Combinatorial approach to Markovian queueing models

Transient Probabilities for Queues with Applications to Hospital Waiting List Management

“…In this section we present three results (without proof) from the theory presented in [1,2]. These results have direct relevance to the type of questions we posed in the introduction but are a small subset of the impressive array of computational results available from the lattice path counting approach.…”

“…(see [1], Result 1). This is a simple application of Pascal's Triangle, it tells you how many routes there are (in a city whose streets are a rectangular grid) from one intersection of streets to another such intersection.…”

“…(see [1], Result 2). The reason for the importance of this result is the physical significance it carries, once a path has reached the line X = Y in the plane an idle period begins since there are no customers left at this point, thus X = Y represents a barrier which evolutionary paths cannot cross.…”

“…In this section we will briefly outline the relevant theory from [1], assuming batch arrivals of size b and that k customers are present in the queue initially. The reader should be aware that b = 1 is admissible, thus we deal with the classical case of single arrivals at the same time.…”

“…However, we emphasise again that we make no assumption on the traffic intensity to ensure steady state equilibrium, thus we are interesting in the transient behaviour of an M b /M/1 queue and we draw upon the theory presented for example in [1][2][3].…”