Non-coincidence probabilities and the time-dependent behavior of tandem queues with deterministic input. (Q1877538)

Let \(\mathbf Q(t)=(Q_1(t),\dots ,Q_r(t))\) be the numbers of customers in a sequence of service stations with waiting rooms of infinite capacity; the output of one station is the input of the next one, the service times are i.i.d. exponential random variables and the customers arrive at the first server at prescribed (deterministic) time instants \(\tau _i\). Let \(\mathbf m, \mathbf k\) be the \(r\)-tuples \(m_i, k_i \in \mathbb N\) and let \(T\) be the time from system startup until a server becomes idle for the first time. The transition function \(P(\mathbf Q(t)=\mathbf k\), \(T>t\mid \mathbf Q(0)=\mathbf m)\) is the probability that \(\mathbf Q(t)\) moves from the state \(\mathbf m\) to \(\mathbf k\) in \((0,t)\) conditionally on no server being idle meanwhile. Two statements are proved: A) The transition functions \(P\) can be interpreted as non-coincidence probabilities associated with a set of dissimilar (i.e. with not necessarily identical transition rates) Poisson processes restricted by a time-dependent boundary induced by the arrival instants \(\tau _1, \tau _2,\dots \) B) Even for such dissimilar restricted Poisson processes, a formula containing a single determinant and analogous to the classical Karlin-McGregor theorem [\textit{S. Karlin} and \textit{J. McGregor}, Pac. J. Math. 9, 1141--1164 (1959; Zbl 0092.34503)] holds. The structure of this determinant is particularly simple if the inter-arrival times \(\tau _{i+1} - \tau _i\) are equal.