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 mu lti-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a cont inuous time Markov phase process. In each phase, the arrivals are a superpo sition of a positive and a negative arrival stream, each of which is a comp ound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Suc h a queueing system is well suited to B-ISDN/ATM networks since it can acco unt for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimen sional (queue length by phase) Markov process that describes the dynamics o f 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 distributi on 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 ne tworks of switching nodes in terms of their internal arrival processes.