A Markov modulated multi-server queue with negative customers the MM CPP/GE/c/L G-queue

We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing. We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.