An analysis of nonstationary coupled queues

We consider a two dimensional time varying tandem queue with coupled processors. We assume that jobs arrive to the first station as a non-homogeneous Poisson process. When each queue is non-empty, jobs are processed separately like an ordinary tandem queue. However, if one of the processors is empty, then the total service capacity is given to the other processor. This problem has been analyzed in the constant rate case by leveraging Riemann Hilbert theory and two dimensional generating functions. Since we are considering time varying arrival rates, generating functions cannot be used as easily. Thus, we choose to exploit the functional Kolmogorov forward equations (FKFE) for the two dimensional queueing process. In order to leverage the FKFE, it is necessary to approximate the queueing distribution in order to compute the relevant expectations and covariance terms. To this end, we expand our two dimensional Markovian queueing process in terms of a two dimensional polynomial chaos expansion using the Hermite polynomials as basis elements. Truncating the polynomial chaos expansion at a finite order induces an approximate distribution that is close to the original stochastic process. Using this truncated expansion as a surrogate distribution, we can accurately estimate probabilistic quantities of the two dimensional queueing process such as the mean, variance, and probability that each queue is empty.