptotic expansions.