Discrete-time single server queues with correlated inputs

A wide variety of queueing systems with a single server can be modeled by the equation b<inf>n+1</inf> = (b<inf>n</inf> − 1)⁺ + z<inf>n</inf>, where b<inf>n</inf> denotes queue length and z<inf>n</inf> the input. The usual assumption about the sequence {z<inf>n</inf>} is that it be a sequence of independent identically distributed (i. i. d.) random variables. However, in many applications, this is not really the case; {z<inf>n</inf>} is a sequence of correlated random variables. We show that with the help of a transformation, a (k + 1)-dimensional Markov process that suffices to describe the queueing system may be found, where k is the memory of the input process. We derive an equation for the steady-state generating function corresponding to the joint distribution of this vector process. We find that a simple set of equations can be obtained for the marginal distributions. In particular, the steady-state distribution of b<inf>n</inf>, the queue length, can be obtained without solving for the joint distribution.