Lattice path counting and \(M/M/c\) queueing systems (Q1892653)

The transient solution of \(M/M/c\) queue in a closed form is obtained for the probability of exactly \(i\) arrivals and \(j\) departures within a time interval of length \(t\). The method is based on the well-known combinatorial considerations: the authors count the paths from the origin to \((i,j)\) that have exactly \(r(d)\) \(x\)-steps whose depth from the line \(y = x\) is \(d = 0, \dots, c - 1\). Some numerical examples are considered.